Geometry of Banach Spaces, Operator Theory, and ...

Journal of Function Spaces and Applications

Geometry of Banach Spaces, Operator Theory, and Their Applications Guest Editors: Genqi Xu, Ji Gao, Pei Liu, and Satit Saejung

Geometry of Banach Spaces, Operator Theory, and Their Applications

Journal of Function Spaces

Geometry of Banach Spaces, Operator Theory, and Their Applications Guest Editors: Genqi Xu, Ji Gao, Pei Liu, and Satit Saejung

Copyright © 2014 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in “Journal of Function Spaces.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Editorial Board John R. Akeroyd, USA Gerassimos Barbatis, Greece Ismat Beg, Pakistan Bjorn Birnir, USA Messaoud Bounkhel, Saudi Arabia Huy Qui Bui, New Zealand Victor I. Burenkov, Italy Jiecheng Chen, China Jaeyoung Chung, Korea Sompong Dhompongsa, Thailand Luisa Di Piazza, Italy Lars Diening, Germany Dragan Djordjevic, Serbia Miroslav Engliˇs, Czech Republic Jose A. Ezquerro, Spain Dashan Fan, USA Xiang Fang, USA Hans G. Feichtinger, Austria Alberto Fiorenza, Italy Ajda Foˇsner, Slovenia Eva A. Gallardo Gutiérrez, Spain Aurelian Gheondea, Turkey Antonio S. Granero, Spain Yongsheng S. Han, USA Seppo Hassi, Finland Stanislav Hencl, Czech Republic

Norimichi Hirano, Japan Henryk Hudzik, Poland Pankaj Jain, India Krzysztof Jarosz, USA Anna Kamińska, USA H. Turgay Kaptanoglu, Turkey Valentin Keyantuo, Puerto Rico Vakhtang M. Kokilashvili, Georgia Alois Kufner, Czech Republic David R. Larson, USA Yuri Latushkin, USA Young Joo Lee, Republic of Korea Hugo Leiva, Venezuela Guozhen Lu, USA Dag Lukkassen, Norway Qiaozhen Ma, China Mark A. McKibben, USA Mihail Megan, Romania Alfonso Montes-Rodriguez, Spain Dumitru Motreanu, France Sivaram K. Narayan, USA Renxing Ni, China Kasso A. Okoudjou, USA ´ Gestur Olafsson, USA Josip E. Peˇcarić, Croatia

´ Peláez, Spain José A. Lars E. Persson, Sweden Adrian Petrusel, Romania Konrad Podczeck, Austria José Rodr´ıguez, Spain Natasha Samko, Portugal Simone Secchi, Italy Naseer Shahzad, Saudi Arabia Mitsuru Sugimoto, Japan Rodolfo H. Torres, USA Wilfredo Urbina, USA Nikolai L. Vasilevski, Mexico P. Veeramani, India Igor E. Verbitsky, USA Dragan Vukotic, Spain Bruce A. Watson, South Africa Anthony Weston, USA Quanhua Xu, France Gen QI Xu, China Dachun Yang, China Kari Ylinen, Finland Chengbo Zhai, China Ruhan Zhao, USA Kehe Zhu, USA William P. Ziemer, USA

Contents Geometry of Banach Spaces, Operator Theory, and Their Applications, Genqi Xu, Ji Gao, Pei Liu, and Satit Saejung Volume 2014, Article ID 635649, 1 page Composition Operators on Cesàro Function Spaces, Kuldip Raj, Suruchi Pandoh, and Seema Jamwal Volume 2014, Article ID 501057, 6 pages 𝐿𝑝 Approximation Strategy by Positive Linear Operators, Wei Xiao Volume 2013, Article ID 856084, 10 pages Stability Analysis of a Repairable System with Warning Device and Repairman Vacation, Yangli Ren, Lina Guo, and Lingling Zhang Volume 2013, Article ID 674191, 19 pages On the Distance between Three Arbitrary Points, Parin Chaipunya and Poom Kumam Volume 2013, Article ID 194631, 7 pages On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules, Guo Tiexin Volume 2013, Article ID 989102, 13 pages The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum’s Sense, Z. Jesús, O. Mej´ıa, N. Merentes, and S. Rivas Volume 2013, Article ID 284389, 13 pages Abstract Semilinear Evolution Equations with Convex-Power Condensing Operators, Lanping Zhu, Qianglian Huang, and Gang Li Volume 2013, Article ID 473876, 9 pages On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator, Dinu Teodorescu and N. Hussain Volume 2013, Article ID 902563, 6 pages 0

A Minimax Theorem for 𝐿 -Valued Functions on Random Normed Modules, Shien Zhao and Yuan Zhao Volume 2013, Article ID 704251, 10 pages New Sequence Spaces and Function Spaces on Interval [0, 1], Cheng-Zhong Xu and Gen-Qi Xu Volume 2013, Article ID 601490, 10 pages Semicommutators and Zero Product of Block Toeplitz Operators with Harmonic Symbols, Puyu Cui, Yufeng Lu, and Yanyue Shi Volume 2013, Article ID 656034, 4 pages On Supra-Additive and Supra-Multiplicative Maps, Jin Xi Chen and Zi Li Chen Volume 2013, Article ID 108535, 3 pages On the Aleksandrov-Rassias Problems on Linear 𝑛-Normed Spaces, Yumei Ma Volume 2013, Article ID 394216, 7 pages

Construction of Frames for Shift-Invariant Spaces, Stevan Pilipović and Suzana Simić Volume 2013, Article ID 163814, 7 pages Suzuki-Type Fixed Point Results in Metric-Like Spaces, Nabiollah Shobkolaei, Shaban Sedghi, Jamal Rezaei Roshan, and Nawab Hussain Volume 2013, Article ID 143686, 9 pages Maximal and Area Integral Characterizations of Bergman Spaces in the Unit Ball of C𝑛 , Zeqian Chen and Wei Ouyang Volume 2013, Article ID 167514, 13 pages On a Class of Anisotropic Nonlinear Elliptic Equations with Variable Exponent, Guoqing Zhang and Hongtao Zhang Volume 2013, Article ID 247628, 9 pages Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set, Jinli Xu, Baodong Zheng, and Hongmei Yao Volume 2013, Article ID 182569, 7 pages Exponential Stability of a Linear Distributed Parameter Bioprocess with Input Delay in Boundary Control, Fu Zheng, LingLing Fu, and MingYan Teng Volume 2013, Article ID 274857, 9 pages On the Gauss Map of Surfaces of Revolution with Lightlike Axis in Minkowski 3-Space, Minghao Jin, Donghe Pei, and Shu Xu Volume 2013, Article ID 130495, 8 pages

Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 635649, 1 page http://dx.doi.org/10.1155/2014/635649

Editorial Geometry of Banach Spaces, Operator Theory, and Their Applications Genqi Xu,1 Ji Gao,2 Peide Liu,3 and Satit Saejung4 1

Department of Mathematics, Tianjin University, Tianjin 300072, China Department of Mathematics, Community College of Philadelphia, Philadelphia, PA 19130-3991, USA 3 College of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 4 Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand 2

Correspondence should be addressed to Genqi Xu; [email protected] Received 2 January 2014; Accepted 2 January 2014; Published 18 February 2014 Copyright © 2014 Genqi Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In recent years, with developments of computer, high technique, and life science, more and more requirements were proposed to functional analysis. There were a large number of research articles published in the relative area and many major discoveries have been obtained. At the present time the ideas, terminology, and methods of functional analysis have penetrated deeply, not only into natural sciences, but also into applied disciplines. Many real world problems were analyzed and solved by using the ideas and results of function analysis. To fit the requirements, we edit this special issue. The special issue represented the recent development in geometric structure of Banach spaces, Bergman spaces, and Minkowski spaces, random normed modules on function space, linear and nonlinear operator theory, and the applications of modern analysis in related areas of mathematics, as well as other disciplines, such as economics, finance and risk management, dynamic systems, the natural and life sciences, medicine, physics, and other real world problems. The special issue consisted of twenty research articles contributed by participants of the 4th International Conference on Modern Analysis and Its Applications which was held at the Tianjin University, China, from August 1 to 4, 2013. With over 240 participants from different countries, the conference proved highly successful in bringing together experts and researchers and provided an opportunity to evaluate and disseminate new ideas and methods in modern analysis. Each paper has been peer-reviewed and the editors thank the referees for their time and efforts to help ensure a high standard for this special issue.

We hope that readers interested in function spaces and operator theory will find in this special issue not only new information about development of functional analysis, but also important questions be resolved and directions for further study. At same time, we hope that in the near future, we can see new results to be published based on this special issue.

Acknowledgment The auditors acknowledge Chinese Mathematical Society, National Nature Science Foundation of China, and the Department of Mathematics of Tianjin University, China. Without their support the conference would not have been possible. Thanks are also to editorial office of the Journal of Function Spaces and Applications. Genqi Xu Ji Gao Peide Liu Satit Saejung

Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 501057, 6 pages http://dx.doi.org/10.1155/2014/501057

Research Article Composition Operators on Cesàro Function Spaces Kuldip Raj, Suruchi Pandoh, and Seema Jamwal School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, India Correspondence should be addressed to Kuldip Raj; [email protected] Received 17 May 2013; Accepted 20 November 2013; Published 30 January 2014 Academic Editor: Satit Saejung Copyright © 2014 Kuldip Raj et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The compact, invertible, Fredholm, and closed range composition operators are characterized. We also make an effort to compute the essential norm of composition operators on the Cesàro function spaces.

1. Introduction and Preliminaries Let (𝑋, 𝑠, 𝜇) be a 𝜎-finite measure space and let 𝐿0 = 𝐿0 (𝑋) denote the set of all equivalence classes of complex valued measurable functions defined on 𝑋, where 𝑋 = [0, 1] or 𝑋 = [0, ∞). Then, for 1 ≤ 𝑝 < ∞, the Cesàro function space is denoted by Ces𝑝 (𝑋) and is defined as Ces𝑝 (𝑋) 𝑝 1 𝑥󵄨 󵄨 = {𝑓 ∈ 𝐿0 (𝑋) : ∫ ( ∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝜇 (𝑡)) 𝑑𝜇 (𝑥) < ∞} . 𝑋 𝑥 0 (1)

The Cesàro function space Ces𝑝 (𝑋) is a Banach space under the norm 1/𝑝

𝑝 1 𝑥󵄨 󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩 = (∫ ( ∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝜇 (𝑡)) 𝑑𝜇 (𝑥)) 𝑋 𝑥 0

;

(2)

see [1]. The Cesàro functions spaces Ces𝑝 [0, ∞) for 1 ≤ 𝑝 ≤ ∞ were considered by Shiue [2], Hassard and Hussein [3], and Sy et al. [4]. The space Ces∞ [0, 1] appeared already in 1948 and it is known as the Korenblyum, Krein, and Levin space 𝐾 (see [5, 6]). Recently, in [7], it is proved that, in contrast to Cesàro sequence spaces, the Cesàro function spaces Ces𝑝 (𝑋) on both 𝑋 = [0, 1] and 𝑋 = [0, ∞) for 1 < 𝑝 < ∞ are not reflexive and they do not have the fixed point property. In [8], Astashkin and Maligranda investigated Rademacher sums in Ces𝑝 [0, 1] for 1 ≤ 𝑝 ≤ ∞. The description is different for 1 ≤ 𝑝 < ∞ and 𝑝 = ∞.

Let 𝑇 : 𝑋 → 𝑋 be a nonsingular measurable transformation; that is, 𝜇𝑇−1 (𝐴) = 𝜇(𝑇−1 (𝐴)) = 0, for each 𝐴 ∈ 𝑠, whenever 𝜇(𝐴) = 0. This condition means that the measure 𝜇𝑇−1 is absolutely continuous with respect to 𝜇. Let 𝑓0 = 𝑑𝜇𝑇−1 /𝑑𝜇 be the Radon-Nikodym derivative. In addition, we assume that 𝑓0 is almost everywhere finite valued or equivalently that (𝑋, 𝑇−1 (s), 𝜇) is 𝜎-finite. An atom of the measure 𝜇 is an element 𝐴 ∈ 𝑠 with 𝜇(𝐴) > 0 such that, for each 𝐹 ∈ 𝑠, if 𝐹 ⊂ 𝐴, then either 𝜇(𝐹) = 0 or 𝜇(𝐹) = 𝜇(𝐴). Let 𝐴 be an atom. Since 𝜇 is 𝜎-finite, it follows that 𝜇(𝐴) < ∞. Also every 𝑠-measurable function 𝑓 on 𝑋 is constant almost everywhere on 𝐴. It is a well-known fact that every sigma finite measure space (𝑋, 𝑠, 𝜇) can be decomposed into two disjoint sets 𝑋1 and 𝑋2 such that 𝜇 is atomic over 𝑋1 and 𝑋2 is a countable collection of disjoint atoms (see [9]). Any nonsingular measurable transformation 𝑇 induces a linear operator 𝐶𝑇 from Ces𝑝 (𝑋) into the linear space of equivalence classes of 𝑠-measurable functions on 𝑋 defined by 𝐶𝑇 𝑓 = 𝑓 ∘ 𝑇, 𝑓 ∈ Ces𝑝 (𝑋). Hence, the nonsingularity of 𝑇 guarantees that the operator 𝐶𝑇 is well defined. If 𝐶𝑇 takes Ces𝑝 (𝑋) into itself, then we call that 𝐶𝑇 is a composition operator on Ces𝑝 (𝑋). By 𝐵(Ces𝑝 (𝑋)), we denote the set of all bounded linear operators from Ces𝑝 (𝑋) into itself. So far as we know, the earliest appearance of a composition transformation was in 1871 in a paper of Schrljeder [10], where it is asked to find a function 𝑓 and a number 𝛼 such that (𝑓 ∘ 𝑇) (𝑧) = 𝛼𝑓 (𝑧) ,

(3)

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Journal of Function Spaces

for every 𝑧, in an appropriate domain, whenever the function 𝑇 is given. If 𝑧 varies in the open unit disk and 𝑇 is an analytic function, then a solution is obtained by Köenigs [11]. In 1925, these operators were employed in Littlewood’s subordination theory [12]. In the early 1930s, the composition operators were used to study problems in mathematical physics and especially classical mechanics; see Koopman [13]. In those days, these operators were known as substitution operators. The systematic study of composition operators has relatively a very short history. It was started by Nordgren in 1968 in his paper [14]. After this, the study of composition operators has been extended in several directions by several mathematicians. For more details on composition operators, see [15–27] and references therein. Associated with each 𝜎-finite subalgebra 𝑠0 ⊂ 𝑠, there exists an operator 𝐸 = 𝐸𝑠0 , which is called conditional expectation operator; on the set of all nonnegative measurable functions 𝑓 or for each 𝑓 ∈ 𝐿0 (𝑋, 𝑠, 𝜇), the operator 𝐸 is uniquely determined by the following conditions:

𝐶𝑇 : Ces𝑝 (𝑋) → Ces𝑝 (𝑋) under certain conditions on 𝑝 and 𝑋. The main purpose of this paper is to characterize the boundedness, compactness, closed range, and Fredholmness of composition operators on Cesàro function spaces. We also make an effort to compute the essential norm of composition operators in Section 3 of this paper.

2. Composition Operators In this section of the paper, we will investigate the necessary and sufficient condition for a composition operator to be bounded. Theorem 1. Let (𝑋, 𝑠, 𝜇) be a 𝜎-finite measure space and let 𝑇 : 𝑋 → 𝑋 be nonsingular measurable transformation. Then, 𝑇 induces a composition operator 𝐶𝑇 on Ces𝑝 (𝑋) if and only if there exists 𝑀 > 0 such that 𝜇𝑇−1 (𝐸) ≤ 𝑀𝜇(𝐸) for every 𝐸 ∈ 𝑠. Moreover,

(i) 𝐸(𝑓) is 𝑠0 -measurable; (ii) if 𝐴 is any 𝑠0 -measurable set for which ∫𝐴 𝑓𝑑𝜇 exists, we have ∫𝐴 𝑓𝑑𝜇 = ∫𝐴 𝐸(𝑓)𝑑𝜇. The expectation operator 𝐸 has the following properties:

󵄩󵄩 󵄩󵄩 󵄩󵄩𝐶𝑇 󵄩󵄩 =

sup ((

0 0 almost everywhere. For deeper study of properties of 𝐸, see [28]. Let B be a Banach space and let K be the set of all compact operators on B. For 𝑈 ∈ L(B), the Banach algebra of all bounded linear operators on B into itself, the essential norm of 𝑈 means the distance from 𝑈 to K in the operator norm; namely, ‖𝑈‖𝑒 = inf {‖𝑈 − 𝑆‖ : 𝑆 ∈ K} .

(5)

(4)

Clearly, 𝑈 is compact if and only if ‖𝑈‖𝑒 = 0. As seen in [29], the essential norm plays an interesting role in the compact problems of concrete operators. Many people have computed the essential norm of various concrete operators. For the study of essential norm of composition operators, see [30–33] and reference therein. The question of actually calculating the norm and essential norm of composition operators on Cesàro function spaces is not a trivial one. In spite of the difficulties associated with computing the essential norm exactly, it is often possible to find upper and lower bound for the essential norm of

𝜇𝑇−1 (𝐸) ≤ 𝑀𝜇 (𝐸) .

(6)

(7)

Conversely, suppose that the condition is true. Then, 𝜇𝑇−1 ≪ 𝜇 and hence the Radon-Nikodym derivative 𝑓0 of 𝜇𝑇−1 with respect to 𝜇 exists and 𝑓0 ≤ 𝑀 a.e. Let 𝑓 ∈ Ces𝑝 (𝑋). Then, 𝑝 1 𝑥󵄨 󵄨 󵄩󵄩 󵄩𝑝 󵄩󵄩𝐶𝑇 𝑓󵄩󵄩󵄩 = ∫ ( ∫ 󵄨󵄨󵄨(𝑓 ∘ 𝑇) (𝑡)󵄨󵄨󵄨 𝑑𝜇 (𝑡)) 𝑑𝜇 (𝑥) 𝑋 𝑥 0 𝑝 1 𝑥󵄨 󵄨 ≤ ∫ ( ∫ 󵄨󵄨󵄨𝑓󵄨󵄨󵄨 𝑑𝜇𝑇−1 (𝑡)) 𝑑𝜇 (𝑥) 𝑋 𝑥 0 𝑝 1 𝑥󵄨 󵄨 = ∫ ( ∫ 󵄨󵄨󵄨𝑓󵄨󵄨󵄨 𝑓0 𝑑𝜇 (𝑡)) 𝑑𝜇 (𝑥) 𝑋 𝑥 0

(8)

𝑝 1 𝑥󵄨 󵄨 ≤ 𝑀𝑝 ∫ ( ∫ 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 𝑑𝜇 (𝑡)) 𝑑𝜇 (𝑥) . 𝑋 𝑥 0

Therefore, ‖𝐶𝑇 𝑓‖ ≤ 𝑀‖𝑓‖. Now, Let 𝑁 = sup0 0, a constant depending sometimes upon 𝑝 as well as the kernels {K𝑛,𝑗 (𝑥)} (e.g., the kernels of the Shepard operators; it contains a parameter 𝜆; see (10)), is called Jackson constant. Since the kernel {K𝑛,𝑗 (𝑥)} can have two types, decided by the summation indices, respectively (see (1)), the Kantorovich type operators are therefore defined by (2) or (3). However, with similar arguments, we can only investigate the positive linear operators of the form of (3). Sometimes, we need to write 𝑄𝑛 (𝑥, 𝑡) = K𝑛,𝑗 (𝑥) for (𝑗 − 1)/𝑛 ≤ 𝑡 < 𝑗/𝑛, 𝑗 = 1, 2, . . . , 𝑛; then the operators (3) can be written as an integral form:

𝑗/𝑛

(𝑗−1)/𝑛

𝑓 (𝑡) 𝑑𝑡,

(3)

0

and the 𝐿 approximation is characterized by (4)

where 𝜖𝑛 , a positive number depending on 𝑛 only (such as 𝑛−1/2 , 𝑛−1 ), is called Jackson order of 𝐿𝑝 approximation; and

(5)

hence, 1

∫ 𝑄𝑛 (𝑥, 𝑡) 𝑑𝑡 = 𝑛−1 . 0

(6)

This means, for Kantorovich type operators, there does not exist any difference whether (3) or (5) is taken. In particular, we give the following examples. When the kernel function 𝑄𝑛 (𝑥, 𝑡) satisfies 𝜕 𝑛 𝑄 (𝑥, 𝑡) (𝑡 − 𝑥) , 𝑄 (𝑥, 𝑡) = 𝜕𝑥 𝑛 𝜙 (𝑥) 𝑛

𝑝

󵄩 󵄩󵄩 󵄩󵄩𝑓 − 𝐿 𝑛 (𝑓)󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝜔(𝑓, 𝜖𝑛 )𝐿𝑝 ,

𝐿 𝑛 (𝑓, 𝑥) = 𝑛 ∫ 𝑄𝑛 (𝑥, 𝑡) 𝑓 (𝑡) 𝑑𝑡;

(7)

where 𝜙(𝑥) > 0 satisfying 𝜙(0) = 𝜙(1) = 0 is a polynomial with a degree at most 2, then the operators 𝐿 𝑛 (𝑓, 𝑥) have exponential type, which have been studied deeply in [1–3], for example.

2

Journal of Function Spaces and Applications 𝑝

and the 𝐿𝑝 modulus of continuity of 𝑓 ∈ 𝐿 [0,1] is defined by

When the kernel function K𝑛,𝑗 (𝑥) in (3) is taken as 𝑅𝑗 (𝑥) =

𝑃𝑗 (𝑥) , 𝑛 ∑𝑘=1 𝑃𝑘 (𝑥)

𝜔(𝑓, 𝛿)𝐿𝑝 = 𝜔(𝑓, 𝛿)𝐿𝑝

[0,1−𝛿]

(8)

𝑘 𝑙 −Δ𝜆 𝑙 𝑃𝑘 (𝑥) = 𝑥𝜆 𝑘 ∏( ) , 𝑛 𝑙=1

where {𝑥𝜆 0 , 𝑥𝜆 1 , . . . , 𝑥𝜆 𝑛 , . . .} is a Müntz system satisfying that Δ𝜆 𝑛 = 𝜆 𝑛 − 𝜆 𝑛−1 ≥ 𝑀𝑛 (𝑀 is a given positive number), then the operator 𝐿 𝑛 (𝑓, 𝑥) becomes the rational Müntz operator: 𝑛

𝑀𝑛 (𝑓, 𝑥) = 𝑛 ∑ 𝑅𝑘 (𝑥) ∫

𝑘/𝑛

(𝑘−1)/𝑛

𝑘=1

𝑓 (𝑡) 𝑑𝑡;

(9)

readers can refer to [4, 5]. When the kernel function K𝑛,𝑗 (𝑥) of (2) is taken as 𝑟𝑗 (𝑥) =

|𝑥 − 𝑗/𝑛|−𝜆 ∑𝑛𝑘=0 |𝑥 − 𝑘/𝑛|−𝜆

(𝜆 > 1) ,

(10)

then the operator 𝐿 𝑛 (𝑓, 𝑥) is the well-known Shepard operator 𝑛

𝑆𝑛,𝜆 (𝑓, 𝑥) = (𝑛 + 1) ∑ 𝑟𝑘 (𝑥) ∫

(𝑘+1)/(𝑛+1)

𝑘/(𝑛+1)

𝑘=0

𝑓 (𝑡) 𝑑𝑡;

(11)

one can check [6, 7], for example. When the kernel K𝑛,𝑗 (𝑥) in (2) is taken as 𝑛 𝑃𝑛,𝑗 (𝑥) = ( ) 𝑥𝑗 (1 − 𝑥)𝑛−𝑗 , 𝑗

(12)

then we obtain the Kantorovich-Bernstein operator 𝑛

𝐵𝑛 (𝑓, 𝑥) = (𝑛 + 1) ∑ 𝑃𝑛,𝑘 (𝑥) ∫

(𝑘+1)/(𝑛+1)

𝑘/(𝑛+1)

𝑘=0

󵄩 󵄩 = sup 󵄩󵄩󵄩𝑓(𝑥 + 𝑡) − 𝑓(𝑥)󵄩󵄩󵄩𝐿𝑝 . [0,1−𝛿]

𝑓 (𝑡) 𝑑𝑡,

(13)

which has been studied most widely among the positive linear operators of the form (2) (see [8–23]). Interested readers could also refer to the related papers for the other similar operators. This paper will take the above three typical operators (9), (11), and (13) as examples to illustrate two quantitative methods on 𝐿𝑝 approximation. Over discussion, we find out that the Jackson order in 𝐿𝑝 spaces to approximate 𝑓(𝑥) ∈ 𝑝 𝐿 [0,1] by the operators in (3) or (5) is decided completely by the kernels {K𝑛,𝑗 (𝑥)}𝑛𝑗=1 , or by the kernel function 𝑄𝑛 (𝑥, 𝑡). Therefore, on applying this idea, we need only to investigate the properties of the kernels to obtain the magnitude of the Jackson order of the corresponding operators, which seems to be a different approach from the past 𝐿𝑝 approximating methods.

(15)

0 0, 𝑘 = 1, 2, . . ., then the kernel sequence K𝑛 (𝑥) is called a globally dominated arithmetic sequence with power 𝜌, or being dominated by a global arithmetic sequence with power 𝜌, or having (global) 𝜌-arithmetic order. Definition 2. For any 𝑥 ∈ [0, 1], if there exist a sequence {Φ(𝑘)}∞ 𝑘=1 with 0 < Φ(𝑘) < 1, 𝑘 = 1, 2, . . . and an integer subset 𝑁𝑥 ⊂ {1, 2, . . . , 𝑛} satisfying that K𝑛,𝑟 (𝑥) ≤ 𝐶Φ (|[𝑛𝑥] − 𝑟| + 1) ,

𝑟 ∈ 𝑁𝑥 ,

(17)

then {Φ(𝑘)}∞ 𝑘=1 is called a local domination of K𝑛 (𝑥). Like in Definition 1, a locally dominated geometrical sequence of K𝑛 (𝑥) and a locally dominated 𝜌-arithmetic sequence can be defined similarly. The conceptions of Definitions 1 and 2 will be illustrated by the following examples. Example 3. The kernel functions 𝑅𝑘 (𝑥) of the rational Müntz operators defined by (9) satisfy 𝑅𝑘 (𝑥) ≤ 𝐶𝑀 exp (−𝐶𝑀 |[𝑛𝑥] − 𝑘|) ,

𝑘 = 0, 1, 2, . . . , 𝑛 (18)

2. Notations and Terminologies

(see [4] or [5, Lemma 1]). The kernel functions 𝑅𝑘 (𝑥) then have geometrical order.

In this section, we give all preliminary notations and termi𝑝 nologies. For 𝑓(𝑥) ∈ 𝐿 [0,1] , the usual 𝐿𝑝 norm is defined by

Example 4. The kernels 𝑟𝑘 (𝑥) of the Shepard operators 𝑆𝑛,𝜆 (𝑓, 𝑥) (𝜆 > 1) defined by (11) satisfy

𝑏

1/𝑝

󵄩󵄩 󵄩󵄩 𝑝 𝑝 󵄩󵄩𝑓󵄩󵄩𝐿 = {∫ |𝑓(𝑥)| 𝑑𝑥} [𝑎,𝑏] 𝑎

,

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑓󵄩󵄩𝐿𝑝 = 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝 , (14) [0,1]

𝑟𝑘 (𝑥) ≤ (

𝜆 2 ) , |[(𝑛 + 1) 𝑥] − 𝑘| + 1

𝑘 = 0, 1, 2, . . . , 𝑛 (19)


3

(see [12, Lemma 1]). The kernels 𝑟𝑘 (𝑥) have 𝜆-arithmetic order.

Remark 10. We make a brief discussion on the above definitions.

Example 5. The kernels 𝑃𝑛,𝑘 (𝑥) of the Kantorovich-Bernstein operators (13) satisfy

(1) 𝑀𝑟 (K𝜖𝑛 ) < ∞ or 𝐷𝑀𝑟 (K𝜖𝑛 ) < ∞ for some 0 < 𝜖 < 1 obviously yields that 𝑀𝑟 (K𝑛 ) < ∞ or 𝐷𝑀𝑟 (K𝑛 ) < ∞; the converse may not be true.

𝑃𝑛,𝑘 (𝑥) ≈ (2𝜋𝑛𝑥(1 − 𝑥))−1/2 ⋅ exp (−

2 𝑘 𝑛 ( − 𝑥) ) , 2𝑥 (1 − 𝑥) 𝑛 (20)

where 𝑥 ∈ (0, 1), and 𝑘 satisfies 󵄨󵄨 𝑘 󵄨󵄨 󵄨󵄨 󵄨 −𝛼 󵄨󵄨 − 𝑥󵄨󵄨󵄨 ≤ 𝑛 󵄨󵄨 𝑛 󵄨󵄨

(3) The kernels having 𝜌-arithmetic order satisfy 𝐷(K𝜖𝑛 ) < ∞ for 𝜌 > 1. (21)

for real number 𝛼 > 1/3 (see [15, Theorem 1.5.2]). That is to say, if the set of all 𝑘 satisfying (21) is written as 𝑁𝑥 , then, while 𝑘 ∈ 𝑁𝑥 , 𝑃𝑛,𝑘 (𝑥) has asymptotic expression (20). Hence, the kernels 𝑃𝑛,𝑘 (𝑥) of Bernstein operators, from Definition 2, have local geometrical order. Definition 6. For 𝑟 = 0, 1, . . ., and any 𝑥 ∈ [0, 1], denote 𝑛

𝑀𝑟 (K𝑛 ) := sup ∑|𝑛𝑥 − 𝑗|𝑟 K𝑛,𝑗 (𝑥) , 𝑛≥1 𝑗=1

(2) The kernels having geometrical order satisfy 𝑀𝑟 (K𝜖𝑛 ) < ∞ and 𝐷𝑀𝑟 (K𝜖𝑛 ) < ∞ for all 𝑟 and 0 < 𝜖 ≤ 1.

(22)

3. Elementary Approximation Technique (I) This section gives one of the approximation techniques in 𝐿𝑝 spaces by the Kantorovich type operators (3). We mainly apply 𝐾-functional and maximum principle to obtain the Jackson type estimation in 𝐿𝑝 spaces. Theorem 11. Let {𝛿𝑛 } be a positive null sequence. For any 𝑝 𝑓(𝑥) ∈ 𝐿 [0,1] , 1 < 𝑝 < ∞, the Kantorovich operators defined by (3) satisfy (i) ‖𝐿 𝑛 (𝑓)‖𝐿𝑝 which is uniformly bounded; (ii) 𝐿 𝑛 (|𝑡 − 𝑥|, 𝑥) = 𝑂(𝛿𝑛 ).

particularly, 𝑀 (K𝑛 ) = 𝑀1 (K𝑛 ) , 𝐷 (K𝑛 ) = 𝑀0 (K𝑛 ) .

(23)

(28)

holds, where 𝐶𝑝 is a positive constant depending upon 𝑝 only.

Definition 7. For 𝑟 = 1, 2, . . ., write

𝑝

𝑛

𝐷𝑀𝑟 (K𝑛 ) := sup ∑ 𝑘𝑟 Φ (𝑘) , 𝑛≥1 𝑘=1

(24)

where {Φ(𝑘)}∞ 𝑘=1 is the global domination of K𝑛 (𝑥) in Definition 1. In particular, 𝐷𝑀 (K𝑛 ) = 𝐷𝑀1 (K𝑛 ) .

(25)

∞

𝐷 (𝑅𝑘𝜖 (𝑥)) ≤ 𝐶𝑀 ∑ exp (−𝐶𝑀𝜖𝑘) < ∞,

(26)

𝑘=0

where the kernels 𝑅𝑘 (𝑥) (8), as well as Φ(𝑘) = 𝑒−𝐶𝑀 𝑘 , from Definitions 6 and 7, satisfy 𝑀𝑟 (K𝜖𝑛 ) < ∞ and 𝐷𝑀𝑟 (K𝜖𝑛 ) < ∞ for all 𝑟 = 1, 2, . . . and any given 0 < 𝜖 ≤ 1. Example 9. Propose that 𝜆 > 1, write 𝜖𝜆 = (𝜆 + 1)/2. In view of Example 4, we get ∞

𝐷 (𝑟𝑘𝜆 (𝑥)) ≤ 2𝜆(𝜆+1)/2 ∑ 𝑘−𝜆(𝜆+1)/2 < ∞.

Proof. For any function 𝑓(𝑥) ∈ 𝐿 [0,1] , from the definition of the 𝐾-functional 󵄩 󵄩 󵄩 󵄩 𝐾(𝑓, ℎ)𝐿𝑝 = inf 󸀠 𝑝 (󵄩󵄩󵄩𝑓 − 𝑔󵄩󵄩󵄩𝐿𝑝 + ℎ󵄩󵄩󵄩󵄩𝑔󸀠 󵄩󵄩󵄩󵄩𝐿𝑝 ) , (29) 𝑔∈𝐴𝐶 ,𝑔 ∈𝐿 [0,1]

where 𝐴𝐶[0,1] indicates the set of all absolute continuous functions on the interval [0, 1], we know that 𝐾(𝑓, ℎ)𝐿𝑝 and 𝜔(𝑓, ℎ)𝐿𝑝 are equivalent (see [24, Theorem 2.1]); that is, 𝜔(𝑓, ℎ)𝐿𝑝 ≈ 𝐾(𝑓, ℎ)𝐿𝑝 ,

Example 8. For any 𝜖 ∈ (0, 1), from Example 3, we have

𝜖

Then, the estimate 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑝 𝜔(𝑓, 𝛿𝑛 )𝐿𝑝

(27)

𝑘=1

Thus, from Definitions 6 and 7, the kernels 𝑟𝑘 (𝑥) (10), as well as Φ(𝑘) = 𝑘−𝜆 , satisfy 𝑀𝑟 (K𝜖𝑛𝜆 ) < ∞ and 𝐷𝑀𝑟 (K𝜖𝑛𝜆 ) < ∞ for some 𝑟.

(30)

where 𝐴 ℎ ≈ 𝐵ℎ we mean there is a constant 𝐶 > 0 independent of ℎ such that 𝐶−1 𝐴 ℎ ≤ 𝐵ℎ ≤ 𝐶𝐴 ℎ . 𝑝 For given 𝑓(𝑥) ∈ 𝐿 [0,1] , the Hardy-Littlewood maximum function is defined by 1 𝛿󵄨 󵄨 𝑀 (𝑓, 𝑥) = sup ∫ 󵄨󵄨󵄨𝑓 (𝑥 + 𝑢)󵄨󵄨󵄨 𝑑𝑢, 𝛿 0 𝛿>0

(31)

where if 𝑥 + 𝑢 ∉ [0, 1], we simply set 𝑓(𝑥 + 𝑢) = 0. It is well known that (see [25]) 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑀(𝑓)󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑝 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝 , 𝑝 > 1. (32) 𝑝

Since for any 𝑔(𝑥) ∈ 𝐴𝐶[0,1] , 𝑔󸀠 (𝑥) ∈ 𝐿 [0,1] , 𝑝 > 1, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 󵄩󵄩󵄩𝐿 𝑛 (𝑓 − 𝑔) − (𝑓 − 𝑔)󵄩󵄩󵄩𝐿𝑝 󵄩 󵄩 + 󵄩󵄩󵄩𝐿 𝑛 (𝑔) − 𝑔󵄩󵄩󵄩𝐿𝑝 ,

(33)

4


then the condition (i) of Theorem 11 induces that 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶󵄩󵄩󵄩𝑓 − 𝑔󵄩󵄩󵄩𝐿𝑝 + 󵄩󵄩󵄩𝐿 𝑛 (𝑔) − 𝑔󵄩󵄩󵄩𝐿𝑝 .

(34)

then, the estimate 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝜔(𝑓, 1/𝑛)𝐿𝑝 [0,1] [0,1] holds, where 𝐶 > 0 is an absolute constant.

It is easy to deduce from definition (3) that 󵄨 󵄨󵄨 󵄨󵄨𝐿 𝑛 (𝑔, 𝑥) − 𝑔 (𝑥)󵄨󵄨󵄨

To prove Theorem 12, we first give two lemmas.

𝑛

≤ 𝑛 ∑K𝑛,𝑗 (𝑥) ∫

𝑗/𝑛

(𝑗−1)/𝑛

𝑗=1

𝑝

Lemma 13. Given ℎ with 0 < ℎ < 1 and 𝑓(𝑥) ∈ 𝐿 [0,1] , write the Steklov function of 𝑓(𝑥) as

󵄨 󵄨󵄨 󵄨󵄨𝑔 (𝑡) − 𝑔 (𝑥)󵄨󵄨󵄨 𝑑𝑡

󵄨󵄨 𝑡 󵄨󵄨 󵄨󵄨 󵄨 󸀠 = 𝑛 ∑K𝑛,𝑗 (𝑥) ∫ 󵄨󵄨∫ 𝑔 (𝑢) 𝑑𝑢󵄨󵄨󵄨 𝑑𝑡 󵄨 󵄨󵄨 (𝑗−1)/𝑛 𝑥 󵄨 𝑗=1 𝑛

ℎ

𝑗/𝑛

󸀠

𝑛

𝑗/𝑛

≤ 𝑛𝑀 (𝑔 , 𝑥) ∑K𝑛,𝑗 (𝑥) ∫ 𝑗=1

(𝑗−1)/𝑛

𝑓ℎ (𝑥) = ℎ−1 ∫ 𝑓 (𝑥 + 𝑢) 𝑑𝑢,

(35)

where 𝑥 + 𝑢 ∈ [0, 1]. Then, the following results hold:

|𝑡 − 𝑥| 𝑑𝑡

𝑓 (𝑥 + ℎ) − 𝑓 (𝑥) ; ℎ 󵄩 󵄩 ≤ 𝜔(𝑓, ℎ)𝐿𝑝 . (ii) 󵄩󵄩󵄩𝑓 − 𝑓ℎ 󵄩󵄩󵄩𝐿𝑝 [0,1−ℎ]

(i) 𝑓ℎ󸀠 (𝑥) =

With the condition (ii) in Theorem 11, we obtain 󵄨󵄨 󵄨 󸀠 󵄨󵄨𝐿 𝑛 (𝑔, 𝑥) − 𝑔 (𝑥)󵄨󵄨󵄨 ≤ 𝐶𝑀 (𝑔 , 𝑥) 𝛿𝑛 .

(36)

(37)

(38)

Finally, from the definition of the 𝐾-functional we achieve that 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑝 𝐾(𝑓, 𝛿𝑛 )𝐿𝑝 .

(39)

Therefore, in view of (30) and (39), we have completed Theorem 11.

4. Elementary Approximation Technique (II) In Section 3, by applying the 𝐾-functional, we obtain the Jackson type estimation in 𝐿𝑝 spaces for 𝑝 > 1. However, the Jackson constant in that case must depend upon 𝑝, and thus we cannot establish corresponding result in 𝐿1 space! In this section, we will exhibit another efficient technique in 𝐿𝑝 spaces which will be used to obtain Jackson constant independent of 𝑝! 𝑝

Theorem 12. Let 𝑓(𝑥) ∈ 𝐿 [0,1] , 1 ≤ 𝑝 < ∞, an 𝜖 be given with 0 < 𝜖 < 1 and the positive linear operators 𝐿 𝑛 (𝑓, 𝑥) defined by (3). If the kernels K𝑛 (𝑥) with (1) are dominated globally by {Φ(𝑘)}∞ 𝑘=1 , and for some 0 < 𝜖 < 1 satisfy the following conditions: (i) (ii)

(43)

Proof. Equations (42) and (43) can be directly verified from the definition of the Steklov function (41).

∫

𝑎+𝛿

𝑎

Together with (34) and (37), we get 󵄩 󸀠󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶󵄩󵄩󵄩𝑓 − 𝑔󵄩󵄩󵄩𝐿𝑝 + 𝐶𝑝 𝛿𝑛 󵄩󵄩󵄩󵄩𝑔 󵄩󵄩󵄩󵄩𝐿𝑝 .

(42)

Lemma 14. Propose that 0 ≤ 𝑎, 𝑏 ≤ 1, 0 < 𝛿 < 1, 𝑎 + 𝛿 ≤ 1, 𝑝 and 𝑏 + 𝛿 ≤ 1, then for any 𝑓(𝑥) ∈ 𝐿 [0,1] , one has

Combining (32) and (36) leads to 󵄩 󸀠󵄩 󵄩󵄩 󵄩 󵄩󵄩𝐿 𝑛 (𝑔) − 𝑔󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑝 𝛿𝑛 󵄩󵄩󵄩󵄩𝑔 󵄩󵄩󵄩󵄩𝐿𝑝 .

(41)

0

= 𝑀 (𝑔󸀠 , 𝑥) 𝐿 𝑛 (|𝑡 − 𝑥| , 𝑥) .

𝐷(K1−𝜖 𝑛 ) < ∞, 𝐷𝑀(K𝜖𝑛 ) < ∞,

(40)

∫

𝑏+𝛿

𝑏

󵄨𝑝 󵄨󵄨 𝑝 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦 ≤ 2𝛿𝜔 (𝑓, |𝑎 − 𝑏| + 𝛿)𝐿𝑝 . (44)

Proof. This Lemma is proved in [7]; we give a sketch here for self-completeness. Due to the symmetries on 𝑎 and 𝑏, as well as on 𝑥 and 𝑦, we need only to prove the lemma under 𝑏 ≥ 𝑎. By calculations, 𝑎+𝛿

∫

𝑎

∫

𝑏+𝛿

𝑏

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 𝑑𝑥 𝑑𝑦

= 2∫

𝑎+𝛿

𝑎

= 2∫

𝑎+𝛿

𝑎

= 2∫

𝑏+𝛿

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 𝑑𝑦) 𝑑𝑥 𝑏−𝑎+𝑥

(∫

𝑏+𝛿−𝑥

(∫

𝑏−𝑎

𝑏−𝑎+𝛿

𝑏−𝑎

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑥 + 𝑦)󵄨󵄨󵄨 𝑑𝑦) 𝑑𝑥

𝑏+𝛿−𝑦

(∫

𝑎

(45)

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑥 + 𝑦)󵄨󵄨󵄨 𝑑𝑥) 𝑑𝑦

≤ 2𝛿𝜔𝑝 (𝑓, 𝑏 − 𝑎 + 𝛿)𝐿𝑝 . Lemma 14 is done. Proof of Theorem 12. Take ℎ = 1/𝑛 in the Steklov function (41); check 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 [1−ℎ,1] 1

= {∫

1−ℎ

𝑛

(𝑛∑K𝑛,𝑟 (𝑥) ∫ 𝑟=1

𝑟/𝑛

(𝑟−1)/𝑛

𝑝

1/𝑝

(𝑓 (𝑡) − 𝑓 (𝑥)) 𝑑𝑡) 𝑑𝑥} (46)


5 Now we verify the case when 𝑥 ∈ [0, 1 − 1/𝑛]. It is

by applying Minkowski inequality, we get 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 [1−ℎ,1] 𝑛

≤ 𝑛∑{∫

1

1−1/𝑛

𝑟=1

K𝑝𝑛,𝑟 (𝑥) (∫

𝑟/𝑛

(𝑟−1)/𝑛

(𝑟+1)/𝑛

𝑝

󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝑡) − 𝑓 (𝑥)󵄨󵄨󵄨 𝑑𝑡) 𝑑𝑥}

1/𝑝

.

𝑓(𝑡)𝑑𝑡 = not difficult to see from Lemma 13 that 𝑛 ∫𝑟/𝑛 𝑓ℎ (𝑟/𝑛). By the definition of Kantorovich type operators, rewrite 𝐿 𝑛 (𝑓, 𝑥) − 𝑓ℎ (𝑥)

(47)

𝑛

= 𝑛∑K𝑛,𝑟 (𝑥) ∫

Then, apply Hölder inequality: 󵄩󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩 𝑝 󵄩𝐿 [1−ℎ,1] 󵄩 𝑛

≤ 𝑛1/𝑝 ∑{∫

1

1−1/𝑛

𝑟=1

K𝑝𝑛,𝑟 (𝑥) ∫

𝑟/𝑛

(𝑟−1)/𝑛

𝑛

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑡) − 𝑓 (𝑥)󵄨󵄨󵄨 𝑑𝑡 𝑑𝑥}

= ∑K𝑛,𝑟 (𝑥) (𝑓ℎ (

1/𝑝

𝑟=1

.

When 𝑥 ∈ [1 − 1/𝑛, 1], [𝑛𝑥] = 𝑛 − 1. From the definition of the domination, we know K𝑛,𝑟 (𝑥) ≤ 𝐶Φ (|[𝑛𝑥] − 𝑟| + 1) = 𝐶Φ (|𝑛 − 𝑟 − 1| + 1) . (49)

𝑛

1−1/𝑛

× {∫

1

1−1/𝑛

1−1/𝑛

0

𝑟=1 𝑟/𝑛

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑡) − 𝑓 (𝑥)󵄨󵄨󵄨 𝑑𝑡 𝑑𝑥} (𝑟−1)/𝑛

∫

≤∫

1/𝑝

.

1−1/𝑛

0

󵄨𝑝 󵄨󵄨 󵄨󵄨𝐿 𝑛 (𝑓, 𝑥) − 𝑓ℎ (𝑥)󵄨󵄨󵄨 𝑑𝑥 󵄨󵄨 𝑛 󵄨󵄨𝑝 (𝑟−1)/𝑛 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨∑K𝑛,𝑟 (𝑥) ∫ 𝑓ℎ (𝑢) 𝑑𝑢󵄨󵄨󵄨 𝑑𝑥 󵄨󵄨 󵄨󵄨 𝑥 󵄨󵄨𝑟=1 󵄨 𝑝−1

𝑛

(∑K𝑝(1−𝜖)/(𝑝−1) (𝑥)) 𝑛,𝑟 𝑟=1

󵄨󵄨 (𝑗−1)/𝑛 󵄨󵄨𝑝 𝑛 󵄨 󵄨 𝑝𝜖 ⋅ ∑K𝑛,𝑗 (𝑥) 󵄨󵄨󵄨󵄨∫ 𝑓ℎ󸀠 (𝑢) 𝑑𝑢󵄨󵄨󵄨󵄨 𝑑𝑥 󵄨 󵄨󵄨 󵄨 𝑥 𝑗=1

(50) Applying Lemma 14, we have 󵄩󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩 𝑝 󵄩 󵄩𝐿 [1−ℎ,1]

𝑓ℎ󸀠 (𝑢) 𝑑𝑢.

[0,1−𝑛−1 ]

=∫

∑Φ (|𝑛 − 𝑟 − 1| + 1)

𝑥

≤∫

1−1/𝑛

0

𝑝

𝑛

(∑K1−𝜖 𝑛,𝑟 (𝑥)) 𝑟=1

𝑛

𝑛−𝑟+1 ≤ 𝐶∑Φ (|𝑛 − 𝑟 − 1| + 1) 𝜔(𝑓, ) 𝑛 𝐿𝑝 𝑟=1

󵄨󵄨𝑝−1 󵄨󵄨 𝑗 − 1 𝑝𝜖 󵄨 󵄨 ⋅ ∑K𝑛,𝑗 (𝑥) 󵄨󵄨󵄨 − 𝑥󵄨󵄨󵄨 󵄨󵄨 󵄨 𝑛 󵄨 𝑗=1 𝑛

𝑛 1 ≤ 𝐶𝜔(𝑓, ) ∑Φ (|𝑛 − 𝑟 − 1| + 1) (𝑛 − 𝑟 + 1) 𝑛 𝐿𝑝 𝑟=1 𝑛 1 ≤ 𝐶𝜔(𝑓, ) ∑ (|𝑛 − 𝑟 − 1| + 1) Φ (|𝑛 − 𝑟 − 1| + 1) . 𝑛 𝐿𝑝 𝑟=1 (51)

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 𝑑𝑥. ⋅ 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝑥 󵄨 Since 𝐷(K1−𝜖 𝑛 ) < ∞, 󵄩󵄩 󵄩𝑝 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓ℎ 󵄩󵄩󵄩𝐿𝑝

[0,1−𝑛−1 ]

Applying (ii), we see that ≤ 𝐶𝑝 ∫

𝑛

󵄨󵄨𝑝−1 󵄨󵄨 𝑗 − 1 𝑝𝜖 󵄨 󵄨 − 𝑥󵄨󵄨󵄨 ∑K𝑛,𝑗 (𝑥) 󵄨󵄨󵄨 󵄨󵄨 󵄨 𝑛 󵄨 𝑗=1

1−1/𝑛 𝑛

0

∑ (|𝑛 − 𝑟 − 1| + 1) Φ (|𝑛 − 𝑟 − 1| + 1) 𝑟=1 𝑛

𝜖

≤ ∑ (|𝑛 − 𝑟 − 1| + 1) Φ (|𝑛 − 𝑟 − 1| + 1)

𝑛−1

= 𝐷𝑀 (K𝜖𝑛 ) < ∞,

𝑛 󵄨󵄨 𝑗 − 1 󵄨󵄨𝑝−1 𝑝𝜖 󵄨 󵄨 − 𝑥󵄨󵄨󵄨 ∑K𝑛,𝑗 (𝑥) 󵄨󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 (𝑙−1)/𝑛 𝑗=1

= 𝐶𝑝 ∑ ∫ 𝑙=1

so that we obtain 1 󵄩󵄩 󵄩 ≤ 𝐶𝜔(𝑓, ) . 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 [1−1/𝑛,1] 𝑛 𝐿𝑝

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 𝑑𝑥 × 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝑥 󵄨

(52)

𝑟=1

(53)

(54)

This leads to 󵄩𝑝 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓ℎ 󵄩󵄩󵄩𝐿𝑝

0

𝑛

(𝑟−1)/𝑛

𝑟=1

𝑓 (𝑡) 𝑑𝑡 − 𝑓ℎ (𝑥)

𝑟−1 ) − 𝑓ℎ (𝑥)) 𝑛

= ∑K𝑛,𝑟 (𝑥) ∫

=∫

Then, 󵄩󵄩 󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 [1−ℎ,1] ≤ 𝐶𝑛

(𝑟−1)/𝑛

𝑟=1

(48)

1/𝑝

𝑟/𝑛

𝑙/𝑛

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 𝑑𝑥 × 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝑥 󵄨

(55)

6

Journal of Function Spaces and Applications 𝑛−1 𝑛

= 𝐶𝑝 ∑ ∑ ∫

1/𝑛

𝑙=1 𝑗=1 0

𝑝𝜖

K𝑛,𝑗 (𝑡 +

𝑙 − 1 󵄨󵄨󵄨󵄨 𝑗 − 𝑙 󵄨󵄨󵄨󵄨𝑝−1 ) 󵄨󵄨 − 𝑡󵄨󵄨 󵄨󵄨 𝑛 󵄨󵄨 𝑛

which leads to 1 󵄩 󵄩󵄩 ≤ 𝐶𝜔(𝑓, ) . 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓ℎ 󵄩󵄩󵄩𝐿𝑝 −1 𝑛 𝐿𝑝 [0,1−𝑛 ]

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 𝑑𝑡 × 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨󵄨 𝑡+((𝑙−1)/𝑛) 󵄨 󵄨

Combining (62) with (43), we get

𝑛−1 𝑛

1 󵄩 󵄩󵄩 ≤ 𝐶𝜔(𝑓, ) . 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 𝑛 𝐿𝑝 [0,1−𝑛−1 ]

󵄨 󵄨 ≤ 𝐶𝑝 𝑛−𝑝 ∑ ∑Φ𝑝𝜖 (󵄨󵄨󵄨𝑙 − 𝑗 − 1󵄨󵄨󵄨 + 1) 𝑙=1 𝑗=1

𝑝−1 󵄨 󵄨 × (󵄨󵄨󵄨𝑙 − 𝑗 − 1󵄨󵄨󵄨 + 1)

For 𝐿1 space, we have the following result while conditions of Theorem 12 can be loosed. (56)

where for 𝑗 > 𝑙,

Proof. The argument of proof is similar, and we can just repeat the corresponding parts of the proof of Theorem 12.

𝑙=1 𝑗=1

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 × 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝜏 󵄨 𝑚=1

Φ𝑝𝜖 (𝑚) 𝑚𝑝−1

|𝑙−𝑗−1|+1=𝑚

1≤𝑙≤𝑛−1,1≤𝑗≤𝑛

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 × 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝜏 󵄨 𝑛

≤ ∑ Φ𝑝𝜖 (𝑚) 𝑚𝑝−1 ⋅ 2𝑚 ⋅ ∫ ≤ 2∫

1−ℎ

0

𝑚=1 1−ℎ

0

(58)

󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢 󵄨 󵄨

𝑚=1

𝑛

∑ 𝑚Φ𝜖 (𝑚) ≤ 𝐷𝑀 (K𝜖𝑛 ) < ∞.

(59)

𝑚=1

This means 1−1/𝑛

󵄩𝑝 󵄩󵄩 ≤ 𝐶𝑝 𝑛−𝑝 ∫ 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓ℎ 󵄩󵄩󵄩𝐿𝑝 [0,1−1/𝑛] 0

󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢. 󵄨 󵄨

(60)

However, from Lemma 13, 1−1/𝑛

0

󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢 󵄨 󵄨 1−1/𝑛

= 𝑛𝑝 ∫

0

󵄨󵄨 󵄨𝑝 󵄨󵄨𝑓 (𝑢 + 1 ) − 𝑓 (𝑢)󵄨󵄨󵄨 𝑑𝑢 󵄨󵄨 󵄨󵄨 𝑛 󵄨 󵄨

1 ≤ 𝑛𝑝 𝜔𝑝 (𝑓, ) , 𝑛 𝐿𝑝

Remark 16. (1) If the kernels possess good properties, the conditions of Theorem 12 can be easily verified on the terminology of domination. For instance, if the kernels have geometric order, then the corresponding conditions of Theorem 12 are obviously satisfied (see the next section). (2) There exists essential difference between Theorem 11 and Theorem 12. Theorem 11 requires weaker conditions than Theorem 12 does, but the latter obtains stronger result (the Jackson order is complete up to 1/𝑛, and the Jackson constant is independent of 𝑝!); we will make further illustrations in the coming section.

5. Applications

𝑝

𝑛 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢 ⋅ ( ∑ 𝑚Φ𝜖 (𝑚)) . 󵄨 󵄨

Furthermore, condition (ii) implies that

∫

(64)

holds.

𝑝−1 󵄨 󵄨 󵄨 󵄨 ∑ ∑ Φ𝑝𝜖 (󵄨󵄨󵄨𝑙 − 𝑗 − 1󵄨󵄨󵄨 + 1) (󵄨󵄨󵄨𝑙 − 𝑗 − 1󵄨󵄨󵄨 + 1)

∑

1 󵄩 󵄩󵄩 󵄩󵄩𝐿 𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿1 ≤ 𝐶𝜔(𝑓, ) 1 𝑛 𝐿

for 𝑗 ≤ 𝑙.

𝑛−1 𝑛

𝑛

Theorem 15. Let 𝑓(𝑥) ∈ 𝐿1[0,1] , 𝐿 𝑛 (𝑓, 𝑥) be defined by (3). If the kernels with (1) are dominated globally by {Φ(𝑘)} and satisfy 𝐷𝑀(K𝑛 ) < ∞, then the estimate

(57)

Note that

= ∑

(63)

This, with (53), finishes Theorem 12.

󵄨󵄨 (𝑗−1)/𝑛 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑝 󵄨󵄨󵄨 󵄨󵄨𝑓ℎ (𝑢)󵄨󵄨 𝑑𝑢󵄨󵄨 , × 󵄨󵄨󵄨󵄨∫ 󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝜏 󵄨

𝑙−1 { { { 𝑛 , 𝜏={ { {𝑙, {𝑛

(62)

(61)

This section illustrates how to apply Theorems 11 and 12 to estimate 𝐿𝑝 approximation. To check the efficiency of two techniques on 𝐿𝑝 approximation by Kantorovich type positive linear operators, three examples will be exhibited. Those positive linear operators come from three different categories: rational Müntz operators from rational Müntz systems; the Shepard operators from general real rational function systems; and Bernstein polynomials from the polynomial system. Moreover, in our point of view, they represent three different types: positive linear operators with kernels of geometric order, positive linear operators of arithmetical order, and positive linear operators of local geometric order. It is because the kernels have different domination properties or different speeds of {Φ(𝑛)} that the 𝐿𝑝 approximations by the corresponding positive linear operators possess different Jackson orders. To show the key role of the global (or local) domination on the kernels, the condition (ii) of Theorem 11 will be further explicated to the following lemma.


7

Lemma 17. For any kernel K𝑛 (𝑥), one has 𝐿 𝑛 (|𝑡 − 𝑥| , 𝑥) = 𝑂 (𝑛−1 ) (𝑀 (K𝑛 ) + 1) .

(65)

5.1. Rational Müntz Approximation. Rational Müntz Approximation has been researched in [5], shows the application of Theorem 12, and simplifies the proof of [5]. Write

Especially, if the kernels K𝑛 (𝑥) are globally dominated, then −1

𝐿 𝑛 (|𝑡 − 𝑥| , 𝑥) = 𝑂 (𝑛 ) 𝐷𝑀 (K𝑛 ) .

(66)

Proof. From definition (3), 𝑛

𝐿 𝑛 (|𝑡 − 𝑥| , 𝑥) = 𝑛∑K𝑛,𝑟 (𝑥) ∫

𝑟/𝑛

(𝑟−1)/𝑛

𝑟=1

|𝑡 − 𝑥| 𝑑𝑡.

Λ 𝑛 = {𝑥𝜆 1 , 𝑥𝜆 2 , . . . , 𝑥𝜆 𝑛 } ,

where span{𝑥𝜆 𝑘 }, 𝜆 𝑘 ∈ Λ 𝑛 , is the class of all linear 𝑛 𝑝 combinations of {𝑥𝜆 𝑘 }𝑘=1 . For 𝑓(𝑥) ∈ 𝐿 [0,1] , define 󵄩 󵄩 𝑅𝑛 (𝑓, Λ) = min 󵄩󵄩󵄩𝑓 − 𝑟󵄩󵄩󵄩𝐿𝑝 . 𝑟∈𝑅(Λ ) 𝑛

(67)

For any 𝑥 ∈ [0, 1], there exists an integer 𝑚 such that 𝑟/𝑛

𝑝

(𝑟−1)/𝑛

|𝑡 − 𝑥| 𝑑𝑡 = ∫

𝑟/𝑛

(𝑟−1)/𝑛

(𝑥 − 𝑡) 𝑑𝑡

1 = 𝑛 (𝑛𝑥 − 𝑟 + ) . 2

(68)

−2

(b) When 𝑚 + 1 ≤ 𝑟 ≤ 𝑛, 𝑟/𝑛

∫

(𝑟−1)/𝑛

|𝑡 − 𝑥| 𝑑𝑡 = ∫

𝑟/𝑛

(𝑟−1)/𝑛

(69)

(𝑚−1)/𝑛

𝑟/𝑛

(𝑟−1)/𝑛

Then, 𝑛

𝑛

𝑘=1

𝑘=1

(78)

∞

≤ 𝐶𝑀 ∑ exp (−𝐶𝑀𝑘 (1 − 𝜖)) < ∞.

|𝑡 − 𝑥| 𝑑𝑡 ≤ ∫

𝑚/𝑛

(𝑚−1)/𝑛

𝑘=0

−1

−2

𝑛 𝑑𝑡 = 𝑛 .

(70)

Combining (a), (b), and (c), we obtain that ∫

(77)

∑ 𝑅𝑘1−𝜖 (𝑥) ≤ 𝐶𝑀 ∑ exp (−𝐶𝑀 |[𝑛𝑥] − 𝑘| (1 − 𝜖))

(c) When 𝑟 = 𝑚, ∫

(76)

Proof. We verify that 𝑀𝑛 (𝑓, 𝑥) satisfies all the requirements of Theorem 12. Take Φ(𝑘) = 𝑒−𝐶𝑀 𝑘 , where 𝐶𝑀 > 0 is the constant appeared in [4], or [5]; see the following inequality (77). Given a fixed 𝜖 with 0 < 𝜖 < 1, we verify that (i) 𝐷(𝑅𝑘1−𝜖 (𝑥)) < ∞. From [4], or [5], 𝑅𝑘 (𝑥) ≤ 𝐶𝑀 exp (−𝐶𝑀 |[𝑛𝑥] − 𝑘|) .

(𝑡 − 𝑥) 𝑑𝑡

1 = 𝑛−2 (𝑟 − 𝑛𝑥 − ) . 2

𝑚/𝑛

1 󵄩 󵄩 𝑅𝑛 (𝑓, Λ) ≤ 󵄩󵄩󵄩𝑀𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑀𝜔(𝑓, ) , 𝑛 𝐿𝑝

where 𝐶𝑀 is an absolute constant depending only upon 𝑀 (independent of 𝑝!).

(a) When 1 ≤ 𝑟 < 𝑚, 𝑟/𝑛

(75)

Corollary 18. Given 𝑓(𝑥) ∈ 𝐿 [0,1] , 1 ≤ 𝑝 < ∞, the rational Müntz operators are defined by (9). If Δ𝜆 𝑛 ≥ 𝑀𝑛, 𝑛 = 1, 2, 3, . . ., where 𝑀 > 0 is an absolute constant, one has

(𝑚 − 1)/𝑛 ≤ 𝑥 < 𝑚/𝑛; then ∫(𝑟−1)/𝑛 |𝑡 − 𝑥|𝑑𝑡 will be calculated according to the following three cases, respectively.

∫

(74)

|𝑡 − 𝑥| 𝑑𝑡 ≤ 𝑛−2 (|[𝑛𝑥] − 𝑟| + 1) .

𝐷𝑀(𝑅𝑘𝜖 (𝑥))

(ii) < ∞. Due to Example 3, we have Φ(𝑘) = exp(−𝐶𝑀𝑘). From Definition 7, 𝑛

𝐷𝑀 (𝑅𝑘𝜖 ) = sup ∑ 𝑘 exp (−𝐶𝑀𝑘𝜖) (71)

𝑛>1 𝑘=1 ∞

From Definition 6, (67) and (71) yield (65). At the same time, since the kernels of 𝐿 𝑛 (𝑓, 𝑥) are globally dominated, we obtain by Definition 1 that K𝑛,𝑟 (𝑥) ≤ 𝐶Φ (|[𝑛𝑥] − 𝑟| + 1) ,

𝑟 = 1, 2, . . . , 𝑛.

(72)

Thus, 𝑛

𝐿 𝑛 (|𝑡 − 𝑥| , 𝑥) ≤ 𝐶𝑛−1 ∑ (|[𝑛𝑥] − 𝑟| + 1) Φ (|[𝑛𝑥] − 𝑟| + 1) ; 𝑟=1

(73) that is, from Definition 7, (66) holds. Lemma 17 is done.

(79)

≤ ∑ 𝑘 exp (−𝐶𝑀𝑘𝜖) < ∞. 𝑘=0

Therefore, Corollary 18 can be deduced from Theorem 12. By the same argument of (ii), we obviously can obtain 𝐷𝑀(𝑅𝑘 ) < ∞. From (66), we know 𝑀𝑛 (|𝑡 − 𝑥|, 𝑥) = 𝑂(𝑛−1 ). Moreover, it is simple to verify ‖𝑀𝑛 (𝑓)‖𝐿𝑝 < ∞ (see, e.g., [7]). Hence, applying Theorem 11, we get Corollary 19. For rational Müntz operator 𝑀𝑛 (𝑓, 𝑥) (9), one has 1 (80) 𝑅𝑛 (𝑓, Λ)𝐿𝑝 ≤ 𝐶𝑀,𝑝 𝜔(𝑓, ) , 𝑛 𝐿𝑝 where 𝐶𝑀,𝑝 is depending on 𝑀 and 𝑝.

8


Remark 20. Since the kernels 𝑅𝑘 (𝑥) (8) have geometrical order (see Example 3), they certainly satisfy the conditions of both Theorems 11 and 12. Hence, 𝐿𝑝 approximation by these rational Müntz operators can always reach the Jackson order by applying both techniques of Theorems 12 and 11. However, for these operators, the conclusion of Corollary 18 surely contains Corollary 19 and makes the latter trivial.

From Definition 6 and Example 4, 𝑛

∑ 𝑟𝑘1−𝜖 (𝑥)

𝑘=1

𝑛

≤ 𝐶𝜆 ∑ (|[(𝑛 + 1) 𝑥] − 𝑘| + 1)−𝜆(1−𝜖) 𝑘=0

(85)

∞

5.2. The Shepard Operators. The approximation of the Shepard operators in the continuous function space 𝐶[0,1] has been studied very deeply (see [13, 26–32]). The 𝐿𝑝 approximation by the Shepard operators is investigated in [6, 7]. 𝑝

Corollary 21. Proposed that 𝑓(𝑥) ∈ 𝐿 [0,1] , 𝑝 > 1, the Shepard operators are defined by (11). Then, 󵄩󵄩 󵄩 󵄩󵄩𝑆𝑛,𝜆 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑝,𝜆 𝜔(𝑓, 𝜖𝑛 )𝐿𝑝 ,

(81)

where 𝑖𝑓 𝜆 > 2, 𝑛−1 , { { −1 𝜖𝑛 = {𝑛 log 𝑛, 𝑖𝑓 𝜆 = 2, { 1−𝜆 𝑖𝑓 1 < 𝜆 < 2. {𝑛 ,

(82)

Proof. By applying Theorem 11, we verify this result. (i) ‖𝑆𝑛,𝜆 (𝑓)‖𝐿𝑝 are uniformly bounded (see [6]). (ii) 𝑆𝑛,𝜆 (|𝑡 − 𝑥|, 𝑥) = 𝜖𝑛 (Lemma 17, Inequality (66)). Due to Example 4, we know that the global dominated sequence is Φ(𝑘) = 𝑘−𝜆 . Then, from Definition 7, 𝑛

𝑛

𝐷𝑀 (𝑟𝑘 (𝑥)) = sup ∑ 𝑘Φ (𝑘) = sup ∑ 𝑘1−𝜆 . 𝑛>1 𝑘=1

𝑛>1 𝑘=1

(83)

≤ 𝐶𝜆 ∑ 𝑘−𝜆(1−𝜖) < ∞. 𝑘=1

Then, (ii) 𝐷𝑀(𝑟𝑘𝜖 (𝑥)) < ∞. Note that the present dominated sequence Φ(𝑘) = 𝑘−𝜆 ; then from Definition 7 𝑛

∞

𝐷𝑀 (𝑟𝑘𝜖 (𝑥)) = sup ∑ 𝑘Φ𝜖 (𝑘) ≤ sup ∑ 𝑘1−𝜆𝜖 < ∞. 𝑛>1 𝑘=1

𝑛>1 𝑘=1

(86)

Corollary 22 is completed from Theorem 12 as (i) and (ii) hold. Remark 23. The difference between 𝐿𝑝 approximation techniques (I) and (II) with respect to Theorems 11 and 12 is fully exhibited on the Shepard operators by Corollaries 21 and 22. Stronger requirements by applying technique (II) than by applying technique (I) are needed. However, if the conditions are satisfied, the former can obtain essentially better result (the Jackson constant is independent of 𝑝!). On this particular case, we can obtain the Jackson type estimation by applying technique (I) for 𝜆 > 1, while achieve the corresponding result by applying technique (II) only for 𝜆 > 3. We still do not know how to deal with the cases when 1 < 𝜆 ≤ 3 by applying technique (II). 5.3. Bernstein Operators. There are many results on 𝐿𝑝 approximation by Bernstein polynomials; interested readers may refer to [8–23]. These results can be classified into the following categories: (1) uniform convergence; see [8, 12, 15, 18]; (2) quantitative estimations; see [10, 11, 14, 19]; (3) equivalence theorems; see [16, 17]; (4) saturation problems; see [9, 16, 20–23].

1−𝜆 < ∞. When 𝜆 > 2, ∑∞ 𝑘=1 𝑘 𝑛+1 1−𝜆 When 𝜆 = 2, ∑𝑘=1 𝑘 ≤ 2 log 𝑛. 1−𝜆 ≤ 𝐶𝜆 𝑛2−𝜆 . When 1 < 𝜆 < 2, ∑𝑛+1 𝑘=1 𝑘 By Theorem 11, Corollary 21 holds. 𝑝

Corollary 22. Propose that 𝑓(𝑥) ∈ 𝐿 [0,1] , 1 ≤ 𝑝 < ∞, the Shepard operators are defined by (11). If 𝜆 > 3, then

Here, we apply our technique (I) to test the corresponding Jackson type estimation as an example. The following proof is simpler than the past proof. 𝑝

1 󵄩󵄩 󵄩 ) , 󵄩󵄩𝑆𝑛,𝜆 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝜆 𝜔(𝑓, 𝑛 + 1 𝐿𝑝

(84)

where 𝐶𝜆 depends on 𝜆 only. Proof. Theorem 12 will be applied to prove this result. The details can be referred in [7]. When 𝜆 > 3, let 𝜖 = (𝜆 + 1)/2𝜆, then 0 < 𝜖 < 1, 𝜆𝜖 > 2 and 𝜆(1 − 𝜖) > 1. Evidently, the Shepard kernels have 𝜆-arithmetic order (see Example 4). We check the corresponding conditions of Theorem 12: (i) 𝐷(𝑟𝑘1−𝜖 (𝑥)) < ∞.

Corollary 24. Given 𝑓(𝑥) ∈ 𝐿 [0,1] , 𝑝 > 1, one has the following estimation: 1 󵄩󵄩 󵄩 ) . 󵄩󵄩𝐵𝑛 (𝑓) − 𝑓󵄩󵄩󵄩𝐿𝑝 ≤ 𝐶𝑝 𝜔(𝑓, √𝑛 𝐿𝑝

(87)

Proof. ‖𝐵𝑛 (𝑓)‖𝐿𝑝 is uniformly bounded which is known from [33]. We only need to evaluate 𝐵𝑛 (|𝑡 − 𝑥|, 𝑥) from Theorem 11. It is known from [15, page 15] and Definition 6 that 𝑛

𝑀 (𝑃𝑛,𝑘 (𝑥)) = sup ∑ |𝑛𝑥 − 𝑘| 𝑃𝑛,𝑘 (𝑥) ≤ 𝐶√𝑛. 𝑛≥1 𝑘=0

(88)


9

From (65) of Lemma 17 and (88), −1

𝐵𝑛 (|𝑡 − 𝑥| , 𝑥) = 𝑂 (𝑛 ) 𝑀 (𝑃𝑛,𝑘 (𝑥)) = 𝑂 (𝑛

−1/2

)

(89)

which, from Theorem 11, leads to Corollary 24. Remark 25. The approximation order of Corollary 24 is sharp which shows that (88) cannot be improved. In other words, 𝑀(𝑃𝑛,𝑘 (𝑥)) is unbounded. Hence, the approximation technique (II) or Theorem 12 cannot be applied in this case. That is to say, we can never obtain the Jackson constant independent of 𝑝 for Bernstein polynomials in 𝐿𝑝 spaces. Furthermore, Corollary 24 also exhibits that Kantorovich type operators (2) or (3) cannot reach Jackson type estimation with the Jackson constant independent of 𝑝 unless their kernels possess good properties such as having globally geometric domination (rational Müntz approximation case) or having global 𝜌-arithmetic domination for sufficiently large 𝜌 (the Shepard operators with 𝜆 > 3).

6. Conclusions On the above discussions, the positive linear operators used in 𝐿𝑝 approximation can be classified according to the properties of their kernels. We have three categories: kernels with geometrical order (such as the rational Müntz operators), kernels with arithmetic order (such as the Shepard operators), and kernels with local arithmetic order (such as the Bernstein operators). In another word, for characterizing the Jackson type estimate in 𝐿𝑝 spaces by the Kantorovich type operators (3) or (5), it always plays an essential role how well the kernels of the operators under study behave.

References [1] M. Felten, “Local and global approximation theorems for positive linear operators,” Journal of Approximation Theory, vol. 94, no. 3, pp. 396–419, 1998. [2] D. H. Mache and D. X. Zhou, “Characterization theorems for the approximation by a family of operators,” Journal of Approximation Theory, vol. 84, no. 2, pp. 145–161, 1996. [3] K. Sat¯o, “Global approximation theorems for some exponentialtype operators,” Journal of Approximation Theory, vol. 32, no. 1, pp. 32–46, 1981. [4] J. Bak, “On the efficiency of general rational approximation,” Journal of Approximation Theory, vol. 20, no. 1, pp. 46–50, 1977. [5] W. Xiao and S. Zhou, “On Müntz rational approximation rate in 𝐿𝑝 spaces,” Journal of Approximation Theory, vol. 111, no. 1, pp. 50–58, 2001. [6] W. Xiao and S. P. Zhou, “A Jackson type estimate for Shepard operators in 𝐿𝑝 spaces for 𝑝 ≥ 1,” Acta Mathematica Hungarica, vol. 95, no. 3, pp. 217–224, 2002. [7] W. Xiao, S. P. Zhou, and L. Y. Zhu, “𝐿𝑝 -approximation for the Shepard operators,” Journal of Zhejiang University, vol. 27, no. 4, pp. 364–371, 2000 (Chinese). [8] M. Becker and R. J. Nessel, “On the global approximation by Kantorovitch polynomials,” in Approximation Theory III, pp. 207–212, Academic Press, New York, NY, USA, 1980. [9] M. Becker and R. J. Nessel, “On global saturation for Kantorovitch polynomials,” in Approximation and Function Spaces, pp. 89–101, North-Holland, Amsterdam, The Netherlands, 1981.

[10] H. Berens and R. A. Devore, “Quantitative Korovkin theorems for 𝐿𝑝 -spaces,” in Approximation Theory II, pp. 289–298, Academic Press, New York, NY, USA, 1976. [11] R. Bojanic and O. Shisha, “Degree of 𝐿 1 approximation to integrable functions by modified Bernstein polynomials,” Journal of Approximation Theory, vol. 13, pp. 66–72, 1975. [12] P. L. Butzer, “Dominated convergence of Kantorovitch polynomials in the space 𝐿𝑝 ,” Transactions of the Royal Society of Canada, vol. 46, pp. 23–27, 1952. [13] Z. Ditzian, “A global inverse theorem for combinations of Bernˇste˘ın polynomials,” Journal of Approximation Theory, vol. 26, no. 3, pp. 277–292, 1979. [14] A. Grundmann, “Güteabschätzungen für den Kantoroviˇcoperator in der 𝐿 1 -norm,” Mathematische Zeitschrift, vol. 150, no. 1, pp. 45–47, 1976. [15] G. G. Lorentz, Bernstein polynomials, University of Toronto Press, Toronto, Canada, 1953. [16] G. G. Lorentz and L. L. Schumaker, “Saturation of positive operators,” Journal of Approximation Theory, vol. 5, pp. 413–424, 1972. [17] D. H. Mache, “Equivalence theorem on weighted simultaneous 𝐿𝑝 -approximation by the method of Kantoroviˇc operators,” Journal of Approximation Theory, vol. 78, no. 3, pp. 321–350, 1994. [18] V. Maier, “𝐿𝑝 -approximation by Kantoroviˇc operators,” Analysis Mathematica, vol. 4, no. 4, pp. 289–295, 1978. [19] M. W. Miller, “Die G¯ute der L𝑝 -approximation durch Kantorovic-polynome,” Mathematische Zeitschrift, vol. 152, pp. 243–247, 1976. [20] S. D. Riemenschneider, “The 𝐿𝑝 -saturation of the Bernˇste˘ınKantorovitch polynomials,” Journal of Approximation Theory, vol. 23, no. 2, pp. 158–162, 1978. [21] V. Totik, “L𝑝 (𝑝 > 1)-approximation by Kantorovich polynomials,” Analysis, vol. 3, no. 1–4, pp. 79–100, 1983. [22] V. Totik, “Problems and solutions concerning Kantorovich operators,” Journal of Approximation Theory, vol. 37, no. 1, pp. 51–68, 1983. [23] V. Totik, “Approximation in 𝐿1 by Kantorovich polynomials,” Acta Scientiarum Mathematicarum, vol. 46, no. 1–4, pp. 211–222, 1983. [24] R. A. Devore, “Degree of approximation,” in Approximation Theory II, pp. 117–161, Academic Press, New York, NY, USA, 1976. [25] E. M. Stein, Singular Integrals, Princeton, New Jersey, NJ, USA, 1970. [26] G. Criscuolo and G. Mastroianni, “Estimates of the Shepard interpolatory procedure,” Acta Mathematica Hungarica, vol. 61, no. 1-2, pp. 79–91, 1993. [27] G. Somorjai, “On a saturation problem,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 32, no. 3-4, pp. 377– 381, 1978. [28] J. Szabados, “Direct and converse approximation theorems for the Shepard operator,” Approximation Theory and Its Applications, vol. 7, no. 3, pp. 63–76, 1991. [29] B. Della Vecchia and G. Mastroianni, “Pointwise simultaneous approximation by rational operators,” Journal of Approximation Theory, vol. 65, no. 2, pp. 140–150, 1991. [30] B. Della Vecchia, “Direct and converse results by rational operators,” Constructive Approximation, vol. 12, no. 2, pp. 271– 285, 1996.

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Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 674191, 19 pages http://dx.doi.org/10.1155/2013/674191

Research Article Stability Analysis of a Repairable System with Warning Device and Repairman Vacation Yangli Ren, Lina Guo, and Lingling Zhang Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China Correspondence should be addressed to Lina Guo; [email protected] Received 19 April 2013; Accepted 7 August 2013 Academic Editor: Gen-Qi Xu Copyright © 2013 Yangli Ren et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper considers a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. By Markov renewal process theory and the probability analysis method, the system is first transformed into a group of integrodifferential equations. Then, the existence and uniqueness as well as regularity of the system dynamic solution are discussed with the functional analysis method. Further, the asymptotic stability, especially the exponential stability of the system dynamic solution, is studied by using the strongly continuous semigroup theory or 𝐶0 semigroup theory. The reliability indices and some applications (such as the comparisons of indices and profit of systems with and without warning device), as well as numerical examples, are presented at the end of the paper.

1. Introduction A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method, rather than the replacement of the entire system. With different repair levels, repair can be broken down into three categories (see [1]): perfect repair, normal repair, and minimal repair. A perfect repair can restore a system to an “as good as new” state, a normal repair is assumed to bring the system to any condition, and a minimal repair, or imperfect repair, can restore the system to the exact state it was before failure. Repairable system is not only a kind of important system discussed in reliability theory but also one of the main objects studied in reliability mathematics. Since the 1960s, various repairable system models have been established and researched. However, in traditional repairable systems, it is assumed that the repairman or server remains idle until a failed component presents. But as Mobley [2] pointed out, onethird of all maintenance costs were wasted as the result of unnecessary or improper maintenance activities. Today, the role of maintenance tends to be a “profit contributor.” Therefore, much more profit can be produced when the repairman in a system might take a sequence of vacations in the idle time.

Repairman’s vacation may literally mean a lack of work or repairman taking another assigned job. From the perspective of rational use of human resources, the introduction of repairman’s vacation makes modeling of the repairable system more realistic and flexible. This is due to the fact that in practice, the vast majority of small-and medium-sized enterprises (SMEs) cannot afford to hire a full-time repairman. So, the repairman in SMEs usually plays two roles: one for looking after the equipment and one for other duties. Under normal circumstances, the repairman has to periodically check the status of the system. If he finds that the system failed, he repairs it immediately after the end of vacation; otherwise, he will leave the system for other duties or for a vacation. Vacation model originally arised in queueing theory and has been well studied in the past three decades and successfully applied in many areas such as manufacturing/service and computer/communication network systems. Excellent surveys on the earlier works of vacation models have been reported by Doshi [3], Takagi [4], and Tian and Zhang [5]. A number of works (e.g., please see [6–10] and references therein) have recently appeared in the queueing literature in which concepts of different control operating politics along with vacations have been discussed. And Ke et al. [11] provided a summary of the most recent research works on

2 vacation queueing systems in the past 10 years, in which a wide class of vacation policies for governing the vacation mechanism is presented. In the past decade, inspired by the vacation queueing theory, some researchers introduced vacation model into repairable systems. The available references concerning repairman vacation in repairable systems can be classified into two categories: one is focused on the system indices and the other is the optimization problems. For the first category, Jain and Rakhee [12] considered the bilevel control policy for a machining system having two repairmen. One turns on when queue size of failed units reaches a preassigned level. The other’s provision in case of long queue of failed units may be helpful in reducing the backlog. The steady state queue size distribution is obtained by applying the recursive method. Hu et al. [13] studied the steady-state availability and the mean up-time of a seriesparallel repairable system consisting of one master control unit, two slave units, and a single repairman who operates single vacation by using the supplementary variable method and the vector Markov process theory. Q. T. Wu and S. M. Wu [14] analyzed some reliability indices of a cold standby system consisting of two repairable units, a switch and a repairman who may not always be at the job site or take vacation. Yuan [15] and Yuan and Cui [16] studied a k-out-of-n:G system and a consecutive-k-out-of-n:F system, respectively, with R repairmen who can take multiple vacations and by using Markov model; the analytical solution of some reliability indices was discussed. Yuan and Xu [17] studied a deteriorating system with a repairman who can have multiple vacations. By means of the geometric process and the supplementary variable techniques, a group of partial differential equations of the system was presented, and some reliability indices were derived. Ke and Wu [18] studied a multiserver machine repair model with standbys and synchronous multiple vacations, and the stationary probability vectors were obtained by using the matrix-analytical approach and the technique of matrix recursive. For the second category, Ke and Wang [19] studied a machine repair problem consisting of M operating machines with two types of spare machines and R servers (repairmen) who can take different vacation policies. The steady-state probabilities of the number of failed machines in the system as well as the performance measures were derived by using the matrix geometric theory, and a direct search algorithm was used to determine the optimal values of the number of two types of spares and the number of servers while maintaining a minimum specified level of system availability. Jia and Wu [20] considered a replacement policy for a repairable system that can not be repaired “as good as new” with a repairman who can have multiple vacations. By using geometric processes, the explicit expression of the expected cost rate was derived, and the corresponding optimal policy was determined analytically and numerically. Yuan and Xu [21, 22] considered, respectively, a deteriorating repairable system and a cold standby repairable system with two different components of different priority in use, both with one repairman who can take multiple vacations. The explicit expression of the expected cost rate was given, and

Journal of Function Spaces and Applications an optimal replacement policy was discussed. Yu et al. [23] analyzed a phase-type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. Employing the theory of renewal reward process, the explicit expression of the long-run average profit rate for the system was derived, and the optimal maintenance policy was also numerically determined. However, to the best knowledge of the authors, whichever the catalogue, the references above only concentrated on the steady state (the steady-state indices or the steady-state optimization problems) of the systems. It is because that the transient behavior of a system is difficult to be studied. Therefore, in reliability study researchers usually substitute the steady-state solution for the instantaneous one of a system, for the steady-state solution can be easily obtained by Laplace transform and a limit theorem. Whereas, Laplace transform should be based on the two hypotheses: (1) the instantaneous solution of the interested system existed and (2) the instantaneous solution of the system is stable. Whether the hypotheses hold or not is still an open question and should be justified. Moreover, the substitution of the steadystate solution for the instantaneous one is not always rational. For detailed information or explanations, please see [24, 25]. Warning systems emerge in the background of repairable systems which are stepping into the times of requiring of both advanced warning and real-time fault detection. The socalled warning system is able to send emergency signals and report dangerous situations prior to disasters, catastrophes and/or other dangers need to watch out based on previous experiences and/or observed possible omens. Real-time warning systems play an important role in fault management in banking, telecommunications, securities, electric power, and other industries. If the warning prompts during system operation, operating staff can choose shut down the system, operate carefully, or repair the system. Warning systems can help users to achieve the 24-hour uninterrupted real-time monitoring and alerting during running of various types of network infrastructure sand application services. Therefore, there is a need to study the repairable systems with warning device. This paper considers a simple repairable system with a warning device and a repairman who can have delayedmultiple vacations. The delayed-multiple vacations mean that the repairman will not leave for a vacation immediately if there is no component failed. However, there is a stochastic vacation-preparing period in which if a failed component appears he will stop the vacation preparing and serve it immediately; otherwise, he will take a rest on the end of the vacation-preparing period. When he returns from a vacation, he will either deal with the failed components waiting in the system or prepare for another vacation. In this paper, we are devoted to studying the asymptotic behavior of the system by strongly continuous semigroup theory and make comparisons of indices (such as reliability, availability, and the probability of the repairman’s vacation) and profit of the two systems with and without warning device. The paper is structured as follows. The coming section introduces the system model specifically and expresses it into a group of integrodifferential equations by Markov


3

renewal process theory and the probability analysis method. Section 3 discusses the existence and uniqueness as well as the regularity of the system dynamic solution by the functional analysis method. Section 4 studies the asymptotic behavior of the system by strongly continuous semigroup theory or 𝐶0 semigroup theory. Section 5 presents some reliability indices of the system, and the steady-state indices are discussed from the viewpoint of eigenfunction of the system operator. In Section 6, comparisons of indices and profit of systems with and without warning device are made. And a brief conclusion is offered in the last section.

2. System Formulation The system model of interest is a simple repairable system (i.e., a repairable system with a unit and a repairman) with repairman vacation and a warning device. It is described specifically as follows: at the initial time 𝑡 = 0, the unit is new, the system begins to work, and the repairman starts to prepare for the vacation. If the unit fails in the delayedvacation period, the repairman deals with it immediately, and the delayed vacation is terminated. Otherwise, he leaves for a vacation after the delayed-vacation period ends. If the warning device sends alerts in the delayed-vacation period, the repairman will stay in the system until the unit fails. Whenever the repairman returns from a vacation, he either prepares for the next vacation if the unit is working or deals with the failed unit immediately or stays in the system if the warning device has sent alerts. The repair facility neither failed nor deteriorated. The unit is repaired as good as new. Further, we assume the following. (1) The distribution function of the working time of the unit is 𝐹(𝑡) = 1 − 𝑒−𝜆𝑡 , 𝑡 ≥ 0, 𝜆 is a positive constant, and the distribution function of its repair time is 𝑡

𝑡 − ∫0

𝐺(𝑡) = ∫0 𝑔(𝑥)d𝑥 = 1 − 𝑒 1/𝑏.

𝜇(𝑥)d𝑥

3: the system is warning, and the repairman is on vacation; 4: the unit failed, and the repairman is on vacation; 5: the repairman is dealing with the failed unit. Then, by using probability analysis method, the system model can be described as the following group of integrodifferential equations: (

∞ d + 𝜀 + 𝛼0 ) 𝑃0 (𝑡) = 𝜇0 𝑃1 (𝑡) + ∫ 𝜇 (𝑥) 𝑃5 (𝑡, 𝑥) d𝑥, d𝑡 0

( (

d + 𝜇0 + 𝛼0 ) 𝑃1 (𝑡) = 𝜀𝑃0 (𝑡) , d𝑡

d + 𝜆) 𝑃2 (𝑡) = 𝛼0 𝑃0 (𝑡) + 𝜇0 𝑃3 (𝑡) , d𝑡 d ( + 𝜇0 + 𝜆) 𝑃3 (𝑡) = 𝛼0 𝑃1 (𝑡) , d𝑡 ( [

(1)

d + 𝜇0 ) 𝑃4 (𝑡) = 𝜆𝑃3 (𝑡) , d𝑡

𝜕 𝜕 + + 𝜇 (𝑥)] 𝑃5 (𝑡, 𝑥) = 0. 𝜕𝑡 𝜕𝑥

The boundary condition is 𝑃5 (𝑡, 0) = 𝜆𝑃2 (𝑡) + 𝜇0 𝑃4 (𝑡) .

(2)

The initial conditions are 𝑃0 (0) = 1, the others equal to 0.

(3)

∞

and ∫0 𝑡d𝐺(𝑡) =

(2) The distribution function of the delayed-vacation time of the repairman is 𝐷(𝑡) = 1 − 𝑒−𝜀𝑡 , 𝑡 ≥ 0, 𝜀 is a positive constant, and the distribution function of his vacation time is 𝑉(𝑡) = 1 − 𝑒−𝜇0 𝑡 , 𝜇0 is a positive constant. (3) The distribution function of the time of the warning device from its beginning to work to its first sending alerts is 𝑈(𝑡) = 1 − 𝑒−𝛼0 𝑡 , 𝑡 ≥ 0; 𝛼0 is a positive constant. (4) The above stochastic variables are independent of each other. Set 𝑁(𝑡) to be the state in which the system is at time 𝑡, and assume all the possible states as follows: 0: the system is working, and the repairman is preparing for the vacation; 1: the system is working, and the repairman is on vacation; 2: the system is warning, and the repairman is in the system;

Here, 𝑃𝑖 (𝑡) represents the probability that the system is in state 𝑖 at time 𝑡, 𝑖 = 0, 1, . . . , 4, and 𝑃5 (𝑡, 𝑥)d𝑥 represents the probability that the system is in state 5 with elapsed repair time lying in [𝑥, 𝑥 + d𝑥) at time 𝑡. Concerning the practical background, we can assume that 𝜇 (𝑥) ≥ 0,

𝜇 = sup 𝜇 (𝑥) < ∞. 𝑥∈[0,∞)

(4)

3. Existence and Uniqueness of System Solution In this section, we will study the existence and uniqueness as well as the regularity of the system solution. Firstly, we will transform the system (1)–(3) into an equivalent integral problem (P) by the method of characteristics. Secondly, the existence and uniqueness of the local solution of problem (P) are discussed by using the fixed point theory. Then, the existence and uniqueness of the global solution of problem (P) is further studied by a uniform priori estimate. Thus, the existence and uniqueness of the solution of system (1)–(3) are obtained. Moreover, the regularity or the 𝐶1 continuity of the system solution is also discussed.

4


3.1. Unique Existence of System Local Solution. For convenience, we will give some notations. Let 𝐿 1 = 𝐿1 [0, ∞) ,

𝑉0 = 𝐶 [0, 𝑇] ,

𝑉1 = 𝐶 ([0, 𝑇] , 𝐿 1 ) (5)

𝑡

𝐾1 (𝑞) (𝑡) = ∫ 𝜀𝑞 (𝑠) 𝑒−(𝜇0 +𝛼0 )(𝑡−𝑠) d𝑠, 0

(9)

𝑡

𝐾2 (𝑞) (𝑡) = ∫ [𝛼0 𝑞 (𝑠) + 𝜇0 𝐾3 (𝑞) (𝑠)] 𝑒−𝜆(𝑡−𝑠) d𝑠, 0

(10)

𝑡

with norm

𝐾3 (𝑞) (𝑡) = ∫ 𝛼0 𝐾1 (𝑞) (𝑠) 𝑒−(𝜇0 +𝜆)(𝑡−𝑠) d𝑠,

∞

0

󵄨 󵄩󵄩 󵄩󵄩 󵄨 󵄩󵄩𝜑󵄩󵄩𝐿 1 = ∫ 󵄨󵄨󵄨𝜑 (𝑥)󵄨󵄨󵄨 d𝑥, 0 󵄩󵄩 󵄩󵄩 󵄨 󵄨 󵄩󵄩𝑓󵄩󵄩𝑉0 = max 󵄨󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 , 𝑡∈[0,𝑇]

(11)

𝑡

𝐾4 (𝑞) (𝑡) = ∫ 𝜆𝐾3 (𝑞) (𝑠) 𝑒−𝜇0 (𝑡−𝑠) d𝑠, (6)

∞

󵄩󵄩 󵄩󵄩 󵄨 󵄨 󵄩󵄩𝑔󵄩󵄩𝑉1 = max ∫ 󵄨󵄨󵄨𝑔 (𝑡, 𝑥)󵄨󵄨󵄨 d𝑥. 𝑡∈[0,𝑇] 0

0

𝐾5 (𝑞) (𝑡, 𝑥) = [𝜆𝐾2 (𝑞) (𝑡 − 𝑥) + 𝜇0 𝐾4 (𝑞) (𝑡 − 𝑥)] 𝑥

× 𝑒− ∫0

Choose

(12)

𝜇(𝜏)d𝜏

.

(13)

Clearly, 𝑀 is a closed subspace of 𝑉0 . By the method of characteristics [26], the following equivalent proposition can be easily obtained [27, 28].

It can be seen that for 𝑞 ∈ 𝑉0 , if the operators 𝐾𝑖 , 𝑖 = 1, 2, . . . , 5 are determined, it needs only to get the fixed point of the operator 𝐾0 in order to get the existence and uniqueness of the solution of the integral problem (P). From (9)–(13), the following two lemmas can be easily obtained.

Theorem 1. For a given constant 𝑇 > 0, 𝑃𝑖 (𝑡) ∈ 𝑉0 , 𝑖 = 0, 1, . . . , 4, 𝑃5 (𝑡, 𝑥) ∈ 𝑉1 are the solution to (1)–(3) if and only if they are the solution to the following integral problem (P):

Lemma 2. For a given constant 𝑇 > 0 such that 𝑡 ∈ [0, 𝑇], then for any 𝑞 ∈ 𝑀, there exist unique and nonnegative 𝐾𝑖 (𝑞) ∈ 𝑉0 , 𝑖 = 1, . . . , 4 and 𝐾5 (𝑞) ∈ 𝑉1 satisfying (9)–(13).

󵄩 󵄩 𝑀 = {𝑞 ∈ 𝑉0 | 𝑞 (0) = 1, 𝑞 ≥ 0, 󵄩󵄩󵄩𝑞󵄩󵄩󵄩 ≤ 2} .

𝑡

∞

0

0

(7)

𝑃0 (𝑡) = 𝑒−(𝜀+𝛼0 )𝑡 + ∫ [𝜇0 𝑃1 (𝑠) + ∫ 𝜇 (𝑥) 𝑃5 (𝑠, 𝑥) d𝑥] × 𝑒−(𝜀+𝛼0 )(𝑡−𝑠) d𝑠, 𝑡

𝑃1 (𝑡) = ∫ 𝜀𝑃0 (𝑠) 𝑒−(𝜇0 +𝛼0 )(𝑡−𝑠) d𝑠, 0

𝑡

𝑃2 (𝑡) = ∫ [𝛼0 𝑃0 (𝑠) + 𝜇0 𝑃3 (𝑠)] 𝑒−𝜆(𝑡−𝑠) d𝑠, 0

𝑡

𝑃3 (𝑡) = ∫ 𝛼0 𝑃1 (𝑠) 𝑒−(𝜇0 +𝜆)(𝑡−𝑠) d𝑠, 0

𝑡

𝑃4 (𝑡) = ∫ 𝜆𝑃3 (𝑠) 𝑒−𝜇0 (𝑡−𝑠) d𝑠, 0

0, 𝑥≥𝑡 𝑥 𝑃5 (𝑡, 𝑥) = { −∫0 𝜇(𝜏) d𝜏 [𝜆𝑃2 (𝑡 − 𝑥) + 𝜇0 𝑃4 (𝑡 − 𝑥)] 𝑒 , 𝑥 < 𝑡. (P) Clearly, to get the existence and uniqueness of the solution of system (1)–(3), it is necessary to study the existence and uniqueness of the solution of the above integral problem (P). To this end, for any 𝑞 ∈ 𝑉0 , we define six operators as follows: −(𝜀+𝛼0 )𝑡

𝐾0 (𝑞) (𝑡) = 𝑒

𝑡

+ ∫ [𝜇0 𝐾1 (𝑞) (𝑠) 0

∞

+ ∫ 𝜇 (𝑥) 𝐾5 (𝑞) (𝑠, 𝑥) d𝑥] 0

−(𝜀+𝛼0 )(𝑡−𝑠)

×𝑒

d𝑠

(8)

Lemma 3. For a given constant 𝑇 > 0 such that 𝑡 ∈ [0, 𝑇], then for any 𝑞, 𝑞̃ ∈ 𝑀, the following estimations hold: 󵄩󵄩 󵄩 󵄩󵄩𝐾1 (𝑞)󵄩󵄩󵄩 ≤ 2𝜀𝑇, 󵄩 󵄩󵄩 󵄩 󵄩 𝑞)󵄩󵄩󵄩 ≤ 𝜀𝑇 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 , 󵄩󵄩𝐾1 (𝑞) − 𝐾1 (̃ 󵄩󵄩 󵄩 2 󵄩󵄩𝐾3 (𝑞)󵄩󵄩󵄩 ≤ 2𝛼0 𝜀𝑇 , 󵄩 󵄩 󵄩󵄩 󵄩 𝑞)󵄩󵄩󵄩 ≤ 𝛼0 𝜀𝑇2 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 , 󵄩󵄩𝐾3 (𝑞) − 𝐾3 (̃ 󵄩󵄩 󵄩 2 󵄩󵄩𝐾2 (𝑞)󵄩󵄩󵄩 ≤ 2𝛼0 𝑇 (1 + 𝜇0 𝜀𝑇 ) , 󵄩 󵄩 󵄩 󵄩󵄩 𝑞)󵄩󵄩󵄩 ≤ 𝛼0 𝑇 (1 + 𝜇0 𝜀𝑇2 ) 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 , 󵄩󵄩𝐾2 (𝑞) − 𝐾2 (̃ 󵄩󵄩 󵄩 3 󵄩󵄩𝐾4 (𝑞)󵄩󵄩󵄩 ≤ 2𝜆𝛼0 𝜀𝑇 ,

(14)

󵄩 󵄩 󵄩󵄩 󵄩 𝑞)󵄩󵄩󵄩 ≤ 𝜆𝛼0 𝜀𝑇3 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 , 󵄩󵄩𝐾4 (𝑞) − 𝐾4 (̃ 󵄩󵄩 󵄩 2 2 󵄩󵄩𝐾5 (𝑞)󵄩󵄩󵄩 ≤ 2𝜆𝛼0 𝑇 (1 + 2𝜇0 𝜀𝑇 ) , 󵄩 󵄩 󵄩 󵄩󵄩 𝑞)󵄩󵄩󵄩 ≤ 𝜆𝛼0 𝑇2 (1 + 2𝜇0 𝜀𝑇2 ) 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 . 󵄩󵄩𝐾5 (𝑞) − 𝐾5 (̃ Theorem 4. There exists a 𝑇 = 𝑇0 > 0, such that 𝐾0 has a unique fixed point on 𝑀. Proof. We prove the theorem in two steps. Firstly, we prove that the operator 𝐾0 is a mapping from 𝑀 to 𝑀. From the definition of 𝐾0 , we can know that if 𝑞 ∈ 𝑀, then 𝐾0 (𝑞) ∈ 𝑉0 and 𝐾0 (𝑞)(0) = 1. Choose 0 < 𝑇0 < 1 satisfying 1 𝑇02 [𝜇0 𝜀 + 𝜇𝜆𝛼0 (1 + 2𝜇0 𝜀)] < . 2

(15)


5 𝑡

Then, from (8) and Lemmas 2 and 3, it can be derived that

≤ 1 + 𝜇0 ∫ 𝑃1 (𝑠) d𝑠 0

󵄨 󵄨 󵄩 󵄩󵄩 2 󵄩󵄩𝐾0 (𝑞)󵄩󵄩󵄩 = max 󵄨󵄨󵄨𝐾0 (𝑞) (𝑡)󵄨󵄨󵄨 ≤ 1 + 2𝑇 𝑡∈[0,𝑇] × [𝜇0 𝜀 + 𝜇𝜆𝛼0 𝑇 (1 + 2𝜇0 𝜀𝑇2 )]

∞

0

0

+ 𝜇 ∫ [∫ 𝑃5 (𝑠, 𝑥) d𝑥] d𝑠, (16)

𝑡

𝑃1 (𝑡) ≤ 𝜀 ∫ 𝑃0 (𝑠) d𝑠,

2

< 1 + 2𝑇 [𝜇0 𝜀 + 𝜇𝜆𝛼0 (1 + 2𝜇0 𝜀)] < 2.

0

This implies that 𝐾0 (𝑞) ∈ 𝑀. Secondly, we prove that the operator 𝐾0 is a strictly compressed mapping on 𝑀. For any 𝑞, 𝑞̃ ∈ 𝑀, from (8) and Lemma 3, we have 󵄩 󵄩󵄩 𝑞)󵄩󵄩󵄩 󵄩󵄩𝐾0 (𝑞) − 𝐾0 (̃ 󵄩 󵄩 ≤ 𝑇2 [𝜇0 𝜀 + 𝜇𝜆𝛼0 𝑇 (1 + 2𝜇0 𝜀𝑇2 )] 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 󵄩 󵄩 < 𝑇2 [𝜇0 𝜀 + 𝜇𝜆𝛼0 (1 + 2𝜇0 𝜀)] 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩

𝑡

𝑡

𝑡

0

0

𝑃2 (𝑡) ≤ 𝛼0 ∫ 𝑃0 (𝑠) d𝑠 + 𝜇0 ∫ 𝑃3 (𝑠) d𝑠, 𝑡

𝑃4 (𝑡) ≤ 𝜆 ∫ 𝑃3 (𝑠) d𝑠,

0

0

∞

∞

0

0

∫ 𝑃5 (𝑡, 𝑥) d𝑥 ≤ 𝜆 ∫ 𝑃2 (𝑡 − 𝑥) d𝑥 ∞

+ 𝜇0 ∫ 𝑃4 (𝑡 − 𝑥) d𝑥

(17)

0

1󵄩 󵄩 < 󵄩󵄩󵄩𝑞 − 𝑞̃󵄩󵄩󵄩 . 2

𝑡

𝑡

0

0

= 𝜆 ∫ 𝑃2 (𝑠) d𝑠 + 𝜇0 ∫ 𝑃4 (𝑠) d𝑠.

This means that 𝐾0 is strictly compressed. According to the Banach contraction mapping principle combining the above two steps, it can be deduced readily that 𝐾0 has a unique fixed point on 𝑀. The proof of Theorem 4 is completed. Theorems 1 and 4 combing Lemma 2 follows the existence and uniqueness of the local solution of system (1)–(3). Theorem 5 (existence and uniqueness of local solution). There exists a 𝑇 = 𝑇0 > 0 such that the system (1)–(3) has a unique nonnegative local solution (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 ) ∈ 𝑉05 ×𝑉1 . 3.2. Unique Existence of System Global Solution. In this section, we will prove the existence and uniqueness of the global solution of system (1)–(3) by a uniform priori estimate and extension theorem. Lemma 6. For a given constant 𝑇 > 0, if (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 ) ∈ 𝑉05 × 𝑉1 is the nonnegative solution of system (1)–(3), then one has: 𝐸 (𝑡) ≤ 𝑒𝑄𝑇 ,

∀𝑡 ∈ [0, 𝑇] ,

4

∞

𝐸 (𝑡) = ∑𝑃𝑖 (𝑡) + ∫ 𝑃5 (𝑡, 𝑥) d𝑥 0

where 𝐸(𝑡) = 𝑃𝑖 (𝑡) + ∫0 𝑃5 (𝑡, 𝑥) d𝑥, 𝑄 = max{𝜀 + 𝛼0 , 𝜇0 + 𝛼0 , 𝜇0 + 𝜆, 𝜇} and 𝜇 is defined in (4). Proof. Because the solution of system (1)–(3) is the solution of problem (P), the estimation of the system solution can be obtained easily as follows: 𝑡

0

𝑡

∞

0

0

+ ∫ [∫ 𝜇 (𝑥) 𝑃5 (𝑠, 𝑥) d𝑥] d𝑠

𝑡

≤ 1 + (𝜀 + 𝛼0 ) ∫ 𝑃0 (𝑠) d𝑠 0 𝑡

𝑡

0

0

𝑡

𝑡

0

0

+ (𝜇0 + 𝛼0 ) ∫ 𝑃1 (𝑠) d𝑠 + 𝜆 ∫ 𝑃2 (𝑠) d𝑠

(20)

+ (𝜇0 + 𝜆) ∫ 𝑃3 (𝑠) d𝑠 + 𝜇0 ∫ 𝑃4 (𝑠) d𝑠 𝑡

∞

0

0

+ 𝜇 ∫ [∫ 𝑃5 (𝑠, 𝑥) d𝑥] d𝑠. Let 𝑄 = max{𝜀 + 𝛼0 , 𝜇0 + 𝛼0 , 𝜇0 + 𝜆, 𝜇}, then 𝐸(𝑡) ≤ 1 + 𝑡 𝑄 ∫0 𝐸(𝑠)d𝑠. The Gronwall Inequality follows the estimation immediately: 𝐸(𝑡) ≤ 𝑒𝑄𝑇 , for all 𝑡 ∈ [0, 𝑇]. The proof of Lemma 6 is completed.

(18)

∞

𝑃0 (𝑡) ≤ 𝑒−(𝜀+𝛼0 )𝑡 + ∫ 𝜇0 𝑃1 (𝑠) d𝑠

(19)

Thus,

𝑖=0

∑4𝑖=0

𝑡

𝑃3 (𝑡) ≤ 𝛼0 ∫ 𝑃1 (𝑠) d𝑠,

From Theorem 5, Lemma 6, and extension theorem, the existence and uniqueness of the system solution can be derived readily as below. Theorem 7 (existence and uniqueness of global solution). For any 𝑇 > 0, the system (1)–(3) has a unique nonnegative solution (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 ) ∈ 𝑉05 × 𝑉1 . 3.3. Regularity of System Solution. In this section, we discuss the regularity or the 𝐶1 continuity of the solution of system (1)–(3). From Theorem 1 and the expressions in problem (P) and noting the assumption (4), the following result is obvious.

6


Theorem 8. For any 𝑇 > 0, if (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 ) ∈ 𝑉05 × 𝑉1 is the nonnegative solution of system (1)–(3), then 𝑃𝑖 ∈ 𝐶[0, 𝑇], 𝑖 = 0, 1, . . . , 4 and 𝑃5 ∈ 𝐶([0, 𝑇] × [0, ∞)). Theorem 9. For any 𝑇 > 0, assume 𝜇(𝑥) is continuous on [0, 𝑇]. If (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 ) ∈ 𝑉05 × 𝑉1 is the nonnegative solution of system (1)–(3), then 𝑃𝑖 ∈ 𝐶1 [0, 𝑇], 𝑖 = 0, 1, . . . , 4, 𝑃5 ∈ 𝐶1 (𝐷), where 𝐷 = {(𝑡, 𝑥) | 0 < 𝑡 ≤ 𝑇, 0 ≤ 𝑥 < 𝑡}. Proof. From Theorem 1 and the expressions in problem (P) combing the assumption (4), it is not difficult to know that for any 𝑇 > 0, if (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 ) ∈ 𝑉05 × 𝑉1 is the nonnegative solution of system (1)–(3), then 𝑃𝑖 (𝑡) is differentiable on [0, 𝑇] and 𝑃𝑖󸀠 (𝑡) ∈ 𝐶[0, 𝑇] by Theorem 8, 𝑖 = 0, 1, . . . , 4. That is, 𝑃𝑖 (𝑡) ∈ 𝐶1 [0, 𝑇], 𝑖 = 0, 1, . . . , 4. And with the expression of 𝑃5 (𝑡, 𝑥) in problem (P), we have 𝑥 𝜕𝑃5 (𝑡, 𝑥) 󸀠 󸀠 −∫0 𝜇(𝜏)d𝜏 { − 𝑥) + 𝜇 − 𝑥)] 𝑒 𝑃 , = [𝜆𝑃 (𝑡 (𝑡 { 0 2 4 { 𝜕𝑡 { 𝑥 𝑡, {

𝑥𝑡 𝑥→𝑡

(23)

∈ 𝑋 | 𝑃5󸀠 (𝑥) ∈ 𝐿1 (R+ ) ;

= 0,

= −𝜆𝛼0 𝑒

𝑖=0

0

𝑥 𝜕𝑃5 (𝑡, 𝑥) 󵄨󵄨󵄨󵄨 = 𝜆𝛼0 𝑒− ∫0 𝜇(𝜏)d𝜏 , 󵄨󵄨 󵄨 𝜕𝑡 󵄨 𝑥𝑥𝑡 𝑥→𝑡

T

𝑋 = {𝑃 = (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 (𝑥)) | 𝑃𝑖 ∈ R, 𝑖 = 0, 1, . . . , 4,

− (𝜀 + 𝛼0 ) 𝑃0 + 𝜇0 𝑃1 + ∫ 𝜇 (𝑥) 𝑃5 (𝑥) d𝑥

𝑥 𝜕𝑃5 (𝑡, 𝑥) 󸀠 󸀠 −∫0 𝜇(𝜏)d𝜏 { 𝑃 = [−𝜆𝑃 − 𝑥) − 𝜇 − 𝑥)] 𝑒 (𝑡 (𝑡 { 0 2 4 { { 𝜕𝑥 𝑥 { { −𝜇 (𝑥) [𝜆𝑃2 (𝑡 − 𝑥) + 𝜇0 𝑃4 (𝑡 − 𝑥)] 𝑒−∫0 𝜇(𝜏)d𝜏 , { { { 𝑥 𝑡. { (21)

𝜕𝑃5 (𝑡, 𝑥) 󵄨󵄨󵄨󵄨 󵄨 𝜕𝑡 󵄨󵄨󵄨

4.1. System Transformation. In this section, we will translate the system equations into an abstract Cauchy problem in a suitable Banach space. First, choose the state space 𝑋 to be

(24)

(22) ,

= 0.

Therefore, by the continuity of 𝑃2󸀠 (𝑡), 𝑃4󸀠 (𝑡), and 𝜇(𝑥) on [0, 𝑇], it can be yielded that 𝑃5 (𝑡, 𝑥) ∈ 𝐶1 (𝐷), where 𝐷 = {(𝑡, 𝑥) | 0 < 𝑡 ≤ 𝑇, 0 ≤ 𝑥 < 𝑡}. The proof of Theorem 9 is completed.

4. Stability of System Solution In this section, we will study the asymptotic stability and exponential stability of the solution of system (1)–(3). For convenience, we will first translate the system equations into an abstract Cauchy problem in a Banach space. Then, the asymptotic stability of the system solution is discussed by analyzing the spectral distributions of the system operator and that of its adjoint operator. Further, the exponential stability of the system solution is studied by analyzing the essential spectrum bound of the system operator.

Then, the system (1)–(3) can be rewritten as an abstract Cauchy problem in the Banach space 𝑋: d𝑃 (𝑡, ⋅) = 𝐴𝑃 (𝑡, ⋅) , d𝑡

𝑡≥0 T

𝑃 (𝑡, ⋅) = (𝑃0 (𝑡) , . . . , 𝑃4 (𝑡) , 𝑃5 (𝑡, 𝑥))

(25)

𝑃 (0, ⋅) ≜ 𝑃0 = (1, 0, . . . , 0)T . 4.2. Properties of System Operator 𝐴. In this section, we will study some properties of the system operator 𝐴. Lemma 10. The system operator 𝐴 is a densely closed dissipative operator. Proof. Firstly, we prove that 𝐴 is a closed operator. Choose 𝑃𝑛 = (𝑃𝑛0 , 𝑃𝑛1 , . . . , 𝑃𝑛4 , 𝑃𝑛5 (𝑥))T ∈ 𝐷(𝐴), 𝑃𝑛 → 𝑃 = (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 (𝑥))T , 𝐴𝑃𝑛 → 𝑄 = (𝑄0 , 𝑄1 , . . . , 𝑄4 , 𝑄5 (𝑥))T , 𝑛 → ∞. By Proposition 1 ([29, II.2.10]), we know that the


7

differential operator D is the infinitesimal generator of a left translation semigroup {𝑇𝑙 (𝑡)}𝑡≥0 with domain 𝐷 (D) = {𝑓 ∈ 𝐿1 (R+ ) | 𝑓

(26)

is absolutely continuous satisfying 𝑓󸀠 ∈ 𝐿1 (R+ )} . Because 𝐷(D) is closed and 𝑃𝑛5 ∈ 𝐷(D), then 𝑃5 ∈ 𝐷(D), that is, 𝑃5󸀠 (𝑥) ∈ 𝐿1 (R+ ), and 𝑃5 (𝑥) is absolutely continuous. Moreover, 𝑃𝑛5 (0) = 𝜆𝑃2𝑛 + 𝜇0 𝑃4𝑛 → 𝜆𝑃2 + 𝜇0 𝑃4 , 𝑛 → ∞. Thus, 𝑃 ∈ 𝐷(𝐴). Therefore, it is not difficult to get that 𝐴𝑃 = 𝑄 by noting the bounded measure of 𝜇(𝑥). This implies that 𝐴 is a closed operator. Next, we prove that 𝐷(𝐴); the domain of 𝐴 is dense in 𝑋. For any 𝐹 = (𝐹0 , 𝐹1 , . . . , 𝐹4 , 𝐹5 (𝑥))T ∈ 𝑋, let 𝑃𝑖 = 𝐹𝑖 , 𝑖 = 0, 1, . . . , 4. Because 𝐹5 (𝑥) ∈ 𝐿1 (R+ ), then for any 𝜀 > 0, there exist 𝛿1 > 0 and 𝐺 > 0 such that 𝛿1

𝜀 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝐹5 (𝑥)󵄨󵄨󵄨 d𝑥 < , 6 0

∞

𝜀 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝐹5 (𝑥)󵄨󵄨󵄨 d𝑥 < . 3 𝐺

(27)

𝑖 = 0, 1, . . . , 4, 𝑄5 (𝑥) = ‖𝑃‖ sgn(𝑃5 (𝑥)). Clearly, 𝑄 ∈ 𝑋∗ = R5 × 𝐿∞ (R+ ), the dual space of 𝑋, and ⟨𝑃, 𝑄⟩ = ‖𝑃‖2 = ‖𝑄‖2 . Moreover, it is not difficult to know that ⟨𝐴𝑃, 𝑄⟩ ≤ 0. This manifests that 𝐴 is a dissipative operator. The proof of Lemma 10 is completed. Lemma 11. {𝛾 ∈ C | Re 𝛾 > 0 or 𝛾 = 𝑖𝑎, 𝑎 ∈ R \ {0}} ⊂ 𝜌(𝐴), the resolvent set of the system operator 𝐴. Proof. For any 𝐺 = (𝐺0 , 𝐺1 , . . . , 𝐺4 , 𝐺5 (𝑥))T ∈ 𝑋, consider the operator equation (𝛾𝐼 − 𝐴)𝑃 = 𝐺. That is, ∞

(𝛾 + 𝜀 + 𝛼0 ) 𝑃0 = 𝐺0 + 𝜇0 𝑃1 + ∫ 𝜇 (𝑥) 𝑃5 (𝑥) d𝑥, 0

(𝛾 + 𝛼0 + 𝜇0 ) 𝑃1 = 𝐺1 + 𝜀𝑃0 ,

(32)

(𝛾 + 𝜆) 𝑃2 = 𝐺2 + 𝛼0 𝑃0 + 𝜇0 𝑃3 ,

(33)

(𝛾 + 𝜆 + 𝜇0 ) 𝑃3 = 𝐺3 + 𝛼0 𝑃1 ,

(34)

(𝛾 + 𝜇0 ) 𝑃4 = 𝐺4 + 𝜆𝑃3 ,

(35)

𝑃5󸀠 (𝑥) + (𝛾 + 𝜇 (𝑥)) 𝑃5 (𝑥) = 𝐺5 (𝑥) ,

(36)

𝑃5 (0) = 𝜆𝑃2 + 𝜇0 𝑃4 .

(37)

Set 𝛿 = min{𝛿1 , 1/6(1 + |𝜆𝑃2 + 𝜇0 𝑃4 |)}, and define 𝜆𝑃 + 𝜇0 𝑃4 , 0 ≤ 𝑥 < 𝛿 { { 2 𝑃5 (𝑥) = {𝑔 (𝑥) , 𝛿≤𝑥≤𝐺 { 𝑥 > 𝐺. {0,

Solving (36) with the help of (37) yields (28)

𝑥

𝑃5 (𝑥) = 𝑃5 (0) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 𝑥

𝜀 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑃5 (𝑥) − 𝑔 (𝑥)󵄨󵄨󵄨 d𝑥 < . 3 𝛿

(29)

(38)

𝑥

= (𝜆𝑃2 + 𝜇0 𝑃4 ) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 + 𝑌5 (𝑥) , 𝑥

𝑥

Choose 𝑃 = (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 (𝑥)) , then 𝑃 ∈ 𝐷(𝐴), and 𝛿

󵄨 󵄨 ‖𝑃 − 𝐹‖ = ∫ 󵄨󵄨󵄨𝑃5 (𝑥) − 𝐹5 (𝑥)󵄨󵄨󵄨 d𝑥 0

∞

𝑥

∫ 𝑒− ∫0

𝜇(𝑠)d𝑠

𝑡

𝐺

󵄨 󵄨 + ∫ 󵄨󵄨󵄨𝑃5 (𝑥) − 𝐹5 (𝑥)󵄨󵄨󵄨 d𝑥

d𝑥 ≤ 𝑁,

∀𝑡 ≥ 0.

(39)

Thus, 𝑃5 (𝑥) ∈ 𝐿1 (R+ ). Substituting (38) into (31) derives

𝛿

∞

𝛿

0

where 𝑌5 (𝑥) = ∫0 𝐺5 (𝜏)𝑒− ∫𝜏 [𝛾+𝜇(𝑠)]d𝑠 d𝜏. By [30], there exists a constant 𝑁, such that

T

󵄨 󵄨 + ∫ 󵄨󵄨󵄨𝑃5 (𝑥) − 𝐹5 (𝑥)󵄨󵄨󵄨 d𝑥 𝐺

𝑥

+ ∫ 𝐺5 (𝜏) 𝑒− ∫𝜏 (𝛾+𝜇(𝑠))d𝑠 d𝜏

Here, 𝑔(𝑥) is continuously differentiable function on [𝛿, 𝐺] satisfying 𝑔(𝛿) = 𝜆𝑃5 + 𝜇𝑃4 , 𝑔(𝐺) = 0 and 𝐺

(31)

(30)

(𝛾 + 𝜀 + 𝛼0 ) 𝑃0 = 𝐺0 + 𝜇0 𝑃1

𝛿

󵄨 󵄨 󵄨 󵄨 < ∫ 󵄨󵄨󵄨𝑃5 (𝑥)󵄨󵄨󵄨 d𝑥 + ∫ 󵄨󵄨󵄨𝐹5 (𝑥)󵄨󵄨󵄨 d𝑥 0 0 𝜀 𝜀 𝜀 2𝜀 + + < (𝜆𝑃2 + 𝜇0 𝑃4 ) 𝛿 + + 3 3 6 3 < 𝜀. This implies that 𝐷(𝐴) is dense in 𝑋. Thirdly, we prove that 𝐴 is a dissipative operator. In fact, for any 𝑃 = (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 (𝑥))T ∈ 𝐷(𝐴), choose 𝑄 = (𝑄0 , 𝑄1 , . . . , 𝑄4 , 𝑄5 (𝑥))T , where 𝑄𝑖 = ‖𝑃‖ sgn(𝑃𝑖 ),

∞

+ ∫ 𝜇 (𝑥) 0

𝑥

× [𝑃5 (0) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 + 𝑌5 (𝑥)] d𝑥 (40) = 𝜇0 𝑃1 + (𝜆𝑃2 + 𝜇0 𝑃4 ) ∞

𝑥

× ∫ 𝜇 (𝑥) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 d𝑥 + 𝑌0 , 0

∞

where 𝑌0 = 𝐺0 + ∫0 𝜇(𝑥)𝑌5 (𝑥)d𝑥.

8


Combing (40) and (32)–(35) follows the following matrix equation: 𝑥

∞

𝑥

∞

𝑃0 𝑌0 −𝜇0 −𝜆∫0 𝜇 (𝑥) 𝑒∫0 (𝛾+𝜇(𝑠))d𝑠 d𝑥 0 −𝜇0 ∫0 𝜇 (𝑥) 𝑒−∫0 (𝛾+𝜇(𝑠))d𝑠 d𝑥 𝛾 + 𝜀 + 𝛼0 𝑃1 𝐺1 −𝜀 𝛾 + 𝛼0 + 𝜇0 0 0 0 ) (𝑃2 ) (𝐺2 ) . ( −𝛼0 0 𝛾+𝜆 −𝜇0 0 𝑃3 𝐺3 0 𝛾 + 𝜆 + 𝜇0 0 0 −𝛼0 𝑃4 𝐺4 0 0 0 −𝜆 𝛾 + 𝜇0

For Re 𝛾 > 0 or 𝛾 = 𝑖𝑎, 𝑎 ∈ R \ {0}, it is not difficult to get the following estimation from the definition of modulus of complex number: 󵄨󵄨 󵄨󵄨 ∞ 𝑥 󵄨 󵄨󵄨 − ∫ (𝛾+𝜇(𝑠))d𝑠 d𝑥󵄨󵄨󵄨 < 1. 󵄨󵄨∫ 𝜇 (𝑥) 𝑒 0 󵄨󵄨 󵄨󵄨 0

(42)

Thus, the coefficient matrix of the matrix equation (41) is a strictly diagonally dominant matrix for column. So, it is inverse, and the matric equation (41) has a unique solution (𝑃0 , 𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 )T . Combing (38), it can be seen that (31)– (37) have a unique solution 𝑃 = (𝑃0 , 𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 , 𝑃5 (𝑥))T ∈

𝐷(𝐴). This means that 𝛾𝐼 − 𝐴 is surjective. Because 𝛾𝐼 − 𝐴 is closed and 𝐷(𝐴) is dense in 𝑋, then (𝛾𝐼 − 𝐴)−1 exists and is bounded by Inverse Operator Theorem, for any Re 𝛾 > 0 or 𝛾 = 𝑖𝑎, 𝑎 ∈ R \ {0}. The proof of Lemma 11 is completed. Lemma 12. 0 is an eigenvalue of the system operator 𝐴 with algebraic multiplicity one. Proof. Consider the operator equation (𝛾𝐼 − 𝐴)𝑃 = 0. Let 𝐷(𝛾) be the determinant of coefficient of the matrix equation (41), then we have

󵄨󵄨𝛾 + 𝜀 + 𝛼 −𝜇0 −𝜆 (1 − 𝛾𝑔 (𝛾)) 0 −𝜇0 (1 − 𝛾𝑔 (𝛾))󵄨󵄨󵄨 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 −𝜀 󵄨󵄨 0 0 0 𝛾 + 𝛼0 + 𝜇0 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 𝛾𝐹 (𝛾) . 0 𝛾+𝜆 −𝜇0 0 𝐷 (𝛾) = 󵄨󵄨󵄨󵄨 −𝛼0 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 𝛾 + 𝜆 + 𝜇 0 0 −𝛼 󵄨󵄨 0 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 0 0 −𝜆 𝛾 + 𝜇0 󵄨 ∞

(41)

𝑥

(43)

+ 𝛾3 (2𝜆 + 𝜀 + 2𝛼0 + 3𝜇0 + 𝛼0 𝜆𝑔 (𝛾))

Here, 𝑔(𝛾) = ∫0 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 and

+ 𝛾4 . (44)

𝐹 (𝛾) = 𝛼0 𝜆𝜇0 𝑔 (𝛾) (𝜆 + 𝜇0 ) (𝜀 + 𝛼0 + 𝜇0 ) Because

+ 𝛼0 (𝜀𝜇02 + 𝜇3 + 𝜇02 𝛼0 )

𝐹 (0) = 𝛼0 𝜆𝜇0 𝑔 (0) (𝜆 + 𝜇0 ) (𝜀 + 𝛼0 + 𝜇0 )

+ 𝜆 (𝛼0 + 𝜇0 ) [𝜆 (𝜇0 + 𝜀) +𝜇0 (𝛼0 + 𝜀 + 𝜇0 )]

+3𝜇02 + 2𝜆𝜇0 + 𝛼0 𝜆) + 𝛾 [ (𝜆𝜇0 + 𝜆𝛼0 + 𝛼0 𝜇0 ) (4𝜇0 + 2𝜀 + 𝜆)

+2𝜇0 𝛼02 + 𝜀𝜆𝜇0 + 𝜆𝛼02 ] + 𝛼0 𝜆𝛾2 𝑔 (𝛾) (3𝜇0 + 𝛼0 + 𝜆) + 𝛾 (5𝜆𝜇0 + 2𝜀𝜆 + 𝜆 +

𝛼02

+

This means 𝛾 = 0 is an eigenvalue of the system operator 𝐴 with algebraic multiplicity one. The proof of Lemma 12 is completed. 4.3. Properties of Adjoint Operator 𝐴∗ . In this section, we will study some properties of 𝐴∗ , the adjoint operator of system operator 𝐴. The dual space of 𝑋 is

+ (𝜆2 + 𝜇02 ) (𝜇0 + 𝜀)

2

(45)

× [𝜆 (𝜇0 + 𝜀) + 𝜇0 (𝛼0 + 𝜀 + 𝜇0 )] > 0.

+ 𝛼0 𝜆𝛾𝑔 (𝛾) (2𝜀𝜇0 + 2𝛼0 𝜇0

2

+ 𝛼0 (𝜀𝜇02 + 𝜇3 + 𝜇02 𝛼0 ) + 𝜆 (𝛼0 + 𝜇0 )

3𝜇02

+𝜀𝛼0 + 5𝜇0 𝛼0 + 3𝜆𝛼0 + 2𝜀𝜇0 )

𝑋∗ = R5 × 𝐿∞ (R+ )

(46)

with norm ‖𝑄‖ = sup{|𝑄𝑖 |, ‖𝑄5 ‖𝐿∞ (R+ ) , 𝑖 = 0, 1, . . . , 4} for 𝑄 = (𝑄0 , 𝑄1 , . . . , 𝑄4 , 𝑄5 (𝑥))T ∈ 𝑋∗ .


9

Lemma 13. 𝐴∗ , the adjoint operator of the system operator 𝐴 is as follows:

(𝛾 + 𝜆) 𝑄2 = 𝜆𝑊5 (0) ,

(52)

(𝛾 + 𝜆 + 𝜇0 ) 𝑄3 = 𝜇0 𝑊2 + 𝜆𝑊4 ,

(53)

(𝛾 + 𝜇0 ) 𝑄4 = 𝜇0 𝑊5 (0) ,

(54)

d𝑄5 (𝑥) = (𝛾 + 𝜇 (𝑥)) 𝑄5 (𝑥) − 𝜇 (𝑥) 𝑊0 . d𝑥

(55)

− (𝜀 + 𝛼0 ) 𝑄0 + 𝜀𝑄1 + 𝛼0 𝑄2 − (𝛼0 + 𝜇0 ) 𝑄1 + 𝜇0 𝑄0 + 𝛼0 𝑄3

( −𝜆𝑄2 + 𝜆𝑄5 (0) ( 𝐴 𝑄=( ( − (𝜆 + 𝜇 ) 𝑄 + 𝜇 𝑄 + 𝜆𝑄 ( 0 3 0 2 4 −𝜇0 𝑄4 + 𝜇0 𝑄5 (0) ∗

(

𝑄5󸀠

) ) ) ≜ (𝐶 + 𝐷) 𝑄 ) )

Solving (50)–(54) yields

(𝑥) + 𝜇 (𝑥) [𝑄0 − 𝑄5 (𝑥)] )

(47)

with domain

𝑄0 =

𝜀𝑊1 + 𝛼0 𝑊2 , 𝛾 + 𝜀 + 𝛼0

𝑄1 =

𝑄2 =

∗

𝐷 (𝐴 ) = {𝑄 = (𝑄0 , 𝑄1 , . . . , 𝑄4 , 𝑄5 (𝑥))

T

𝑄3 =

∈ 𝑋∗ | 𝑄5󸀠 (𝑥) ∈ 𝐿∞ (R+ ) , 𝑄5 (𝑥)

𝜇0 𝑊0 + 𝛼0 𝑊3 , 𝛾 + 𝛼0 + 𝜇0

𝜆𝑊5 (0) , 𝛾+𝜆

𝜇0 𝑊2 + 𝜆𝑊4 , 𝛾 + 𝜆 + 𝜇0

𝑄4 =

(56) 𝜇0 𝑊5 (0) . 𝛾 + 𝜇0

(48) Solving (55) derives

is absolutely continuous satisfying 𝑄5 (∞) < ∞} .

𝑥

𝑥

𝜏

𝑄5 (𝑥) = 𝑒∫0 (𝛾+𝜇(𝑠))d𝑠 [𝑄5 (0) − ∫ 𝑊0 𝜇 (𝜏) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 d𝜏] 0

Here, 𝐶𝑄 = diag ( − (𝜀 + 𝛼0 ) , − (𝛼0 + 𝜇0 ) , −𝜆, − (𝜆 + 𝜇0 ) , −𝜇0 , 𝜀 𝛼0 0 0

0

𝜇0

0 0 𝛼0 0

0

( ( 0 𝐷𝑄 = ( ( 0 ( 0

𝑥

0 0

0 0 𝜇0 𝜃 (⋅) 0 0

0

𝜏

0

(49)

0 𝜇0

(𝜇 (𝑥) 0 0

∞

𝑄5 (0) = ∫ 𝑊0 𝜇 (𝜏) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 d𝜏.

) 0 0 𝜆𝜃 (⋅) ) )𝑄 0 𝜆 0 ) )

0 0

multiplying 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 to the two sides of (57) and letting 𝑥 → ∞ by noting 𝑄5 (∞) < ∞ follows

d − 𝜇 (𝑥)) 𝑄, d𝑥

0

(57)

Substituting (58) into (57) yields ∞

𝑥

𝜏

𝑄5 (𝑥) = 𝑒∫0 (𝛾+𝜇(𝑠))d𝑠 ∫ 𝑊0 𝜇 (𝜏) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 d𝜏. 𝑥

)

Thus, we can get the following estimation:

and 𝜃(⋅) : 𝐿∞ (R+ ) → C, satisfying 𝜃(𝑓) = 𝑓(0). Proof. For any 𝑃 ∈ 𝐷(𝐴) and 𝑄 ∈ 𝑋∗ , 𝐴∗ and its domain 𝐷(𝐴∗ ) can be readily derived by the equality ⟨𝐴𝑃, 𝑄⟩ = ⟨𝑃, 𝐴∗ 𝑄⟩. The proof of Lemma 13 is completed. Lemma 14. 𝑆 = {𝛾 ∈ C | sup{(𝜀 + 𝛼0 )/|𝛾 + 𝜀 + 𝛼0 |, (𝜇0 + 𝛼0 )/|𝛾 + 𝜇0 + 𝛼0 |, 𝜆/|𝛾 + 𝜆|, (𝜆 + 𝜇0 )/|𝛾 + 𝜆 + 𝜇0 |, 𝜇0 /|𝛾 + 𝜇0 |, 𝑀/(Re 𝛾 + 𝑀)} < 1} ⊂ 𝜌(𝐴∗ ), the resolvent set of 𝐴∗ , where 𝑀 = sup𝑥≥0 𝜇(𝑥). Proof. For any 𝑊 = (𝑊0 , 𝑊1 , . . . , 𝑊4 , 𝑊5 (𝑥))T consider the equation (𝛾𝐼 − 𝐶)𝑄 = 𝐷𝑊. That is, (𝛾 + 𝜀 + 𝛼0 ) 𝑄0 = 𝜀𝑊1 + 𝛼0 𝑊2 , (𝛾 + 𝛼0 + 𝜇0 ) 𝑄1 = 𝜇0 𝑊0 + 𝛼0 𝑊3 ,

(58)

󵄩󵄩 󵄩󵄩 󵄩󵄩𝑄5 󵄩󵄩𝐿∞ [0,∞) 󵄨 󵄨 = sup 󵄨󵄨󵄨𝑄5 (𝑥)󵄨󵄨󵄨 𝑥∈[0,∞)

󵄨󵄨 𝑥 󵄨 = sup 󵄨󵄨󵄨𝑒∫0 (𝛾+𝜇(𝑠))d𝑠 𝑥∈[0,∞) 󵄨󵄨 󵄨󵄨 ∞ 𝜏 󵄨 × ∫ 𝑊0 𝜇 (𝜏) 𝑒− ∫0 (𝛾+𝜇(𝑠))d𝑠 d𝜏󵄨󵄨󵄨 󵄨󵄨 𝑥

∈ 𝑋∗ ,

𝑥

≤ ‖𝑊‖ sup 𝑒∫0 (Re 𝛾+𝜇(𝑠))d𝑠 (50) (51)

𝑥∈[0,∞)

∞

𝜏

× ∫ −𝑒− Re 𝛾𝜏 d𝑒− ∫0 𝜇(𝑠)d𝑠 𝑥

(59)

10

Journal of Function Spaces and Applications 𝑥

𝜇0 𝑀 󵄨󵄨 󵄨󵄨 , Re 𝛾 + 𝑀 } ‖𝑊‖ 󵄨󵄨𝛾 + 𝜇0 󵄨󵄨

= ‖𝑊‖ sup 𝑒∫0 (Re 𝛾+𝜇(𝑠))d𝑠 𝑥∈[0,∞) 𝑥

< ‖𝑊‖ .

× [𝑒− ∫0 (Re 𝛾+𝜇(𝑠))d𝑠 ∞

(62)

𝜏 − ∫0 (Re 𝛾+𝜇(𝑠))d𝑠

− Re 𝛾 ∫ 𝑒 𝑥

This implies that ‖(𝛾𝐼 − 𝐶)−1 𝐷‖ < 1. Thus, [𝐼 − (𝛾𝐼 − 𝐶)−1 𝐷] is invertible. Therefore, (𝛾𝐼 − 𝐴∗ ) is invertible and

d𝜏]

= ‖𝑊‖ sup [1 − Re 𝛾𝑒Re 𝛾𝑥

(𝛾𝐼 − 𝐴∗ )

𝑥∈[0,∞)

∞

−1

−1

= [𝛾𝐼 − (𝐶 + 𝐷)]

−1

−1

−1

= [𝐼 − (𝛾𝐼 − 𝐶) 𝐷] (𝛾𝐼 − 𝐶) ,

𝜏

× ∫ 𝑒− ∫𝑥 (Re 𝛾+𝜇(𝑠))d𝑠 d𝜏] 𝑥

(63)

𝛾 ∈ 𝑆. Re 𝛾𝑥

≤ ‖𝑊‖ sup [1 − Re 𝛾𝑒

The proof of Lemma 14 is completed.

𝑥∈[0,∞)

∞

𝜏

× ∫ 𝑒− ∫𝑥 (Re 𝛾+𝑀)d𝑠 d𝜏]

(60)

𝑥

Lemma 15. 0 is an eigenvalue of operator 𝐴∗ with algebraic multiplicity one. Proof. We prove Lemma 15 in two steps. Firstly, we prove that 0 is an eigenvalue of operator 𝐴∗ . For any 𝑄 ∈ 𝐷(𝐴∗ ), consider the operator equation 𝐴∗ 𝑄 = 0, that is,

= ‖𝑊‖ sup [1 − Re 𝛾𝑒(Re 𝛾+𝑀)𝑥 𝑥∈[0,∞)

∞

× ∫ 𝑒−(Re 𝛾+𝑀)𝜏 d𝜏]

(𝜀 + 𝛼0 ) 𝑄0 = 𝜀𝑄1 + 𝛼0 𝑄2 ,

(64)

(𝛼0 + 𝜇0 ) 𝑄1 = 𝜇0 𝑄0 + 𝛼0 𝑄3 ,

(65)

𝜆𝑄2 = 𝜆𝑄5 (0) ,

(66)

(𝜆 + 𝜇0 ) 𝑄3 = 𝜇0 𝑄2 + 𝜆𝑄4 ,

(67)

𝜇0 𝑄4 = 𝜇0 𝑄5 (0) ,

(68)

𝑄5󸀠 (𝑥) = 𝜇 (𝑥) 𝑄5 (𝑥) − 𝑄0 𝜇 (𝑥) .

(69)

𝑥

= ‖𝑊‖ sup [1 − Re 𝛾𝑒(Re 𝛾+𝑀)𝑥 𝑥∈[0,∞)

×

−(Re 𝛾+𝑀)𝑥

𝑒 ] Re 𝛾 + 𝑀

𝑀 = ‖𝑊‖ , Re 𝛾 + 𝑀

Solving (69) yields

where 𝑀 = sup𝑥≥0 𝜇(𝑥). Equation (56) derive the following estimations: 𝜀 + 𝛼0 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑄0 󵄨󵄨 ≤ 󵄨󵄨 󵄨 ‖𝑊‖ , 󵄨󵄨𝛾 + 𝜀 + 𝛼0 󵄨󵄨󵄨

𝑥

𝜇 + 𝛼0 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑄1 󵄨󵄨 ≤ 󵄨󵄨 0 󵄨 ‖𝑊‖ , 󵄨󵄨𝛾 + 𝛼0 + 𝜇0 󵄨󵄨󵄨

𝜆 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑄2 󵄨󵄨 ≤ 󵄨󵄨 󵄨 ‖𝑊‖ , 󵄨󵄨𝛾 + 𝜆󵄨󵄨󵄨

𝜆 + 𝜇0 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑄3 󵄨󵄨 ≤ 󵄨󵄨 󵄨 ‖𝑊‖ , 󵄨󵄨𝛾 + 𝜆 + 𝜇0 󵄨󵄨󵄨

𝜇 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑄4 󵄨󵄨 ≤ 󵄨󵄨 0 󵄨󵄨 ‖𝑊‖ , 󵄨󵄨𝛾 + 𝜇0 󵄨󵄨

𝑀 󵄨󵄨 󵄨󵄨 ‖𝑊‖ . 󵄨󵄨𝑄5 󵄨󵄨 ≤ Re 𝛾 + 𝑀

Then, for 𝛾 ∈ 𝑆, we have 4

󵄨 󵄨 󵄩 󵄩 ‖𝑄‖ = ∑ 󵄨󵄨󵄨𝑄𝑖 󵄨󵄨󵄨 + 󵄩󵄩󵄩𝑄5 󵄩󵄩󵄩 𝑖=0

𝜇0 + 𝛼0 𝜀 + 𝛼0 ≤ sup { 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨, 󵄨󵄨𝛾 + 𝜀 + 𝛼0 󵄨󵄨 󵄨󵄨𝛾 + 𝛼0 + 𝜇0 󵄨󵄨󵄨 𝜆 + 𝜇0 𝜆 󵄨 , 󵄨󵄨 󵄨󵄨 󵄨, 󵄨 󵄨󵄨𝛾 + 𝜆󵄨󵄨 󵄨󵄨𝛾 + 𝜆 + 𝜇0 󵄨󵄨󵄨

𝑄5 (𝑥) = 𝑒∫0

𝜇(𝑠)d𝑠

𝑥

𝜏

[𝑄5 (0) − 𝑄0 ∫ 𝜇 (𝜏) 𝑒− ∫0 𝜇(𝑠)d𝑠 d𝜏] (70) 0

𝑥 − ∫0

multiplying 𝑒 𝜇(𝑠)d𝑠 to the two sides of (70) and letting 𝑥 → ∞ by noting 𝑄5 (∞) < ∞ derive ∞

𝜏

𝑄5 (0) = 𝑄0 ∫ 𝜇 (𝜏) 𝑒− ∫0 𝜇(𝑠)d𝑠 d𝜏 = 𝑄0 . 0

(61)

(71)

Combing (64)–(68) with (71) follows 𝑄0 = 𝑄1 = 𝑄2 = 𝑄3 = 𝑄4 = 𝑄5 (0) .

(72)

Substituting (71) into (70) yields 𝑥

𝑄5 (𝑥) = 𝑄0 𝑒∫0

𝜇(𝑠)d𝑠

𝑥

𝜏

[1 + ∫ d𝑒− ∫0 𝜇(𝑠)d𝑠 ] = 𝑄0 . 0

(73)

This implies that 𝜂𝑄∗ (𝜂 ≠ 0) is the eigenfunction corresponding to eigenvalue 0 of operator 𝐴∗ , where 𝑄∗ = (1, 1, . . . , 1)T . Next, we prove that the algebraic multiplicity of 0 in 𝑋∗ is one. From the above step, we can see that the geometric


11

multiplicity of 0 in 𝑋∗ is one. Then, we only need to verify the algebraic index of 0 in 𝑋∗ which is also one according to [31]. We use the reduction to absurdity. Suppose that the algebraic index of 0 in 𝑋∗ is 2 without loss of generality, then there exists 𝑌 ∈ 𝑋∗ , such that 𝐴∗ 𝑌 = 𝑄∗ . It is obvious that 0 = ⟨𝐴𝑃, 𝑌⟩ = ⟨𝑃, 𝐴∗ 𝑌⟩ = ⟨𝑃, 𝑄∗ ⟩ ,

(74)

where 𝑃 is the positive eigenfunction corresponding to eigenvalue 0 of 𝐴. However, ∗

4

∞

⟨𝑃, 𝑄 ⟩ = ∑𝑃𝑖 + ∫ 𝑃5 (𝑥) d𝑥 > 0 𝑖=0

0

(75)

which contradicts (74). Thus, the algebraic index of 0 in 𝑋∗ is one. Therefore, the algebraic multiplicity of 0 in 𝑋∗ is one. The proof of Lemma 15 is completed. 4.4. Asymptotic Stability of System Solution. In this section, we will present the asymptotic stability of the system solution by using 𝐶0 semigroup theory. Recalling Phillips Theorem (see [32]) together with Lemmas 10, 11, and 3, we can obtain the following results. Theorem 16. The system operator 𝐴 generates a positive 𝐶0 semigroup of contraction 𝑇(𝑡). Theorem 17. The system (25) has a unique nonnegative timedependent solution 𝑃(𝑡, ⋅) which satisfies ‖𝑃 (𝑡, ⋅)‖ = 1,

∀𝑡 ∈ [0, ∞) .

∀𝑡 ∈ [0, ∞) .

(77)

Because 𝑃(𝑡, ⋅) satisfies (1)–(3), it is not difficult to know that d ‖𝑃 (𝑡, ⋅)‖ = 0. d𝑡

(78)

Therefore, 󵄩 󵄩 󵄩 󵄩 ‖𝑃 (𝑡, ⋅)‖ = 󵄩󵄩󵄩𝑇 (𝑡) 𝑃0 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑃0 󵄩󵄩󵄩 = 1,

∀𝑡 ∈ [0, ∞) .

̂ lim 𝑃 (𝑡, ⋅) = ⟨𝑃0 , 𝑄∗ ⟩ 𝑃̂ = 𝑃,

𝑡→∞

(80)

where 𝑃0 is the initial value of the system. 4.5. Exponential Stability of System Solution. In Section 4.4, we have obtained the asymptotic stability of the system. In other words, the dynamic solution of the system asymptotically converges to its steady-state solution. However, there are still two problems: first, the convergence rate is unknown; second, the convergence is subject to some factors such as failure rate and repair rate. Both can be well settled if the system is exponentially stable. For this purpose, in this section, we will discuss the exponential stability of the system. For simplicity, we will divide the system operator 𝐴 into two operators. The one is a compact operator, and the other generates a quasicompact semigroup. Then, by the perturbation of compact operator, it is derived readily that the system operator also generates a quasicompact 𝐶0 semigroup. Therefore, the system solution is exponentially stable. For convenience, we will introduce three operators first: ∞

𝐵𝑃 = (∫ 𝜇 (𝑥) 𝑃5 (𝑥) d𝑥 + 𝜇0 𝑃1 , 𝜀𝑃0 , 𝛼0 𝑃0 0

T

(76)

Proof. From Theorem 16 and [32], it can be derived that the system (25) has a unique nonnegative solution 𝑃(𝑡, ⋅) which can be expressed as 𝑃 (𝑡, ⋅) = 𝑇 (𝑡) 𝑃0 ,

Theorem 19. Let 𝑃̂ be the eigenfunction corresponding to ̂ = 1, and let eigenvalue 0 of the system operator 𝐴 satisfying ‖𝑃‖ ∗ 𝑄 be defined in Lemma 15, then the time-dependent solution 𝑃(𝑡, ⋅) of the system (25) converges to the nonnegative steadŷ That is, state solution 𝑃.

(79)

This just reflects the physical meaning of 𝑃(𝑡, ⋅). The proof of Theorem 17 is completed. Remark 18. Because the initial value 𝑃0 of the system (25) belongs to the domain 𝐷(𝐴) of the system operator 𝐴, then the nonnegative time-dependent solution of the system expressed in (77) is the strong solution of the system (25). Noting that the 𝐶0 semigroup 𝑇(𝑡) generated by 𝐴 is contractive according to Theorem 16, it is uniformly bounded certainly. Thus, recalling [33] combining Lemmas 11, 12, 14, and 15, we can know that the time-dependent solution of the system strongly converges to its steady-state solution. That is the following result.

+𝜇0 𝑃3 , 𝛼0 𝑃1 , 𝜆𝑃3 , 0)

with 𝐷 (𝐵) = 𝑋,

𝐴 = 𝐴 − 𝐵 with 𝐷 (𝐴) = 𝐷 󸀠 1 = {𝑃 = (𝑃0 , 𝑃1 , . . . , 𝑃4 , 𝑃5 (𝑥)) ∈ 𝑋 | 𝑃5 (𝑥) ∈ 𝐿 (R+ ),} 𝑃5 (𝑥) is an absolutely continuous function T

𝐴 0 = 𝐴 with 𝐷 (𝐴 0 ) = {𝑃 ∈ 𝐷 | 𝑃5 (0) = 0} . (81) It is easy to know that 𝐴 and 𝐴 0 are both closed operators with dense domains in 𝑋. And with the perturbation of 𝐶0 semigroup, it is clear that 𝐴 also generates a 𝐶0 semigroup 𝑆(𝑡). Lemma 20. Assume that the mean of the repair rate exists and greater than zero, that is, 1 𝑥 ∫ 𝜇 (𝑠) d𝑠. 𝑥→∞𝑥 0

0 < 𝜇̂ = lim

(82)

Then, 𝐴 0 generates a quasicompact semigroup 𝑇0 (𝑡). Proof. Firstly, we will prove that 𝐴 0 generates a 𝐶0 semigroup 𝑇0 (𝑡). Consider the following abstract Cauchy problem: d𝑃 (𝑡, ⋅) = 𝐴 0 𝑃 (𝑡, ⋅) , d𝑡 𝑃 (0, ⋅) = Φ,

𝑡 ≥ 0,

(83)

12


where Φ = (𝜑0 , 𝜑1 , . . . , 𝜑4 , 𝜑5 (𝑥))T ∈ 𝑋. That is, d + 𝜀 + 𝛼0 ) 𝑃0 (𝑡) = 0, d𝑡

(84)

d + 𝛼0 + 𝜇0 ) 𝑃1 (𝑡) = 0, d𝑡

(85)

d + 𝜆) 𝑃2 (𝑡) = 0, d𝑡

(86)

d + 𝜆 + 𝜇0 ) 𝑃3 (𝑡) = 0, d𝑡

(87)

d ( + 𝜇0 ) 𝑃4 (𝑡) = 0, d𝑡

(88)

( (

With the help of (97), it is not difficult to deduce that

( (

󵄩 󵄨 󵄨 −(𝜀+𝛼0 )𝑡 󵄨󵄨 󵄨󵄨 −(𝛼0 +𝜇0 )𝑡 󵄩󵄩 + 󵄨󵄨𝜑1 󵄨󵄨 𝑒 󵄩󵄩𝑇0 (𝑡) Φ󵄩󵄩󵄩 = 󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨 𝑒 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝜑2 󵄨󵄨󵄨 𝑒−𝜆𝑡 + 󵄨󵄨󵄨𝜑3 󵄨󵄨󵄨 𝑒−(𝜆+𝜇0 )𝑡 󵄨 󵄨 + 󵄨󵄨󵄨𝜑4 󵄨󵄨󵄨 𝑒−𝜇0 𝑡 ∞

𝑥 󵄨 󵄨 + ∫ 󵄨󵄨󵄨𝜑5 (𝑥 − 𝑡)󵄨󵄨󵄨 𝑒− ∫𝑥−𝑡 𝜇(𝜏)d𝜏 d𝑥 𝑡

󵄨 󵄨 󵄨 󵄨 < 󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨 𝑒−(𝜀+𝛼0 )𝑡 + 󵄨󵄨󵄨𝜑1 󵄨󵄨󵄨 𝑒−(𝛼0 +𝜇0 )𝑡 󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝜑2 󵄨󵄨󵄨 𝑒−𝜆𝑡 + 󵄨󵄨󵄨𝜑3 󵄨󵄨󵄨 𝑒−(𝜆+𝜇0 )𝑡 󵄨 󵄨 + 󵄨󵄨󵄨𝜑4 󵄨󵄨󵄨 𝑒−𝜇0 𝑡

(98)

𝜕 𝜕 [ + + 𝜇 (𝑥)] 𝑃5 (𝑡, 𝑥) = 0, 𝜕𝑡 𝜕𝑥

(89)

󵄨 ̂ 󵄨 + ∫ 󵄨󵄨󵄨𝜑5 (𝑥 − 𝑡)󵄨󵄨󵄨 𝑒−(𝜇−𝜀)𝑡 d𝑥 𝑡

𝑃5 (𝑡, 0) = 0,

(90)

󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨𝜑0 󵄨󵄨󵄨 𝑒−(𝜀+𝛼0 )𝑡 + 󵄨󵄨󵄨𝜑1 󵄨󵄨󵄨 𝑒−(𝛼0 +𝜇0 )𝑡

(91)

󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝜑2 󵄨󵄨󵄨 𝑒−𝜆𝑡 + 󵄨󵄨󵄨𝜑3 󵄨󵄨󵄨 𝑒−(𝜆+𝜇0 )𝑡

𝑃𝑖 (0) = 𝜑𝑖 ,

𝑖 = 0, 1, . . . , 4,

𝑃5 (0, 𝑥) = 𝜑5 (𝑥) .

∞

(92)

󵄨 󵄨 + 󵄨󵄨󵄨𝜑4 󵄨󵄨󵄨 𝑒−𝜇0 𝑡

Solving (84)–(88) with the help of (91) yields 𝑃0 (𝑡) = 𝜑0 𝑒−(𝜀+𝛼0 )𝑡 ,

∞

󵄨 󵄨 ̂ + 𝑒−(𝜇−𝜀)𝑡 ∫ 󵄨󵄨󵄨𝜑5 (𝑥)󵄨󵄨󵄨 d𝑥 0

𝑃1 (𝑡) = 𝜑1 𝑒−(𝛼0 +𝜇0 )𝑡 ,

𝑃2 (𝑡) = 𝜑2 𝑒−𝜆𝑡 , 𝑃3 (𝑡) = 𝜑3 𝑒−(𝜆+𝜇0 )𝑡 ,

𝑃4 (𝑡) = 𝜑4 𝑒−𝜇0 𝑡 .

This manifests that

Solving (89) with the help of (90) and (92) by the method of characteristics yields 𝑥

−∫0 𝜇(𝜏)d𝜏

= 0, 𝑃 (𝑡 − 𝑥, 0) 𝑒 { { { 5 𝑡 −∫ 𝜇(𝑥−𝑡+𝜏)d𝜏 𝑃5 (𝑡, 𝑥) = {𝑃5 (0, 𝑥 − 𝑡) 𝑒 0 { 𝑥 { = 𝜑5 (𝑥 − 𝑡) 𝑒−∫𝑥−𝑡 𝜇(𝜏)d𝜏 , {

𝑡>𝑥 (94) 𝑡 ≤ 𝑥.

T

𝑥 0, there exists 𝑡0 > 0 such that 𝑥 ≥ 𝑡 ≥ 𝑡0 .

󵄩󵄩 󵄩 ̂ − min{𝜀+𝛼0 ,𝜆,𝜇0 ,𝜇−𝜀}𝑡 . 󵄩󵄩𝑇0 (𝑡)󵄩󵄩󵄩 ≤ 𝑒

(99)

󵄩 󵄩 ln 󵄩󵄩󵄩𝑇0 (𝑡)󵄩󵄩󵄩 𝑡→∞ 𝑡

(100)

Then, 𝑊ess (𝐴 0 ) ≤ 𝑊 (𝐴 0 ) = lim

≤ − min {𝜀 + 𝛼0 , 𝜆, 𝜇0 , 𝜇̂ − 𝜀} < 0.

Therefore, it is easy to prove that 𝐴 0 generates a 𝐶0 semigroup 𝑇0 (𝑡) satisfying

1 𝑥 ∫ 𝜇 (𝑠) d𝑠 > 𝜇̂ − 𝜀, 𝑡 𝑥−𝑡

̂ ≤ 𝑒− min{𝜀+𝛼0 ,𝛼0 +𝜇0 ,𝜆,𝜆+𝜇0 ,𝜇0 ,𝜇−𝜀}𝑡 ‖Φ‖ .

(93)

(97)

Therefore, 𝐴 0 generates a quasicompact 𝐶0 semigroup 𝑇0 (𝑡). The proof of Lemma 20 is completed. For 𝛾 > 0, 𝑃 ∈ 𝑋, let Φ𝛾 (𝑃) (𝑥) = [diag (0, 0, 0, 0, 0, 𝜆𝑃2 + 𝜇0 𝑃4 )] ⋅ 𝐸𝛾 (𝑥) , (101) 𝑥

where 𝐸𝛾 (𝑥) = (0, 0, 0, 0, 0, 𝑒− ∫0 [𝛾+𝜇(𝑠)]d𝑠 )T ∈ Ker(𝛾𝐼 − 𝐴) and Φ𝛾 is a compact operator. Then, it is not difficult to obtain the following result. Lemma 21. 𝐼 + Φ𝛾 is a bijection from 𝐷(𝐴 0 ) to 𝐷(𝐴) and [𝛾𝐼 − (𝐴 − 𝐵)] (𝐼 + Φ𝛾 ) = 𝛾𝐼 − 𝐴 0 .

(102)

Lemma 22. 𝑆(𝑡) − 𝑇0 (𝑡) is a compact operator, for any 𝑡 ≥ 0. Here 𝑆(𝑡) is the 𝐶0 semigroup generated by 𝐴. Proof. From Lemma 21, we can see that 𝑅(𝛾, 𝐴 − 𝐵) ≥ 𝑅(𝛾, 𝐴 0 ), for any 𝛾 > 0. Therefore 𝑆(𝑡) ≥ 𝑇0 (𝑡), for any 𝑡 ≥ 0.


13

For 𝑃 ∈ 𝐷(𝐴 0 ), set

Theorem 24. The time-dependent solution of the system (1)– (3) strongly converges to its steady-state solution. That is,

Ψ (𝑠) 𝑃 = 𝑆 (𝑡 − 𝑠) (𝐼 + Φ𝛾 ) 𝑇0 (𝑠) 𝑃,

(103)

where 0 ≤ 𝑠 ≤ 𝑡, 𝛾 > 0. Recalling the properties of 𝐶0 semigroup and Lemma 21, we can obtain Ψ󸀠 (𝑠) 𝑃 = −𝑆 (𝑡 − 𝑠) (𝐴 − 𝐵) (𝐼 + Φ𝛾 ) 𝑇0 (𝑠) 𝑃 + 𝑆 (𝑡 − 𝑠) (𝐼 + Φ𝛾 ) 𝐴 0 𝑇0 (𝑠) 𝑃 = 𝑆 (𝑡 − 𝑠) [𝛾𝐼 − (𝐴 − 𝐵)] (𝐼 + Φ𝛾 ) 𝑇0 (𝑠) 𝑃 + 𝑆 (𝑡 − 𝑠) (𝐼 + Φ𝛾 ) [−𝛾𝐼 + 𝐴 0 ] 𝑇0 (𝑠) 𝑃

(104)

= 𝑆 (𝑡 − 𝑠) [𝛾𝐼 − 𝐴 0 ] 𝑇0 (𝑠) 𝑃 + 𝑆 (𝑡 − 𝑠) (𝐼 + Φ𝛾 ) (−𝛾𝐼 + 𝐴 0 ) 𝑇0 (𝑠) 𝑃 = 𝑆 (𝑡 − 𝑠) Φ𝛾 (−𝛾𝐼 + 𝐴 0 ) 𝑇0 (𝑠) 𝑃. 𝑡

̂ lim 𝑃 (𝑡, ⋅) = 𝑃.

𝑡→∞

Moreover, there exist 𝐶 > 0 and 𝜀 > 0 such that 󵄩 󵄩󵄩 󵄩󵄩𝑃 (𝑡, ⋅) − 𝑃̂󵄩󵄩󵄩 ≤ 𝐶𝑒−𝜀𝑡 . 󵄩 󵄩 Here, 𝑃̂ is defined in Theorem 19.

𝑃 (𝑡, ⋅) = 𝑇 (𝑡) 𝑃0 = (𝑃0 + 𝑅 (𝑡)) 𝑃0

Since [Ψ(𝑡) − Ψ(0)]𝑃 = ∫0 Ψ (𝑠)𝑃d𝑠, then

= ⟨𝑃0 , 𝑄∗ ⟩ 𝑃̂ + 𝑅 (𝑡) 𝑃0 = 𝑃̂ + 𝑅 (𝑡) 𝑃0 ,

𝑡

0

(105)

That is,

(110)

Proof. Recalling Theorem 2.10 (see [35]) combing Theorem 23, we can derive that 𝐶0 semigroup 𝑇(𝑡) generated by the system operator 𝐴 can be decomposed as 𝑇(𝑡) = 𝑃0 + 𝑅(𝑡), where 𝑃0 is the residue corresponding to eigenvalue 0 and ‖𝑅(𝑡)‖ ≤ 𝐶𝑒−𝜀𝑡 for suitable constants 𝜀 > 0 and 𝐶 > 0. However, by Theorem 19, the nonnegative solution of the system (1)–(3) can be expressed as 𝑃(𝑡, ⋅) = 𝑇(𝑡)𝑃0 , 𝑡 ∈ [0, ∞). Then, combing Theorem 12.3 in [36], we can derive that

󸀠

[Ψ (𝑡) − Ψ (0)] 𝑃 = ∫ 𝑆 (𝑡 − 𝑠) Φ𝛾 (−𝛾𝐼 + 𝐴 0 ) 𝑇0 (𝑠) 𝑃d𝑠.

(109)

where 𝑄∗ is defined in Lemma 15. Hence, we can get 󵄩 󵄩󵄩 󵄩󵄩𝑃 (𝑡, ⋅) − 𝑃̂󵄩󵄩󵄩 ≤ 𝐶𝑒−𝜀𝑡 . 󵄩 󵄩 The proof of Theorem 24 is completed.

(111)

(112)

𝑡

5. Reliability Indices

0

In this section, we will discuss some reliability indices of the system. Noting that the eigenfunction corresponding to eigenvalue 0 of the system operator 𝐴 is just the steady-state solution of system (1)–(3), we are dedicated to studying some primary steady-state indices of the system from the point of eigenfunction. We first analyze the eigenfunction corresponding to eigenvalue 0 of the system operator 𝐴. In the proof process of Lemma 11, solving (31)–(36) with the help of (37) by letting 𝛾 = 0 and 𝐺 = 0 derives 𝜀 𝑃, 𝑃1 = (113) 𝛼0 + 𝜇0 0

𝑆 (𝑡) 𝑃 − 𝑇0 (𝑡) 𝑃 = − ∫ 𝑆 (𝑡 − 𝑠) Φ𝛾 (−𝛾𝐼 + 𝐴 0 ) × 𝑇0 (𝑠) 𝑃d𝑠 + Φ𝛾 𝑇0 (𝑡) 𝑃 − 𝑆 (𝑡) Φ𝛾 𝑃. (106) Therefore, 𝑆(𝑡) − 𝑇0 (𝑡) (𝑡 ≥ 0) is compact because the righthand side of the above equation is the sum of three compact operators for the compactness of Φ𝛾 . The proof of Lemma 22 is completed. From the above preparations, we can present the main results of this section. Theorem 23. 𝐶0 semigroup 𝑇(𝑡) generated by the system operator 𝐴 is quasicompact.

𝑃2 =

Proof. According to Proposition 9.20 (see [34]) combing Lemmas 22 and 20, we can deduce that 𝑊ess (𝐴) ≤ 𝑊 (𝐴 0 ) < 0.

𝑃3 =

𝛼0 𝜀𝛼0 𝑃 = 𝑃, 𝜆 + 𝜇0 1 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 ) 0

(115)

𝑃4 =

𝜆𝜀𝛼0 𝜆 𝑃 = 𝑃, 𝜇0 3 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 ) 0

(116)

(107)

This shows that 𝑆(𝑡), the 𝐶0 semigroup generated by 𝐴 is quasicompact. Because 𝐵 is a compact operator, then according to [35], it is evident that 𝑊ess (𝐴) = 𝑊ess (𝐴 − 𝐵) < 0.

𝜇 𝜀𝛼0 𝜇0 𝛼0 𝛼 ] 𝑃 , (114) 𝑃0 + 0 𝑃3 = [ 0 + 𝜆 𝜆 𝜆 𝜆 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 ) 0

(108)

This implies that 𝑇(𝑡) is quasicompact. The proof of Theorem 23 is completed.

𝑃5 (𝑥) = 𝛼0 (1 +

𝑥 𝜀 ) 𝑃0 𝑒− ∫0 𝜇(𝑠)d𝑠 . 𝛼0 + 𝜇0

(117)

Let ∞

𝑃5 = ∫ 𝑃5 (𝑥) d𝑥 = 𝛼0 (1 + 0

𝜀 ) 𝑃 𝑔 (0) , 𝛼0 + 𝜇0 0

(118)

14

Journal of Function Spaces and Applications ∞

𝑥

where 𝑔(0) = ∫0 𝑒− ∫0

𝜇(𝑠)d𝑠

1

d𝑥 and

5

𝑆 = ∑𝑃𝑖

0.8

𝑖=0

𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 𝑃0 . 𝜆𝜇0 (𝛼0 + 𝜇0 ) (𝜆 + 𝜇0 ) (119)

Theorem 25. The steady-state availability of the system is

0.6 P0 (t)

=

0.4

𝜇0 (𝜆 + 𝛼0 ) (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) 𝐴V = . 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 (120)

0.2

Proof. The instantaneous availability of the system at time 𝑡 is 0

3

𝐴 V (𝑡) = ∑𝑃𝑖 (𝑡) .

(121)

0

2

4

𝑖=0

𝜆 = 0.01 𝜆 = 0.05

Let 𝑡 → ∞, then the steady-state availability of the system is obtained as follows: 𝐴V = =

∑3𝑖=0 𝑃𝑖 𝑆

8

10

𝜆 = 0.5 𝜆 = 0.1

Figure 1: Instantaneous probabilities of the system without warning device in good state with different 𝜆.

𝜇0 (𝜆 + 𝛼0 ) (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) . 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 (122)

1

The proof of Theorem 25 is completed.

0.8

Theorem 26. The steady-state probability of the repairman vacation is 𝜆𝜀 (𝛼0 + 𝜇0 ) (𝜆 + 𝜇0 ) . 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 (123)

0.6 P0 (t)

𝑃V =

6 t

0.4

Proof. The instantaneous probability of the repairman vacation at time 𝑡 is (124)

0.2

Letting 𝑡 → ∞ derives the steady-state probability of the repairman vacation:

0

𝑃V (𝑡) = 𝑃1 (𝑡) + 𝑃3 (𝑡) + 𝑃4 (𝑡) .

𝑃V =

𝑃1 + 𝑃3 + 𝑃4 𝑆

𝜆𝜀 (𝛼0 + 𝜇0 ) (𝜆 + 𝜇0 ) = . 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 (125)

0

2

4

6

8

10

t 𝜆 = 0.01 𝜆 = 0.05

𝜆 = 0.5 𝜆 = 0.1

Figure 2: Instantaneous probabilities of the system with warning device in good state with different 𝜆.

The proof of Theorem 26 is completed. Theorem 27. The steady-state probability of the system in warning state is 𝑃𝑤 =

𝛼0 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) . 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 (126)

Proof. The instantaneous probability of the system in warning state at time 𝑡 is 𝑃𝑤 (𝑡) = 𝑃2 (𝑡) + 𝑃3 (𝑡) .

(127)


15 1

0.8

0.8

0.6

0.6

P0 (t)

P0 (t)

1

0.4

0.4

0.2

0.2

0

0

2

4

6

8

0

10

0

2

𝜇 = 0.5 𝜇=1

𝜇 = 0.1 𝜇 = 0.3

4

6

8

10

t

t

𝜇 = 0.5 𝜇=1

𝜇 = 0.1 𝜇 = 0.3

Figure 3: Instantaneous probabilities of the system without warning device in good state with different 𝜇.

Figure 4: Instantaneous probabilities of the system with warning device in good state with different 𝜇.

Letting 𝑡 → ∞ yields the steady-state probability of the system in warning state:

Let 𝑡 → ∞, then the steady-state failure frequency is immediate

𝑃𝑤 = =

𝑃2 + 𝑃3 𝑆

𝑊𝑓 =

𝛼0 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) . 𝜇0 (𝜆 + 𝜇0 ) (𝛼0 + 𝜇0 + 𝜀) (𝜆 + 𝛼0 + 𝜆𝛼0 𝑔 (0)) + 𝜆2 𝜀𝛼0 (128)

The proof of Theorem 27 is completed. Theorem 28. The steady-state failure frequency of the system is 𝑊𝑓 = 𝜆𝑃𝑤 . Proof. Let 𝑃5 (𝑡) ∞

=

∞

∫0 𝑃5 (𝑡, 𝑥)d𝑥 and 𝜇(𝑡)

(129) =

∫0 𝜇(𝑥)𝑃5 (𝑡, 𝑥)d𝑥/𝑃5 (𝑡). Then, the matrix of the transition probability of the system (1)–(3) can be obtained by (1)-(2) as follows: −𝜀 − 𝛼0 𝜇0 𝜀 −𝛼0 − 𝜇0 0 𝛼0 𝑇=( 0 𝛼0 0 0 0 0 (

0 0 0 𝜇 (𝑡) 0 0 0 0 −𝜆 𝜇0 0 0 ) . 0 −𝜆 − 𝜇0 0 0 0 0 𝜆 −𝜇0 𝜆 0 𝜇0 −𝜇 (𝑡)) (130)

Thus, by [37] the instantaneous failure frequency of the system at time 𝑡 can be derived as 𝑊𝑓 (𝑡) = 𝜆 [𝑃2 (𝑡) + 𝑃3 (𝑡)] .

(131)

𝜆 (𝑃2 + 𝑃3 ) = 𝜆𝑃𝑤 . 𝑆

(132)

The proof of Theorem 28 is completed.

6. Applications and Numerical Examples Reference [38] discussed the effects of the delayed vacation and vacation policies on a system. That is, the shorter the delayed vacation time, the larger the reliability and failure frequency of a system; and the reliability of a system with multiple vacations is smaller than that of a system with single vacation, while the profit of a system with multiple vacations is larger than that of a system with single vacation. Therefore, in this section, we only concentrate on that how the warning device will affect the system. Specifically, we will compare the reliability, availability, and profit of the system with warning device and those of the system without warning device and present some numerical examples. 6.1. System without Warning Device. The simple repairable system without warning device and with a repairman following delayed-multiple vacations policy is as follows: (

∞ d + 𝜀 + 𝜆) 𝑃0 (𝑡) = 𝜇0 𝑃1 (𝑡) + ∫ 𝜇 (𝑥) 𝑃5 (𝑡, 𝑥) d𝑥, d𝑡 0 (133)

(

d + 𝜇0 + 𝜆) 𝑃1 (𝑡) = 𝜀𝑃0 (𝑡) , d𝑡

(134)

16

Journal of Function Spaces and Applications 1.0

0.20

0.9 0.15

Wf (t)

A (t)

0.8 0.10

0.7

0.05

0.6

0.5 0

5

10 t

15

20

0

0

2

4

6

8

10

t

A 6 A 4

Wf6 Wf4

Figure 5: Instantaneous availabilities of the systems with and without warning device.

d + 𝜇0 ) 𝑃4 (𝑡) = 𝜆𝑃1 (𝑡) , d𝑡

(135)

𝜕 𝜕 + + 𝜇 (𝑥)] 𝑃5 (𝑡, 𝑥) = 0 𝜕𝑡 𝜕𝑥

(136)

( [

with boundary condition 𝑃5 (𝑡, 0) = 𝜆𝑃0 (𝑡) + 𝜇0 𝑃4 (𝑡)

(137)

and initial conditions 𝑃0 (0) = 1,

the others equal to 0.

(138)

Then, by the same method of Theorems 25, 26, and 28, the corresponding reliability indices of the system (133)–(138) are as follows. Theorem 29. The steady-state availability of the system (133)– (138) is 𝜇0 (𝜇0 + 𝜆 + 𝜀) ̃V = 𝐴 , 𝜇0 (𝜇0 + 𝜆 + 𝜀) (1 + 𝜆𝑔 (0)) + 𝜆𝜀

(139)

where 𝑔(0) is defined in Section 5. Theorem 30. The steady-state probability of the repairman vacation of the system (133)–(138) is 𝑃̃V =

𝜀 (𝜇0 + 𝜆) . 𝜇0 (𝜇0 + 𝜆 + 𝜀) (1 + 𝜆𝑔 (0)) + 𝜆𝜀

(140)

Theorem 31. The steady-state failure frequency of the system (133)–(138) is ̃𝑓 = 𝜆𝐴 V . 𝑊

(141)

Figure 6: Instantaneous failure frequencies of the systems with and without warning device with 𝛼 = 0.01.

6.2. Numerical Examples. By comparing the two groups of (120) and (139) and (129) and (141), it is not difficult to deduce the following results. (i) The steady-state availability of the system with warning device (i.e., system (1)–(3)) is larger than that of the system without warning device (i.e., system (133)– ̃V . (138)). That is, 𝐴 V > 𝐴 (ii) If 𝛼0 ≤ 𝜆, then the steady-state failure frequency of the system with warning device is less than that of the ̃𝑓 . system without warning device. That is, 𝑊𝑓 < 𝑊 ̃ While if 𝛼0 > 𝜆, the magnitude of 𝑊𝑓 and 𝑊𝑓 cannot be determined. Let 𝐼 and 𝐼̃ be the total profit of the system with and without warning device, respectively. That is, 𝐼 = 𝑐1 𝐴 V − 𝑐2 𝑊𝑓 + 𝑐3 𝑃V , ̃V − 𝑐2 𝑊 ̃𝑓 + 𝑐3 𝑃̃V . 𝐼̃ = 𝑐1 𝐴

(142)

Here, 𝑐1 , 𝑐2 , and 𝑐3 represent the income of the system for working unit per unit time, the loss of the system for failed unit per unit time, and the income of the system for the repairman vacation per unit time, respectively. Given 𝜀 = 1, ̃ 𝜇0 = 0.5, 𝛼0 = 0.2, 𝑐1 = 50, 𝑐2 = 15, 𝑐3 = 30, let 𝐷 = 𝐼 − 𝐼. Then, 𝐷 is a function of 𝜆 and 𝜇. From Figure 10, we can know that the profit of the system with warning device is larger than that of the system without warning device. From the above discussions, we can deduce that because both the availability and profit of a system with warning device are larger than those of a system without warning

17

0.20

0.20

0.15

0.15

Wf (t)

Wf (t)


0.10

0.05

0.05

0

0.10

0

2

4

6

8

0

10

0

2

4

Wf6 Wf4

device. Then, a system with warning device is better than a system without warning device in practice. In the following, we will present some numerical examples to illustrate the conclusions. (1) Let 𝜀 = 1, 𝜇0 = 𝜇 = 0.5, 𝛼0 = 0.2, 𝜆 = 0.01, 0.05, 0.1, and 0.5, respectively. Figures 1 and 2 present the instantaneous probabilities of the systems with and without warning device in good state with different values of 𝜆. Let 𝜀 = 1, 𝜆 = 0.1, 𝜇0 = 0.5, 𝛼0 = 0.2, 𝜇 = 0.1, 0.3, 0.5, and 1, respectively. Figures 3 and 4 present the instantaneous probabilities of the systems with and without warning device in good state with different values of 𝜇. From the four figures, we can see that the instantaneous probabilities of the system without warning device in good state decrease with the increasing of 𝜆, while the instantaneous probabilities of the system with warning device in good state increase with the increasing of 𝜆. But both the instantaneous probabilities of the systems with and without warning device in good state increase with the increasing of 𝜇 in general. (2) Let 𝜀 = 1, 𝜆 = 0.1, 𝜇0 = 0.5, 𝜇 = 0.2, 𝛼0 = 0.2. Figure 5 presents the instantaneous availabilities of the systems with and without warning device. From the figure, we can see that the availability of the system with warning device is larger than that of the system without warning device. (3) Let 𝜀 = 1, 𝜆 = 0.1, 𝜇0 = 0.5, 𝜇 = 0.2 and 𝛼0 = 0.01, 0.2, 1, and 5, respectively. Figures 6, 7, 8, and 9 present the instantaneous failure frequencies of the systems with and without warning device. It can be derived from the four figures that when 𝛼0 < 𝜆, the instantaneous failure frequency of the system with warning device is less than that of the system without warning device. But when 𝛼0 ≥

8

10

Wf6 Wf4

Figure 8: Instantaneous failure frequencies of the systems with and without warning device with 𝛼 = 1.

0.20

0.15

Wf (t)

Figure 7: Instantaneous failure frequencies of the systems with and without warning device with 𝛼 = 0.2.

6 t

t

0.10

0.05

0

0

2

4

6

8

10

t Wf6 Wf4

Figure 9: Instantaneous failure frequencies of the systems with and without warning device with 𝛼 = 5.

1 ≫ 𝜆, the instantaneous failure frequency of the system with warning device is larger than that of the system without warning device.

18


Acknowledgments This work is supported by the TianYuan Special Funds of the National Natural Science Foundation of China under Grant no. 11226249, the Foundation Research Project of Shanxi Province (The Youth) under Grant no. 2012021015-5, the Youth Foundation of Taiyuan University of Technology under Grant no. 2012L031, the Special Funds of the National Natural Science Foundation of China under Grant no. 61250011, and the Foundation Research Project of Shanxi Province (The Natural Science) under Grant no. 2012011004-4.

60 50 40 30 D 20 10 0 −10

0

0 2

2 4

4 𝜇

6

6 8

8 10

𝜆

10

Figure 10: Profit difference between the systems with and without warning device.

(4) Let 𝜀 = 1, 𝜇0 = 0.5, 𝛼0 = 0.2, 𝑐1 = 50, 𝑐2 = 15, 𝑐3 = 30. Figure 10 presents the profit difference between the systems with and without warning device. From the figure, it can be derived easily that the profit the system with warning device is more than that of the system without warning device.

7. Conclusion In this paper, we proposed a simple repairable system with a warning device and a repairman who can have delayedmultiple vacations. Because the two hypotheses used for Laplace transform in order to obtain the steady-state solution of a repairable system in traditional reliability research that needs to be verified, and the substitution of steady-state solution for the dynamic one that should be based on some conditions, the study of well-posedness of the time-dependent solution of a system is in demand in terms of theory and practice. In this paper, we first transformed the system model into a group of operator equations and obtained the existence and uniqueness as well as 𝐶1 continuity of the system solution by functional analysis method. Then to study the stability of the system, we translated the system model into an abstract Cauchy problem in a suitable Banach space. The asymptotic stability and further the exponential stability of the system solution were derived by using 𝐶0 semigroup theory and compact operator disturbance theorem. Because the stable solution of the system is just the eigenfunction corresponding to eigenvalue 0 of the system operator, we also presented some reliability indices, such as reliability, failure frequency, probabilities of repairman vacation, and system in warning state of the system in the viewpoint of eigenfunction. At the end of the paper, by the theoretical and numerical analyses, we give the conclusion that the system with warning device is better than the system without warning device in practice.

References [1] M. Yaez, F. Joglar, and M. Modarres, “Generalized renewal process for analysis of repairable systems with limited failure experience,” Reliability Engineering and System Safety, vol. 77, no. 2, pp. 167–180, 2002. [2] R. K. Mobley, An Introduction to Predictive Maintenance, Elsevier Science, New York, NY, USA, 2nd edition, 2002. [3] B. T. Doshi, “Queueing systems with vacations—a survey,” Queueing Systems, vol. 1, no. 1, pp. 29–66, 1986. [4] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, vol. 1, NorthHolland, Amsterdam, The Netherlands, 1991. [5] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, Springer, New York, NY, USA, 2006. [6] J. Ke, “Batch arrival queues under vacation policies with server breakdowns and startup/closedown times,” Applied Mathematical Modelling, vol. 31, no. 7, pp. 1282–1292, 2007. [7] B. K. Kumar, R. Rukmani, and V. Thangaraj, “An M/M/C retrial queueing system with Bernoulli vacations,” Journal of Systems Science and Systems Engineering, vol. 18, no. 2, pp. 222–242, 2009. [8] Z. G. Zhang and N. Tian, “Discrete time 𝐺𝑒𝑜/𝐺/1 queue with multiple adaptive vacations,” Queueing Systems, vol. 38, no. 4, pp. 419–429, 2001. [9] Z. G. Zhang, “On the three threshold policy in the multi-server queueing system with vacations,” Queueing Systems, vol. 51, no. 1-2, pp. 173–186, 2005. [10] J. Wu, Z. Liu, and Y. Peng, “On the BMAP/G/1 𝐺-queues with second optional service and multiple vacations,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 33, no. 12, pp. 4314–4325, 2009. [11] J. C. Ke, C. U. Wu, and Z. G. Zhang, “Recent developments in vacation models: a short survey,” International Journal of Operations Research, vol. 7, no. 4, pp. 3–8, 2010. [12] M. Jain and M. S. Rakhee, “Bilevel control of degraded machining system with warm standbys, setup and vacation,” Applied Mathematical Modelling, vol. 28, no. 12, pp. 1015–1026, 2004. [13] L. M. Hu, D. Q. Yue, and J. D. Li, “Probabilistic analysis of a series-parallel repairable system with three units and vacation,” Applied Mathematical Modelling, vol. 34, no. 10, pp. 2711–2721, 2010. [14] Q. T. Wu and S. M. Wu, “Reliability analysis of two-unit cold standby repairable systems under Poisson shocks,” Applied Mathematics and Computation, vol. 218, no. 1, pp. 171–182, 2011.

Journal of Function Spaces and Applications [15] L. Yuan, “Reliability analysis for a k-out-of-n: G system with redundant dependency and repairmen having multiple vacations,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11959–11969, 2012. [16] L. Yuan and Z. D. Cui, “Reliability analysis for the consecutivek-out-of-n:F system with repairmen taking multiple vacations,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 4685–4697, 2013. [17] W. Z. Yuan and G. Q. Xu, “Modelling of a deteriorating system with repair satisfying general distribution,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6340–6350, 2012. [18] J. C. Ke and C. H. Wu, “Multi-server machine repair model with standbys and synchronous multiple vacation,” Computers and Industrial Engineering, vol. 62, no. 1, pp. 296–305, 2012. [19] J. C. Ke and K. H. Wang, “Vacation policies for machine repair problem with two type spares,” Applied Mathematical Modelling, vol. 31, no. 5, pp. 880–894, 2007. [20] J. S. Jia and S. M. Wu, “A replacement policy for a repairable system with its repairman having multiple vacations,” Computers and Industrial Engineering, vol. 57, no. 1, pp. 156–160, 2009. [21] L. Yuan and J. Xu, “A deteriorating system with its repairman having multiple vacations,” Applied Mathematics and Computation, vol. 217, no. 10, pp. 4980–4989, 2011. [22] L. Yuan and J. Xu, “An optimal replacement policy for a repairable system based on its repairman having vacations,” Reliability Engineering and System Safety, vol. 96, no. 7, pp. 868– 875, 2011. [23] M. M. Yu, Y. H. Tang, L. P. Liu, and J. Cheng, “A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations,” European Journal of Operational Research, vol. 225, no. 2, pp. 310–323, 2013. [24] L. N. Guo, H. B. Xu, C. Gao, and G. T. Zhu, “Stability analysis of a new kind series system,” IMA Journal of Applied Mathematics, vol. 75, no. 3, pp. 439–460, 2010. [25] L. N. Guo, H. B. Xu, C. Gao, and G. T. Zhu, “Stability analysis of a new kind 𝑛-unit series repairable system,” Applied Mathematical Modelling, vol. 35, no. 1, pp. 202–217, 2011. [26] M. E. Gurtin and R. C. MacCamy, “Non-linear age-dependent population dynamics,” Archive for Rational Mechanics and Analysis, vol. 54, pp. 281–300, 1974. [27] Y. Yao, “Global existence and uniqueness of solutions to epidemic dynamic systems of equations,” Chinese Annals of Mathematics A, vol. 12, no. 2, pp. 218–230, 1991. [28] P. Yan, “Well-posedness of a model for the dynamics of infectious diseases without immunity and with latent period,” Journal of Biomathematics, vol. 23, no. 2, pp. 245–256, 2008. [29] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000. [30] H. B. Xu, W. H. Guo, J. Y. Yu, and G. T. Zhu, “The asymptotic stability of a series repairable system,” Acta Mathematicae Applicatae Sinica, vol. 29, no. 1, pp. 46–52, 2006. [31] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Wiley Classics Library, John Wiley & Sons, New York, NY, USA, 1958. [32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. [33] G. Geni, X. Z. Li, and G. T. Zhu, Functional Analysis Method in Queueing Theory, Research Information Limited, Hertfordshire, UK, 2001.

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Research Article On the Distance between Three Arbitrary Points Parin Chaipunya and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam; [email protected] Received 25 March 2013; Accepted 1 August 2013 Academic Editor: Satit Saejung Copyright © 2013 P. Chaipunya and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We point out some equivalence between the results in (Sedghi et al., 2012) and (Khamsi, 2010). Then, we introduce the notion of a general distance between three arbitrary points and study some of its properties. In the final section, some fixed point results are proposed.

1. Introduction

2. Preliminaries

The literature of a distance for any triple of points in a space was first considered during the sixties by Gähler [1, 2]. It is known as a 2-metric, the concept of which was later extended by Dhage [3] into a 𝐷-metric. Both notions are in no easy ways related to the classical concept of a metric. This led to the 𝐺-metric due to Mustafa and Sims [4]. On the other hands, Sedghi et al. [5] had introduced the notion of a 𝐷∗ -metric, which has later been generalized by Sedghi et al. [6] into an 𝑆-metric. These developments confirm that this kind of measurement is recently of many mathematicians’ interests. One of the area that exploited these establishments largely is the fixed point theory, especially the ones involving some generalized contractions; see, for example, Sedghi et al. [5], Sedghi et al. [6], Mustafa et al. [7], Mustafa and Sims [8], Aydi et al. [9], Alghamdi and Karapinar [10], Chandok et al. [11], and Abbas et al. [12]. In this present paper, we divide our interests into three parts. Firstly, we give a remark on the existing fixed point result endowed in a 𝑆-metric space. Secondly, we propose and study a very general principle in measuring the distance between three arbitrary points called a 𝑔 − 3𝑝𝑠. Thirdly, we construct some fixed point theorems by utilizing the 𝑔 − 3𝑝𝑠 and its properties.

This section is devoted to the recollection of important definitions and lemmas. We start with a sequence of definitions of 𝐺-, 𝐷∗ -, and 𝑆-metric spaces. Definition 1 (see [4]). Let 𝑋 be a nonempty set. A function 𝐺 : 𝑋 × 𝑋 × 𝑋 → R+ is said to be a 𝐺-metric if the following conditions are satisfied: (1) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝐺(𝑥, 𝑦, 𝑧) = 0 if 𝑥 = 𝑦 = 𝑧; (2) for 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≠𝑦, 𝐺(𝑥, 𝑥, 𝑦) > 0; (3) for 𝑥, 𝑦, 𝑧 ∈ 𝑋 with 𝑦 ≠𝑧, 𝐺(𝑥, 𝑥, 𝑦) ≤ 𝐺(𝑥, 𝑦, 𝑧); (4) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝐺(𝑥, 𝑦, 𝑧) = 𝐺(𝜋(𝑥, 𝑦, 𝑧)), where 𝜋(𝑥, 𝑦, 𝑧) is any permutation of (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋; (5) for 𝑥, 𝑦, 𝑧, V ∈ 𝑋, 𝐺(𝑥, 𝑦, 𝑧) ≤ 𝐺(𝑥, V, V) + 𝐺(V, 𝑦, 𝑧). The pair (𝑋, 𝐺) is called a 𝐺-metric space. Moreover, if 𝐺(𝑥, 𝑥, 𝑦) = 𝐺(𝑦, 𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋, then 𝐺 is said to be symmetric. Definition 2 (see [5]). Let 𝑋 be a nonempty set. A function 𝐷∗ : 𝑋 × 𝑋 × 𝑋 → R+ is said to be a 𝐷∗ -metric if the following conditions are satisfied: (1) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝐷∗ (𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧;

2

Journal of Function Spaces and Applications (2) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝐷∗ (𝑥, 𝑦, 𝑧) = 𝐷∗ (𝜋(𝑥, 𝑦, 𝑧)), where 𝜋(𝑥, 𝑦, 𝑧) is any permutation of (𝑥, 𝑦, 𝑧) ∈ 𝑋 × 𝑋 × 𝑋; (3) for 𝑥, 𝑦, 𝑧, V ∈ 𝑋, 𝐷∗ (𝑥, 𝑦, 𝑧) ≤ 𝐷∗ (𝑥, 𝑦, V) + 𝐷∗ (V, 𝑧, 𝑧). The pair (𝑋, 𝐷∗ ) is called a 𝐷∗ -metric space.

Definition 3 (see [6]). Let 𝑋 be a nonempty set. A function 𝑆 : 𝑋 × 𝑋 × 𝑋 → R+ is said to be a 𝑆-metric if the following conditions are satisfied: (1) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑆(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧; (2) for 𝑥, 𝑦, 𝑧, V ∈ 𝑋, 𝑆(𝑥, 𝑦, 𝑧) ≤ 𝑆(𝑥, 𝑥, V) + 𝑆(𝑦, 𝑦, V) + 𝑆(𝑧, 𝑧, V). The pair (𝑋, 𝑆) is called a 𝑆-metric space.

It can be seen that each symmetric 𝐺-metric is a 𝐷∗ metric and that each 𝐷∗ -metric is a 𝑆-metric. In case of nonsymmetric 𝐺-metric, the concepts of 𝐺-metric and 𝑆-metric are independent. Definition 5 (see [6]). Let (𝑋, 𝑆) be a 𝑆-metric space. For 𝑟 > 0 and 𝑥 ∈ 𝑋 we define the open ball 𝐵𝑆 (𝑥; 𝑟) as follows: (1)

As in [6], one may consider the topology 𝜏 for 𝑋 which is generated from the base containing all open balls in 𝑋. Some concepts are also introduced. Definition 6 (see [6]). Let (𝑋, 𝑆) be a 𝑆-metric space. A sequence (𝑥𝑛 ) in 𝑋 is called (i) Cauchy if for any 𝜖 > 0, we may find 𝑁 ∈ N such that 𝑛 ∈ N, 𝑛 ≥ 𝑁 󳨐⇒ 𝑆 (𝑥𝑚 , 𝑥𝑚 , 𝑥𝑛 ) < 𝜖;

(2)

lim 𝑆 (𝑥, 𝑥, 𝑥𝑛 ) = 0.

Definition 8 (see [13]). Let 𝑋 be a nonempty set and let 𝐷 : 𝑋 × 𝑋 → R+ be a function satisfying the following conditions: (1) for 𝑥, 𝑦 ∈ 𝑋, 𝐷(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦; (2) for 𝑥, 𝑦 ∈ 𝑋, 𝐷(𝑥, 𝑦) = 𝐷(𝑦, 𝑥); (3) there exists a constant 𝐾 > 0 such that for 𝑥, 𝑦, 𝑧1 , 𝑧2 , . . . , 𝑧𝑛 ∈ 𝑋,

The triple (𝑋, 𝐷, 𝐾) is called a metric type space. In particular, if 𝐾 ≤ 1, then (𝑋, 𝐷) is a metric space. A self-mapping operator 𝑓 on a metric type space (𝑋, 𝐷, 𝐾) is called Lipschizian if there exists 𝜆 ≥ 0 such that 𝐷 (𝑓𝑥, 𝑓𝑦) ≤ 𝜆𝐷 (𝑥, 𝑦) ,

(6)

for all 𝑥, 𝑦 ∈ 𝑋. The smallest 𝜆 > 0 satisfying such condition is denoted by Lip(𝑓). Moreover, the following fixed point theorem was proposed. Theorem 9 (see [13]). Let (𝑋, 𝐷, 𝐾) be a complete metric type space and let 𝑓 : 𝑋 → 𝑋 be an operator such that the composition 𝑓𝑛 is Lipschizian for each 𝑛 ∈ N and ∑𝑛∈N Lip(𝑓𝑛 ) < ∞. Then, 𝑓 has a unique fixed point. It follows that if 𝑓 is a Lipschizian operator with Lip(𝑓) < 1, then 𝑓 has a unique fixed point. Now, let (𝑋, 𝑆) be a 𝑆-metric space. Suppose that a function 𝐷 : 𝑋 × 𝑋 → R+ is given by 𝐷𝑆 (𝑥, 𝑦) := 𝑆 (𝑥, 𝑥, 𝑦) ,

∀𝑥, 𝑦 ∈ 𝑋,

(3)

(7)

𝐷𝑆 (𝑥, 𝑦) = 𝑆 (𝑥, 𝑥, 𝑦)

Moreover, if every Cauchy sequence in 𝑋 is also convergent, 𝑋 is said to be complete.

≤ 2𝑆 (𝑥, 𝑥, 𝑧1 ) + 𝑆 (𝑦, 𝑦, 𝑧1 )

3. A 𝑆-Metric Space as a Metric Type Space

≤ 2𝐷𝑆 (𝑥, 𝑧1 ) + 2𝐷𝑆 (𝑧1 , 𝑧2 ) + 𝐷𝑆 (𝑧2 , 𝑦)

In this section, we shall be giving a small remark on a fixed point theorem due to [6]. According to [6], a self-operator 𝑓 on a 𝑆-metric space (𝑋, 𝑆) is called a contraction if it satisfies the following inequality: 𝑆 (𝑓𝑥, 𝑓𝑥, 𝑓𝑦) ≤ 𝜆𝑆 (𝑥, 𝑥, 𝑦) ,

(5)

for 𝑥, 𝑦 ∈ 𝑋, it is obvious that 𝐷(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦 and 𝐷(𝑥, 𝑦) = 𝐷(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋. Now, observe for each 𝑧1 , 𝑧2 , . . . , 𝑧𝑛 ∈ 𝑋 that

(ii) convergent if there is a point 𝑥 ∈ 𝑋 in which 𝑛→∞

To expedite our remark, we shall first recall the notion of a metric type space, which was introduced by Khamsi in [13].

𝐷 (𝑥, 𝑦) ≤ 𝐾 [𝐷 (𝑥, 𝑧1 ) + 𝐷 (𝑧1 , 𝑧2 ) + ⋅ ⋅ ⋅ + 𝐷 (𝑧𝑛 , 𝑦)] .

Lemma 4 (see [6]). Let (𝑋, 𝑆) be a 𝑆-metric space, then 𝑆(𝑥, 𝑥, 𝑦) = 𝑆(𝑦, 𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋.

𝐵𝑆 (𝑥; 𝑟) = {𝑦 ∈ 𝑋; 𝑆 (𝑥, 𝑥, 𝑦) < 𝑟} .

Theorem 7 (see [6]). Let (𝑋, 𝑆) be a complete 𝑆-metric space and let 𝑓 be a contraction on 𝑋. Then, 𝑓 has a unique fixed point.

(4)

for all 𝑥, 𝑦 ∈ 𝑋, where 0 ≤ 𝜆 < 1. The following result was introduced subsequently.

= 2𝐷𝑆 (𝑥, 𝑧1 ) + 𝐷𝑆 (𝑧1 , 𝑦)

.. .

(8) ≤ 2𝐷𝑆 (𝑥, 𝑧1 ) + 2𝐷𝑆 (𝑧1 , 𝑧2 ) + ⋅ ⋅ ⋅ + 2𝐷𝑆 (𝑧𝑛−1 , 𝑧𝑛 ) + 𝐷𝑆 (𝑧𝑛 , 𝑦) ≤ 2 [𝐷𝑆 (𝑥, 𝑧1 ) + 𝐷𝑆 (𝑧1 , 𝑧2 ) + ⋅ ⋅ ⋅ + 𝐷𝑆 (𝑧𝑛−1 , 𝑧𝑛 ) + 𝐷𝑆 (𝑧𝑛 , 𝑦)] .


3

Thus, (𝑋, 𝐷𝑆 , 2) is a metric type space. Moreover, the balls 𝐵𝑆 (𝑥; 𝑟) and 𝐵𝐷𝑆 (𝑥; 𝑟) coincide. Notice that we may rewrite the inequality (4) as follows: 𝐷𝑆 (𝑓𝑥, 𝑓𝑦) ≤ 𝜆𝐷𝑆 (𝑥, 𝑦) ,

(9)

for all 𝑥, 𝑦 ∈ 𝑋 with the same 𝜆. Also notice that the definition of Cauchyness, convergence and completeness in a 𝑆-metric space (𝑋, 𝑆) may be rewriten in terms of metric type spaces. So, these notions are transferred to the corresponding metric type space (𝑋, 𝐷𝑆 , 2). Now, if 𝑓 is an operator satisfying (9), then each 𝑓𝑛 , where 𝑛 ∈ N, is Lipschizian with Lip(𝑓𝑛 ) = 𝜆𝑛 . Therefore, Theorem 7 is obtained via Theorem 9. Even though we set a new condition for an operator 𝑓, where 0 ≤ 𝜆 < 1, to be 𝑆 (𝑓𝑥, 𝑓𝑦, 𝑓𝑧) ≤ 𝜆𝑆 (𝑥, 𝑦, 𝑧) ,

(10)

for each 𝑥, 𝑦, 𝑧 ∈ 𝑋, the unique fixed point can still be obtained via Theorem 9, anyway. Note that not only the mentioned theorem, but also many theorems in the literature may be proved via this concept in metric type spaces. We shall give some results which seem more general than the setting of Theorem 7 but however equivalent. Beforehand, we give the following useful lemma without proof since it is straight forward. Lemma 10. Let (𝑋, 𝑆) be a 𝑆-metric space and let 𝑥, 𝑦 ∈ 𝑋. Then, the following inequalities hold: (i) 𝑆(𝑥, 𝑦, 𝑦) ≤ 𝑆(𝑥, 𝑥, 𝑦) = 𝑆(𝑦, 𝑦, 𝑥); (ii) 𝑆(𝑥, 𝑦, 𝑥) ≤ 𝑆(𝑥, 𝑥, 𝑦) = 𝑆(𝑦, 𝑦, 𝑥). Theorem 11. Let (𝑋, 𝑆) be a complete 𝑆-metric space and let 𝑓 : 𝑋 → 𝑋 be an operator such that there exists a sequence (𝜆 𝑛 ) of nonnegative reals satisfying the condition: 𝑆 (𝑓𝑥, 𝑓𝑦, 𝑓𝑧) ≤ ∑ 𝜆 𝑛 𝑆 (𝜋𝑛 (𝑥, 𝑦, 𝑧)) , 𝑛∈N

(11)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, where Λ := ∑𝑛∈N 𝜆 𝑛 < 1 and for each 𝑛 ∈ N, 𝜋𝑛 is a fixed permutation in 𝑋3 . Then, 𝑓 has a unique fixed point. Proof. We shall show that (11) implies that 𝑓 is Lipschizian with Lip(𝑓) < 1 in metric type space (𝑋, 𝐷𝑆 , 2). For each 𝑥, 𝑦 ∈ 𝑋, it is easy to verify that 𝑆(𝜋𝑛 (𝑥, 𝑥, 𝑦)) ≤ 𝑆(𝑥, 𝑥, 𝑦) no matter which permutations are defined. Thus, we obtain that 𝐷𝑆 (𝑓𝑥, 𝑓𝑦) = 𝑆 (𝑓𝑥, 𝑓𝑥, 𝑓𝑦) ≤ ∑ 𝜆 𝑛 𝑆 (𝜋𝑛 (𝑥, 𝑥, 𝑦)) 𝑛∈N

(12)

≤ Λ𝑆 (𝑥, 𝑥, 𝑦) = Λ𝐷𝑆 (𝑥, 𝑦) . Apply Theorem 9 (or Theorem 7) to obtain the desired result.

4. A General Distance between Three Arbitrary Points In this section, we shall be dealing with a new concept of a general distance between three arbitrary points (or 𝑔 − 3𝑝𝑠). To be able to define the 𝑔 − 3𝑝𝑠, we first consider a nonempty set 𝑋 together with a function 𝑔 : 𝑋×𝑋×𝑋 → R+ for which 𝑔(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧. Given 𝑥 ∈ 𝑋 and 𝑟 > 0, we define an open ball in the usual sense: 𝐵𝑔 (𝑥; 𝑟) := {𝑦 ∈ 𝑋; 𝑔 (𝑥, 𝑥, 𝑦) < 𝑟} .

(13)

To be natural, we say that a subset 𝐴 ⊂ 𝑋 is bounded if sup𝑥,𝑦,𝑧∈𝐴𝑔(𝑥, 𝑦, 𝑧) < ∞. Certainly, the assertion “𝑔(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧” is not enough to guarantee that open balls in 𝑋 are bounded. We shall illustrate in the following. Example 12. Let 𝑋 := [0, 1] and let 𝑔 : 𝑋 × 𝑋 × 𝑋 → R+ be a function given by 0, { { 󵄨󵄨 󵄨󵄨 𝑥 + 𝑦 { { 󵄨 󵄨 { { {󵄨󵄨󵄨󵄨( 2 ) − 𝑧󵄨󵄨󵄨󵄨 , { { { { { { { 󵄨󵄨 𝑥 + 𝑦 󵄨󵄨 { { 󵄨󵄨( { ) − 𝑧󵄨󵄨󵄨󵄨 , { 󵄨󵄨 { 2 { 󵄨 󵄨 { { { { 𝑔 (𝑥, 𝑦, 𝑧) := { 1 {󵄨 󵄨, { 󵄨 { 󵄨󵄨((𝑥 + 𝑦) /2) − 𝑧󵄨󵄨󵄨 { { { { { { { { 1 { { { 󵄨, 󵄨󵄨 { { 󵄨((𝑥 + 𝑦) /2) − 𝑧󵄨󵄨󵄨 { 󵄨 { { { { { { {1,

if 𝑥 = 𝑦 = 𝑧, if 𝑥, 𝑦 ∈ Q ∩ 𝑋 but 𝑧 ∈ Q∁ ∩ 𝑋, if 𝑥, 𝑦 ∈ Q∁ ∩ 𝑋 but 𝑧 ∈ Q ∩ 𝑋, if 𝑥, 𝑦, 𝑧 ∈ Q ∩ 𝑋 and [𝑥 ≠𝑦 or 𝑦 ≠𝑧] , if 𝑥, 𝑦, 𝑧 ∈ Q∁ ∩ 𝑋 and [𝑥 ≠𝑦 or 𝑦 ≠𝑧] , otherwise. (14)

It is clear that 𝑔(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧. Let 𝑥 ∈ 𝑋 and let 𝑟 > 0. Note that if 𝑥 ∈ Q ∩ 𝑋, then Q∁ ∩ (𝑥 − 𝑟, 𝑥 + 𝑟) ∩ 𝑋 ⊂ 𝐵𝑔 (𝑥; 𝑟) .

(15)

Let (𝑧𝑛 ) be a sequence in Q∁ ∩ (𝑥 − 𝑟, 𝑥 + 𝑟) ∩ 𝑋 such that |𝑥 − 𝑧𝑛 | < 𝑟/(1 + 𝑛𝑟) for each 𝑛 ∈ N. Since |𝑧𝑛 − 𝑧𝑛+1 | < 2𝑟/(1 + 𝑛𝑟), we have 𝑔(𝑧𝑛 , 𝑧𝑛 , 𝑧𝑛+1 ) > (1/2𝑟)+(𝑛/2) for all 𝑛 ∈ N. The same conclusion can be deduced also when 𝑥 ∈ Q∁ . Therefore, 𝐵𝑔 (𝑥; 𝑟) is not bounded at each 𝑥 ∈ 𝑋 and 𝑟 > 0. This is not quite natural and does not meet the requirements we would like to have. So, we may add one more assumption at this stage and define the 𝑔 − 3𝑝𝑠 space as follows. Definition 13. Let 𝑋 be a nonempty set. A function 𝑔 : 𝑋 × 𝑋 × 𝑋 → R+ is said to be a 𝑔−3𝑝𝑠 if the following conditions are satisfied: (g1) for 𝑥, 𝑦, 𝑧 ∈ 𝑋, 𝑔(𝑥, 𝑦, 𝑧) = 0 if and only if 𝑥 = 𝑦 = 𝑧; (g2) there exists some 𝑟0 > 0 such that the balls 𝐵𝑔 (𝑥; 𝑟0 ) are bounded for all 𝑥 ∈ 𝑋. The pair (𝑋, 𝑔) is called a 𝑔 − 3𝑝𝑠 space.

4

Journal of Function Spaces and Applications Next, we shall give a characterization of a 𝑔 − 3𝑝𝑠 space.

Lemma 14. Let 𝑋 be a nonempty set and let 𝑔 : 𝑋 × 𝑋 × 𝑋 → R+ be a function satisfying (g1). Then, the following are equivalent: (i) 𝑔 satisfies (g2); (ii) there exist some constants 𝛿, 𝜂 > 0 such that for any 𝑥, 𝑢, V, 𝑤 ∈ 𝑋, one has 𝑔 (𝑥, 𝑥, 𝑢) + 𝑔 (𝑥, 𝑥, V) + 𝑔 (𝑥, 𝑥, 𝑤) < 𝛿 󳨐⇒ 𝑔 (𝑢, V, 𝑤) < 𝜂. (16) Proof. [(i) ⇒ (ii)] Assume that (i) holds. Set 𝛿 := 𝑟/2 and let 𝑥, 𝑢, V, 𝑤 ∈ 𝑋 arbitrarily. If 𝑔(𝑥, 𝑥, 𝑢) + 𝑔(𝑥, 𝑥, V) + 𝑔(𝑥, 𝑥, 𝑤) < 𝛿, then 𝑢, V, 𝑤 ∈ 𝐵𝑔 (𝑥; 𝑟). Thus, setting 𝜂 := 1 + sup𝑥∈𝑋 sup𝑎,𝑏,𝑐∈𝐵𝑔 (𝑥;𝑟) 𝑔(𝑎, 𝑏, 𝑐) < ∞ and it follows that 𝑔(𝑢, V, 𝑤) < 𝜂. [(ii) ⇒ (i)] Assume that (ii) holds. Let 𝑥 ∈ 𝑋 and suppose that 𝑢, V, 𝑤 ∈ 𝐵𝑔 (𝑥, 𝛿/3). Therefore, we have 𝑔 (𝑥, 𝑥, 𝑢) + 𝑔 (𝑥, 𝑥, V) + 𝑔 (𝑥, 𝑥, 𝑤) < 𝛿.

Remark 15. Suppose that (𝑋, 𝐺) is a 𝐺-metric space and 𝑥, 𝑢, V, 𝑤 ∈ 𝑋. Then, we have

We shall now explicate an example of a 𝑔 − 3𝑝𝑠 space. In particular, this next example will even show that a 𝑔 − 3𝑝𝑠 space is no need to be 𝑇2 -separable. Example 19. Let 𝑋 = [0, 1] and define a function 𝑔 : 𝑋 × 𝑋 × 𝑋 → R+ in the following: 0, { { 𝑔 (𝑥, 𝑦, 𝑧) := {𝑧, { {1,

(18)

𝑋 ∩ [{𝑥} ∪ (0, 𝑟)] , if 𝑟 ≤ 1, 𝐵𝑔 (𝑥; 𝑟) = { 𝑋, if 𝑟 > 1.

(21)

Therefore, any two balls intersect one another, implying that 𝑋 is not 𝑇2 -separable. We next introduce a new concept of convergence and compare it with the classical topological ones.

(1) Cauchy if for any 𝜖 > 0, there exists 𝑁 ∈ N such that 𝑚, 𝑛 ≥ 𝑁 󳨐⇒ 𝑔 (𝑥𝑚 , 𝑥𝑚 , 𝑥𝑛 ) < 𝜖;

Thus, we can choose any 𝛿 > 0 and let 𝜂 := 𝛿 to fulfill the assumptions in Lemma 14. Hence, every 𝐺-metric space is in turn a 𝑔 − 3𝑝𝑠 space. Remark 16. Suppose that (𝑋, 𝑆) is a 𝑆-metric space and 𝑥, 𝑢, V, 𝑤 ∈ 𝑋. Then, we have

= 𝑆 (𝑥, 𝑥, 𝑢) + 𝑆 (𝑥, 𝑥, V) + 𝑆 (𝑥, 𝑥, 𝑤) .

(20)

It is clear that any subset in this space is always bounded. Hence, (𝑋, 𝑔) is a 𝑔 − 3𝑝𝑠 space. Observe that for 𝑥 ∈ 𝑋 and 𝑟 > 0, we have

= 𝐺 (𝑥, 𝑥, 𝑢) + 𝐺 (𝑥, 𝑥, V) + 𝐺 (𝑥, 𝑥, 𝑤) .

𝑆 (𝑢, V, 𝑤) ≤ 𝑆 (𝑢, 𝑢, 𝑥) + 𝑆 (V, V, 𝑥) + 𝑆 (𝑤, 𝑤, 𝑥)

if 𝑥 = 𝑦 = 𝑧, if 𝑥 = 𝑦 ≠𝑧, 𝑧 ≠0, otherwise.

Definition 20. Let (𝑋, 𝑔) be a 𝑔 − 3𝑝𝑠 space. A sequence (𝑥𝑛 ) in 𝑋 is said to be

𝐺 (𝑢, V, 𝑤) ≤ 𝐺 (𝑢, 𝑥, 𝑥) + 𝐺 (𝑥, V, 𝑤)

≤ 𝐺 (𝑥, 𝑥, 𝑢) + 𝐺 (V, 𝑥, 𝑥) + 𝐺 (𝑥, 𝑥, 𝑤)

Proof. Let 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≠𝑦. So, we have 𝑔(𝑥, 𝑥, 𝑦) = 𝑟1 and 𝑔(𝑦, 𝑦, 𝑥) = 𝑟2 , for some 𝑟1 , 𝑟2 > 0. Observe that 𝑦 ∉ 𝐵𝑔 (𝑥; 𝑟1 /2) and 𝑥 ∉ 𝐵𝑔 (𝑦; 𝑟2 /2). The desired result is then followed.

(17)

From (ii), we obtain that 𝑔(𝑢, V, 𝑤) < 𝜂. Thus, sup𝑎,𝑏,𝑐∈𝐵𝑔 (𝑋;𝛿/3) 𝑔(𝑎, 𝑏, 𝑐) ≤ 𝜂. Since 𝑥 ∈ 𝑋 is arbitrary, the balls 𝐵𝑔 (𝑥; 𝛿/3) are bounded for every 𝑥 ∈ 𝑋.

= 𝐺 (𝑥, 𝑥, 𝑢) + 𝐺 (V, 𝑥, 𝑤)

Proposition 18. The topology T for a 𝑔 − 3𝑝𝑠 space (𝑋, 𝑔) is 𝑇1 -separable.

(19)

The same argument is to be considered as in the previous remark. So, a 𝑆-metric space is a 𝑔 − 3𝑝𝑠 space. This immediately implies that a 𝐷∗ -metric space is also a 𝑔 − 3𝑝𝑠 space. Denoted by U the family of all open balls in 𝑋. Throughout this paper, we shall assume that T := T(U) represents the topology having U as its subbase. Also, we write U∗ to denote the base generated by U. Remark 17. The topology here is defined using different idea from those given in symmetric spaces or in semimetric spaces (see e.g., [14–17]).

(22)

(2) g-convergent if we can find a point 𝑥 ∈ 𝑋 in which for any 𝜖 > 0, there exists 𝑁 ∈ N satisfying 𝑛 ≥ 𝑁 󳨐⇒ 𝑔 (𝑥, 𝑥, 𝑥𝑛 ) < 𝜖.

(23)

In this case, we say that (𝑥𝑛 ) 𝑔-converges to 𝑥 and write

𝑔

𝑥𝑛 󳨀 → 𝑥. Remark 21. Given a 𝑔 − 3𝑝𝑠 space (𝑋, 𝑔), a sequence (𝑥𝑛 ) in 𝑋 𝑔-converges to 𝑥 ∈ 𝑋 if and only if for any set 𝑈 ∈ U with 𝑈 ∋ 𝑥, there exists 𝑁 ∈ N such that 𝑛 ≥ 𝑁 󳨐⇒ 𝑥𝑛 ∈ 𝑈.

(24)

Lemma 22. Let (𝑋, 𝑔) be a 𝑔 − 3𝑝𝑠 space and let (𝑥𝑛 ) be a sequence in 𝑋. Then, the following are equivalent: (i) (𝑥𝑛 ) converges to 𝑥 ∈ 𝑋 in the topology T; (ii) for any neighborhood 𝑉 ∈ T of 𝑥, on can find 𝑁 ∈ N such that 𝑛 ≥ 𝑁 󳨐⇒ 𝑥𝑛 ∈ 𝑉;

(25)


5

(iii) for any set 𝑈 ∈ U∗ containing 𝑥, we can find 𝑁 ∈ N such that 𝑛 ≥ 𝑁 󳨐⇒ 𝑥𝑛 ∈ 𝑈.

(26)

Proof. (i) ⇔ (ii) is by definition. So, we only need to show that (ii) ⇔ (iii). [(ii) ⇒ (iii)] Since U∗ ∈ T, we again apply (ii) to obtain our desired result. [(iii) ⇒ (ii)] By the definition of T, for every 𝑧 ∈ 𝐹 ∈ T, we can find 𝐸 ∈ U∗ in which 𝑥 ∈ 𝐸 ⊂ 𝐹. Suppose that 𝑉 ∈ T is a neighborhood of 𝑥, then we can find some 𝑈0 ∈ U∗ such that 𝑥 ∈ 𝑈0 ⊂ 𝑉. Applying (ii), we obtain that 𝑥𝑛 ∈ 𝑉 for every 𝑛 ≥ 𝑁 for some fixed 𝑁 ∈ N. We conclude the following lemma immediately from Remark 21 and Lemma 22.

Proof. Let 𝑥 ∈ 𝑋 be arbitrary. We shall consider the permutation 𝜋 case-by-case. (i) Case I: 𝜋(𝑥, 𝑦, 𝑧) := (𝑥, 𝑦, 𝑧) or 𝜋(𝑥, 𝑦, 𝑧) := (𝑦, 𝑥, 𝑧). Observe that 𝑔 (𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) ≤ 𝜆𝑔 (𝑓𝑛−1 𝑥, 𝑓𝑛−1 𝑥, 𝑓𝑛 𝑥) ≤ 𝜆2 𝑔 (𝑓𝑛−2 𝑥, 𝑓𝑛−2 𝑥, 𝑓𝑛−1 𝑥)

≤ 𝜆𝑛 𝑔 (𝑥, 𝑥, 𝑓𝑥) . (ii) Case II: 𝜋(𝑥, 𝑦, 𝑧) := (𝑥, 𝑧, 𝑦). Observe that 𝑔 (𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) ≤ 𝜆𝑔 (𝑓𝑛−1 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛−1 𝑥) ≤ 𝜆2 𝑔 (𝑓𝑛−2 𝑥, 𝑓𝑛−2 𝑥, 𝑓𝑛−1 𝑥)

Lemma 23. Let (𝑋, 𝑔) be a 𝑔 − 3𝑝𝑠 space and let (𝑥𝑛 ) be a sequence in 𝑋. If (𝑥𝑛 ) converges to some point 𝑥 ∈ 𝑋 in the topology T, then it also 𝑔-converges to 𝑥. This lemma shows that the concept of 𝑔-convergence is weaker than convergence in topology. However, in case when (𝑋, 𝑔) is a 𝐺-metric space or when (𝑋, 𝑔) is a 𝑆-metric space, the two concepts coincide. Along this paper, we shall deal with this new kind of convergence, rather than those topological ones.

(28)

.. .

(29)

.. . ≤ 𝜆𝑛 Γ1 , where ∞ > Γ1 ≥ max{𝑔(𝑥, 𝑥, 𝑓𝑥), 𝑔(𝑥, 𝑓𝑥, 𝑥)}. (iii) Case III: 𝜋(𝑥, 𝑦, 𝑧) := (𝑦, 𝑧, 𝑥). Observe that 𝑔 (𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) ≤ 𝜆𝑔 (𝑓𝑛−1 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛−1 𝑥) ≤ 𝜆2 𝑔 (𝑓𝑛−1 𝑥, 𝑓𝑛−2 𝑥, 𝑓𝑛−2 𝑥)

Definition 24. A 𝑔 − 3𝑝𝑠 space (𝑋, 𝑔) is called (i) 𝑔-Hausdorff if every 𝑔-convergent sequence 𝑔converges to at most one point;

≤ 𝜆3 𝑔 (𝑓𝑛−3 𝑥, 𝑓𝑛−3 𝑥, 𝑓𝑛−2 𝑥)

(ii) Σ-complete if every sequence (𝑥𝑛 ) satisfying ∑𝑛∈N 𝑔(𝑥𝑛 , 𝑥𝑛 , 𝑥𝑛+1 ) < ∞ 𝑔-converges; (iii) Cauchy-complete if every Cauchy sequence 𝑔converges. Remark 25. A 𝑔 − 3𝑝𝑠 space (𝑋, 𝑔) is 𝑔-Hausdorff if and only if for any two distinct points 𝑥, 𝑦 ∈ 𝑋, there exist two disjoint sets 𝑈, 𝑉 ∈ U such that 𝑈 ∋ 𝑥 and 𝑉 ∋ 𝑦.

.. . ≤ 𝜆𝑛 Γ2 , where ∞ > Γ2 ≥ max{𝑔(𝑥, 𝑥, 𝑓𝑥), 𝑓(𝑥, 𝑓𝑥, 𝑥), 𝑓(𝑓𝑥, 𝑥, 𝑥)}. (iv) Case IV: 𝜋(𝑥, 𝑦, 𝑧) := (𝑧, 𝑥, 𝑦) or 𝜋(𝑥, 𝑦, 𝑧) := (𝑧, 𝑦, 𝑥). Observe that 𝑔 (𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) ≤ 𝜆𝑔 (𝑓𝑛 𝑥, 𝑓𝑛−1 𝑥, 𝑓𝑛−1 𝑥)

5. Some Fixed Point Theorems

≤ 𝜆2 𝑔 (𝑓𝑛−2 𝑥, 𝑓𝑛−2 𝑥, 𝑓𝑛−1 𝑥)

Under this section, we propose some fixed point theorems in the framework of a 𝑔 − 3𝑝𝑠 space. We first introduce a lemma which will be used in our main theorems. Lemma 26. Let (𝑋, 𝑔) be a 𝑔 − 3𝑝𝑠 space. Suppose that 𝑓 : 𝑋 → 𝑋 be an operator such that 𝑔 (𝑓𝑥, 𝑓𝑦, 𝑧) ≤ 𝜆𝑔 (𝜋 (𝑥, 𝑦, 𝑧)) ,

(27)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, where 0 ≤ 𝜆 < 1 and 𝜋 is a fixed permutation on 𝑋3 . Then, the following hold for every 𝑥 ∈ 𝑋: 𝑛

𝑛

(i) ∑𝑛∈N 𝑔(𝑓 𝑥, 𝑓 𝑥, 𝑓 (ii) ∑𝑛∈N 𝑔(𝑓

𝑛+1

𝑥, 𝑓

𝑛+1

𝑛+1

𝑥) < ∞;

𝑥, 𝑓𝑛 𝑥) < ∞.

(30)

(31)

.. . ≤ 𝜆𝑛 Γ1 . In each case, we may conclude that ∑ 𝑔 (𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) ≤ Γ2 ∑ 𝜆𝑛 < ∞.

𝑛∈N

𝑛∈N

(32)

Similarly, we may prove that ∑ 𝑔 (𝑓𝑛+1 𝑥, 𝑓𝑛+1 𝑥, 𝑓𝑛 𝑥) ≤ Γ3 ∑ 𝜆𝑛 < ∞,

𝑛∈N

𝑛∈N

(33)

where ∞ > Γ3 ≥ max{𝑔(𝑓𝑥, 𝑓𝑥, 𝑥), 𝑔(𝑓𝑥, 𝑥, 𝑓𝑥), 𝑔(𝑥, 𝑓𝑥, 𝑓𝑥)}.

6


Now, we consider our fixed point results, which exploited the above lemma.

for some 𝑛0 ∈ N. Then, observe that 2𝑔 (𝑓𝑁+1 𝑥, 𝑓𝑁+1 𝑥, 𝑓𝑁𝑥) + 𝑔 (𝑓𝑁+1 𝑥, 𝑓𝑁+1 𝑥, 𝑓𝑁+𝑛0 +1 𝑥)

Theorem 27. Let (𝑋, 𝑔) be a 𝑔-Hausdorff Σ-complete 𝑔 − 3𝑝𝑠 space. Suppose that 𝑓 : 𝑋 → 𝑋 is a 𝑔-sequentially continuous

≤ 2𝑔 (𝑓𝑁+1 𝑥, 𝑓𝑁+1 𝑥, 𝑓𝑁𝑥) + 𝜆𝑔 (𝑓𝑁𝑥, 𝑓𝑁𝑥, 𝑓𝑁+𝑛0 𝑥)

→ 𝑥 ⇒ 𝑓𝑥𝑛 󳨀 → 𝑓𝑥, for every sequence (𝑥𝑛 ) operator (i.e., 𝑥𝑛 󳨀 in 𝑋) satisfying

< 2𝑔 (𝑓𝑁+1 𝑥, 𝑓𝑁+1 𝑥, 𝑓𝑁𝑥) + 𝜆𝜂

𝑔

𝑔

𝑔 (𝑓𝑥, 𝑓𝑦, 𝑧) ≤ 𝜆𝑔 (𝜋 (𝑥, 𝑦, 𝑧)) ,

(34)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, where 0 ≤ 𝜆 < 1 and 𝜋 is a fixed permutation on 𝑋3 . Then, 𝑓 has exactly one fixed point. Proof. Let 𝑥 ∈ 𝑋 be arbitrary. By (i) in Lemma 26, we have ∑ 𝑔 (𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) < ∞. 𝑛∈N

𝑓

Hence, from Lemma 14, we have 𝑔(𝑓𝑁𝑥, 𝑓𝑁𝑥, 𝑓𝑁+𝑛0 +1 𝑥) < 𝜂. Therefore, we have 𝑔(𝑓𝑁𝑥, 𝑓𝑁𝑥, 𝑓𝑁+𝑛 𝑥) < 𝜂 for all 𝑛 ∈ N. Let 𝜖 > 0 be given and let 𝑚, 𝑛 ∈ N with 𝑁 ≤ 𝑚 < 𝑛. Thus, we may write 𝑚 = 𝑁 + ℓ + 𝑝,

𝑔

𝑥󳨀 → 𝑓𝑥∗ . Since 𝑋 is 𝑔-Hausdorff, we have 𝑓𝑥∗ = 𝑥∗ . Assume that 𝑦∗ ∈ 𝑋 is also a fixed point of 𝑓. Observe that 𝑔 (𝑥∗ , 𝑥∗ , 𝑦∗ ) = 𝑔 (𝑓𝑥∗ , 𝑓𝑥∗ , 𝑓𝑦∗ ) ≤ 𝜆𝑔 (𝑥∗ , 𝑥∗ , 𝑦∗ ) .

(41)

(35)

Since 𝑋 is Σ-complete, (𝑓𝑛 𝑥)𝑔-converges to some point 𝑥∗ ∈ 𝑋. Now, since 𝑓 is 𝑔-sequentially continuous, 𝑓𝑓𝑛 𝑥 = 𝑛+1

< 𝛿.

(36)

for some ℓ, 𝑝, 𝑞 ∈ N. Note that 𝑝 < 𝑞. It follows that 𝑔 (𝑓𝑚 𝑥, 𝑓𝑚 𝑥, 𝑓𝑛 𝑥) = 𝑔 (𝑓𝑁+ℓ+𝑝 𝑥, 𝑓𝑁+ℓ+𝑝 𝑥, 𝑓𝑁+ℓ+𝑞 𝑥) ≤ 𝜆𝑔 (𝑓𝑁+ℓ+𝑝−1 𝑥, 𝑓𝑁+ℓ+𝑝−1 𝑥, 𝑓𝑁+ℓ+𝑞−1 𝑥)

𝑔 (𝑓𝑥, 𝑓𝑦, 𝑓𝑧) ≤ 𝜆𝑔 (𝜋 (𝑥, 𝑦, 𝑧)) ,

(37)

for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, where 0 ≤ 𝜆 ≤ 𝛿/3𝜂 and 𝛿, 𝜂 > 0 are given as in Lemma 14. Then, 𝑓 has exactly one fixed point. Proof. Let 𝑥 ∈ 𝑋. From Lemma 26, we also have 𝑔(𝑓𝑛 𝑥, 𝑓𝑛 𝑥, 𝑓𝑛+1 𝑥) → 0 and 𝑔(𝑓𝑛+1 𝑥, 𝑓𝑛+1 𝑥, 𝑓𝑛 𝑥) → 0. So, we may choose 𝑁 ∈ N such that 𝑁

𝑁

𝑔 (𝑓 𝑥, 𝑓 𝑥, 𝑓

𝑁+1

𝛿 𝑥) < , 3

𝑔 (𝑓𝑁+1 𝑥, 𝑓𝑁+1 𝑥, 𝑓𝑁𝑥)
0 on Ω}, where “𝜉 > 0 on Ω” means that 𝜉0 (𝜔) > 0 a.s. for an arbitrarily chosen representative 𝜉0 of 𝜉. Definition 1 (see [3]). An ordered pair (𝑆, ‖ ⋅ ‖) is called a random normed space (briefly, an RN space) over 𝐾 with base (Ω, F, 𝜇) if 𝑆 is a linear space over 𝐾 and ‖ ⋅ ‖ is a mapping from 𝑆 to 𝐿0+ (F) such that the following three conditions are satisfied: (RN-1) ‖𝑥‖ = 0 implies 𝑥 = 𝜃 (the null in 𝑆), (RN-2) ‖𝛼𝑥‖ = |𝛼|‖𝑥‖, for all 𝛼 ∈ 𝐾 and 𝑥 ∈ 𝑆, (RN-3) ‖𝑥 + 𝑦‖ ≤ ‖𝑥‖ + ‖𝑦‖, for all 𝑥, 𝑦 ∈ 𝑆, where ‖𝑥‖ is called the random norm of 𝑥. If ‖ ⋅ ‖ only satisfies (RN-2) and (RN-3), then it is called a random seminorm. Furthermore, if, in addition, 𝑆 is a left module over the algebra 𝐿0 (F, 𝐾) (briefly, an 𝐿0 (F, 𝐾)-module) and the following additional condition is also satisfied: (RNM-1) ‖𝜉𝑥‖ = |𝜉|‖𝑥‖, for all 𝜉 ∈ 𝐿0 (F, 𝐾) and 𝑥 ∈ 𝑆, then (𝑆, ‖ ⋅ ‖) is called a random normed module (briefly, an RN module) over 𝐾 with base (Ω, F, 𝜇), at which time ‖ ⋅ ‖ is called an 𝐿0 norm on 𝑆. Similarly, if ‖ ⋅ ‖ only satisfies (RN-3) and (RNM-1), then it is called an 𝐿0 -seminorm on 𝑆. Remark 2. In [1], the original definition of an RN space was given by only requiring (Ω, F, 𝜇) to be a probability space and defining ‖𝑥‖ to be a nonnegative random variable; the corresponding (RN-1) to (RN-3) are given in the following way: (RN-1)󸀠 ‖𝑥‖(𝜔) = 0 a.s. implies 𝑥 = 𝜃, (RN-2)󸀠 ‖𝛼𝑥‖(𝜔) = |𝛼|‖𝑥‖(𝜔) a.s., for all 𝛼 ∈ 𝐾 and 𝑥 ∈ 𝑆, (RN-3)󸀠 ‖𝑥 + 𝑦‖(𝜔) ≤ ‖𝑥‖(𝜔) + ‖𝑦‖(𝜔) a.s., for all 𝑥, 𝑦 ∈ 𝑆. This definition is natural and intuitive from probability theory, but (RN-1)󸀠 is difficult to satisfy when we construct examples. Thus we essentially have employed Definition 1 since our work [2] by saying that measurable functions or random variables that are equal a.s. are identified; in particular since 1999 we strictly distinguish between measurable functions and their equivalence classes in writings; compare [3]. Remark 3. At outset we consider both the real and complex cases in the study of RN spaces, whereas they only consider the real case in [6, 7] because of their special interests; an RN

Journal of Function Spaces and Applications module over 𝑅 is termed as a randomly normed 𝐿0 -module in [6] and an 𝐿0 -normed module in [7]. We still would like to continue to employ the terminology “an RN module over 𝐾 with base (Ω, F, 𝜇)” in order to keep concordance with the earliest terminology used in [1]. Definition 4 (see [3, 5, 14]). Let (𝑆, ‖ ⋅ ‖) and (𝑆1 , ‖ ⋅ ‖1 ) be two RN spaces over 𝐾 with base (Ω, F, 𝜇). A linear operator 𝑇 from 𝑆 to 𝑆1 is said to be a.s. bounded if there exists 𝜉 ∈ 𝐿0+ (F) such that ‖𝑇𝑥‖1 ≤ 𝜉‖𝑥‖, for all 𝑥 ∈ 𝑆, in which case ‖𝑇‖ is defined to be ∧{𝜉 ∈ 𝐿0+ (F) | ‖𝑇𝑥‖1 ≤ 𝜉‖𝑥‖, ∀𝑥 ∈ 𝑆}, called the random norm of 𝑇. Denote the linear space of a.s. bounded linear operators from 𝑆 to 𝑆1 by 𝐵(𝑆, 𝑆1 ); then (𝐵(𝑆, 𝑆1 ), ‖ ⋅ ‖) still becomes an RN space over 𝐾 with base (Ω, F, 𝜇) when ‖𝑇‖ is defined as above for every 𝑇 ∈ 𝐵(𝑆, 𝑆1 ). In particular, when 𝑆1 = 𝐿0 (F, 𝐾) and ‖ ⋅ ‖1 = | ⋅ | (namely, the absolute value mapping), 𝑆∗ := 𝐵(𝑆, 𝑆1 ) is called the random conjugate space of 𝑆 and an element in 𝑆∗ is called an a.s. bounded random linear functional on 𝑆. Remark 5. When (𝑆1 , ‖ ⋅ ‖1 ) in Definition 4 is an RN module, 𝐵(𝑆, 𝑆1 ) automatically becomes an RN module under the module operation (𝜉 ⋅ 𝑇)(𝑥) = 𝜉 ⋅ (𝑇(𝑥)), for all 𝜉 ∈ 𝐿0 (F, 𝐾), 𝑇 ∈ 𝐵(𝑆, 𝑆1 ), and 𝑥 ∈ 𝑆. When 𝑆 and 𝑆1 are both RN modules, in [6] 𝐵(𝑆, 𝑆1 ) is used to stand for the 𝐿0 (F, 𝐾)module of a.s. bounded module homomorphisms from 𝑆 to 𝑆1 ; we will show that in the special case an a.s. bounded linear operator must be a module homomorphism. Therefore, the two implications of 𝐵(𝑆, 𝑆1 ) coincide in this case. As in the classical functional analysis, we can similarly define a conjugate operator 𝑇∗ : 𝑆2∗ → 𝑆1∗ for an a.s. bounded linear operator 𝑇 from (𝑆1 , ‖ ⋅ ‖1 ) to (𝑆2 , ‖ ⋅ ‖2 ) as follows: (𝑇∗ 𝑓)(𝑥) = 𝑓(𝑇𝑥), for all 𝑓 ∈ 𝑆2∗ , 𝑥 ∈ 𝑆1 . From the Hahn-Banach theorem for a.s. bounded random linear functional established in [2] (also see [8]), one has that ‖𝑇∗ ‖ = ‖𝑇‖. For the sake of convenience, let us recall some notation and terminology in the theory of probabilistic normed spaces. Definition 6 (see [1]). A function 𝑇 : [0, 1] × [0, 1] → [0, 1] is called a weak 𝑡-norm if the following are satisfied: (t-1) 𝑇(𝑎, 𝑏) = 𝑇(𝑏, 𝑎), for all 𝑎, 𝑏 ∈ [0, 1], (t-2) 𝑇(𝑎, 𝑏) ≤ 𝑇(𝑐, 𝑑), for all 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0, 1] with 𝑎 ≤ 𝑐, 𝑏 ≤ 𝑑, (t-3) 𝑇(1, 0) = 0, 𝑇(1, 1) = 1. A weak 𝑡-norm 𝑇 : [0, 1] × [0, 1] → [0, 1] is called a 𝑡-norm if the following two additional conditions are satisfied: (t-4) 𝑇(𝑎, 𝑇(𝑏, 𝑐)) = 𝑇(𝑇(𝑎, 𝑏), 𝑐), for all 𝑎, 𝑏, 𝑐 ∈ [0, 1], (t-5) 𝑇(1, 𝑎) = 𝑎, for all 𝑎 ∈ [0, 1].

3 Although 𝑡-norms are widely used in the theory of probabilistic metric spaces, weak 𝑡-norms have their own advantages, for example, for a family {𝑇𝛼 , 𝛼 ∈ ∧} of weak 𝑡norms, 𝑇 defined by 𝑇(𝑎, 𝑏) = sup{𝑇𝛼 (𝑎, 𝑏) : 𝛼 ∈ ∧}, for all 𝑎, 𝑏 ∈ [0, 1], is still a weak 𝑡-norm, whereas this is not true for 𝑡-norms. Throughout this paper, Δ = {𝐹 : [−∞, +∞] → [0, 1] | 𝐹(−∞) = 0, 𝐹(+∞) = 1, 𝐹 is nondecreasing and left continuous on (−∞, +∞)}, 𝐷 = {𝐹 ∈ Δ | lim𝑥 → −∞ 𝐹(𝑥) = 0 and lim𝑥 → +∞ 𝐹(𝑥) = 1}, Δ+ = {𝐹 ∈ Δ | 𝐹(0) = 0}, and 𝐷+ = {𝐹 ∈ 𝐷 | 𝐹(0) = 0}. For extended real random variable 𝜉 on a probability space (Ω, F, 𝑃), its (left continuous) distribution function 𝑁𝜉 is defined by 𝑁𝜉 (𝑡) = 𝑃{𝜔 ∈ Ω | 𝜉(𝜔) < 𝑡}, for all 𝑡 ∈ [−∞, +∞) and 𝑁𝜉 (+∞) = 1. In particular, 𝜖0 stands for the distribution function defined by 𝜖0 (𝑡) = 1 when 𝑡 > 0 and 𝜖0 (𝑡) = 0 when 𝑡 ≤ 0; namely, 𝜖0 is the distribution function of the constant 0. Definition 7 (see [1]). A triple (𝑆, N, 𝑇) is called a Menger probabilistic normed space (briefly, an M-PN space) over 𝐾 if 𝑆 is a linear space over 𝐾, N is a mapping from 𝑆 to 𝐷+ , and 𝑇 is a weak 𝑡-norm such that the following are satisfied: (MPN-1) 𝑁𝑥 = 𝜖0 if and only if 𝑥 = 𝜃 (the null element in 𝑆), (MPN-2) 𝑁𝛼𝑥 (𝑡) = 𝑁𝑥 (𝑡/|𝛼|), for all 𝑡 ≥ 0, {0} and 𝑥 ∈ 𝑆,

𝛼 ∈ 𝐾\

(MPN-3) 𝑁𝑥+𝑦 (𝑢+V) ≥ 𝑇(𝑁𝑥 (𝑢), 𝑁𝑦 (V)), for all 𝑢, V ≥ 0 and 𝑥, 𝑦 ∈ 𝑆. Here 𝑁𝑥 := N(𝑥) is called the probabilistic norm of 𝑥. ̃|𝑇 ̃ is a weak 𝑡For an M-PN space (𝑆, N, 𝑇), let T = {𝑇 ̃ ̂ : norm such that (𝑆, N, 𝑇) is an M-PN space}, and define 𝑇 ̂ ̃ ̃ [0, 1] × [0, 1] → [0, 1] by 𝑇(𝑎, 𝑏) = sup{𝑇(𝑎, 𝑏) | 𝑇 ∈ T}, ̂ ∈ T. 𝑇 ̂ for all 𝑎, 𝑏 ∈ [0, 1]; then it is very easy to see that 𝑇 is called the largest weak 𝑡-norm of (𝑆, N) such that (𝑆, N) is ̂ From now on, for an M-PN space an M-PN space under 𝑇. (𝑆, N, 𝑇), we always assume that 𝑇 is the largest weak 𝑡-norm of (𝑆, N). In [15], LaSalle introduced the notion of a pseudonormed linear space: let 𝑆 be a linear space over 𝐾 and {𝑝𝛼 }𝛼∈∧ a family of mappings from 𝑆 to 𝑅+ := [0, +∞) and indexed by a directed set ∧; then (𝑆, {𝑝𝛼 }𝛼∈∧ ) is called a pseudonormed linear space if the following are satisfied: (PNS-1) 𝑝𝛼 (𝛽𝑥) = |𝛽|𝑝𝛼 (𝑥), for all 𝛽 ∈ 𝐾, 𝑥 ∈ 𝑆, and 𝛼 ∈ ∧, (PNS-2) 𝑝𝛼1 (𝑥) ≤ 𝑝𝛼2 (𝑥), for all 𝛼1 , 𝛼2 ∈ ∧ such that 𝛼1 ≤ 𝛼2 , 𝑥 ∈ 𝑆,

(PNS-3) for any 𝛼 ∈ ∧, there exists 𝛼󸀠 ∈ ∧ such that 𝑝𝛼 (𝑥 + 𝑦) ≤ 𝑝𝛼󸀠 (𝑥) + 𝑝𝛼󸀠 (𝑦), for all 𝑥, 𝑦 ∈ 𝑆.

Let (𝑆, {𝑝𝛼 }𝛼∈∧ ) be a pseudonormed linear space. For any 𝜖 > 0 and 𝛼 ∈ ∧, let 𝑈𝜃 (𝛼, 𝜖) = {𝑥 ∈ 𝑆 | 𝑝𝛼 (𝑥) < 𝜖}. Then U𝜃 = {𝑈𝜃 (𝛼, 𝜖) | 𝛼 ∈ ∧, 𝜖 > 0} is a local base at the null element 𝜃 of some linear topology for 𝑆, called the linear topology induced by {𝑝𝛼 }𝛼∈∧ . Conversely any linear topology

4


for 𝑆 can be induced by some {𝑝𝛼 }𝛼∈∧ such that (𝑆, {𝑝𝛼 }𝛼∈∧ ) is a pseudonormed linear space. To connect an M-PN space (𝑆, N, 𝑇) to a pseudonormed linear space, for each 𝑟 ∈ (0, 1), define 𝑝𝑟 : 𝑆 → [0, +∞) by 𝑝𝑟 (𝑥) = sup{𝑡 ≥ 0 | 𝑁𝑥 (𝑡) < 𝑟}, for all 𝑥 ∈ 𝑆. Then we have the following. Theorem 8 (see [16]). Let (𝑆, N, 𝑇) be an M-PN space. Then one has the following statements. (1) sup0 𝜂]. We can now prove Theorem 48.

The version of Theorem 48 under the locally 𝐿0 -convex topology, namely, Corollary 50 below, was given in [8]. Corollary 49 below is the version of Theorem 47 under the locally 𝐿0 -convex topology.

Proof of Theorem 48. Let Ran(𝑓) = {𝑓(𝑥) | 𝑥 ∈ 𝑆} and 𝑁(𝑓) = {𝑥 ∈ 𝑆 | 𝑓(𝑥) = 0}; then Ran(𝑓) is a submodule of 𝐿0 (F, 𝐾) and 𝑁(𝑓) a T𝜖,𝜆 -closed submodule of 𝑆. By Proposition 17 {|𝑓(𝑥)| | 𝑥 ∈ 𝑆} is upward directed and hence there exists a sequence {𝑥𝑛 , 𝑛 ∈ 𝑁} in 𝑆 such that {|𝑓(𝑥𝑛 )|, 𝑛 ∈ 𝑁} converges a.s. to 𝜉 := ∨{|𝑓(𝑥)| | 𝑥 ∈ 𝑆} in a nondecreasing manner. Denote 𝜉𝑛 = |𝑓(𝑥𝑛 )|, for all 𝑛 ∈ 𝑁; let 𝜉0 and 𝜉𝑛0 be any chosen representatives of 𝜉 and 𝜉𝑛 , respectively, 𝐵 = {𝜔 ∈ Ω | 𝜉0 (𝜔) > 0}, and 𝐵𝑛 = {𝜔 ∈ Ω | 𝜉𝑛0 (𝜔) > 0}, for all 𝑛 ≥ 1. We can, without loss of generality, assume that 𝐵𝑛 ⊂ 𝐵𝑛+1 , for all 𝑛 ≥ 1, and ⋃∞ 𝑛=1 𝐵𝑛 = 𝐵. Further, let 𝐵0 = ⌀ and 𝐴 𝑛 = 𝐵𝑛 \ 𝐵𝑛−1 , for all 𝑛 ≥ 1; then 𝐴 𝑖 ∩ 𝐴 𝑗 = ⌀, where 𝑖 ≠𝑗, 0 ̃ and 𝐵 = ⋃∞ 𝑛=1 𝐴 𝑛 . Since 𝜉𝑛 > 0 on 𝐴 𝑛 , we have 𝑓(𝑧𝑛 ) = 𝐼𝐴 𝑛 , −1 where 𝑧𝑛 = 𝐼̃𝐴 𝑛 (𝑓(𝑥𝑛 )) ⋅ 𝑥𝑛 , for all 𝑛 ≥ 1. If 𝜇(𝐵) = 0, then taking 𝑦𝑓 = 𝜃 ends the proof. If 𝜇(𝐵) > 0, then 𝑓 ≠0, in which case we can assume that 𝜇(𝐴 𝑛 ) > 0, for all 𝑛 ∈ 𝑁. By Theorem 47 there exists a unique 𝑧𝑛 ∈ 𝑁(𝑓)⊥ such that 𝑧𝑛 −𝑧𝑛 ∈ 𝑁(𝑓), and hence 𝑓(𝑧𝑛 ) = 𝑓(𝑧𝑛 ) = 𝐼̃𝐴 𝑛 , for all 𝑛 ∈ 𝑁. Since 𝐼̃𝐴 𝑛 𝑥 − 𝐼̃𝐴 𝑛 𝑓(𝑥)𝑧𝑛 ∈ 𝑁(𝑓), ⟨𝐼̃𝐴 𝑛 𝑥 − 𝐼̃𝐴 𝑛 𝑓(𝑥)𝑧𝑛 , 𝑧𝑛 ⟩ = 0, for all 𝑛 ∈ 𝑁 and 𝑥 ∈ 𝑆. Since 𝐼̃𝐴 𝑛 = 𝑓(𝑧𝑛 ) = |𝑓(𝑧𝑛 )| ≤ ‖𝑓‖‖𝑧𝑛 ‖, ‖𝑧𝑛 ‖ > 0 on 𝐴 𝑛 and 𝐼̃𝐴 𝑛 𝑓(𝑥) = ⟨𝑥, 𝑧𝑛∗ ⟩, for all 𝑥 ∈ 𝑆, where 𝑧𝑛∗ = 𝐼̃𝐴 𝑛 (‖𝑧𝑛 ‖)−1 𝑧𝑛 , for all 𝑛 ∈ 𝑁. Let 𝑦𝑛 = ∑𝑛𝑖=1 𝑧𝑖∗ , for all 𝑛 ∈ 𝑁. By noticing that 𝐼𝐵 𝑓(𝑥) = 𝑓(𝑥), for all 𝑥 ∈ 𝑆, and 𝐼̃𝐵 = lim𝑛 → ∞ 𝐼̃𝐵𝑛 = lim𝑛 → ∞ ∑𝑛𝑖=1 𝐼̃𝐴 𝑖 , 𝑓(𝑥) = lim𝑛 → ∞ (𝐼̃𝐵𝑛 𝑓(𝑥)) = lim𝑛 → ∞ (∑𝑛𝑖=1 𝐼̃𝐴 𝑖 𝑓(𝑥)) = lim𝑛 → ∞ ⟨𝑥, 𝑦𝑛 ⟩ (where convergence means the a.s. convergence). We can, without loss of generality, assume that (Ω, F, 𝜇) is a probability space; then 𝜇{𝜔 ∈ Ω | ‖𝑦𝑛+𝑚 − 𝑦𝑛 ‖(𝜔) > 𝜖} ≤ ∑𝑛+𝑚 𝑖=𝑛+1 𝜇(𝐴 𝑖 ), for any 𝜖 > 0 and 𝑛, 𝑚 ∈ 𝑁, which implies that {𝑦𝑛 , 𝑛 ∈ 𝑁} is T𝜖,𝜆 -Cauchy and hence convergent to some 𝑦𝑓 ∈ 𝑆, so 𝑓(𝑥) = ⟨𝑥, 𝑦𝑓 ⟩, for all 𝑥 ∈ 𝑆. ‖𝑓‖ ≤ ‖𝑦𝑓 ‖ is obvious. Now, let 𝐴 = [‖𝑦𝑓 ‖ > 0] and 𝐷 = [|𝑓(𝑦𝑓 )| ≤ ‖𝑓‖‖𝑦𝑓 ‖]; then 𝜇(𝐷) = 1 and 𝐼𝐴‖𝑓‖ ≥ 𝐼𝐴‖𝑦𝑓 ‖, where 𝐼𝐴 = 𝐼̃𝐴0 with 𝐴0 being any representative of 𝐴. On

Corollary 49. Let (𝑆, ‖ ⋅ ‖) be a T𝑐 -complete RIP module over 𝐾 with base (Ω, F, 𝜇) and 𝑀 a T𝑐 -closed submodule of 𝑆 such that both 𝑆 and 𝑀 have the countable concatenation property. Then 𝑆 = 𝑀 ⊕ 𝑀⊥ . Proof. It follows from Propositions 34 and 35 and Theorem 47. Corollary 50. Let (𝑆, ⟨⋅, ⋅⟩) be a T𝑐 -complete RIP module over 𝐾 with base (Ω, F, 𝜇) such that 𝑆 has the countable concatenation property. Then for each 𝑓 ∈ 𝑆∗ there exists a unique 𝑦𝑓 ∈ 𝑆 such that 𝑓(𝑥) = ⟨𝑥, 𝑦𝑓 ⟩, for all 𝑥 ∈ 𝑆, and ‖𝑓‖ = ‖𝑦𝑓 ‖. Proof. It follows from Proposition 35 and Theorem 48. Remark 51. Based on Theorem 48, we can establish the spectral representation theorem for random self-adjoint operators on complete complex random inner product modules, which has been used to establish the Stone’s representation theorem for a group of random unitary operators in [24]. 󸀠

5. (𝐿𝑝 (𝑆)) ≅ 𝐿𝑞 (𝑆∗ ) In this section, let (𝑆, ‖ ⋅ ‖) be a given RN module over 𝐾 with base (Ω, F, 𝜇) and 𝑆∗ its random conjugate space. Further, let 1 ≤ 𝑝 < +∞ and 1 < 𝑞 ≤ +∞ be a pair of Hölder conjugate numbers. Let 𝑟 be an extended nonnegative number with 1 ≤ 𝑟 ≤ +∞ and (𝐿𝑟 (F), | ⋅ |𝑟 ) the Banach space of equivalence classes of 𝑟-integrable (when 𝑟 < +∞) or essentially bounded (when 𝑟 = +∞) real F-measurable functions on (Ω, F, 𝜇) with the usual 𝐿𝑟 -norm | ⋅ |𝑟 . Further, let 𝐿𝑟 (𝑆) = {𝑥 ∈ 𝑆 | ‖𝑥‖ ∈ 𝐿𝑟 (F)} and let ‖ ⋅ ‖𝑟 : 𝐿𝑟 (𝑆) → [0, +∞) be defined by ‖𝑥‖𝑟 = |‖𝑥‖|𝑟 , for all 𝑥 ∈ 𝐿𝑟 (𝑆); then (𝐿𝑟 (𝑆), ‖ ⋅ ‖𝑟 ) is a normed space over 𝐾 and T𝜖,𝜆 -dense in 𝑆; compare [22, 25]. Similarly, one can understand the implication of 𝐿𝑟 (𝑆∗ ). Theorem 52 below is proved in [25], a more general result is proved in [6] with 𝐿𝑟 (F) replaced by a Köthe function space, but the two proofs both only give the key idea of them. Here, we give a detailed proof of Theorem 52. Since our aim is to look for the tool for the development of the theory of RN modules together with their random conjugate spaces, Theorem 52 is enough for the aim. 󸀠

Theorem 52. 𝐿𝑞 (𝑆∗ ) ≅ (𝐿𝑝 (𝑆)) under the canonical mapping 󸀠 𝑇, where (𝐿𝑝 (𝑆)) denotes the classical conjugate space of 𝐿𝑝 (𝑆)


11

and for each 𝑓 ∈ 𝐿𝑞 (𝑆∗ ), 𝑇𝑓 (denoting 𝑇(𝑓)): 𝐿𝑝 (𝑆) → 𝐾 is defined by 𝑇𝑓 (𝑥) = ∫Ω 𝑓(𝑥)𝑑𝜇, for all 𝑥 ∈ 𝐿𝑝 (𝑆). We will divide the proof of Theorem 52 into the following two Lemmas—Lemmas 53 and 54. Lemma 53. 𝑇 is isometric. Proof. |𝑇𝑓 (𝑥)| = | ∫Ω 𝑓(𝑥)𝑑𝜇| ≤ ∫Ω |𝑓(𝑥)|𝑑𝜇 ≤ ∫Ω ‖𝑓‖ ‖𝑥‖ 𝑑𝜇 ≤ ‖𝑓‖𝑞 ‖𝑥‖𝑝 , for all 𝑥 ∈ 𝐿𝑝 (𝑆), so ‖𝑇𝑓 ‖ ≤ ‖𝑓‖𝑞 . As for ‖𝑓‖𝑞 ≤ ‖𝑇𝑓 ‖, we can, without loss of generality, assume that (Ω, F, 𝜇) is a probability space. Let {𝑥𝑛 , 𝑛 ∈ 𝑁} be a sequence in 𝑆(1) := {𝑥 ∈ 𝑆 | ‖𝑥‖ ≤ 1} such that {|𝑓(𝑥𝑛 )|, 𝑛 ∈ 𝑁} converges a.s. to ‖𝑓‖ in a nondecreasing manner (such a sequence {𝑥𝑛 , 𝑛 ∈ 𝑁} does exist!). When 𝑝 = 1 and 𝑞 = +∞, for any positive number 𝜖 let 𝐴(𝜖) = [‖𝑓‖ > ‖𝑓‖∞ − 𝜖]; then 𝜇(𝐴(𝜖)) > 0, and hence there exists some 𝑛0 ∈ 𝑁 such that 𝐵(𝜖) := [|𝑓(𝑥𝑛0 )| > ‖𝑓‖∞ −𝜖] has a positive measure. Let 𝑥𝜖 = 1/𝜇(𝐵(𝜖))⋅𝐼𝐵(𝜖) ⋅sgn(𝑓(𝑥𝑛0 ))⋅𝑥𝑛0 ; then ‖𝑥𝜖 ‖1 ≤ 1 and |𝑇𝑓 (𝑥𝜖 )| > ‖𝑓‖∞ − 𝜖, which shows that ‖𝑇𝑓 ‖ ≥ ‖𝑓‖∞ . When 𝑝 > 1, 󵄨 󵄨 󵄨𝑞 󵄨𝑞−1 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 𝑑𝜇 = ∫ 󵄨󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 󵄨󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 𝑑𝜇 Ω Ω 󵄨𝑞−1 󵄨 = ∫ 󵄨󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 ⋅ sgn (𝑓 (𝑥𝑛 )) ⋅ 𝑓 (𝑥𝑛 ) 𝑑𝜇 Ω 󵄨 󵄨𝑞−1 = ∫ 𝑓 (󵄨󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 ⋅ sgn (𝑓 (𝑥𝑛 )) ⋅ 𝑥𝑛 ) 𝑑𝜇 Ω 󵄨 󵄨𝑞−1 = 𝑇𝑓 (󵄨󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 ⋅ sgn (𝑓 (𝑥𝑛 )) ⋅ 𝑥𝑛 ) 󵄩 󵄩 ≤ 󵄩󵄩󵄩󵄩𝑇𝑓 󵄩󵄩󵄩󵄩 (∫

󵄨󵄨 󵄨𝑞 󵄨󵄨𝑓 (𝑥𝑛 )󵄨󵄨󵄨 𝑑𝜇) Ω

1/𝑝

, (1)

then ‖𝑓(𝑥𝑛 )‖𝑞 ≤ ‖𝑇𝑓 ‖, for all 𝑛 ∈ 𝑁. By the Levy theorem we have that ‖𝑓‖𝑞 ≤ ‖𝑇𝑓 ‖. Lemma 54. 𝑇 is surjective. 󸀠

Proof. For any fixed 𝑙 ∈ (𝐿𝑝 (𝑆)) and 𝑥 ∈ 𝐿∞ (𝑆), define the scalar measure 𝐺𝑥 : F → 𝐾 and the vector measure 𝐺 : 󸀠 F → (𝐿∞ (𝑆)) as follows: 𝐺𝑥 (𝐸) = 𝑙(𝐼̃𝐸 ⋅ 𝑥), for all 𝐸 ∈ F, 𝐺(𝐸)(𝑦) = 𝑙(𝐼̃𝐸 ⋅ 𝑦), for all 𝑦 ∈ 𝐿∞ (𝑆) and 𝐸 ∈ F. Since |𝐺(𝐸)(𝑦)| = |𝑙(𝐼̃𝐸 ⋅ 𝑦)| ≤ ‖𝑙‖ ⋅ ‖𝐼̃𝐸 ⋅ 𝑦‖𝑝 ≤ ‖𝑙‖ ⋅ ‖𝑦‖∞ ⋅ ‖𝐼̃𝐸 ‖𝑝 , for all 𝐸 ∈ F, 𝑦 ∈ 𝐿∞ (𝑆), both 𝐺 and 𝐺𝑥 are countably additive. Now, for any finite partition {𝐸1 , 𝐸2 , . . . , 𝐸𝑛 } of Ω to F and finitely many points 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , in the closed unit ball of 𝐿∞ (𝑆), we have | ∑𝑛𝑖=1 𝐺(𝐸𝑖 )(𝑥𝑖 )| = |𝑙(∑𝑛𝑖=1 𝐼̃𝐸𝑖 ⋅ 𝑥𝑖 )| ≤

‖𝑙‖ ⋅ | ∑𝑛𝑖=1 𝐼̃𝐸𝑖 |𝑝 = ‖𝑙‖. Similarly, we have that ∑𝑛𝑖=1 |𝐺𝑥 (𝐸𝑖 )| =

∑𝑛𝑖=1 sgn(𝐺𝑥 (𝐸𝑖 )) ⋅ 𝐺𝑥 (𝐸𝑖 ) = ∑𝑛𝑖=1 𝐺(𝐸𝑖 )(sgn(𝐺𝑥 (𝐸𝑖 )) ⋅ 𝑥) ≤ ‖𝑙‖ ⋅ ‖𝑥‖∞ . So |𝐺|(Ω) ≤ ‖𝑙‖ and |𝐺𝑥 |(Ω) ≤ ‖𝑙‖ ⋅ ‖𝑥‖∞ ; namely, 𝐺 and 𝐺𝑥 are both of bounded variation and they are both absolutely continuous with respect to 𝜇. By the classical Radon-Nikodým theorem there exists a unique 𝑔(𝑥) ∈ 𝐿1 (F, 𝐾) for each 𝑥 ∈ 𝐿∞ (𝑆) such that 𝐺𝑥 (𝐸) = ∫𝐸 𝑔(𝑥)𝑑𝜇, for all 𝐸 ∈ F, and |𝐺𝑥 |(𝐸) = ∫𝐸 |𝑔(𝑥)|𝑑𝜇, for all 𝐸 ∈ F, so we can obtain a mapping 𝑔 : 𝐿∞ (𝑆) → 𝐿1 (𝜇, 𝐾) such that (1) 𝑔(𝛼𝑥 + 𝛽𝑦) = 𝛼𝑔(𝑥) + 𝛽𝑔(𝑦), for all 𝛼, 𝛽 ∈ 𝐾, and 𝑥, 𝑦 ∈ 𝐿∞ (𝑆); (2) 𝑔(𝜉𝑥) = 𝜉 ⋅ 𝑔(𝑥) for each simple element 𝜉 in 𝐿0 (F, 𝐾), 𝑥 ∈ 𝐿∞ (𝑆).

We can now assert that 𝑔(𝜉𝑥) = 𝜉𝑔(𝑥), for all 𝜉 ∈ 𝐿∞ (F, 𝐾) and 𝑥 ∈ 𝐿∞ (𝑆). In fact, for any 𝜉 ∈ 𝐿∞ (F, 𝐾) there are always a sequence {𝜉𝑛 , 𝑛 ∈ 𝑁} of simple elements in 𝐿0 (F, 𝐾) such that {‖𝜉𝑛 − 𝜉‖∞ , 𝑛 ∈ 𝑁} converges to 0; then ‖𝑔(𝜉𝑥) − 𝑔(𝜉𝑛 𝑥)‖1 = |𝐺(𝜉−𝜉𝑛 )𝑥 |(Ω) ≤ ‖𝑙‖ ⋅ ‖𝜉 − 𝜉𝑛 ‖∞ ⋅ ‖𝑥‖∞ → 0 (𝑛 → ∞). On the other hand, ‖𝜉𝑔(𝑥) − 𝜉𝑛 𝑔(𝑥)‖1 ≤ ‖𝜉 − 𝜉𝑛 ‖∞ ‖𝑔(𝑥)‖1 → 0 (𝑛 → ∞), so 𝑔(𝜉𝑥) = 𝐿1 − lim𝑛 → ∞ 𝑔(𝜉𝑛 𝑥) = 𝐿1 − lim𝑛 → ∞ (𝜉𝑛 𝑔(𝑥)) = 𝜉𝑔(𝑥). We prove that {|𝑔(𝑥)| |𝑥 ∈ 𝑆(1)|} is upward directed as follows: for any 𝑥 and 𝑦 ∈ 𝑆(1), let 𝐸 = [|𝑔(𝑥)| ≤ |𝑔(𝑦)|]; then |𝑔(𝑥)|∨|𝑔(𝑦)| = 𝐼𝐸 |𝑔(𝑦)|+(1−𝐼𝐸 )|𝑔(𝑥)| = 𝑔(sgn(𝑔(𝑦))⋅𝐼𝐸 ⋅𝑦+ sgn(𝑔(𝑥))⋅(1−𝐼𝐸 )⋅𝑥) = 𝑔(𝑧) = |𝑔(𝑧)|, where 𝑧 = sgn(𝑔(𝑦))⋅𝐼𝐸 ⋅ 𝑦+sgn(𝑔(𝑥))⋅(1−𝐼𝐸 )⋅𝑥 ∈ 𝑆(1). Hence, there exists a sequence {𝑥𝑛 , 𝑛 ∈ 𝑁} in 𝑆(1) such that {|𝑔(𝑥𝑛 )|, 𝑛 ∈ 𝑁} converges a.s. to 𝜉 := ∨{|𝑔(𝑥)| | 𝑥 ∈ 𝑆(1)} in a nondecreasing manner. By the Levy theorem ‖𝜉‖1 = lim𝑛 → ∞ ‖𝑔(𝑥𝑛 )‖1 = lim𝑛 → ∞ |𝐺𝑥𝑛 |(Ω) ≤ |𝐺|(Ω) ≤ ‖𝑙‖ < +∞, so 𝜉 ∈ 𝐿1 (F, 𝐾). Since for any positive number 𝜖 and 𝑥 ∈ 𝐿∞ (𝑆), it is clear that |𝑔((1/(‖𝑥‖ + 𝜖))𝑥)| ≤ 𝜉; namely, |𝑔(𝑥)| ≤ 𝜉(‖𝑥‖ + 𝜖), which implies that |𝑔(𝑥)| ≤ 𝜉‖𝑥‖, for all 𝑥 ∈ 𝐿∞ (𝑆). By the Hahn-Banach theorem for a.s. bounded random linear functionals (see [2, 8]) there is an a.s. bounded random linear functional 𝑓 ∈ 𝑆∗ such that 𝑓|𝐿∞ (𝑆) = 𝑔; further 𝑓 is unique since 𝐿∞ (𝑆) is T𝜖,𝜆 -dense in 𝑆 and ‖𝑓‖ ≤ 𝜉; we also have that ‖𝑓‖ ∈ 𝐿1 (F, 𝐾). By the definition of 𝐺𝑥 , 𝑙(𝑥) = 𝐺𝑥 (Ω) = ∫Ω 𝑔(𝑥)𝑑𝜇 = ∫Ω 𝑓(𝑥), for all 𝑥 ∈ 𝐿∞ (𝑆). We prove that 𝑓 ∈ 𝐿𝑞 (𝑆∗ ) as follows: let 𝐸𝑛 = [‖𝑓‖ ≤ 𝑛] and 𝑓𝑛 = 𝐼𝐸𝑛 ⋅ 𝑓; then 𝑓𝑛 ∈ 𝐿𝑞 (𝑆∗ ) and ‖𝑓𝑛 ‖ = 𝐼𝐸𝑛 ‖𝑓‖ (here, we can assume that 𝜇 is a probability measure). Since ∫𝐸 𝑓(𝑥)𝑑𝜇 = 𝑙(𝐼𝐸𝑛 ⋅ 𝑥), for all 𝑥 ∈ 𝐿∞ (𝑆) and 𝑛 𝑛 ∈ 𝑁, then 𝑇𝑓𝑛 |𝐿∞ (𝑆) = 𝑙|𝐼𝐸 ⋅𝐿∞ (𝑆) . From the fact that 𝐿∞ (𝑆) is dense in (𝐿𝑝 (𝑆), ‖ ⋅ ‖𝑝 ), 𝑇𝑓𝑛 |𝐿𝑝 (𝑆) = 𝑙|𝐼𝐸 ⋅𝐿𝑝 (𝑆) , so ‖𝑇𝑓𝑛 ‖ ≤ ‖𝑙‖; 𝑛 letting 𝑛 → +∞ yields that |‖𝑓‖|𝑞 ≤ lim𝑛 → ∞ ‖𝑓𝑛 ‖𝑞 = lim𝑛 → ∞ ‖𝑇𝑓𝑛 ‖ ≤ ‖𝑙‖ < +∞; namely, 𝑓 ∈ 𝐿𝑞 (𝑆∗ ). Further, we also have that 𝑙 = 𝑇𝑓 since they coincide on the dense subspace 𝐿∞ (𝑆) of 𝐿𝑝 (𝑆). Remark 55. Let (𝐵, ‖ ⋅ ‖) be a normed space over 𝐾 and 𝐿0 (F, 𝐵) the 𝐿0 (F, 𝐾)-module of equivalence classes of 𝐵-

12 valued F-strongly measurable functions on (Ω, F, 𝜇). Let 𝐵󸀠 be the classical conjugate space of 𝐵 and 𝐿0 (F, 𝐵󸀠 , 𝑤∗ ) the 𝐿0 (F, 𝐾)-module of 𝑤∗ -equivalence classes of 𝐵󸀠 -valued 𝑤∗ measurable functions on (Ω, F, 𝜇). For any 𝑥 ∈ 𝐿0 (F, 𝐵) with a representative 𝑥0 , the 𝐿0 -norm of 𝑥 is defined to be the equivalence class of ‖𝑥0 ‖, still denoted by ‖𝑥‖; then (𝐿0 (F, 𝐵), ‖ ⋅ ‖) is an RN module over 𝐾 with base (Ω, F, 𝜇). For any 𝑦 ∈ 𝐿0 (F, 𝐵󸀠 , 𝑤∗ ) with a representative 𝑦0 , the 𝐿0 -norm of 𝑦 is defined to be the equivalence class of esssup{|𝑦0 (𝑏)| | 𝑏 ∈ 𝐵 and ‖𝑏‖ ≤ 1} (namely, the essential supremum of {|𝑦0 (𝑏)| | 𝑏 ∈ 𝐵 and ‖𝑏‖ ≤ 1}); then 𝐿0 (F, 𝐵󸀠 , 𝑤∗ ) is also an RN module over 𝐾 with base (Ω, F, 𝜇). In [26] it is proved that (𝐿0 (F, 𝐵))∗ = 𝐿0 (F, 𝐵󸀠 , 𝑤∗ ), so if we take 𝑆 = 𝐿0 (F, 𝐵) in Theorem 52 then 𝐿𝑝 (F, 𝐵)󸀠 ≅ 𝐿𝑞 (F, 𝐵󸀠 , 𝑤∗ ). Generally speaking, 𝜎-finite measure spaces are enough for various kinds of problems in analysis, but some more general measure spaces are sometimes necessary; for example, strictly localizable measure spaces are considered in [6]. Even in [27] we introduced the notion of an RN module with base being an arbitrary measure space (Ω, F, 𝜇) by defining it to be a projective limit of a family of RN modules with base (𝐴, 𝐴 ∩ F, 𝜇|𝐴∩F ), where 𝐴 ∈ F satisfies 0 < 𝜇(𝐴) < +∞, and further proved that Theorem 52 remains true for any measure space, so Theorem 52 unifies all the representation theorems of the dual spaces of Lebesgue-Bochner function spaces. As said in [6, 25], it is more interesting that Theorem 52 establishes the key connection between random conjugate spaces and classical conjugate spaces, which has played a crucial role in the subsequent development of random conjugate spaces; compare [22, 28, 29]. Remark 56. Since the Lebesgue-Bochner function space 𝐿𝑝 (F, 𝐵) (or is written as 𝐿𝑝 (𝜇, 𝐵)) has the target space 𝐵, the simple functions in 𝐿𝑝 (F, 𝐵) always play an active role in the study of the dual of 𝐿𝑝 (F, 𝐵), whereas we do not have the counterparts of simple elements in 𝐿𝑝 (F, 𝐵) for abstract spaces 𝐿𝑝 (𝑆), so we are forced to replace simple elements in 𝐿𝑝 (F, 𝐵) with elements in 𝐿∞ (𝑆) in order to complete the proof of Theorem 52. In [26] we prove that a Banach space 𝐵 is reflexive if and only if 𝐿0 (F, 𝐵) is random reflexive; the original motivation of Theorem 52 is to establish the following characterization. Theorem 57. Let (𝑆, ‖ ⋅ ‖) be a T𝜖,𝜆 -complete RN module over 𝐾 with base (Ω, F, 𝜇) and 𝑝 any given positive number such that 1 < 𝑝 < +∞. Then 𝑆 is random reflexive if and only if 𝐿𝑝 (𝑆) is reflexive. Proof. Let 𝐽 : 𝑆 → 𝑆∗∗ and 𝑗 : 𝐿𝑝 (𝑆) → (𝐿𝑝 (𝑆))󸀠󸀠 be the corresponding canonical embedding mappings. (1) Necessity. Since 1 < 𝑝 < +∞, its Hölder conjugate number 󸀠 𝑞 satisfies 1 < 𝑞 < +∞; then (𝐿𝑝 (𝑆))󸀠󸀠 = (𝐿𝑞 (𝑆∗ )) = 𝑝 ∗∗ 𝑝 𝐿 (𝑆 ) = 𝐿 (𝑆). (2) Sufficiency. Let 𝑓∗∗ be any given element in 𝑆∗∗ and 𝐸𝑛 = [𝑛 − 1 ≤ ‖𝑓∗∗ ‖ < 𝑛] for any 𝑛 ∈ 𝑁. We can, without loss of generality, assume that 𝜇 is a probability measure; then

Journal of Function Spaces and Applications ∗∗ ∈ 𝐿𝑝 (𝑆∗∗ ) = (𝐿𝑝 (𝑆))󸀠󸀠 , there ∑∞ 𝑛=1 𝜇(𝐸𝑛 ) = 1. Since 𝐼𝐸𝑛 𝑓 𝑝 exists 𝑥𝑛 ∈ 𝐿 (𝑆) such that 𝑗(𝑥𝑛 ) = 𝐼𝐸𝑛 𝑓∗∗ ; namely, for each 𝑓∗ ∈ 𝐿𝑞 (𝑆∗ ), we have that ∫Ω 𝑓∗ (𝑥𝑛 )𝑑𝜇 = 𝑗(𝑥𝑛 )(𝑓∗ ) = ∫Ω 𝐼𝐸𝑛 𝑓∗∗ (𝑓∗ )𝑑𝜇. By replacing 𝑓∗ with 𝐼𝐸 𝑓∗ we can obtain that ∫𝐸 𝑓∗ (𝑥𝑛 )𝑑𝜇 = ∫𝐸 𝐼𝐸𝑛 𝑓∗∗ (𝑓∗ )𝑑𝜇, for all 𝐸 ∈ F and 𝑓∗ ∈ 𝐿𝑞 (𝑆∗ ), which implies that 𝑓∗ (𝑥𝑛 ) = 𝐼𝐸𝑛 𝑓∗∗ (𝑓∗ ), for all 𝑓∗ ∈ 𝐿𝑞 (𝑆∗ ). Since 𝐿𝑞 (𝑆∗ ) is T𝜖,𝜆 -dense in 𝑆∗ , 𝐽(𝑥𝑛 ) = 𝐼𝐸𝑛 𝑓∗∗ , for all 𝑛 ∈ 𝑁. Let 𝑦𝑛 = ∑𝑛𝑘=1 𝐼𝐸𝑘 ⋅ 𝑥𝑘 , for all 𝑛 ∈ 𝑁; then {𝑦𝑛 , 𝑛 ∈ 𝑁} is T𝜖,𝜆 -Cauchy in 𝑆 and hence convergent to some 𝑦 ∈ 𝑆, which shows that 𝐽(𝑦) = lim𝑛 → ∞ 𝐽(𝑦𝑛 ) = lim𝑛 → ∞ ∑𝑛𝑘=1 𝐼𝐸𝑘 ⋅ 𝑓∗∗ = ∗∗ = 𝑓∗∗ ; namely, 𝐽 is surjective. (∑∞ 𝑛=1 𝐼𝐸𝑛 ) ⋅ 𝑓

Remark 58. Concerning Theorem 57, a similar and more general result was given in [6] where 𝐿𝑝 (F) is replaced by a reflexive Köthe function space.

Acknowledgment The author would like to thank Professor Quanhua Xu for kindly providing the reference [6] in February 2012, which makes us know, for the first time, the excellent work of Haydon, Levy, and Raynaud.

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Research Article The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum’s Sense Z. Jesús,1 O. Mejía,1 N. Merentes,1 and S. Rivas2 1 2

Departamento de Matemática, Universidad Central de Venezuela, Caracas 1220A, Venezuela Instituto Universitario Tecnológico El´ıas Calixto Pompa, Guatire, Venezuela

Correspondence should be addressed to O. Mej´ıa; odalism [email protected] Received 20 March 2013; Accepted 20 July 2013 Academic Editor: Gen-Qi Xu Copyright © 2013 Z. Jesús et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We show that the composition operator 𝐻, associated with ℎ : [𝑎, 𝑏] → R, maps the spaces Lip[𝑎, 𝑏] on to the space 𝜅BV𝜙 [𝑎, 𝑏] of functions of bounded variation in Schramm-Korenblum’s sense if and only if ℎ is locally Lipschitz. Also, verify that if the composition operator generated by ℎ : [𝑎, 𝑏] × R → R maps this space into itself and is uniformly bounded, then regularization of ℎ is affine in the second variable.

1. Introduction The composition operator problem (or COP, for short) refers to determining the conditions on a function ℎ : R → R, such that the composition operator, associated with the function ℎ, maps a space X of functions 𝑢 : [𝑎, 𝑏] → R into itself [1, 2]. There are several spaces where the COP has been resolved. For example, in 1961, Babaev [3] showed that the composition operator 𝐻, associated with the function ℎ : R → R, maps the space Lip[𝑎, 𝑏] of the Lipschitz functions into itself if and only if ℎ is locally Lipschitz; in 1967, Mukhtarov [4] obtained the same result for the space Lip𝛼 [𝑎, 𝑏] of the Hölder functions of order 𝛼 (0 < 𝛼 < 1). The first work on the COP in the space of functions of bounded variation BV[𝑎, 𝑏] was made by Josephy in 1981 [5]. In 1986, Ciemnoczołowski and Orlicz [6] got the same result for the space of the functions of bounded 𝜑variation in Wiener’s sense. In 1974, Chaika and Waterman [7] reached a similar result for the space of functions of bounded harmonic variation HBV[𝑎, 𝑏]. In the years 1991 and 1995 Merentes showed a similar result for the spaces of absolutely continuous functions AC[𝑎, 𝑏] and the space of function of bounded 𝜑-variation in Riesz’s sense RV𝜑 [𝑎, 𝑏] (see [8, 9]), and in 1998 Merentes and Rivas achieved the same result when the composition operator maps the space RV𝑝 [𝑎, 𝑏] of the functions of bounded 𝑝-variation in Riesz’s sense (1 < 𝑝 < ∞) into the space BV[𝑎, 𝑏] [10]. In 2003, Pierce

and Waterman solved the COP for the spaces 𝜙BV[𝑎, 𝑏] and ΛBV[𝑎, 𝑏] [11]. More recently, in 2011, Appell et al. [1] conclude the same results verifying when the composition operator maps Lip[𝑎, 𝑏] into BV[𝑎, 𝑏]. Finally, Appell and Merentes verify the same result for the space of functions of bounded 𝜅-variation [12]. There exist spaces X of real functions defined on an interval [𝑎, 𝑏], such that 𝐻 maps X into itself and ℎ is not locally Lipschitz. For example, in the case the space of continuous functions 𝐶[𝑎, 𝑏] it follows from the TietzeUrysohn theorem that the composition operator 𝐻 acts from 𝐶[𝑎, 𝑏] into itself and the function ℎ must be continuous; that is, ℎ does not need to be Lipschitz. A similar result was obtained in the space of regulated functions [13]. A first objective of this work is to demonstrate that the composition operator, associated with the function ℎ, maps the space Lip[𝑎, 𝑏] of the Lipschitz functions into the space 𝜅BV𝜙 [𝑎, 𝑏] of functions of bounded variation in SchrammKorenblum’s sense or into the space 𝜅BV[𝑎, 𝑏] of functions of bounded variation in Korenblum’s sense if and only if ℎ is locally Lipschitz. We also extend this result to function spaces X, Y , such that Lip[𝑎, 𝑏] ⊂ X ⊂ Y , where Y ⊂ 𝜅BV[𝑎, 𝑏] or Y ⊂ BV𝜙 [𝑎, 𝑏]. In a seminal article of 1982, Matkowski [14] showed that if the composition operator 𝐻, associated with the function ℎ : [𝑎, 𝑏] × R → R, maps the space Lip[𝑎, 𝑏] of the Lipschitzian

2


functions into itself and is a globally Lipschitzian map, then the function ℎ has the form ℎ (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R,

(1)

for some 𝛼, 𝛽 ∈ Lip[𝑎, 𝑏]. There are a variety of spaces besides Lip[𝑎, 𝑏] that verify this result [15]. The spaces of Banach (X, ‖ ⋅ ‖) that fulfill this property are said to satisfy the Matkowski property [1]. In 1984, Matkowski and Mi´s [16] considered the same hypotheses on the operator 𝐻 for the space BV[𝑎, 𝑏] of the function of bounded variation and concluded that (1) is true for the regularization ℎ− of the function ℎ with respect of the first variable; that is, ℎ− (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R,

(2)

where 𝛼, 𝛽 ∈ BV− [a, b]. Spaces that satisfy this conditions said to be verified Weak Matkowski Property [1]. A second objective of this paper is to show that if function ℎ(𝑡, ⋅) is continuous in the second variable, for each 𝑡 ∈ [𝑎, 𝑏], and the composition operator 𝐻, associated with the function ℎ, is uniformly bounded, then ℎ satisfies (2).

2. Preliminaries Let [𝑎, 𝑏] be a closed interval of the real line R (𝑎, 𝑏 ∈ R, 𝑎 < 𝑏). From now on, for a function 𝑢 : [𝑎, 𝑏] → R denote by 𝐿𝑏𝑎 (𝑢) the Lipschitz constant of 𝑢; that is, 𝐿𝑏𝑎 (𝑢) = sup {

|𝑢 (𝑡) − 𝑢 (𝑠)| : 𝑡, 𝑠 ∈ [𝑎, 𝑏] , 𝑠 ≠𝑡} . |𝑠 − 𝑡|

(3)

By Lip[𝑎, 𝑏] = {𝑢 : [𝑎, 𝑏] → R : 𝐿𝑏𝑎 (𝑢) < ∞} we will denote the space of the Lipschitz functions. It is well known that the space Lip[𝑎, 𝑏] is a Banach space endowed with the norm ‖𝑢‖Lip := |𝑢 (𝑎)| + 𝐿𝑏𝑎 (𝑢)

(𝑢 ∈ Lip [𝑎, 𝑏]) .

(4)

Ever since the notion of a function of bounded variation appeared, it has led to an incredible number of generalizations. In 1881, Jordan [17] introduced the definition of function of bounded variation for a function 𝑢 : [𝑎, 𝑏] → R and showed that these kinds of function can be decomposed as the difference of two monotone functions. As a consequence of this result we have that those functions satisfy the Dirichlet criterion, that is, the functions that have pointwise convergent Fourier series. Jordan defined such functions in the following way. Definition 1. Let 𝑢 : [𝑎, 𝑏] → R and 𝜋 : 𝑎 ≤ 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛 ≤ 𝑏 be a partition of the interval [𝑎, 𝑏]. Consider 𝑛−1 󵄨 󵄨 𝜎 (𝑢, 𝜋) := ∑ 󵄨󵄨󵄨󵄨𝑢 (𝑡𝑗+1 ) − 𝑢 (𝑡𝑗 )󵄨󵄨󵄨󵄨 , 𝑗=1

𝑉 (𝑢; [𝑎, 𝑏]) = sup 𝜎 (𝑢, 𝜋) , 𝜋

where the supremum is taken over all partitions 𝜋 of the interval [𝑎, 𝑏]. If 𝑉(𝑢; [𝑎, 𝑏]) < ∞, then 𝑢 has bounded variation on the interval [𝑎, 𝑏] and this number is called the variation in Jordan’s sense on [𝑎, 𝑏]. This space of function is denoted by BV[𝑎, 𝑏]. The concept of bounded variation has been the subject of intensive research, and many applications, generalizations, and improvements of them can be found in the literature (see for instance [18–20]). Some generalizations have been introduced by De La Vallée Poussin, F. Riesz, N. Wiener, L. C. Young, Yu. T. Medvedev, D. Waterman, B. Korenblum and M. Schramm. In 1975, Korenblum in [21] considered a new kind of variation, called 𝜅-variation, introducing a function 𝜅 for distorting the expression |𝑡𝑗 − 𝑡𝑗−1 | in the partition itself rather than the expression |𝑓(𝑡𝑗 ) − 𝑓(𝑡𝑗−1 )| in the range. Subsequently, this class of functions has been studied in detail by Cyphert and Kelingos [22]. One advantage of this alternate approach is that a function of bounded 𝜅-variation may be decomposed into the difference of two simpler functions called 𝜅-decreasing functions (for the precise definition see the following). Definition 2. A function 𝜅 : [0, 1] → [0, 1] is said to be a 𝜅-function or distortion function if it satisfies the following properties: (1) 𝜅 is continuous with 𝜅(0) = 0 and 𝜅(1) = 1, (2) 𝜅 is concave, increasing, and (3) lim𝑡 → 0+ (𝜅(𝑡)/𝑡) = ∞. Simple examples of distortion functions are 𝜅 (𝑡) = 𝑡𝛼 (0 < 𝛼 < 1) ,

𝜅 (𝑡) = 𝑡 (1 − log 𝑡) .

(6)

From Definition 2 we can see that 𝜅 is subadditive; that is, 𝜅 (𝑠 + 𝑡) ≤ 𝜅 (𝑠) + 𝜅 (𝑡) and since lim𝑡 → 0+ (𝜅(𝑡)/𝑡) = generality we can assume that 𝜅 (𝑡) ≥ 𝑡

(0 ≤ 𝑠, 𝑡 ≤ 1) ,

(7)

∞, then without loss of

(𝑡 ∈ [0, 1]) .

(8)

Furthermore Korenblum introduces the following concept of variation. Definition 3. Let 𝜅 : [0, 1] → [0, 1] be a distortion function and 𝑢 : [𝑎, 𝑏] → R and 𝜋 : 𝑎 ≤ 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛 ≤ 𝑏 a partition of the interval [𝑎, 𝑏]. Consider 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ∑𝑛−1 𝑗=1 󵄨󵄨𝑢 (𝑡𝑗+1 ) − 𝑢 (𝑡𝑗 )󵄨󵄨 , 𝜅 (𝑢, 𝜋) := 𝑛−1 ∑𝑗=1 𝜅 ((𝑡𝑗+1 − 𝑡𝑗 ) / (𝑏 − 𝑎)) (9) 𝜅𝑉 (𝑢) = 𝜅𝑉 (𝑢; [𝑎, 𝑏]) := sup 𝜅 (𝑢, 𝜋) ,

(5)

𝜋

where the supremum is taken over all partitions 𝜋 of the interval [𝑎, 𝑏], called the 𝜅-variation of 𝑢 on [𝑎, 𝑏]. In the case,


3

𝜅𝑉(𝑢; [𝑎, 𝑏]) < ∞ we say that 𝑢 has 𝜅-variation on [𝑎, 𝑏] and we will denote by 𝜅BV[𝑎, 𝑏] the space of functions of 𝜅variation on [𝑎, 𝑏]. In the case that [𝑎, 𝑏] = [0, 1] the equality (9) becomes 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ∑𝑛−1 𝑗=1 󵄨󵄨𝑢 (𝑡𝑗+1 ) − 𝑢 (𝑡𝑗 )󵄨󵄨 . (10) 𝜅 (𝑢, 𝜋) := 𝑛−1 ∑𝑗=1 𝜅 (𝑡𝑗+1 − 𝑡𝑗 ) Some properties of the functions with bounded 𝜅variation are summarized in the following theorem. Theorem 4 (see [22]). Let 𝜅 : [0, 1] → [0, 1] be a distortion function. Then (1) 𝜅BV[𝑎, 𝑏] is a Banach space endowed with the norm ‖𝑢‖𝜅 = |𝑢 (𝑎)| + 𝜅𝑉 (𝑢; [𝑎, 𝑏])

(𝑢 ∈ 𝜅BV [𝑎, 𝑏]) .

(11)

(2) If the function 𝑢 is monotone, then 𝜅𝑉(𝑢; [𝑎, 𝑏]) = |𝑢(𝑏) − 𝑢(𝑎)|. (3) If 𝑢 ∈ 𝜅BV[𝑎, 𝑏], then 𝑢 is bounded and ‖𝑢‖∞ ≤ 2 ‖𝑢‖𝜅 . (4) Lip[𝑎, 𝑏] ⊂ BV[𝑎, 𝑏] ⊂ 𝜅BV[𝑎, 𝑏] ⊂ 𝑅[𝑎, 𝑏], where 𝑅[𝑎, 𝑏] denotes the space of regulated functions. (5) If 𝑢 ∈ 𝜅BV[𝑎, 𝑏], then 𝑢 can be decomposed as difference of two 𝜅-decreasing functions, that is, there exist functions V : [𝑎, 𝑏] → R such that sup𝑠 0. We denote by 𝐹N [𝑎, 𝑏] the collection of finite or numerable family of nonoverlapping interval {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 , such that ⋃𝑛≥1 [𝑎𝑛 , 𝑏𝑛 ] = [𝑎, 𝑏]. In 1985, Schramm [23] introduced a new concept of variation as follows. Definition 5. Let 𝜙 = {𝜑𝑛 }𝑛≥1 be a 𝜙-sequence, {𝐼𝑛 = [𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏], and 𝑢 : [𝑎, 𝑏] → R. We define ∞

󵄨 󵄨 𝜎𝜙 (𝑢, 𝐼𝑛 ) := ∑ 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨) , 𝑛=1

𝑉𝜙 (𝑢) = 𝑉𝜙 (𝑢; [𝑎, 𝑏]) :=

(12) sup

𝐼𝑛 ∈𝐹N [𝑎,𝑏]

𝜎𝜙 (𝑢, 𝐼𝑛 ) .

If 𝑉𝜙 (𝑢; [𝑎, 𝑏]) < ∞, we say that 𝑢 has bounded 𝜙-variation in the interval [𝑎, 𝑏] and this number denotes the 𝜙-variation of 𝑢 in Schramm’s sense in [𝑎, 𝑏]. The class of functions that have bounded 𝜙-variation in the interval [𝑎, 𝑏] is denoted by 𝑉𝜙 [𝑎, 𝑏]. The vectorial space generated by this class is denoted by BV𝜙 [𝑎, 𝑏].

The next lemma is useful for building the space generated by several classes of functions. Lemma 6. Let X be a vector space and 𝐴 ⊂ X a nonempty and symmetric set. Then (1) 0 ∈ 𝐴. (2) The vector space generated for 𝐴 is equal to ⟨𝐴⟩ = {𝑥 ∈ X : ∃𝜆 > 0 such that 𝜆𝑥 ∈ 𝐴} = ⋃ 𝜆𝐴. 𝜆>0

(13)

Some properties of functions of bounded 𝜙-variation in Schramm’s sense are given in the following theorem. Theorem 7 (see [23]). Let 𝜙 = {𝜑𝑛 }𝑛≥1 be 𝜙-sequence then (1) BV𝜙 [𝑎, 𝑏] is a Banach space endowed with the norm ‖𝑢‖𝜙 := |𝑢 (𝑎)| + 𝜇 (𝑢)

(𝑢 ∈ BV𝜙 [𝑎, 𝑏]) ,

(14)

where 𝜇(𝑢) := inf 𝜆>0 {𝜆 > 0 : 𝑉𝜙 (𝑢/𝜆) ≤ 1}. (2) If 𝑢 is monotone, then 𝑉𝜙 (𝑢) = 𝜑1 (|𝑢(𝑏) − 𝑢(𝑎)|). (3) If 𝑢 ∈ 𝑉𝜙 [𝑎, 𝑏], then 𝑢 is bounded and ‖𝑢‖∞ ≤ |𝑢(𝑎)| + 2𝜑1−1 (𝑉𝜙 (𝑢)). (4) 𝑉𝜙 [𝑎, 𝑏] is a symmetrical and convex set. (5) BV𝜙 [𝑎, 𝑏] = {𝑢 : [𝑎, 𝑏] → R : ∃𝜆 > 0 such that 𝑉𝜙 (𝜆𝑢) < ∞}. (6) 𝑢 ∈ 𝑉𝜙 [𝑎, 𝑏], then 𝑢 has lateral limits at each point of [𝑎, 𝑏]. In 1986, S. K. Kim and J. Kim [24] combined the concepts of 𝜅-variation and 𝜙-variation introduced by Korenblum and Schramm to create the concept of 𝜅𝜙-variation or variation in Schramm-Korenblum’s sense. Definition 8. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, {𝐼𝑛 = [𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏], and 𝑢 : [𝑎, 𝑏] → R. We define 𝜅𝜎𝜙 (𝑢, 𝐼𝑛 ) :=

󵄨󵄨 󵄨󵄨 ∑∞ 𝑛=1 𝜑𝑛 (󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨) , ∑∞ 𝑛=1 𝜅 ((𝑏𝑛 − 𝑎𝑛 ) / (𝑏 − 𝑎))

(15)

𝜅𝑉𝜙 (𝑢) = 𝜅𝑉𝜙 (𝑢; [𝑎, 𝑏]) := sup𝐼𝑛 ∈𝐹N [𝑎,𝑏] 𝜅𝜎𝜙 (𝑢, 𝐼𝑛 ) . If 𝜅𝑉𝜙 (𝑢; [𝑎, 𝑏]) < ∞, we say that 𝑢 has bounded 𝜅𝜙variation in the interval [𝑎, 𝑏] and this number denotes the 𝜅𝜙-variation of 𝑢 in Schramm-Korenblum’s sense in [𝑎, 𝑏]. The class of functions that have bounded 𝜅𝜙-variation in the interval [𝑎, 𝑏] is denoted by 𝜅𝑉𝜙 [𝑎, 𝑏]. The vectorial space generated by this class is denoted by 𝜅BV𝜙 [𝑎, 𝑏]. A particular case of 𝜙-sequence is when all the functions 𝜑𝑛 , 𝑛 ∈ N are equal to a fixed 𝜑-function 𝜑. In this situation the class 𝜅𝑉𝜑 [𝑎, 𝑏] is the class of the functions that have bounded 𝜅𝜑-variation in Wiener-Korenblum’s sense. This class of functions is denoted by 𝜅𝑉𝜑 [𝑎, 𝑏] and the vectorial

4


space generated by this class of function is denoted by 𝜅BV𝜑 [𝑎, 𝑏]. Some properties of functions of bounded 𝜅𝜙-variation in Schramm-Korenblum’s sense are given in the following theorem. Theorem 9. Let 𝜅 : [0, 1] → [0, 1] be a distortion function and 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, then (1) 𝜅BV𝜙 [𝑎, 𝑏] is a Banach space endowed with the norm ‖𝑢‖𝜅𝜙 := |𝑢 (𝑎)| + 𝜇𝜙 (𝑢)

(𝑢 ∈ 𝜅BV𝜙 [𝑎, 𝑏]) ,

(16)

From the Definition of 𝑘𝑉𝜙 (𝑢) we have the reciprocal inequality. Part (3). We consider 𝑎 < 𝑡 ≤ 𝑏, then 𝜑1 (|𝑢 (𝑡) − 𝑢 (𝑎)|) ≤ 𝜅𝑉𝜙 (𝑢) . (19) 𝜅 ((𝑡 − 𝑎) / (𝑏 − 𝑎)) + 𝜅 ((𝑏 − 𝑡) / (𝑏 − 𝑎)) Then 𝜑1 (|𝑢(𝑡) − 𝑢(𝑎)|) ≤ 2𝜅𝑉𝜙 (𝑢) and from this inequality we have the required relation. Part (4). We get Part (4) by the convexity of functions 𝜑𝑛 , 𝑛 ∈ N and the definition of 𝜅𝜙-variation. Part (5). It follows from part (4) and Lemma 6.

where 𝜇𝜙 (𝑢) := inf 𝜆>0 {𝜆 > 0 : 𝜅𝑉𝜙 (𝑢/𝜆) ≤ 1}. (2) If 𝑢 is monotone, then 𝜅𝑉𝜙 (𝑢) = 𝜑1 (|𝑢(𝑏) − 𝑢(𝑎)|). (3) If 𝑢 ∈ 𝜅𝑉𝜙 [𝑎, 𝑏], then 𝑢 is bounded and ‖𝑢‖∞ ≤ |𝑢(𝑎)|+ 2𝜑1−1 (𝜅𝑉𝜙 (𝑢)). (4) 𝜅𝑉𝜙 [𝑎, 𝑏] is a symmetrical and convex set. (5) 𝜅BV𝜙 [𝑎, 𝑏] = {𝑢 : [𝑎, 𝑏] → R : ∃𝜆 > 0 such that 𝜅𝑉𝜙 (𝜆𝑢) < ∞}.

Part (6). Let 𝑢 ∈ BV[𝑎, 𝑏] and consider {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏]. Define 󵄨 󵄨 𝐴 := {𝑛 ∈ N : 󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 ≤ 1} . (20) Then ∞

󵄨 󵄨 ∑ 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨)

(6) BV[𝑎, 𝑏] ⊂ 𝑉𝜙 [𝑎, 𝑏] ⊂ 𝜅𝑉𝜙 [𝑎, 𝑏], and therefore BV[𝑎, 𝑏] ⊂ BV𝜙 [𝑎, 𝑏] ⊂ 𝜅BV 𝜙 [𝑎, 𝑏].

𝑛=1

󵄨 󵄨 ≤ 𝜑1 (1) ∑ 󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨

(7) 𝜅BV[𝑎, 𝑏] ⊂ 𝜅BV𝜙 [𝑎, 𝑏]. (8) 𝑢 ∈ 𝜅𝑉𝜙 [𝑎, 𝑏], then 𝑢 has lateral limits at each point of [𝑎, 𝑏]. Proof. Part (1). See [24]. Part (2). We take {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏], then since the functions 0 < 𝑡 → 𝜑𝑛 (𝑡)/𝑡 are increasing, we obtain 󵄨󵄨 󵄨󵄨 ∑∞ 𝑛=1 𝜑𝑛 (󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨) ∑∞ 𝑛=1 𝜅 ((𝑏𝑛 − 𝑎𝑛 ) / (𝑏 − 𝑎)) 󵄨 󵄨 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨) 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 𝑢 (𝑏 ) − 𝑢 (𝑎 ) 󵄨 󵄨 𝑛 󵄨 𝑛=1 󵄨 𝑛

𝑛=1

≤

𝑏𝑛 − 𝑎𝑛 )) 𝑏−𝑎

󵄨 󵄨 𝑉 (𝑢) ≥ ∑ 󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 ≥ [𝑉 (𝑢)] + 1.

(22)

∞

󵄨 󵄨 ∑ 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨)

𝑛=1

󵄨 󵄨 ≤ 𝜑1 (1) 𝑉 (𝑢) + ∑ 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨)

(23)

𝑛∉𝐴

𝜑 (|𝑢 (𝑏) − 𝑢 (𝑎)|) 󵄨󵄨 󵄨 ≤∑ 𝑛 󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 − 𝑢 (𝑏) (𝑎)| |𝑢 𝑛=1 ∞

The last sum has at most [𝑉(𝑢)] terms, where [𝑥] = max{𝑛 : 𝑛 ≤ 𝑥}. Because otherwise it has at least [𝑉(𝑢)] + 1 summands. Accordingly

−1

∞

× (∑𝜅 (

𝑛∉𝐴

Which is a contradiction. Therefore

∞

𝑏 − 𝑎𝑛 × (∑𝜅 ( 𝑛 )) 𝑏−𝑎 𝑛=1

󵄨 󵄨 + ∑ 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨) .

𝑛∉𝐴

=∑

∞

(21)

𝑛∈𝐴

(17)

−1

This concludes that BV[𝑎, 𝑏] ⊂ 𝑉𝜙 [𝑎, 𝑏]. Let us show that 𝑉𝜙 [𝑎, 𝑏] ⊂ 𝜅𝑉𝜙 [𝑎, 𝑏]. In Fact let 𝑢 ∈ 𝑉𝜙 [𝑎, 𝑏] and {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏], then ∞

󵄨 󵄨 ∑ 𝜑𝑛 (󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨)

𝜑1 (|𝑢 (𝑏) − 𝑢 (𝑎)|) ∞ 󵄨󵄨 󵄨 ∑ 󵄨𝑢 (𝑏 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 |𝑢 (𝑏) − 𝑢 (𝑎)| 𝑛=1 󵄨 𝑛

𝑛=1

∞

𝑏 − 𝑎𝑛 ≤ 𝑉𝜙 (𝑢) 𝜅 (1) ≤ 𝑉𝜙 (𝑢) 𝜅 ( ∑ 𝑛 ). 𝑛=1 𝑏 − 𝑎

= 𝜑1 (|𝑢 (𝑏) − 𝑢 (𝑎)|) ,

(24)

Hence, we get that 𝜅𝑉𝜙 (𝑢) ≤ 𝑉𝜙 (𝑢).

from which is obtained 𝜅𝑉𝜙 (𝑢) ≤ 𝜑1 (|𝑢 (𝑏) − 𝑢 (𝑎)|) .

≤ 𝜑1 (1) 𝑉 (𝑢) + 𝜑1 (2‖𝑢‖∞ ) [𝑉 (𝑢)] .

(18)

Part (7). Let 𝑢 ∈ 𝜅BV[𝑎, 𝑏], then by part (3) 𝑢 is bounded in [𝑎, 𝑏]. Let us fix 𝜆 > 0, such that ‖𝜆𝑢‖∞ ≤ 1/2. Let


5

{[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏], then from the convexity of the functions 𝜑𝑛 , 𝑛 ∈ N, we have 󵄨󵄨 󵄨󵄨 ∑∞ 𝑛=1 𝜑𝑛 (󵄨󵄨𝜆𝑢 (𝑏𝑛 ) − 𝜆𝑢 (𝑎𝑛 )󵄨󵄨) ∑∞ 𝑛=1 𝜅 ((𝑏𝑛 − 𝑎𝑛 ) / (𝑏 − 𝑎)) 󵄨 󵄨 ∑∞ 𝜑𝑛 (1) 𝜆 󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 (25) ≤ 𝑛=1 ∞ ∑𝑛=1 𝜅 ((𝑏𝑛 − 𝑎𝑛 ) / (𝑏 − 𝑎)) ≤ 𝜑1 (1) 𝜆𝜅𝑉 (𝑢) . Thus Lemma 6 concludes that 𝑢 ∈ 𝜅BV𝜙 [𝑎, 𝑏]. Part (8). Suppose that there is 𝑡∗ ∈ (𝑎, 𝑏] such that lim𝑡↑𝑡∗ 𝑢(𝑡) does not exist. By part (3) 𝑢 is bounded then 𝛼 = lim∗ inf 𝑢 (𝑡) < lim∗ sup 𝑢 (𝑡) = 𝛽, 𝑡↑𝑡

𝑡↑𝑡

𝜅BV −𝜙 [𝑎, 𝑏] := {𝑢− : 𝑢 ∈ 𝜅BV 𝜙 [𝑎, 𝑏]} .

Using the definition of 𝜅𝜙-variation, we have 󵄨 󵄨 𝜑1 (󵄨󵄨󵄨󵄨𝑢 (𝑡𝑛 ) − 𝑢 (𝑡𝑛󸀠 )󵄨󵄨󵄨󵄨) 𝑡𝑛󸀠

𝑡𝑛󸀠

Definition 11. Let 1 < 𝑝 < ∞, 𝜅 : [0, 1] → [0, 1] be a distortion function and 𝑢 : [𝑎, 𝑏] → R and 𝜋 : 𝑎 ≤ 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛 ≤ 𝑏 a partition of the interval [𝑎, 𝑏]. We define 𝜅𝑉𝑝𝑅 (𝑢) = 𝜅𝑉𝑝𝑅 (𝑢; [a, 𝑏]) := sup 𝜅𝜎𝑝 (𝑢, 𝜋) , 𝜋

− 𝑡𝑛 𝑏− 𝑡𝑛 − 𝑎 ) + 𝜅( ) + 𝜅( )) 𝑏−𝑎 𝑏−𝑎 𝑏−𝑎

(28)

≤ 𝜅𝑉𝜙 (𝑢) . Therefore,

(33)

where 󵄨

󵄨𝑝 − 𝑢 (𝑡𝑗 )󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨𝑝−1 ) 󵄨󵄨𝑡𝑗+1 − 𝑡𝑗 󵄨󵄨󵄨 󵄨 󵄨

𝑛−1 󵄨󵄨𝑢 (𝑡 󵄨󵄨 𝑗+1 )

𝜅𝜎𝑝 (𝑢, 𝜋) := ( ∑

𝑗=1 𝑛−1

× (∑𝜅 ( 𝑗=1

−1

(32)

Similarly, we defined 𝜅BV𝜙+ [𝑎, 𝑏]. Recently Castillo et al. [25] introduced the concept of 𝜅𝑝variation in Riesz-Korenblum’s sense in the following way.

(𝛼, 𝛽 ∈ R) . (26)

For each integer 𝑛 (large enough) we can choose 𝑡𝑛 , 𝑡𝑛󸀠 such that 1 𝑡∗ − < 𝑡𝑛 < 𝑡𝑛󸀠 < 𝑡∗ , 𝑛 (27) 󵄨󵄨 󵄨 󵄨󵄨𝑢 (𝑡𝑛 ) − 𝑢 (𝑡𝑛󸀠 )󵄨󵄨󵄨 ≥ 𝛽 − 𝛼. 󵄨 󵄨

× (𝜅 (

The function 𝑢− is called the left regularization of the function 𝑢 and the function 𝑢+ the right regularization of the function 𝑢. Applying the previous definition and the last part of Theorem 9, we can define

𝑡𝑗+1 − 𝑡𝑗 𝑏−𝑎

−1

(34)

))

and the supremum is taken on the set of all partitions of [𝑎, 𝑏]. If 𝜅𝑉𝑝𝑅 (𝑢; [𝑎, 𝑏]) < ∞, we say that 𝑢 has bounded 𝜅𝑝-variation in the interval [𝑎, 𝑏]. The number 𝜅𝑉𝑝𝑅 (𝑢; [𝑎, 𝑏]) denoted the 𝜅𝑝-variation of 𝑢 in Riesz-Korenblum’s sense in [𝑎, 𝑏]. The space of functions that have bounded 𝜅𝑝-variation in the interval [𝑎, 𝑏] is denoted by 𝜅RV𝑝 [𝑎, 𝑏].

(29)

Some properties of these functions are exposed in the following theorem.

By taking limit when 𝑛 → ∞, we obtain 𝜑1 (𝛽 − 𝛼) = 0, which is absurd. From each it follows that

Theorem 12 (see [25]). Let 1 < 𝑝 < ∞ and let 𝜅 : [0, 1] → [0, 1] be a distortion function, then

𝜑1 (𝛽 − 𝛼) ≤ 𝜅𝑉𝜙 (𝑢) (2 + 𝜅 (

1 )) . 𝑛 (𝑏 − 𝑎)

lim∗ inf 𝑢 (𝑡) = lim∗ sup 𝑢 (𝑡) . 𝑡↑𝑡

𝑡↑𝑡

(30)

By a similar argument it follows that there exist lim𝑡↓𝑡∗ 𝑢(𝑡), 𝑡∗ ∈ [𝑎, 𝑏). Assuming that lim𝑡 → 0 (𝑘(𝑡)/𝑡) is finite, then V𝜙 [𝑎, 𝑏] = 𝜅𝑉[𝑎, 𝑏]. For the last part of Theorem 9, we can give the definition of left and right regularizations of the function 𝑢 ∈ 𝜅𝑉𝜙 [𝑎, 𝑏]. Definition 10. Let 𝑢 ∈ 𝜅BV𝜙[𝑎, 𝑏], then {lim 𝑢 (𝑠) , 𝑡 ∈ (𝑎, 𝑏] 𝑢− (𝑡) := { 𝑠↑𝑡 𝑢 , 𝑡 = 𝑎, { (𝑎) {lim 𝑢 (𝑠) , 𝑡 ∈ [𝑎, 𝑏) 𝑢+ (𝑡) := { 𝑠↓𝑡 𝑢 , 𝑡 = 𝑏. { (𝑏)

(1) 𝜅RV[𝑎, 𝑏] is a Banach space endowed with the norm ‖𝑢‖𝜅𝑝 := |𝑢 (𝑎)| + (𝜅𝑉𝑝𝑅 (𝑢))

1/𝑝

(𝑢 ∈ 𝜅RV𝑝 [𝑎, 𝑏]) . (35)

(2) Lip[𝑎, 𝑏] ⊂ RV𝑝 [𝑎, 𝑏] ⊂ 𝜅RV𝑝 [𝑎, 𝑏] ⊂ 𝜅BV[𝑎, 𝑏], where RV𝑝 [𝑎, 𝑏] denote the space of the functions 𝑢 that have bounded 𝑝-variation in Riesz’s sense [20]. (3) 𝜅RV𝑝 [𝑎, 𝑏] is an algebra. This concept was generalized by Castillo et al. [26] as stated in the following definition.

(31)

Definition 13. Let 𝜑 be a 𝜑-function, 𝜅 : [0, 1] → [0, 1] be a distortion function, and 𝑢 : [𝑎, 𝑏] → R and 𝜋 : 𝑎 ≤ 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛 ≤ 𝑏 a partition of the interval [𝑎, 𝑏]. We define 𝜅𝑉𝜑𝑅 (𝑢) = 𝜅𝑉𝜑𝑅 (𝑢; [𝑎, 𝑏]) := sup 𝜅𝜎𝜑 (𝑢, 𝜋) , 𝜋

(36)

6


where 𝜅𝜎𝜑 (𝑢, 𝜋) :=

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑛−1 𝑗=1 𝜑 (󵄨󵄨𝑢 (𝑡𝑗+1 ) − 𝑢 (𝑡𝑗 )󵄨󵄨 / 󵄨󵄨𝑡𝑗+1 − 𝑡𝑗 󵄨󵄨) 󵄨󵄨𝑡𝑗+1 − 𝑡𝑗 󵄨󵄨 ∑𝑛−1 𝑗=1 𝜅 ((𝑡𝑗+1 − 𝑡𝑗 ) / (𝑏 − 𝑎))

(37) and the supremum is taken over all partitions of [𝑎, 𝑏]. If 𝜅𝑉𝜑𝑅 (𝑢; [𝑎, 𝑏]) < ∞, we say that 𝑢 has bounded 𝜅𝜑variation in the interval [𝑎, 𝑏] and this number denotes the 𝜅𝜑-variation of 𝑢 in Riesz-Korenblum’s sense in [𝑎, 𝑏]. The class of functions that have bounded 𝜅𝜑-variation in the interval [𝑎, 𝑏] is denoted by 𝜅𝑉𝜑 [𝑎, 𝑏]. The vectorial space generate by this class is denoted by 𝜅RV𝜑 [𝑎, 𝑏]. The space of all functions that have bounded 𝜅𝜑-variation on [𝑎, 𝑏] is denoted by 𝜅RV 𝜑 [𝑎, 𝑏]. Some properties of these functions are exposed in the following theorem. Theorem 14 (see [26]). Let 𝜑 be a convex 𝜑-function and 𝜅 : [0, 1] → [0, 1] a distortion function, then (1) Lip[𝑎, 𝑏] ⊂ 𝜅𝑉𝜑𝑅 [𝑎, 𝑏] ⊂ 𝜅BV[𝑎, 𝑏]. (2) RV𝜑 [𝑎, 𝑏] ⊂ 𝜅RV𝜑 [𝑎, 𝑏] ⊂ 𝜅BV[𝑎, 𝑏], where RV𝜑 [𝑎, 𝑏] denote the space generated by the class of functions of bounded 𝜑-variation in Riesz’s sense [20]. (3) If lim𝑡 → ∞ (𝜑(𝑡)/𝑡) < ∞, then

𝜅𝑉𝜑𝑅 [𝑎, 𝑏]

= 𝜅BV[𝑎, 𝑏].

(4) If 𝑢 ∈ 𝜅𝑉𝜑𝑅 [𝑎, 𝑏], then 𝑢 is bounded. (5) 𝜅𝑉𝜑𝑅 [𝑎, 𝑏] is a convex and symmetric set. (6) 𝜅RV𝜑 [𝑎, 𝑏] = {𝑢 : [𝑎, 𝑏] → R : ∃𝜆 > 0 such that 𝜅𝑉𝜙𝑅 (𝜆𝑢) < ∞}. (7) 𝜅RV 𝜑 [𝑎, 𝑏] is a Banach space endowed with the norm ‖𝑢‖𝜅𝜑 := |𝑢 (𝑎)| + 𝜇 (𝑢)

(𝑢 ∈ 𝜅RV𝜑 [𝑎, 𝑏]) ,

(38)

where 𝜇𝜑 (𝑢) := inf 𝜆>0 {𝜆 > 0 : 𝜅𝑉𝜑𝑅 (𝑢/𝜆) ≤ 1}.

3. Composition Operator between Lip[𝑎,𝑏] and 𝜅BV[𝑎, 𝑏] or 𝜅BV𝜙 [𝑎, 𝑏] Given a function ℎ : R → R, the composition operator 𝐻, associated to the function ℎ (case autonomous), maps each function 𝑢 : [𝑎, 𝑏] → R into the composition function 𝐻𝑢 : [𝑎, 𝑏] → R defined by 𝐻𝑢 (𝑡) := ℎ (𝑢 (𝑡)) ,

(𝑡 ∈ [𝑎, 𝑏]) .

(39)

More generally, given ℎ : [𝑎, 𝑏] × R → R, we consider operator 𝐻 defined by

will refer to (39) as the autonomous case and to (40) as the nonautonomous case. A problem related with this operator is to establish necessary and sufficient conditions of function ℎ so that the operators 𝐻 map the space X of real functions defined on [𝑎, 𝑏] into itself, that is, 𝐻(X) ⊂ X, or in more general way that operator 𝐻 maps the space X into space of functions Y (𝐻(X) ⊂ Y ). This problem is sometimes referred to as the composition operator problem (or COP). The solution to this problem for given X is sometimes very easy and sometimes highly nontrivial. As we mentioned in the introduction of this paper in a variety of spaces the required condition is that function ℎ is locally Lipschitz. Another interesting problem is to determine the smallest space of functions X and the bigger space Y such that 𝐻(X) ⊂ Y . In order to obtain the main result of this section, we will use a function of the zig-zig type such as the employed by Appell et al. in [1, 15]. In this section we will show that the locally Lipschitz condition of the function ℎ is a necessary and sufficient condition such that 𝐻(Lip[𝑎, 𝑏]) ⊂ 𝜅BV[𝑎, 𝑏] and that in this situation 𝐻 is bounded. The following lemma will be useful in the proof of our main theorem (Theorem 17). Lemma 15. Let 𝑢 : [𝑎, 𝑏] → R, 𝑎 ≤ 𝑠 < 𝜂 < 𝑡 ≤ 𝑏, then 󵄨 󵄨 󵄨 󵄨 |𝑢 (𝑡) − 𝑢 (𝑠)| 󵄨󵄨󵄨𝑢 (𝜂) − 𝑢 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑢 (𝑡) − 𝑢 (𝜂)󵄨󵄨󵄨 ≤ + . 𝑡−𝑠 𝜂−𝑠 𝑡−𝜂

(41)

Proof. Let 𝑎 ≤ 𝑠 < 𝜂 < 𝑡 ≤ 𝑏. Then |𝑢 (𝑡) − 𝑢 (𝑠)| 𝑡−𝑠 󵄨 󵄨 󵄨 󵄨󵄨 󵄨𝑢 (𝜂) − 𝑢 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑢 (𝑡) − 𝑢 (𝜂)󵄨󵄨󵄨 ≤󵄨 + 𝑡−𝑠 𝑡−𝑠 (42) 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨𝑢 (𝜂) − 𝑢 (𝑠)󵄨󵄨 𝜂 − 𝑠 󵄨𝑢 (𝑡) − 𝑢 (𝜂)󵄨󵄨󵄨 𝑡 − 𝜂 =󵄨 ( )+ 󵄨 ( ) 𝜂−𝑠 𝑡−𝑠 𝑡−𝜂 𝑡−𝑠 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨𝑢 (𝜂) − 𝑢 (𝑠)󵄨󵄨 󵄨󵄨𝑢 (𝑡) − 𝑢 (𝜂)󵄨󵄨 + . ≤󵄨 𝜂−𝑠 𝑡−𝜂 Lemma 16. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, 𝑢 ∈ 𝜅BV𝜙 [𝑎, 𝑏], and 𝜆 > 0. Then 𝜇𝜙 (𝑢) < 𝜆 if and only if 𝜅𝑉𝜙 (𝑢/𝜆) < 1. Proof. Let 𝑢 ∈ 𝜅BV 𝜙 [𝑎, 𝑏]. Suppose that 𝜇𝜙 (𝑢) < 𝜆; then, by definition of 𝜇𝜙 (𝑢) there exists 𝑘 such that 𝜆 > 𝑘 > 𝜇𝜙 (𝑢) and 𝜅𝑉𝜙 (𝑢/𝑘) ≤ 1. Hence, by the convexity of the functions 𝜑𝑛 , we have 𝑢 𝑢𝑘 𝑢 𝑘 𝑘 𝜅𝑉𝜙 ( ) = 𝜅𝑉𝜙 ( ) ≤ 𝜅𝑉𝜙 ( ) ≤ ≤ 1. 𝜆 𝑘𝜆 𝜆 𝑘 𝜆

(43)

(40)

Conversely, assume 𝜅𝑉𝜙 (𝑢/𝜆) < 1, then 𝜆 ∈ {𝜆 > 0 : 𝜅𝑉𝜙 (𝑢/𝜆) ≤ 1}; hence 𝜇𝜙 (𝑢) < 𝜆.

This operator is also called superposition operator or substitution operator or Nemytskii operator. In what follows,

Theorem 17. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, ℎ : R → R, and 𝐻 the composition

𝐻𝑢 (𝑡) := ℎ (𝑡, 𝑢 (𝑡)) ,

(𝑡 ∈ [𝑎, 𝑏]) .


7

operator associated to ℎ. 𝐻 maps the space Lip[0, 1] into the space 𝜅BV𝜙 [0, 1] or 𝜅BV[0, 1] if and only if ℎ is locally Lipschitz. Furthermore operator 𝐻 is bounded. Proof. Let 𝑢 ∈ Lip[0, 1], 𝑟 = ‖𝑢‖∞ and suppose that ℎ is locally Lipschitz, then there exist 𝑘 = 𝑘(𝑟), such that |ℎ (𝑡) − ℎ (𝑠)| ≤ 𝑘 (𝑟) |𝑠 − 𝑡| ,

(𝑠, 𝑡 ∈ R, |𝑠| ≤ 𝑟, |𝑡| ≤ 𝑟) . (44)

Let {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏] and 𝜆 > 0, such that 𝜆 < 1/(2𝑘(𝑟)‖𝑢‖Lip ‖𝑢‖∞ + 1), then 󵄨󵄨 󵄨󵄨 ∑∞ 𝑛=1 𝜑𝑛 (𝜆 󵄨󵄨ℎ (𝑢 (𝑏𝑛 )) − ℎ (𝑢 (𝑎𝑛 ))󵄨󵄨) ∑∞ 𝑛=1 𝜅 ((𝑏𝑛 − 𝑎𝑛 ) / (𝑏 − 𝑎)) 󵄨 󵄨 ∑∞ 𝜑 (𝜆𝑘 (𝑟) 󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨) ≤ 𝑛=1 𝑛 (45) 𝜅 (1) ∞

Consider the new sequence {𝑚𝑛 }𝑛≥1 defined by 𝑚𝑛 :=

Then by Lemma 15 and Theorem 9, we have 𝐻(𝑢) ∈ 𝜅BV𝜙 [𝑎, 𝑏]. The proof of the only if direction will be by contradiction, that is, we assume 𝐻(Lip[0, 1]) ⊂ 𝜅BV𝜙 [0, 1] and ℎ is not locally Lipschitz. Since the identity function 𝐼𝑑 : [0, 1] → [0, 1] belongs to Lip[0, 1], then ℎ ∘ 𝐼𝑑 ∈ 𝜅BV𝜙 [0, 1] and therefore ℎ is bounded in the interval [0, 1]. Without loss of generality we may assume that 1 󵄩 󵄩󵄩 (46) 󵄩󵄩ℎ|[0,1] 󵄩󵄩󵄩∞ ≤ . 4 Since ℎ is not locally Lipschitz in R, there is a closed interval 𝐼 such that ℎ does not satisfy any Lipschitz condition. In order to simplify the proof we can assume that 𝐼 = [0, 1]. In this way for any increasing sequence of positive real numbers {𝑘𝑛 }𝑛≥1 that converge to infinite that we will define later, we can choose sequences {𝑎𝑛 }𝑛≥1 , {𝑏𝑛 }𝑛≥1 , such that 󵄨󵄨󵄨ℎ (𝑏𝑛 ) − ℎ (𝑎𝑛 )󵄨󵄨󵄨 > 𝑘𝑛 󵄨󵄨󵄨𝑏𝑛 − 𝑎𝑛 󵄨󵄨󵄨 , (𝑛 ∈ N) . (47) 󵄨 󵄨 󵄨 󵄨 In addition we can choose 𝑎𝑛 , 𝑏𝑛 such that (48)

Considering subsequences if necessary, we can assume that the sequence {𝑎𝑛 }𝑛≥1 is monotone. We can assume without loss of generality that sequence {𝑎𝑛 }𝑛≥1 is increasing. Since [0, 1] is compact, from inequality (47) we have that there exist subsequences of {𝑎𝑛 }𝑛≥1 and {𝑏𝑛 }𝑛≥1 that we will denote in the same way, and that converge to 𝑎∞ ∈ [0, 1]. Since the sequence {𝑎𝑛 }𝑛≥1 is a Cauchy sequence, we can assume (taking subsequence if necessary) that 󵄨󵄨 󵄨 1 󵄨󵄨𝑎𝑚 − 𝑎𝑛 󵄨󵄨󵄨 < , 𝑘𝑛

(𝑚 > 𝑛) .

Consider the sequence defined recursively {𝑡𝑛 }𝑛≥1 by 𝑡1 := 0,

𝑡𝑛+1 := 𝑡𝑛 + 𝑎𝑛+1 − 𝑎𝑛 + 2 [𝑚𝑛 ] (𝑏𝑛 − 𝑎𝑛 ) , (𝑛 ∈ N) .

∞

∞

𝑛=1

𝑛=1

= ∑ (𝑎𝑛+1 − 𝑎𝑛 ) + 2 ∑ [𝑚𝑛 ] (𝑏𝑛 − 𝑎𝑛 ) ∞

1 . 𝑛=1 𝑘𝑛

Then to ensure that 𝑡∞ ∈ [0, 1], it is sufficient to suppose that ∑∞ 𝑛=1 (1/𝑘𝑛 ) ≤ 1/3. We define the continuous zig-zag function 𝑢 : [0, 1] → R, as shown in the following: 𝑢 (𝑡) 𝑎𝑛 , { { { 𝑏𝑛 , := { { { 𝑎∞ , { affine,

𝑡 = 𝑡𝑛 + 2𝑖 (𝑏𝑛 − 𝑎𝑛 ) , 𝑖 = 0, . . . , [𝑚𝑛 ] 𝑡 = 𝑡𝑛 + (2𝑖 + 1) (𝑏𝑛 − 𝑎𝑛 ) , 𝑖 = 0, . . . , [𝑚𝑛 ] − 1 𝑡∞ ≤ 𝑡 ≤ 1 other case. (55)

Put 𝑡𝑛,𝑖 := 𝑡𝑛 + 𝑖 (𝑏𝑛 − 𝑎𝑛 ) ,

𝑛 ∈ N, 𝑖 = 0, . . . , 2 [𝑚𝑛 ] .

𝐼𝑛,𝑖 := [𝑡𝑛.𝑖 , 𝑡𝑛,𝑖+1 ] ,

𝑖 = 0, . . . , [𝑚𝑛 ] − 1,

𝐼𝑛,2[𝑚𝑛 ] := [𝑡𝑛,2[𝑚𝑛 ] , 𝑡𝑛+1 ] .

(57)

And function 𝑢 is defined on 𝐼𝑛,𝑖 , 𝑖 = 0, . . . , 2[𝑚𝑛 ] as follows:

𝑢 (𝑡) = 𝑡 − 𝑡𝑛 + (2𝑖 + 1) (𝑏𝑛 − 𝑎𝑛 ) + 𝑏𝑛 ,

(50)

(56)

We can write each interval 𝐼𝑛 = [𝑡𝑛 , 𝑡𝑛+1 ], 𝑛 ∈ N, as the union of the family of nonoverlapping intervals

Again considering subsequences if needed using the properties of the function 𝜅 we can assume that (𝑛 ∈ N, 𝑚 ≥ 𝑛) .

(54)

≤ 3∑

𝑢 (𝑡) = 𝑡 − (𝑡𝑛 + 2𝑖 (𝑏𝑛 − 𝑎𝑛 )) + 𝑎𝑛 ,

1 , 𝑘𝑛

(53)

This sequence is strictly increasing and from the relations (49) and (50), we get

(49)

max {𝜅 (𝑏𝑛 − 𝑎𝑛 ) , 𝜅 (𝑎𝑚 − 𝑎𝑛 )}
2; therefore 𝑚𝑛 < [𝑚𝑛 ] ≤ 𝑚𝑛 , (𝑛 ∈ N) . (52) 2

𝑛=1

(𝑛 ∈ N) .

(𝑛 ∈ N) .

𝑡𝑛 󳨀→ 𝑡∞ := ∑ (𝑡𝑛+1 − 𝑡𝑛 )

≤ 𝜆𝑘 (𝑟) 𝜑1 (1) ‖𝑢‖Lip ∑ (𝑏𝑛 − 𝑎𝑛 ) < ∞.

𝑎𝑛 < 𝑏𝑛 ,

1 , 𝑘𝑛 (𝑏𝑛 − 𝑎𝑛 )

𝑢 (𝑡) = 𝑡 − 𝑡𝑛+1 + 𝑎𝑛 ,

(𝑡 ∈ 𝐼𝑛,2𝑖 ) ,

(58)

(𝑡 ∈ 𝐼𝑛,2𝑖+1 ) , (59)

(𝑡 ∈ 𝐼𝑛,2[𝑚] ) .

(60)

In all these situations, the slopes of the segments of lines are equal to 1.

8


Hence, we have for 𝑛 ∈ N, the absolute value of the slope of the line segments in these ranges are bounded by 1, as shown below 󵄨󵄨 󵄨 󵄨𝑏 − 𝑎𝑛 󵄨󵄨󵄨 ≤ 2−𝑛 𝑘𝑛 (𝑏𝑛 − 𝑎𝑛 ) ≤ 1, 2−𝑛 −1󵄨 𝑛 𝜅 (𝑏𝑛 − 𝑎𝑛 ) (61) 𝑎𝑛+1 − 𝑎𝑛 −(𝑛+1) 2 ≤ 1. 𝑡𝑛 + 𝑎𝑛+1 − 𝑎𝑛

Case 4. If 𝑠 ∈ 𝐼𝑛 , 𝑛 ∈ N, 𝑡 = 𝑡∞ . Then from Lemma 15 󵄨󵄨 󵄨 󵄨󵄨𝑢 (𝑡∞ ) − 𝑢 (𝑠)󵄨󵄨󵄨 𝑡∞ − 𝑠 󵄨 󵄨 󵄨󵄨 󵄨 󵄨𝑢 (𝑡𝑛,𝑖+1 ) − 𝑢 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑎∞ − 𝑢 (𝑡𝑛,𝑖+1 )󵄨󵄨󵄨 ≤󵄨 + 𝑡𝑛,𝑖+1 − 𝑠 𝑏𝑛 − 𝑎𝑛

We will show that 𝑢 ∈ Lip[0, 1]. Let 0 ≤ 𝑠 < 𝑡 ≤ 1, and then there are the following possibilities for the location of 𝑠 and 𝑡 on [0, 1]. Case 1. If 𝑠, 𝑡 ∈ 𝐼𝑛 , (𝑛 ∈ N) and are in the same interval 𝐼𝑛,𝑖 , 𝑖 = 0, . . . , 2[𝑚𝑛 ]. From relations (58), (59), and (60) it follows that |𝑢(𝑡) − 𝑢(𝑠)|/|𝑡 − 𝑠| = 1. Case 2. If 𝑠, 𝑡 ∈ 𝐼𝑛 , (𝑛 ∈ N) and are in two different intervals 𝐼𝑛,𝑖 , 𝑖 = 0, . . . , 2[𝑚𝑛 ]. There are several possibilities. (a) 𝑠 ∈ 𝐼𝑛.𝑖 , 𝑡 ∈ 𝐼𝑛,𝑗 , 𝑖 < 𝑗 < 2[𝑚𝑛 ]. (a1 ) 𝑗 = 𝑖 + 1. By Lemma 15 and relations (58) and (59) we have |𝑢 (𝑡) − 𝑢 (𝑠)| |𝑡 − 𝑠| 󵄨 󵄨 󵄨 󵄨󵄨 󵄨𝑢 (𝑡𝑛,𝑖+1 ) − 𝑢 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑢 (𝑡) − 𝑢 (𝑡𝑛,𝑖+1 )󵄨󵄨󵄨 ≤󵄨 ≤ 2. + 𝑡𝑛,𝑖+1 − 𝑠 𝑡 − 𝑡𝑛,𝑖+1

(62)

Case 5. If 𝑠 < 𝑡∞ < 𝑡 ≤ 1. From Lemma 15 and Case 4, 󵄨 󵄨 |𝑢 (𝑡) − 𝑢 (𝑠)| 󵄨󵄨󵄨𝑢 (𝑡∞ ) − 𝑢 (𝑠)󵄨󵄨󵄨 ≤ ≤ 2. 𝑡−𝑠 𝑡∞ − 𝑠

(63)

(b) 𝑠 ∈ 𝐼𝑛.𝑖 , 𝑡 ∈ 𝐼𝑛,𝑗 , 𝑖 < 𝑗 = 2[𝑚𝑛 ]. If 𝑗 = 𝑖 + 1, proceed as (𝑎1 ). If 𝑗 > 𝑖 + 1, again using the Lemma 15 and relations (58), (59), and (60) we obtain

󵄨 󵄨󵄨 󵄨󵄨𝑢 (𝑡) − 𝑢 (𝑡𝑛,2[𝑚𝑛 ] )󵄨󵄨󵄨 󵄨 󵄨 𝑡𝑛,2[𝑚𝑛 ] − 𝑡

|𝑢 (𝑡) − 𝑢 (𝑠)| ≤ |𝑡 − 𝑠| ,

(𝑠, 𝑡 ∈ [0, 1]) .

(68)

So 𝑢 is Lipschitz in [0, 1]. Moreover, for each partition of the interval [0, 1] of the form

(69)

< ⋅ ⋅ ⋅ < 𝑡𝑘 + 2 [𝑚𝑘 ] (𝑏𝑘 − 𝑎𝑘 ) < 1 and 𝑐 > 0, using the inequality (47), convexity of the function 𝜑𝑛 , 𝑛 ≥ 1, and definition of 𝑚𝑛 , 𝑛 ∈ N, we have 𝜅𝑉𝜙 (𝑐 (ℎ ∘ 𝑢) ; [0, 1]) ≥

≥ (64) ≥

𝑏𝑛 − 𝑎𝑛 + 1 ≤ 2. 𝑡𝑛,2[𝑚𝑛 ] − 𝑡𝑛,2[𝑚𝑛 ]−1

󵄨 󵄨 ∑𝑘𝑛=1 2 [𝑚𝑛 ] 𝜑𝑛 (𝑐) 󵄨󵄨󵄨ℎ (𝑏𝑛 ) − ℎ (𝑎𝑛 )󵄨󵄨󵄨

∑𝑘𝑛=1 [2 [𝑚𝑛 ] 𝜅 (𝑏𝑛 − 𝑎𝑛 ) + 𝜅 (𝑎𝑛+1 − 𝑎𝑛 )] ∑𝑘𝑛=1 2 [𝑚𝑛 ] 𝜑𝑛 (𝑐) 𝑘𝑛 (𝑏𝑛 − 𝑎𝑛 )

(70)

∑𝑘𝑛=1 [2 [𝑚𝑛 ] 𝜅 (𝑏𝑛 − 𝑎𝑛 ) + 𝜅 (𝑎𝑛+1 − 𝑎𝑛 )] ∑𝑘𝑛=1 2 [𝑚𝑛 ] 𝑘𝑛 (𝑏𝑛 − 𝑎𝑛 ) 𝜑𝑛 (𝑐) ∑𝑘𝑛=1

(1/𝑘𝑛 )

𝑘

≥ ∑ 𝜑𝑛 (𝑐) . 𝑛=1

As the series ∑∞ 𝑛=1 𝜑𝑛 (𝑐) diverge, 𝑐ℎ ∘ 𝑢 ∉ 𝜅BV𝜙 [0, 1], which is a contradiction. Let us see that ℎ ∘ 𝑢 ∉ 𝜅BV[𝑎, 𝑏]. In fact, as in the case of 𝜅-variation we have

Case 3. If 𝑠 ∈ 𝐼𝑛 , 𝑡 ∈ 𝐼𝑚 , 𝑛, 𝑚 ∈ N, 𝑛 < 𝑚. From Lemma 15 and Case 2, we conclude that |𝑢 (𝑡) − 𝑢 (𝑠)| 𝑡−𝑠 󵄨 󵄨 󵄨 󵄨󵄨 󵄨𝑢 (𝑡 ) − 𝑢 (𝑠)󵄨󵄨󵄨 󵄨󵄨󵄨𝑢 (𝑡) − 𝑢 (𝑡𝑚 )󵄨󵄨󵄨 ≤ 4. ≤ 󵄨 𝑛+1 + 𝑡𝑛+1 − 𝑠 𝑡 − 𝑡𝑚

(67)

Case 6. If 𝑡∞ ≤ 𝑠 < 𝑡 ≤ 1. In this circumstance 𝑢(𝑠) = 𝑢(𝑡) = 𝑎∞ and the situation is trivial. Therefore we have that

< 𝑡2 < 𝑡2 + (𝑏2 − 𝑎2 ) < ⋅ ⋅ ⋅ < 𝑡𝑘

𝑏𝑛 − 𝑎𝑛 |𝑢 (𝑡) − 𝑢 (𝑠)| = 1. ≤ 𝑡𝑛,𝑖+2 − 𝑡𝑛,,𝑖+1 |𝑡 − 𝑠|

≤

𝑎∞ − 𝑎𝑛 ≤ 2. 𝑏𝑛 − 𝑎𝑛

𝜋 : 0 = 𝑡1 < 𝑡1 + (𝑏1 − 𝑎1 ) < ⋅ ⋅ ⋅ < 𝑡1 + 2 [𝑚1 ] (𝑏1 − 𝑎2 )

(a2 ) 𝑗 > 𝑖 + 1. Then

|𝑢 (𝑡) − 𝑢 (𝑠)| |𝑡 − 𝑠| 󵄨󵄨 󵄨 󵄨󵄨𝑢 (𝑡𝑛,2[𝑚𝑛 ] ) − 𝑢 (𝑠)󵄨󵄨󵄨 󵄨 󵄨+ ≤ 𝑡𝑛,2[𝑚𝑛 ] − 𝑠

≤1+

(66)

𝑘

(65)

𝑉𝜅 (ℎ ∘ 𝑢; [0, 1]) ≥ ∑ 2 [𝑚𝑘 ] 𝑘𝑛 (𝑏𝑛 − 𝑎𝑛 ) ≥ 𝑘. 𝑛=1

Therefore, ℎ ∘ 𝑢 ∉ 𝜅BV[0, 1].

(71)


9

To prove that the operator 𝐻 is bounded, let 𝑟 > 0 and 𝑢 ∈ Lip[0, 1], such that ‖𝑢‖Lip ≤ 𝑟, then from the definition of ‖ ⋅ ‖Lip[0,1] , we have ‖𝑢‖∞ ≤ |𝑢 (0)| + 𝐿10 (𝑢) ≤ 𝑟.

(72)

Without loss of generality assume that 𝑟 = 1. As ℎ is locally Lipschitz, there exists 𝑘(1) > 0, such that 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨ℎ (𝑥) − ℎ (𝑦)󵄨󵄨󵄨 ≤ 𝑘 (1) 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 , |𝑥| ≤ 1, 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ≤ 1. (73) As the identity function 𝐼𝑑 : [0, 1] → [0, 1] belongs to Lip[0, 1], then ℎ ∘ 𝐼𝑑 ∈ 𝜅BV𝜙 [0, 1]. By Theorem 9 we have that ‖ℎ ∘ 𝐼𝑑 ‖∞ < ∞. Let {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [0, 1] and choose 𝜆 such that 𝑘(1) 𝜅𝑉𝜙 (𝐼𝑑 ) < 𝜆, then

∞

𝑛=1

𝑘 (1) (𝑏𝑛 − 𝑎𝑛 ) ≤ ∑ 𝜑𝑛 ( ) 𝜆 𝑛=1

(74)

∞

󵄨 = ( ∑ 󵄨󵄨󵄨𝑢 (] ((𝑏𝑛 − 𝑎) / (𝑏 − 𝑎))) 𝑛=1

(78)

− (𝑎𝑛 − 𝑎) / (𝑏 − 𝑎)) )

so 𝑢 ∈ 𝜅BV𝜙 [𝑎, 𝑏]. (2) The proof is similar from part (1).

From Lemma 6 we have 𝜇(ℎ∘𝑢) < 𝜆(𝑟). Thus we conclude that (75)

And so operator 𝐻 is bounded. In the case that 𝐻(Lip[0, 1]) ⊂ 𝜅BV[0, 1], proceed similarly. In the following result we give a Lemma of invariance. Lemma 18. Let 𝜅 be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙sequence, and ] : [0, 1] → [𝑎, 𝑏] affine function that maps [0, 1] on [𝑎, 𝑏] (V(𝑡) := (𝑏 − 𝑎)𝑡 + 𝑎, 𝑡 ∈ [0, 1]). (1) 𝑢 ∈ 𝜅BV𝜙 [𝑎, 𝑏] if and only if 𝑢 ∘ ] ∈ 𝜅BV𝜙 [0, 1]. (2) 𝑢 ∈ 𝜅BV[𝑎, 𝑏] if and only if 𝑢 ∘ ] ∈ 𝜅BV𝜙 [0, 1]. Proof. (1) Let 𝑢 ∈ 𝜅BV𝜙 [𝑎, 𝑏] and {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [0, 1], then {[](𝑎𝑛 ), ](𝑏𝑛 )]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏] and

≤ 𝜅𝑉𝜙 (𝑢; [𝑎, 𝑏]) .

∑∞ 𝑛=1 𝜅 ((𝑏𝑛 − 𝑎𝑛 ) / (𝑏 − 𝑎))

≤ 𝜅𝑉𝜙 (𝑢 ∘ ]; [0, 1]) ,

𝑘 (1) 𝜅𝑉𝜙 (𝐼𝑑 ) < 1. 𝜆

󵄨󵄨 󵄨󵄨 ∑∞ 𝑛=1 󵄨󵄨𝑢 (] (𝑏𝑛 )) − 𝑢 (] (𝑎𝑛 ))󵄨󵄨 ∞ ∑𝑛=1 𝜅 (𝑏𝑛 − 𝑎𝑛 ) 󵄨 󵄨 ∑∞ 󵄨󵄨𝑢 (] (𝑏𝑛 )) − 𝑢 (] (𝑎𝑛 ))󵄨󵄨󵄨 = 𝑛−1𝑛=1 󵄨 ∑𝑛=1 𝜅 ((] (𝑏𝑛 ) − ] (𝑎𝑛 )) / (𝑏 − 𝑎))

󵄨󵄨 󵄨󵄨 ∑𝑛−1 𝑗=1 󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨

−1

𝑘 (1) 𝜑𝑛 (𝑏𝑛 − 𝑎𝑛 ) 𝑛=1 𝜆

≤ ‖ℎ ∘ 𝑢‖∞ + 𝜆 (𝑟) .

We have {𝐼𝑛 }𝑛≥1 ∈ 𝐹N [0, 1] and

𝑛=1

∞

‖𝐻 (𝑢)‖𝜅𝜙 = |ℎ (𝑢 (𝑎))| + 𝜇 (ℎ ∘ 𝑢)

(77)

× ( ∑ 𝜅 ((𝑏𝑛 − 𝑎) / (𝑏 − 𝑎)

≤∑ ≤

(𝑛 ∈ N) .

∞

󵄨 󵄨 𝑘 (1) 󵄨󵄨󵄨𝑢 (𝑏𝑛 ) − 𝑢 (𝑎𝑛 )󵄨󵄨󵄨 ) 𝜆

∞

𝐼𝑛 := []−1 (𝑏𝑛 ) , ]−1 (𝑎𝑛 )]

󵄨 −𝑢 (] ((𝑎𝑛 − 𝑎) / (𝑏 − 𝑎)))󵄨󵄨󵄨 )

󵄨󵄨 󵄨󵄨 ∑∞ 𝑛=1 𝜑𝑛 (󵄨󵄨ℎ (𝑢 (𝑏𝑛 )) − ℎ (𝑢 (𝑎𝑛 ))󵄨󵄨 /𝜆) ∑∞ 𝑛=1 𝜅 (𝑏𝑛 − 𝑎𝑛 ) ≤ ∑ 𝜑𝑛 (

Hence 𝑢 ∘ ] ∈ 𝜅BV𝜙 [0, 1]. Reciprocally, let us suppose that 𝑢 ∘ ] ∈ 𝜅BV𝜙 [0, 1] and let {[𝑎𝑛 , 𝑏𝑛 ]}𝑛≥1 ∈ 𝐹N [𝑎, 𝑏] be a partition of the interval [𝑎, 𝑏]. Define

As consequence of Lemma 18 we have the following results Lemma 19. Let 𝜅 be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, ℎ : R → R, and 𝐻 the composition operator associated to the function ℎ. (1) 𝐻(Lip[𝑎, 𝑏]) ⊂ 𝜅BV𝜙 [𝑎, 𝑏] if and only if 𝐻(Lip[0, 1]) ⊂ 𝜅BV𝜙 [0, 1]. (2) 𝐻(Lip[𝑎, 𝑏]) ⊂ 𝜅BV[𝑎, 𝑏] if and only if 𝐻(Lip[0, 1]) ⊂ 𝜅BV[0, 1]. Corollary 20. Let 𝜅 be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙sequence, and ℎ : R → R. Then the composition operator 𝐻, associated with function ℎ, maps the space Lip[𝑎, 𝑏] on 𝜅BV 𝜙 [𝑎, 𝑏] or on 𝜅BV[𝑎, 𝑏] if and only if ℎ is locally Lipschitz. Corollary 21. Let 𝜅 be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, ℎ : R → R, and X, Y normed spaces such that Lip[𝑎, 𝑏] ⊂ X ⊂ Y , where Y ⊂ 𝜅BV 𝜙 [𝑎, 𝑏] or Y ⊂ 𝜅BV[𝑎, 𝑏]. Then the composition operator 𝐻 associated with the function ℎ maps the space X in the space Y if and only if ℎ is locally Lipschitz. Some particular cases of Corollary 21 are the following

(76)

(1) X = Y , where X is one of the following spaces Lip[𝑎, 𝑏] [3], 𝐻𝛼 [𝑎, 𝑏] (0 < 𝛼 < 1) [4], BV 𝜑 [𝑎, 𝑏] [6], BV[𝑎, 𝑏] [5], HBV [7], AC [𝑎, 𝑏] [8], RV 𝜑 [𝑎, 𝑏] [9], 𝜙BV [23], ΛBV [11], and 𝜅BV[𝑎, 𝑏] [12].

10

Journal of Function Spaces and Applications (2) X = RV𝜑 [𝑎, 𝑏], Y = BV[𝑎, 𝑏]. See [10].

From inequality (79) and Lemma 16, we get

(3) X = Lip[𝑎, 𝑏], Y = BV[𝑎, 𝑏]. See [1]. 𝜅𝑉𝜙 (

From Corollary 21 we get the following new cases (1) X is one of the following spaces: Lip[𝑎, 𝑏], 𝐻𝛼 (0 < 𝛼 < 1), RV𝜑 [𝑎, 𝑏], AC[𝑎, 𝑏], BV[𝑎, 𝑏], BV 𝜙 [𝑎, 𝑏], and 𝜅RV 𝜑 [𝑎, 𝑏]; Y is one of the following spaces: 𝜅BV[𝑎, 𝑏], 𝜅BV 𝜙 [𝑎, 𝑏]. (2) X = 𝜅RV𝑝 [𝑎, 𝑏], Y = 𝜅RV𝑞 [𝑎, 𝑏], 1 < 𝑝, 𝑞 < ∞.

(1) X = 𝜅1 BV[𝑎, 𝑏], Y = 𝜅2 RV𝜙 [𝑎, 𝑏]; (2) X = 𝜅1 RV𝜑 [𝑎, 𝑏], Y = 𝜅2 RV𝜓 [𝑎, 𝑏]; (3) X = 𝜅1 RV𝜙 [𝑎, 𝑏], Y = 𝜅2 RV𝜙 [𝑎, 𝑏];

(𝑢1 − 𝑢2 ) (𝑡) =

In many problems solving equation where the composition operator appears to guarantee the existence of solution it is necessary to apply a Fixed Point Theorem. To ensure the application this type of results is necessary to request the condition of global Lipschitz operator 𝐻. In several works Matkowski and M´ıs have shown that this condition implies that the function ℎ has the form (1) or (2) (see, e.g., [16, 27]). This means that we may apply the Banach contraction mapping principle only if the underlying problems are actually 𝑙𝑖𝑛𝑒𝑎𝑟 and therefore are not interesting. More recently, Matkowski and other researchers have replaced the condition of global Lipschitz by uniform continuity conditions or uniform boundedness composition operator (see e.g., [14]). In this section we present results in this direction for the space 𝜅BV𝜙 [𝑎, 𝑏]. Theorem 22. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, and ℎ : [𝑎, 𝑏] × R → R such that the function ℎ(𝑡, ⋅) : R → R is continuous with respect to the second variable, for each 𝑡 ∈ (𝑎, 𝑏]. If the composition operator H, associated with ℎ, maps 𝜅BV𝜙 [𝑎, 𝑏] into itself and satisfies the inequality

𝑥1 − 𝑥2 , 2

(𝑡 ∈ [𝑎, 𝑏]) .

𝑚

󵄨 ( ∑ 𝜑𝑛 ( (󵄨󵄨󵄨ℎ (𝑏𝑛 , 𝑢1 (𝑏𝑛 )) − ℎ (𝑏𝑛 , 𝑢2 (𝑏𝑛 )) 𝑛=1

󵄨 −ℎ (𝑎𝑛 , 𝑢1 (𝑎𝑛 )) + ℎ (𝑎𝑛 , 𝑢2 (𝑎𝑛 ))󵄨󵄨󵄨) −1 󵄩 󵄩 ×(𝛾 (󵄩󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩𝜅𝜙 )) ) )

× (𝜅 (

(84)

𝑚

𝑏 − 𝑎𝑛 𝑎1 − 𝑎 ) + ∑𝜅 ( 𝑛 ) 𝑏−𝑎 𝑏−𝑎 𝑛=1 −1

𝑚−1

𝑎 −𝑏 𝑏 − 𝑏𝑚 + ∑ 𝜅 ( 𝑛+1 𝑛 ) + 𝜅 ( )) 𝑏−𝑎 𝑏−𝑎 𝑛=1

≤ 1.

By the construction of the 𝑢𝑘 , 𝑘 = 1, 2, we get 𝑚

( ∑ 𝜑𝑛 ((2ℎ (𝑏𝑛 , 𝑥1 ) − ℎ (𝑏𝑛 , 𝑛=1

(79)

𝑥1 + 𝑥2 ) 2

𝑥1 + 𝑥2 ) + ℎ (𝑎𝑛 , 𝑥2 )) 2

󵄨 󵄨 −1 ×(𝛾 (2−1 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨)) ) )

(80)

Proof. As for each 𝑥 ∈ R fixed the function 𝑢(𝑡) = 𝑥, 𝑡 ∈ [𝑎, 𝑏] is in 𝜅BV𝜙 [𝑎, 𝑏], then 𝐻(𝑢) = ℎ(⋅, 𝑥) ∈ 𝜅BV𝜙 [𝑎, 𝑏]. There is left regularization of ℎ− (⋅, 𝑥).

(83)

From inequality (81) and the definition of 𝑘𝜙-variation, we have

−ℎ (𝑎𝑛 ,

then there exists 𝛼, 𝛽 ∈ 𝜅BV −𝜙 [𝑎, 𝑏], such that (𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R) .

(81)

Let 𝑎 < 𝑟 < 𝑎1 < 𝑏1 < 𝑎2 < ⋅ ⋅ ⋅ < 𝑎𝑚 < 𝑏𝑚 = 𝑠 < 𝑏, 𝑥1 , 𝑥2 ∈ R, 𝑥1 ≠𝑥2 and put the zig-zag continuous functions 𝑢𝑘 : [𝑎, 𝑏] → R, 𝑘 = 1, 2, as

4. Uniformly Continuous Composition Operator in the Space 𝜅BV𝜙 [𝑎,𝑏]

ℎ (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

(𝑢, V ∈ 𝜅BV𝜙 [𝑎, 𝑏]) .

The functions 𝑢𝑘 , 𝑘 = 1, 2 are Lipschitz and therefore belongs to 𝜅BV𝜙 [𝑎, 𝑏]. Furthermore

where 𝜅1 , 𝜅2 are distortion functions.

−

) < 1,

𝑥𝑘 + 𝑥2 , 𝑎 ≤ 𝑡 ≤ 𝑎1 or 𝑡 = 𝑎𝑖 , 𝑖 = 1, . . . , 𝑚 { { { 2 = { 𝑥𝑘 + 𝑥1 , 𝑏 ≤ 𝑡 ≤ 𝑏 or 𝑡 = 𝑏 , 𝑖 = 1, . . . , 𝑚, 𝑘 = 1, 2 { 𝑚 𝑖 { 2 other case. { affine, (82)

More generally

(𝑢, V ∈ 𝜅BV𝜙 [𝑎, 𝑏]) ,

𝛾 (‖𝑢 − V‖𝜅𝜙 )

𝑢𝑘 (𝑡)

(3) X = 𝜅BV[𝑎, 𝑏], Y = 𝜅BV𝜙 [𝑎, 𝑏],

‖𝐻 (𝑢) − 𝐻 (V)‖𝜅𝜙 ≤ 𝛾 (‖𝑢 − V‖𝜅𝜙 ) ,

𝐻 (𝑢) − 𝐻 (V)

× (𝜅 ( 𝑚−1

(85)

𝑚 𝑏 − 𝑎𝑛 𝑎1 − 𝑎 ) + ∑𝜅 ( 𝑛 ) 𝑏−𝑎 𝑏−𝑎 𝑛=1

𝑎 −𝑏 𝑏 − 𝑏𝑚 + ∑ 𝜅 ( 𝑛+1 𝑛 ) + 𝜅 ( )) 𝑏 − 𝑎 𝑏−𝑎 𝑛=1

−1

≤ 1.


11

Let 𝑟 tend to 𝑠 in the above inequality; we obtain 󵄨 󵄨󵄨 − 𝑚 󵄨ℎ (𝑠, 𝑥1 ) − 2ℎ− (𝑠, (𝑥1 + 𝑥2 ) /2) + ℎ− (𝑠, 𝑥2 )󵄨󵄨󵄨 ) ∑ 𝜑𝑛 ( 󵄨 󵄨 󵄨 𝛾 (2−1 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨) 𝑛=1 ≤ 𝜅(

𝑠−𝑎 𝑏−𝑠 ) + 𝜅( ), 𝑏−𝑎 𝑏−𝑎

(𝑠 ∈ (𝑎, 𝑏]) . (86)

Passing the limit as 𝑚 → ∞, 󵄨 󵄨󵄨 − ∞ 󵄨ℎ (𝑠, 𝑥1 ) − 2ℎ− (𝑠, (𝑥1 + 𝑥2 ) /2) + ℎ− (𝑠, 𝑥2 )󵄨󵄨󵄨 ) ∑ 𝜑𝑛 ( 󵄨 󵄨 󵄨 𝛾 (2−1 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨) 𝑛=1 ≤ 𝜅(

𝑠−𝑎 𝑏−𝑠 ) + 𝜅( ), 𝑏−𝑎 𝑏−𝑎

(𝑠 ∈ (𝑎, 𝑏]) . (87)

As the series ∑∞ 𝑛=1 𝜑𝑛 (𝑥) is divergent for each 𝑥 > 0, necessarily 󵄨 󵄨󵄨 − 󵄨󵄨ℎ (𝑠, 𝑥 ) − 2ℎ− (𝑠, 𝑥1 + 𝑥2 ) + ℎ− (𝑠, 𝑥 )󵄨󵄨󵄨 = 0, 󵄨󵄨 1 2 󵄨󵄨 2 󵄨 󵄨

(88)

(𝑠 ∈ (𝑎, 𝑏]) . So we conclude that ℎ− (𝑠, ⋅) satisfies the Jensen equation in R (see [28], page 315). The continuity of ℎ with respect of the second variable implies that for every 𝑡 ∈ [𝑎, 𝑏] there exist 𝛼, 𝛽 : [𝑎, 𝑏] → R, such that −

ℎ (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

(𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R) .

Corollary 23. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, and ℎ : [𝑎, 𝑏] × R → R. If the composition operator 𝐻, associated with ℎ, maps 𝜅BV𝜙 [𝑎, 𝑏] in itself and satisfies the inequality

(𝑢, V ∈ 𝜅BV 𝜙 [𝑎, 𝑏]) ,

(90)

for any function 𝛾 : [0, ∞) → [0, ∞), verifying 𝛾(𝑡) → 𝛾(0) = 0, when 𝑡 ↓ 0, then there exists 𝛼, 𝛽 ∈ 𝜅BV −𝜙 [𝑎, 𝑏], such that −

ℎ (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

(𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R) .

(91)

Proof. Let us fix 𝑥, 𝑦 ∈ R, 𝑥 ≠𝑦, and define 𝑢(𝑡) := 𝑥, V(𝑡) := 𝑦, 𝑡 ∈ [𝑎, 𝑏]. Then 𝑢, V ∈ 𝜅BV 𝜙 [𝑎, 𝑏] and ‖𝑢 − V‖𝜅𝜙 = |𝑥 − 𝑦|. Then from inequality (90) and Lemma 16, we have 󵄨 󵄨󵄨 󵄨ℎ (𝑡, 𝑥) − ℎ (𝑡, 𝑦) − ℎ (𝑎, 𝑥) + ℎ (𝑎, 𝑦)󵄨󵄨󵄨 ) 𝜑1 ( 󵄨 󵄨 󵄨 𝛾 (󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨) 𝑡−𝑎 𝑏−𝑡 ≤ 𝜅( ) + 𝜅( ), 𝑏−𝑎 𝑏−𝑎

(𝑡 ∈ (𝑎, 𝑏]) .

󵄨 󵄨 ≤ 𝜑1−1 (2) 𝛾 (󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨) .

(93)

Proceeding as in the proof of Theorem 22 we have that there is left regularization of ℎ− (⋅, 𝑥), for each 𝑥 ∈ R and if 𝑎 ≤ 𝑟 < 𝑎1 < 𝑏1 < 𝑎2 < ⋅ ⋅ ⋅ < 𝑎𝑚 < 𝑏𝑚 = 𝑠 ≤ 𝑏, 𝑥, 𝑦 ∈ R, 𝑥 ≠𝑦 we defined the zig-zag continuous functions 𝑢, V : [𝑎, 𝑏] → R, as show below 𝑎 ≤ 𝑡 ≤ 𝑎1 or 𝑡 = 𝑎𝑖 , 𝑖 = 1, . . . , 𝑚 { 𝑥, 𝑏𝑚 ≤ 𝑡 ≤ 𝑏 or 𝑡 = 𝑏𝑖 , 𝑖 = 1, . . . , 𝑚 𝑢 (𝑡) = { 𝑦, { affine, other case, 𝑎 ≤ 𝑡 ≤ 𝑎1 or 𝑡 = 𝑎𝑖 , 𝑖 = 1, . . . , 𝑚 { 𝑦, 𝑏𝑚 ≤ 𝑡 ≤ 𝑏 or 𝑡 = 𝑏𝑖 , 𝑖 = 1, . . . , 𝑚 𝑢 (𝑡) = { 𝑥, { affine, other case,

(94)

then 𝑢, V ∈ 𝜅BV𝜙 [𝑎, 𝑏], (𝑢 − V)(𝑡) = 𝑥 − 𝑦, 𝑡 ∈ [𝑎, 𝑏] and for 𝑠 ∈ (𝑎, 𝑏] 󵄨 󵄨 2 󵄨󵄨ℎ− (𝑠, 𝑥) − ℎ− (𝑠, 𝑦)󵄨󵄨󵄨 ) 𝜑𝑚 ( 󵄨 󵄨󵄨 󵄨󵄨 𝛾 (󵄨󵄨𝑥 − 𝑦󵄨󵄨) 󵄨 󵄨 𝑚 2 󵄨󵄨ℎ− (𝑠, 𝑥) − ℎ− (𝑠, 𝑦)󵄨󵄨󵄨 ≤ ∑ 𝜑𝑛 ( 󵄨 ) 󵄨󵄨 󵄨󵄨 𝛾 (󵄨󵄨𝑥 − 𝑦󵄨󵄨) 𝑛=1 ≤ 𝜅(

(89)

Since 𝛽(𝑡) = ℎ− (𝑡, 0), 𝑡 ∈ [𝑎, 𝑏], 𝛼(𝑡) = ℎ− (𝑡, 1) − 𝛽(𝑡) and ℎ− (⋅, 𝑥) ∈ 𝜅BV−𝜙 [𝑎, 𝑏], for each 𝑥 ∈ R, we obtain that 𝛼, 𝛽 ∈ 𝜅BV𝜙 [𝑎, 𝑏].

‖𝐻 (𝑢) − 𝐻 (V)‖𝜅𝜙 ≤ 𝛾 (‖𝑢 − V‖𝜅𝜙 ) ,

And therefore 󵄨 󵄨󵄨 󵄨󵄨ℎ (𝑡, 𝑥) − ℎ (𝑡, 𝑦) − ℎ (𝑎, 𝑥) + ℎ (𝑎, 𝑦)󵄨󵄨󵄨

(95)

𝑠−𝑎 𝑏−𝑠 ) + 𝜅( ) ≤ 2. 𝑏−𝑎 𝑏−𝑎

Hence, −1 󵄨 󵄨 𝜑 (2) 󵄨󵄨 󵄨󵄨 − − 𝛾 (󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨) , 󵄨󵄨ℎ (𝑠, 𝑥) − ℎ (𝑠, 𝑦)󵄨󵄨󵄨 ≤ 𝑚 2

(96)

(𝑠 ∈ (𝑎, 𝑏]) . And so ℎ− (⋅, 𝑥) is continuous in (𝑎, 𝑏]. Now the result is a consequence of Theorem 22. Corollary 24. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, ℎ : [𝑎, 𝑏] × R → R, and 𝐻 the composition operator associated with ℎ. Suppose that 𝐻 maps 𝜅BV 𝜙 [𝑎, 𝑏] into itself and is uniformly continuous, then there exist 𝛼, 𝛽 ∈ 𝜅BV −𝜙 [𝑎, 𝑏], such that ℎ− (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

(𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R) ,

(97)

where ℎ− (⋅, 𝑥) is the left regularization of ℎ(⋅, 𝑥) for all 𝑥 ∈ R. Proof. Consider the modulus of continuity associated with 𝐻; that is, 𝛾 (𝑡) := sup {‖𝐻 (𝑢) − 𝐻 (V)‖𝜅𝜙 : ‖𝑢 − V‖𝜅𝜙

(92)

≤ 𝑡, 𝑢, V ∈ 𝜅BV𝜙 [𝑎, 𝑏]} , (𝑡 ≥ 0) .

(98)

12


Then 𝛾(𝑡) ≥ 0, 𝛾(0) = 0. Furthermore, if 𝑡 > 0 and ‖𝑢 − V‖𝜅𝜙 ≤ 𝑡, we obtain ‖𝐻 (𝑢) − 𝐻 (V)‖𝜅𝜙 ≤ 𝛾 (𝑡) .

(99)

Particularly, in the case where 𝑡 = ‖𝑢 − V‖𝜅𝜙 , we have ‖𝐻 (𝑢) − 𝐻 (V)‖𝜅𝜙 ≤ 𝛾 (‖𝑢 − V‖𝜅V ) , (𝑢, V ∈ 𝜅BV𝜙 [𝑎, 𝑏]) .

(100)

From Corollary 23 we obtain the conclusion. Matkowski [27] introduced the notion of a uniformly bounded operator and proved that the generator of any uniformly bounded composition operator acting between general Lipschitz function normed spaces must be affine with respect to the function variable. Definition 25. Let Y and Z be two metric (or normed) spaces. We say that a mapping 𝐹 : Y → Z is uniformly bounded, if for any 𝑡 > 0, there exists a nonnegative real number 𝛾(𝑡) such that for any nonempty set 𝐵 ⊂ Y we have diam 𝐵 ≤ 𝑡 󳨐⇒ diam 𝐹 (𝐵) ≤ 𝛾 (𝑡) .

(101)

Remark 26. Every uniformly continuous operator or Lipschitzian operator is uniformly bounded. Corollary 27. Let 𝜅 : [0, 1] → [0, 1] be a distortion function, 𝜙 = {𝜑𝑛 }𝑛≥1 a 𝜙-sequence, and ℎ : [𝑎, 𝑏] × R → R, such that function ℎ(𝑡, ⋅) : R → R is continuous in respect to the second variable, for each 𝑡 ∈ (𝑎, 𝑏]. If the composition operator 𝐻, associated with ℎ, maps 𝜅BV𝜙 [𝑎, 𝑏] in itself and is uniformly bounded, then exists 𝛼, 𝛽 ∈ 𝜅BV −𝜙 [𝑎, 𝑏], such that ℎ− (𝑡, 𝑥) = 𝛼 (𝑡) 𝑥 + 𝛽 (𝑡) ,

(𝑡 ∈ [𝑎, 𝑏] , 𝑥 ∈ R) .

(102)

Proof. Take any 𝑡 > 0 and 𝑢, V ∈ 𝜅BV𝜙 [𝑎, 𝑏] such that ‖𝑢 − V‖𝜅𝜙 ≤ 𝑡. Since diam {𝑢, V} ≤ 𝑡, by uniform boundedness of 𝐻, we have diam 𝐻({𝑢, V}) ≤ 𝛾(𝑡); that is, ‖𝐻 (𝑢) − 𝐻 (V)‖𝜅𝜙 = diam 𝐻 ({𝑢, V}) ≤ 𝛾 (‖𝑢 − V‖𝜅𝜙 ) . (103) From Theorem 22 we have (102).

Acknowledgment This research has been party supported by the Central Bank of Venezuela. The authors want to thank the library staff of BCV for compiling the references.

References [1] J. Appell, N. Guanda, and M. Väth, “Function spaces with the Matkowski property and degeneracy phenomena for composition operators,” Fixed Point Theory, vol. 12, no. 2, pp. 265–284, 2011. [2] J. Appell, Z. Jesús, and O. Mej´ıa, “Some remarks on nonlinear composition operators in spaces of differentiable functions,” Bollettino Della Unione Matematica Italiana, vol. 4, no. 3, pp. 321–336, 2011.

[3] A. A. Babaev, “On the structure of a certain nonlinear operator and itsapplication,” Uchen. Zap. Azerbajdzh. Gos. Univ, vol. 4, pp. 13–16, 1961 (Russian). [4] K. S. Mukhtarov, “On the properties of theoperator 𝐹𝑢 = 𝑓(𝑢(𝑥)) in the space 𝐻𝜑 ,” Sbornik Nauchm. Rabot Mat. Kaf. Dagestan Univ, pp. 145–150, 1967 (Russian). [5] M. Josephy, “Composing functions of bounded variation,” Proceedings of the American Mathematical Society, vol. 83, no. 2, pp. 354–356, 1981. [6] J. Ciemnoczołowski and W. Orlicz, “Composing functions of bounded 𝜙-variation,” Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 431–436, 1986. [7] M. Chaika and D. Waterman, “On the invariance of certain classes of functions under composition,” Proceedings of the American Mathematical Society, vol. 43, pp. 345–348, 1974. [8] N. Merentes, “On the composition operator in 𝐴𝐶[𝑎, 𝑏],” Collectanea Mathematica, vol. 42, no. 3, pp. 231–238, 1991. [9] N. Merentes, “On the composition operator in 𝑅𝑉𝜑 [𝑎, 𝑏],” Collectanea Mathematica, vol. 46, no. 3, pp. 231–238, 1995. [10] N. Merentes and S. Rivas, “On the composition operator between 𝑅𝑉𝑝 [𝑎, 𝑏] and 𝐵𝑉[𝑎, 𝑏],” Studia Mathematica, vol. 1, no. 3, pp. 237–292, 1998. [11] P. B. Pierce and D. Waterman, “On the invariance of classes 𝜙𝐵𝑉, Λ𝐵𝑉 under composition,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 755–760, 2004. [12] J. Appell and N. Merentes, “Composing functions of bounded Koremblum variation,” Dynamic Systems and Applications, vol. 22, pp. 197–206, 2013. [13] W. Aziz, N. Merentes, and J. L. Sánchez, “A note on the composition of regularfunctions,” Zeitschrift für Analysis und ihre Anwendungen. In press. [14] J. Matkowski, “Functional equations and Nemytski˘ı operators,” Fako de l’Funkcialaj Ekvacioj Japana Matematika Societo, vol. 25, no. 2, pp. 127–132, 1982. [15] J. Appell, N. Guanda, N. Merentes, and J. L. Sánchez, “Some boundedness and continuity properties of nonlinear composition operators: a survey,” Communications on Pure and Applied Analysis, vol. 15, no. 2–4, pp. 153–182, 2011. [16] J. Matkowski and J. Mi´s, “On a characterization of Lipschitzian operators of substitution in the space 𝐵𝑉[𝑎, 𝑏],” Mathematische Nachrichten, vol. 117, pp. 155–159, 1984. [17] C. Jordan, “Sur la série de Fourier,” Comptes Rendus De L’Académie Des Sciences, pp. 228–230, 1881. [18] J. Appell, J. Banas, and N. Merentes, Bounded variation and around. Gruyter. Series in nonlinear analysis and application. [19] M. Avdispahić, “Concepts of generalized bounded variation and the theory of Fourier series,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 2, pp. 223– 244, 1986. [20] N. Merentes and S. Rivas, El Operador De Composición En Espacios Con Algún Tipo De Variación Acotada, IX Escuela Venezolana de Matemáticas, Mérida, Venezuela, 1996. [21] B. Korenblum, “An extension of the Nevanlinna theory,” Acta Mathematica, vol. 135, no. 3-4, pp. 187–219, 1975. [22] D. S. Cyphert and J. A. Kelingos, “The decomposition of functions of bounded 𝜅-variation into differences of 𝜅-decreasing functions,” Polska Akademia Nauk, vol. 81, no. 2, pp. 185–195, 1985. [23] M. Schramm, “Functions of Φ-bounded variation and Riemann-Stieltjes integration,” Transactions of the American Mathematical Society, vol. 287, no. 1, pp. 49–63, 1985.

Journal of Function Spaces and Applications [24] S. K. Kim and J. Kim, “Functions of 𝜅𝜙-bounded variation,” Bulletin of the Korean Mathematical Society, vol. 23, no. 2, pp. 171–175, 1986. [25] M. Castillo, M. Sanoja, and I. Zea, “The spaces of functions of bounded 𝜅-variation in the sense ofRiesz-Koremblum,” Journal of Mathematical Control Science and Applications. In press. [26] M. Castillo, S. Rivas, M. Sanoja, and I. Zea, “functions of bounded 𝐾𝜑 -variation in the sense of Riesz-Korenblum,” Journal of Function Spaces and Applications, vol. 2013, Article ID 718507, 12 pages, 2013. [27] J. Matkowski, “Uniformly bounded composition operators between general Lipschitz function normed spaces,” Topological Methods in Nonlinear Analysis, vol. 38, no. 2, pp. 395–406, 2011. [28] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Editors and Silesian University, Warszawa, Poland, 1885.

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Research Article Abstract Semilinear Evolution Equations with Convex-Power Condensing Operators Lanping Zhu, Qianglian Huang, and Gang Li School of Mathematics, Yangzhou University, Yangzhou 225002, China Correspondence should be addressed to Lanping Zhu; [email protected] Received 14 May 2013; Revised 16 August 2013; Accepted 18 August 2013 Academic Editor: Ji Gao Copyright © 2013 Lanping Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using the techniques of convex-power condensing operators and fixed point theorems, we investigate the existence of mild solutions to nonlocal impulsive semilinear differential equations. Two examples are also given to illustrate our main results.

1. Introduction This paper is concerned with the existence of mild solutions for the following impulsive semilinear differential equations with nonlocal conditions 𝑢󸀠 (𝑡) = 𝐴𝑢 (𝑡) + 𝑓 (𝑡, 𝑢 (𝑡)) ,

𝑡 ∈ [0, 𝑇] , 𝑡 ≠𝑡𝑖 ,

𝑢 (0) = 𝑔 (𝑢) , Δ𝑢 (𝑡𝑖 ) = 𝐼𝑖 (𝑢 (𝑡𝑖 )) ,

𝑖 = 1, 2, . . . , 𝑝,

(1)

0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑝 < 𝑇, where 𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 𝑋 is the infinitesimal generator of strongly continuous semigroup 𝑆(𝑡) for 𝑡 > 0 in a real Banach space 𝑋 and Δ𝑢(𝑡𝑖 ) = 𝑢(𝑡𝑖+ ) − 𝑢(𝑡𝑖− ) constitutes an impulsive condition. 𝑓 and 𝑔 are 𝑋-valued functions to be given later. As far as we know, the first paper dealing with abstract nonlocal initial value problems for semilinear differential equations is due to [1]. Because nonlocal conditions have better effect in the applications than the classical initial ones, many authors have studied the following type of semilinear differential equations under various conditions on 𝑆(𝑡), 𝑓, and 𝑔: 𝑢󸀠 (𝑡) = 𝐴𝑢 (𝑡) + 𝑓 (𝑡, 𝑢 (𝑡)) , 𝑢 (0) = 𝑔 (𝑢) .

𝑡 ∈ [0, 𝑇] ,

(2)

For instance, Byszewski and Lakshmikantham [2] proved the existence and uniqueness of mild solutions for nonlocal

semilinear differential equations when 𝑓 and 𝑔 satisfy Lipschitz type conditions. In [3], Ntouyas and Tsamatos studied the case with compactness conditions. Byszewski and Akca [4] established the existence of solution to functionaldifferential equation when the semigroup is compact and 𝑔 is convex and compact on a given ball. Subsequently, Benchohra and Ntouyas [5] discussed the second-order differential equation under compact conditions. The fully nonlinear case was considered by Aizicovici and McKibben [6], Aizicovici and Lee [7], Aizicovici and Staicu [8], Garc´ıa-Falset [9], Paicu and Vrabie [10], Obukhovski and Zecca [11], and Xue [12, 13]. Recently, the theory of impulsive differential inclusions has become an important object of investigation because of its wide applicability in biology, medicine, mechanics, and control theory and in more and more fields. Cardinali and Rubbioni [14] studied the multivalued impulsive semilinear differential equation by means of the Hausdorff measure of noncompactness. Liang et al. [15] investigated the nonlocal impulsive problems under the assumptions that 𝑔 is compact, Lipschitz, and 𝑔 is not compact and not Lipschitz, respectively. All these studies are motivated by the practical interests of nonlocal impulsive Cauchy problems. For a more detailed bibliography and exposition on this subject, we refer to [14– 18]. The present paper is motivated by the following facts. Firstly, the approach used in [9, 12, 13, 19, 20] relies on the assumption that the coefficient 𝑙 of the function 𝑓 about the measure of noncompactness satisfies a strong inequality, which is difficult to be verified in applications. Secondly, in

2


[21], it seems that authors have considered the inequality restriction on coefficient function 𝑙(𝑡) of 𝑓 may be relaxed for impulsive nonlocal differential equations. However, in fact, they only solve the classical initial value problems 𝑢(0) = 𝑢0 rather than the nonlocal initial problems 𝑢(0) = 𝑢0 + 𝑔(𝑢). For more details, one can refer to the proof of Theorem 3.1 in [21] (see the inequalities (3.3) and (3.4) in page 5 and the estimations of the measure of noncompactness in page 6 and page 7 of [21]). Therefore, we will continue to discuss the impulsive nonlocal differential equations under more general assumptions. Throughout this work, we mainly use the property of convexpower condensing operators and fixed point theorems to obtain the main result (Theorem 10). Indeed, the fixed point theorem about the convex-power condensing operators is an extension for Darbo-Sadovskii’s fixed point theorem. But the former seems more effective than the latter at times for some problems. For example, in [22] we ever applied the former to study the nonlocal Cauchy problem and obtained more general and interesting existence results. Based on the results obtained, we discuss the impulsive nonlocal differential equations. Fortunately, applying the techniques of convexpower condensing operators and fixed point theorems solves the difficulty involved by coefficient restriction that is, the constraint condition for the coefficient function 𝑙(𝑡) of 𝑓 is unnecessary (see Theorem 10). Therefore, our results generalize and improve many previous ones in this field, such as [9, 12, 13, 19, 20]. The outline of this paper is as follows. In Section 2, we recall some concepts and facts about the measure of noncompactness, fixed point theorems, and impulsive semilinear differential equations. In Section 3, we obtain the existence results of (1) when 𝑔 is compact in 𝑃𝐶([0, 𝑇]; 𝑋). In Section 4, we discuss the existence result of (1) when 𝑔 is Lipschitz continuous, while Section 5 contains two illustrating examples.

2. Preliminaries Let 𝐸 be a real Banach space, we introduce the Hausdorff measure of noncompactness 𝛼 defined on each bounded subset Ω of 𝐸 by

(3) 𝛼(Ω1 ) ≤ 𝛼(Ω2 ) when Ω1 ⊂ Ω2 ; (4) 𝛼(Ω1 ⋃ Ω2 ) ≤ max{𝛼(Ω1 ), 𝛼(Ω2 )}; (5) 𝛼(𝜆Ω) = |𝜆|𝛼(Ω), for any 𝜆 ∈ 𝑅; (6) 𝛼(Ω1 + Ω2 ) ≤ 𝛼(Ω1 ) + 𝛼(Ω2 ), where Ω1 + Ω2 = {𝑥 + 𝑦; 𝑥 ∈ Ω1 , 𝑦 ∈ Ω2 };

(7) if {𝑊𝑛 }+∞ 𝑛=1 is a decreasing sequence of nonempty bounded closed subsets of 𝐸 and lim𝑛 → ∞ 𝛼(𝑊𝑛 ) = 0, then ∩+∞ 𝑛=1 𝑊𝑛 is nonempty and compact in 𝐸.

The map 𝑄 : 𝐷 ⊂ 𝐸 → 𝐸 is said to be 𝛼-condensing if for every bounded and not relatively compact 𝐵 ⊂ 𝐷, we have 𝛼(𝑄𝐵) < 𝛼(𝐵) (see [23]). Lemma 2 (see [9]: Darbo-Sadovskii). If 𝐷 ⊂ 𝐸 is bounded closed and convex, the continuous map 𝑄 : 𝐷 → 𝐷 is 𝛼condensing, then the map 𝑄 has at least one fixed point in 𝐷. In the sequel, we will continue to generalize the definition of condensing operator. First of all, we give some notations. Let 𝐷 ⊂ 𝐸 be bounded closed and convex, the map 𝑄 : 𝐷 → 𝐷, and 𝑥0 ∈ 𝐷 for every 𝐵 ⊂ 𝐷, set 𝑄(1,𝑥0 ) (𝐵) = 𝑄 (𝐵) , 𝑄(𝑛,𝑥0 ) 𝐵 = 𝑄 (co {𝑄(𝑛−1,𝑥0 ) 𝐵, 𝑥0 }) ,

𝑛 = 2, 3, . . . ,

(4)

where co means the closure of the convex hull. Now we give the definition of a kind of new operator. Definition 3. Let 𝐷 ⊂ 𝐸 be bounded closed and convex, the map 𝑄 : 𝐷 → 𝐷 is said to be 𝛼-convex-power condensing if there exist 𝑥0 ∈ 𝐷, 𝑛0 ∈ 𝑁 and for every bounded and not relatively compact 𝐵 ⊂ 𝐷, we have 𝛼 (𝑄(𝑛0 ,𝑥0 ) (𝐵)) < 𝛼 (𝐵) .

(5)

From this definition, if 𝛼(𝑄(𝑛0 ,𝑥0 ) (𝐵)) = 𝛼(𝐵), one obtains 𝐵 ⊂ 𝐸 as relatively compact. Subsequently, we give the fixed point theorem about the convex-power condensing operator. Lemma 4 (see [23]). If 𝐷 ⊂ 𝐸 is bounded closed and convex, the continuous map 𝑄 : 𝐷 → 𝐷 is 𝛼-convex-power condensing, then the map 𝑄 has at least one fixed point in 𝐷.

𝛼 (Ω) = inf {𝑟 > 0; there are finite points 𝑛

𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝐸 with Ω ⊂ ⋃𝐵 (𝑥𝑖 , 𝑟)} . 𝑖=1

(3) Now we recall some basic properties of the Hausdorff measure of noncompactness. Lemma 1. For all bounded subsets Ω, Ω1 , and Ω2 of 𝐸, the following properties are satisfied: (1) Ω is precompact if and only if 𝛼(Ω) = 0; (2) 𝛼(Ω) = 𝛼(Ω) = 𝛼(co Ω), where Ω and co Ω mean the closure and convex hull of Ω, respectively;

Throughout this paper, let (𝑋, ‖ ⋅ ‖) be a real Banach space. We denote by 𝐶([0, 𝑇]; 𝑋) the Banach space of all continuous functions from [0, 𝑇] to 𝑋 with the norm ‖𝑢‖ = sup {‖𝑢(𝑡)‖, 𝑡 ∈ [0, 𝑇]} and by 𝐿1 ([0, 𝑇]; 𝑋) the Banach space of all 𝑋-valued Bochner integrable functions defined on 𝑇 [0, 𝑇] with the norm ‖𝑢‖1 = ∫0 ‖𝑢(𝑡)‖𝑑𝑡. Let 𝑃𝐶([0, 𝑇]; 𝑋) = {𝑢 : 𝑢 is a function from [0, 𝑇] into 𝑋 such that 𝑢(𝑡) is continuous at 𝑡 ≠𝑡𝑖 and the left continuous at 𝑡 = 𝑡𝑖 and the right limit 𝑢(𝑡𝑖+ ) exists for 𝑖 = 1, 2, ..., 𝑝}. It is easy to check that 𝑃𝐶([0, 𝑇]; 𝑋) is a Banach space with the norm ‖𝑢‖𝑃𝐶 = sup{‖𝑢(𝑡)‖, 𝑡 ∈ [0, 𝑇]} and 𝐶([0, 𝑇]; 𝑋) ⊆ 𝑃𝐶([0, 𝑇]; 𝑋) ⊆ 𝐿1 ([0, 𝑇]; 𝑋). Moreover, we denote 𝛽 by the


3

Hausdorff measure of noncompactness of 𝑋, denote 𝛽𝑐 by the Hausdorff measure of noncompactness of 𝐶([0, 𝑇]; 𝑋) and denote 𝛽𝑝𝑐 by the Hausdorff measure of noncompactness of 𝑃𝐶([0, 𝑇]; 𝑋). Throughout this work, we suppose the following (𝐻𝐴 ) The linear operator 𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 𝑋 generates an equicontinuous 𝐶0 -semigroup {𝑆(𝑡) : 𝑡 ≥ 0}. Hence, there exists a positive number 𝑀 such that ‖𝑆(𝑡)‖ ≤ 𝑀. For further information about the theory of semigroup of operators, we may refer to some classic books, such as [24– 26]. To discuss the problem (1), we also need the following lemma. Lemma 5. If 𝑊 ⊆ 𝑃𝐶([0, 𝑇]; 𝑋) is bounded, then one has sup 𝛽 (𝑊 (𝑡)) ≤ 𝛽𝑝𝑐 (𝑊) ,

𝑡∈[0,𝑇]

(6)

where 𝑊(𝑡) = {𝑢(𝑡); 𝑢 ∈ 𝑊} ⊂ 𝑋. Lemma 6 (see [27]). If 𝑊 ⊆ 𝐶([0, 𝑇]; 𝑋) is bounded, then for all 𝑡 ∈ [0, 𝑇], 𝛽 (𝑊 (𝑡)) ≤ 𝛽𝑐 (𝑊) .

(7)

Furthermore, if 𝑊 is equicontinuous on [0, 𝑇], then 𝛽(𝑊(𝑡)) is continuous on [0, 𝑇] and

(2) for finite positive constant 𝑟 > 0, there exists a function 𝛼𝑟 ∈ 𝐿1 (0, 𝑇; 𝑅) such that 󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑡, 𝑥)󵄩󵄩󵄩 ≤ 𝛼𝑟 (𝑡)

(10)

for a.e. 𝑡 ∈ [0, 𝑇] and 𝑥 ∈ 𝐵𝑟 ; (3) there exists a function 𝑙(𝑡) ∈ 𝐿1 (0, 𝑇; 𝑅) such that 𝛽 (𝑓 (𝑡, 𝐷)) ≤ 𝑙 (𝑡) 𝛽 (𝐷)

(11)

for a.e. 𝑡 ∈ [0, 𝑇] and every bounded subset 𝐷 ⊂ 𝐵𝑟 ; (𝐻𝑔 ) 𝑔 : 𝑃𝐶([0, 𝑇]; 𝑋) → 𝑋 is a continuous and compact mapping; furthermore, there exists a positive number 𝑁1 such that ‖𝑔(𝑢)‖ ≤ 𝑁1 , for any 𝑢 ∈ 𝑊𝑟 ; (𝐻𝐼 ) 𝐼𝑖 : 𝑋 → 𝑋 is a continuous and compact mapping for every 𝑖 = 1, 2, . . . , 𝑝; 𝑝

(𝐻𝑟 ) 𝑀(𝑁1 + ‖𝛼𝑟 ‖𝐿1 + sup𝑢∈𝑊𝑟 ∑𝑖=1 ‖𝐼𝑖 (𝑢(𝑡𝑖 ))‖) ≤ 𝑟. Remark 9. The mapping 𝑓 is said to be 𝐿1 -Carathéodory if the assumption (𝐻𝑓 )(1)(2) is satisfied.

(8)

Theorem 10. If the hypotheses (𝐻𝐴), (𝐻𝑓 )(1)(2)(3), (𝐻𝑔 ), (𝐻𝐼 ), and (𝐻𝑟 ) are satisfied, then the nonlocal problem (1) has at least one mild solution on [0, 𝑇].

Since 𝐶0 -semigroup 𝑆(𝑡) is said to be equicontinuous, the following lemma is easily checked.

To prove the above theorem, we need the following lemma.

Lemma 7. If the semigroup 𝑆(𝑡) is equicontinuous and 𝑤 ∈ 𝑡 𝐿1 ([0, 𝑇]; 𝑅+ ), then the set {∫0 𝑆(𝑡 − 𝑠)𝑢(𝑠)𝑑𝑠, ‖𝑢(𝑠)‖ ≤ 𝑤(𝑠) for a.e. 𝑠 ∈ [0, 𝑇]} is equicontinuous for 𝑡 ∈ [0, 𝑇].

Lemma 11. If the condition (𝐻𝑟 ) holds, then for arbitrary bounded set 𝐵 ⊂ 𝑊𝑟 , we have

𝛽𝑐 (𝑊) = sup {𝛽 (𝑊 (𝑡)) : 𝑡 ∈ [0, 𝑇]} .

Definition 8. A function 𝑢 ∈ 𝑃𝐶([0, 𝑇]; 𝑋) is said to be a mild solution of the nonlocal problem (1), if it satisfies

0

(12)

𝑡

≤ 4𝑀 ∫ 𝛽 (𝑓 (𝑠, 𝐵 (𝑠))) 𝑑𝑠,

𝑡

0

𝑢 (𝑡) = 𝑆 (𝑡) 𝑔 (𝑢) + ∫ 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑢 (𝑠)) 𝑑𝑠 0

𝑡

𝛽 (∫ 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝐵 (𝑠)) 𝑑𝑠) 𝑡 ∈ [0, 𝑇] .

(9)

This proof is quite similar to that of Lemma 3.1 in [20]; we omit it.

In addition, let 𝑟 be a finite positive constant, and set 𝐵𝑟 := {𝑥 ∈ 𝑋 : ‖𝑥‖ ≤ 𝑟} and 𝑊𝑟 := {𝑢 ∈ 𝐶([0, 𝑇]; 𝑋) : 𝑢(𝑡) ∈ 𝐵𝑟 , for all 𝑡 ∈ [0, 𝑇]}.

Proof of Theorem 10. We consider the operator 𝑄 : 𝑃𝐶([0, 𝑇]; 𝑋) → 𝑃𝐶([0, 𝑇]; 𝑋) defined by

+ ∑ 𝑆 (𝑡 − 𝑡𝑖 ) 𝐼𝑖 (𝑢 (𝑡𝑖 )) ,

0 ≤ 𝑡 ≤ 𝑇.

0 0 such that 𝛽 (𝑓 (𝑡, 𝐷)) ≤ 𝑙𝛽 (𝐷)

(34)

(35)

for any 𝐵 ⊂ 𝑊. Thus, there exists a large enough positive integral 𝑛0 such that 𝑇𝑛0 4 𝑀 𝑙 < 1; 𝑛0 ! 𝑛0

𝑛0 𝑛0

󵄩 󵄩󵄩 󵄩󵄩𝑔 (𝑢) − 𝑔 (V)󵄩󵄩󵄩 ≤ 𝑘 ‖𝑢 − V‖ ,

for 𝑢, V ∈ 𝑃𝐶 ([0, 𝑇] ; 𝑋) ; (38)

(𝐻𝐼󸀠 ) 𝐼𝑖 : 𝑋 → 𝑋 is Lipschitz continuous with Lipschitz constant 𝑘𝑖 , for 𝑖 = 1, 2, . . . , 𝑝. Theorem 15. If the hypotheses (𝐻𝐴), (𝐻𝑓 )(1)(2)(3), (𝐻𝑔󸀠 ), (𝐻𝐼󸀠 ), and (𝐻𝑟 ) are satisfied, then the nonlocal problem (1) has 𝑝 at least one mild solution on [0, 𝑇] provided that 𝑀(𝑘+∑𝑖=1 𝑘𝑖 + 4‖𝑙‖𝐿1 ) < 1. Proof of Theorem 15. Given 𝑥 ∈ 𝑊, let’s first consider the following Cauchy initial problem: 𝑢󸀠 (𝑡) = 𝐴𝑢 (𝑡) + 𝑓 (𝑡, 𝑢 (𝑡)) , Δ𝑢 (𝑡𝑖 ) = 𝐼𝑖 (𝑥 (𝑡𝑖 )) ,

𝑡 ∈ [0, 𝑇] , 𝑡 ≠𝑡𝑖 , 𝑖 = 1, 2, . . . , 𝑝,

0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑝 < 𝑇,

(36)

𝑢 (0) = 𝑔 (𝑥) . From the proof of Theorem 10, we can easily see that there exists at least one mild solution to (39). Define 𝐺 : 𝑊 → 𝑊 by that 𝐺𝑥 is the mild solution to (39). Then (𝐺𝑥) (𝑡) = 𝑆 (𝑡) 𝑔 (𝑥) + ∑ 𝑆 (𝑡 − 𝑡𝑖 ) 𝐼𝑖 (𝑥 (𝑡𝑖 )) 0 0, there exist two integrable functions 𝜑𝑟 , 𝜔 : [0, 𝑇] → 𝑅 such that ‖𝑓(𝑡, 𝑦)‖ ≤ 𝜑𝑟 (𝑡) and 𝛽(𝑓(𝑡, 𝐷)) ≤ 𝜛(𝑡)𝛽(𝐷) for a.e. 𝑡 ∈ [0, 𝑇], 𝑦 ∈ 𝐵𝑟 , and every bounded subset 𝐷 ⊂ 𝐵𝑟 ; (2) 𝑔 : 𝑃𝐶([0, 𝑇]; 𝑋) → 𝑋 is defined by 𝑞

𝑔 (𝑢) (𝑥) = ∑ 𝑐𝑗 𝑢 (𝑠𝑗 , 𝑥) , 𝑗=1

(51)

0 < 𝑠1 < 𝑠2 < ⋅ ⋅ ⋅ < 𝑠𝑞 < 𝑇, 𝑥 ∈ Ω. Then 𝑔 is Lipschitz continuous with constant 𝑘 = 𝑞 ∑𝑗=1 |𝑐𝑗 |; that is, the assumption 𝐻𝑔󸀠 is satisfied. (3) 𝐼𝑖 : 𝑋 → 𝑋 is a continuous function for each 𝑖 = 1, 2, . . . , 𝑝, defined by 𝐼𝑖 (𝑢) (𝑥) = 𝐼𝑖 (𝑢 (𝑥)) .

(52)

Here we take 𝐼𝑖 (𝑢(𝑥)) = (𝛼𝑖 |𝑢(𝑥)| + 𝑡𝑖 )−1 , 𝛼𝑖 > 0, 𝑖 = 1, 2, . . . , 𝑝, 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑞 < 𝑇, 𝑥 ∈ Ω. Then 𝐼𝑖 is Lipschitz continuous with constant 𝑘𝑖 = 𝛼𝑖 /𝑡𝑖2 , 𝑖 = 1, 2, . . . , 𝑝; that is, the assumption 𝐻𝐼󸀠 is satisfied.

The authors would like to thank the referees for their careful reading and their valuable comments and suggestions to improve their results. This research is supported by the Natural Science Foundation of China (11201410 and 11271316), the Natural Science Foundation of Jiangsu Province (BK2012260), and the Natural Science Foundation of Jiangsu Education Committee (10KJB110012).

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Research Article On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator Dinu Teodorescu1 and N. Hussain2 1 2

Department of Mathematics, Valahia University of Targoviste, Boulevard Unirii 18, 130024 Targoviste, Romania Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Dinu Teodorescu; [email protected] Received 5 May 2013; Accepted 13 September 2013 Academic Editor: Satit Saejung Copyright © 2013 D. Teodorescu and N. Hussain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper we study semilinear equations of the form 𝐴𝑢 + 𝜆𝐹(𝑢) = 𝑓, where 𝐴 is a linear self-adjoint operator, satisfying a strong positivity condition, and 𝐹 is a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided.

1. Introduction and Preliminaries The study of abstract operator equations involving linear or nonlinear operators has generated over time useful instruments in the approach of some concrete equations. Therefore, we consider as interesting to present some aspects regarding the semilinear abstract operator equations in Hilbert spaces. Let 𝐻 be a real Hilbert space endowed with the inner product ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖. In [1, 2] semilinear equations of the form 𝐴𝑢 − 𝐹(𝑢) = 0 were studied, where 𝐴 : 𝐷(𝐴) ⊂ 𝐻 → 𝐻 is a self-adjoint linear operator with the resolvent set 𝑅(𝐴) and 𝐹 : 𝐻 → 𝐻 is a Gateaux differentiable gradient operator. If there exist real numbers 𝑎 < 𝑏 such that [𝑎, 𝑏] ⊂ 𝑅(𝐴) and 𝑎≤

⟨𝐹 (𝑢) − 𝐹 (V) , 𝑢 − V⟩ ≤ 𝑏, ‖𝑢 − V‖2

(1)

for all 𝑢, V ∈ 𝐻, 𝑢 ≠V (i.e., 𝐹 interacts suitably with the spectrum of 𝐴), then it is proved in [2] that the equation 𝐴𝑢 − 𝐹(𝑢) = 0 has exactly one solution. The author in [3] presented an existence and uniqueness result for the semilinear equation 𝐴𝑢 + 𝐹(𝑢) = 𝑓, where 𝐴 : 𝐷(𝐴) ⊂ 𝐻 → 𝐻 is a linear maximal monotone operator, satisfying a strong positivity condition, and the nonlinearity 𝐹 : 𝐻 → 𝐻 is a Lipschitz operator.

Let 𝑋 be a real Banach space, ordered by a cone 𝐾. A cone 𝐾 is a closed convex subset of 𝑋 with 𝜆𝐾 ⊆ 𝐾 (𝜆 ≥ 0), and 𝐾 ∩ (−𝐾) = {0}. As usual 𝑥 ≤ 𝑦 ⇔ 𝑦 − 𝑥 ∈ 𝐾. Definition 1. Let 𝑀 be a nonempty subset of an ordered Banach space 𝑋 with order ≤. Two mappings 𝑆, 𝑇 : 𝑀 → 𝑀 are said to be weakly isotone increasing if 𝑆𝑥 ≤ 𝑇𝑆𝑥 and 𝑇𝑥 ≤ 𝑆𝑇𝑥 hold for all 𝑥 ∈ 𝑀. Similarly, we say that 𝑆 and 𝑇 are weakly isotone decreasing if 𝑇𝑥 ≥ 𝑆𝑇𝑥 and 𝑆𝑥 ≥ 𝑇𝑆𝑥 hold for all 𝑥 ∈ 𝑀. The mappings 𝑆 and 𝑇 are said to be weakly isotone if they are either weakly isotone increasing or weakly isotone decreasing. In our considerations, the following definition will play an important role. Let B(𝑋) denote the collection of all nonempty bounded subsets of 𝑋 and W(𝑋) the subset of B(𝑋) consisting of all weakly compact subsets of 𝑋. Also, let 𝐵𝑟 denote the closed ball centered at 0 with radius 𝑟. Definition 2 (see [4]). A function 𝜓 : B(𝑋) → R+ is said to be a measure of weak noncompactness if it satisfies the following conditions. (1) The family ker(𝜓) = {𝑀 ∈ B(𝑋) : 𝜓(𝑀) = 0} is nonempty, and ker(𝜓) is contained in the set of relatively weakly compact sets of 𝑋.

2

Journal of Function Spaces and Applications (2) 𝑀1 ⊆ 𝑀2 ⇒ 𝜓(𝑀1 ) ≤ 𝜓(𝑀2 ). (3) 𝜓(co(𝑀)) = 𝜓(𝑀), where co(𝑀) is the closed convex hull of 𝑀. (4) 𝜓(𝜆𝑀1 + (1 − 𝜆)𝑀2 ) ≤ 𝜆𝜓(𝑀1 ) + (1 − 𝜆)𝜓(𝑀2 ) for 𝜆 ∈ [0, 1]. (5) If (𝑀𝑛 )𝑛≥1 is a sequence of nonempty weakly closed subsets of 𝑋 with 𝑀1 bounded and 𝑀1 ⊇ 𝑀2 ⊇ ⋅ ⋅ ⋅ ⊇ 𝑀𝑛 ⊇ ⋅ ⋅ ⋅ such that lim𝑛 → ∞ 𝜓(𝑀𝑛 ) = 0, then 𝑀∞ := ⋂∞ 𝑛=1 𝑀𝑛 is nonempty.

The family ker 𝜓 described in (1) is said to be the kernel of the measure of weak noncompactness 𝜓. Note that the intersection set 𝑀∞ from (5) belongs to ker 𝜓 since 𝜓(𝑀∞ ) ≤ 𝜓(𝑀𝑛 ) for every 𝑛, and lim𝑛 → ∞ 𝜓(𝑀𝑛 ) = 0. Also, it can be easily verified that the measure 𝜓 satisfies 𝜓 (𝑀𝑤 ) = 𝜓 (𝑀) ,

(2)

where 𝑀𝑤 is the weak closure of 𝑀. A measure of weak noncompactness 𝜓 is said to be regular if 𝜓 (𝑀) = 0

iff 𝑀 is relatively weakly compact,

(3)

subadditive if 𝜓 (𝑀1 + 𝑀2 ) ≤ 𝜓 (𝑀1 ) + 𝜓 (𝑀2 ) ,

(4)

homogeneous if 𝜓 (𝜆𝑀) = |𝜆| 𝜓 (𝑀) ,

𝜆 ∈ R,

(5)

and set additive (or has the maximum property) if 𝜓 (𝑀1 ∪ 𝑀2 ) = max (𝜓 (𝑀1 ) , 𝜓 (𝑀2 )) .

(6)

The first important example of a measure of weak noncompactness has been defined by de Blasi [5] as follows: 𝑤 (𝑀) = inf {𝑟 > 0 : there exists 𝑊 ∈ W (𝑋) with 𝑀 ⊆ 𝑊 + 𝐵𝑟 } ,

(7)

for each 𝑀 ∈ B(𝑋). Notice that 𝑤(⋅) is regular, homogeneous, subadditive, and set additive (see [5]). By a measure of noncompactness on a Banach space 𝑋, we mean a map 𝜓 : B(𝑋) → R+ which satisfies conditions (1)–(5) in Definition 2 relative to the strong topology instead of the weak topology. Definition 3. Let 𝑋 be a Banach space and 𝜓 a measure of (weak) noncompactness on 𝑋. Let 𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 𝑋 be a mapping. If 𝐴(𝐷(𝐴)) is bounded and for every nonempty bounded subset 𝑀 of 𝐷(𝐴) with 𝜓(𝑀) > 0, we have 𝜓(𝐴(𝑀)) < 𝜓(𝑀); then 𝐴 is called 𝜓-condensing. If there exists 𝑘, 0 ≤ 𝑘 ≤ 1, such that 𝐴(𝐷(𝐴)) is bounded and for each nonempty bounded subset 𝑀 of 𝐷(𝐴), we have 𝜓(𝐴(𝑀)) ≤ 𝑘𝜓(𝑀); then 𝐴 is called 𝑘-𝜓-contractive.

Definition 4 (see [6]). A map 𝐴 : 𝐷(𝐴) → 𝑋 is said to be ws-compact if it is continuous, and for any weakly convergent sequence (𝑥𝑛 )𝑛∈N in 𝐷(𝐴) the sequence (𝐴𝑥𝑛 )𝑛∈N has a strongly convergent subsequence in 𝑋. Definition 5. A map 𝐴 : 𝐷(𝐴) → 𝑋 is said to be wwcompact if it is continuous, and for any weakly convergent sequence (𝑥𝑛 )𝑛∈N in 𝐷(𝐴) the sequence (𝐴𝑥𝑛 )𝑛∈N has a weakly convergent subsequence in 𝑋. Definition 6. Let 𝑋 be a Banach space. A mapping 𝑇 : 𝐷(𝑇) ⊆ 𝑋 → 𝑋 is called a nonlinear contraction if there exists a continuous and nondecreasing function 𝜑 : R+ → R+ such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 𝜑 (󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩) ,

(8)

for all 𝑥, 𝑦 ∈ 𝐷(𝑇), where 𝜑(𝑟) < 𝑟 for 𝑟 > 0. In this paper we consider the semilinear equation 𝐴𝑢 + 𝜆𝐹 (𝑢) = 𝑓,

(9)

where 𝐴 : 𝐻 → 𝐻 is a linear self-adjoint operator, satisfying a strong positivity condition, 𝐹 : 𝐻 → 𝐻 is a nonlinear Lipschitz operator, and 𝜆 is a positive parameter. Using the Banach fixed point theorem, we prove an existence and uniqueness result about the considered equation. Thus, we obtain here the same type of result as in [2], replacing the maximal monotonicity of linear part 𝐴 of the semilinear equation with the hypothesis that 𝐴 is self-adjoint. So, the principal result of this paper can be applied in the study of nonlinear Lipschitz perturbations of a linear integral operator with symmetric kernel. Further a result of continuous dependence on the free term and a fixed point theorem are presented. As applications we present some common fixed point theorems and approximation results for a pair of nonlinear mappings. Finally, the existence of solution of an integral equation is provided to illustrate the usability of the obtained results.

2. Results Theorem 7. Let 𝐴 : 𝐻 → 𝐻 be a linear self-adjoint operator and 𝐹 : 𝐻 → 𝐻 nonlinear, satisfying the following conditions: (i) 𝐹 is a Lipschitz operator, that is, there is a constant 𝑀 > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝐹 (𝑥) − 𝐹 (𝑦)󵄩󵄩󵄩 ≤ 𝑀 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,

(10)

for all 𝑥, 𝑦 ∈ 𝐻; (ii) 𝐴 is a strongly positive operator, that is, there is a constant 𝑐 > 0 such that ⟨𝐴𝑥, 𝑥⟩ ≥ 𝑐‖𝑥‖2 ,

(11)

for all 𝑥 ∈ 𝐻. Then the equation 𝐴𝑢 + 𝜆𝐹(𝑢) = 𝑓 has a unique solution for all 𝑓 ∈ 𝐻 and 𝜆 ∈ (0; 𝑐/𝑀).


3

Proof. Let us choose 𝑠 in the spectrum of 𝐴. We have 𝑠 ≥ inf {⟨𝐴𝑥, 𝑥⟩ | 𝑥 ∈ 𝐻, ‖𝑥‖ = 1} ≥ 𝑐,

(12)

and we obtain that every real number 𝜔 ∈ (−∞; 𝑐) is in the resolvent set of the operator 𝐴. Consequently, we have Rg (𝐴 − 𝜔𝐼) = 𝐻,

Theorem 8. Under the assumptions from the hypothesis of Theorem 7, let 𝑖 ∈ {1, 2}, and let 𝑢𝑖 be the unique solution of the equation

(13)

for all 𝜔 < 𝑐, where 𝐼 is the identity of 𝐻. Let 𝛿 < 0. We write (9) in the equivalent form 𝐴 𝛿 𝑢 + 𝐹𝛿 (𝑢) = 𝑓,

𝐴𝑢 + 𝜆𝐹 (𝑢) = 𝑓𝑖 ,

1 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩 ≤ 󵄩𝑓 − 𝑓2 󵄩󵄩󵄩 . 𝑐 − 𝜆𝑀 󵄩 1

(14)

for all 𝑥, 𝑦 ∈ 𝐻. Also

≥ (𝑐 − 𝛿) ‖𝑥‖2 = (𝑐 + |𝛿|) ‖𝑥‖2 From (16) we obtain 󵄩󵄩󵄩𝐴 𝛿 𝑥󵄩󵄩󵄩 ≥ (𝑐 + |𝛿|) ‖𝑥‖ 󵄩 󵄩

∀𝑥 ∈ 𝐻.

(16)

(24)

󵄩 󵄩󵄩 󵄩󵄩 −1 󵄩󵄩 −1 󵄩 + 󵄩󵄩𝐴 𝛿 𝑓1 − 𝐴−1 󵄩 ≤ 󵄩󵄩󵄩󵄩𝐴−1 𝛿 𝐹𝛿 𝑢1 − 𝐴 𝛿 𝐹𝛿 𝑢2 󵄩 𝛿 𝑓2 󵄩 󵄩 󵄩 󵄩 (25) 1 󵄩󵄩 𝜆𝑀 + |𝛿| 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩 + 󵄩𝑓 − 𝑓2 󵄩󵄩󵄩 . 𝑐 + |𝛿| 𝑐 + |𝛿| 󵄩 1

It results that 󵄩 󵄩 󵄩 󵄩 (𝑐 − 𝜆𝑀) 󵄩󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑓1 − 𝑓2 󵄩󵄩󵄩 ,

(26)

and thus our assertion is proved. ∀𝑥 ∈ 𝐻.

(17)

𝐴−1 𝛿

Consequently, there exists : 𝐻 → 𝐻 which is linear −1 and continuous, that is 𝐴 𝛿 ∈ L(𝐻), the Banach space of all linear and bounded operators from 𝐻 to 𝐻. Moreover, we have 1 󵄩󵄩 −1 󵄩󵄩 󵄩󵄩𝐴 𝛿 󵄩󵄩 󵄩L(𝐻) ≤ 𝑐 + |𝛿| . 󵄩

(18)

Now (14) can be equivalently written as −1 𝑢 + 𝐴−1 𝛿 𝐹𝛿 𝑢 = 𝐴 𝛿 𝑓.

(19)

We consider the operator 𝑇 : 𝐻 → 𝐻 defined by −1 𝑇𝑢 = −𝐴−1 𝛿 𝐹𝛿 𝑢 + 𝐴 𝛿 𝑓.

(20)

Therefore (19) becomes 𝑢 = 𝑇𝑢,

(21)

and so, the problem of the solvability of (9) is reduced to the study of fixed points of the operator 𝑇. We have 󵄩 󵄩 󵄩 −1 󵄩󵄩 −1 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝐴 𝛿 𝐹𝛿 𝑥 − 𝐴 𝛿 𝐹𝛿 𝑦󵄩󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 −1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩𝐴 𝛿 󵄩󵄩 = 󵄩󵄩󵄩󵄩𝐴−1 𝛿 (𝐹𝛿 𝑥 − 𝐹𝛿 𝑦)󵄩 󵄩L(𝐻) 󵄩󵄩𝐹𝛿 𝑥 − 𝐹𝛿 𝑦󵄩󵄩 󵄩 󵄩 𝜆𝑀 + |𝛿| 󵄩󵄩 󵄩 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 𝑐 + |𝛿|

(23)

Proof. According to the equivalent form (19) of (9), we have 󵄩󵄩 󵄩 󵄩 −1 −1 −1 −1 󵄩 󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩−𝐴 𝛿 𝐹𝛿 𝑢1 + 𝐴 𝛿 𝑓1 + 𝐴 𝛿 𝐹𝛿 𝑢2 − 𝐴 𝛿 𝑓2 󵄩󵄩󵄩󵄩

≤

⟨𝐴 𝛿 𝑥, 𝑥⟩ = ⟨𝐴𝑥, 𝑥⟩ − 𝛿‖𝑥‖2

𝑖 ∈ {1, 2} ; 𝑓1 , 𝑓2 ∈ 𝐻.

Then

where 𝐴 𝛿 = 𝐴 − 𝛿𝐼 and 𝐹𝛿 = 𝜆𝐹 + 𝛿𝐼. We have Rg(𝐴 𝛿 ) = 𝐻 and 󵄩 󵄩 󵄩󵄩 󵄩 (15) 󵄩󵄩𝐹𝛿 (𝑥) − 𝐹𝛿 (𝑦)󵄩󵄩󵄩 ≤ (𝜆𝑀 + |𝛿|) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,

≤

Let us consider now the dependence of solution of (9) on the data 𝑓.

∀𝑥, 𝑦 ∈ 𝐻. (22)

It results that 𝑇 is a strict contraction from 𝐻 to 𝐻 because 𝜆𝑀 < 𝑐. According to the Banach fixed point theorem, 𝑇 has a unique fixed point, and thus the proof of Theorem 7 is complete.

In fact, Theorem 8 establishes the continuous dependence of the solution of (9) on the free term and signifies the stability of the solution.

3. Consequences of Principal Result The previous results prove that, under the considered assumptions, the operator 𝐴 + 𝜆𝐹 is invertible for all 𝜆 ∈ (0; 𝑐/𝑀) and the inverse 𝐶 = (𝐴 + 𝜆𝐹)−1 is a Lipschitz operator. From (24) it results that the operator 𝐶 satisfies 1 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝐶 (𝑓1 ) − 𝐶 (𝑓2 )󵄩󵄩󵄩 ≤ 󵄩𝑓 − 𝑓2 󵄩󵄩󵄩 𝑐 − 𝜆𝑀 󵄩 1

∀𝑓1 , 𝑓2 ∈ 𝐻. (27)

Suppose now that 𝑐 > 1 and 𝜆 ∈ (0; (𝑐−1)/𝑀) ⊂ (0; 𝑐/𝑀). We obtain 𝑀𝜆 < 𝑐 − 1 or 1/(𝑐 − 𝜆𝑀) < 1, and, consequently, 𝐶 is a strict contraction. It results that there exists an unique element 𝑢∗ ∈ 𝐻 such that 𝐶(𝑢∗ ) = 𝑢∗ which is equivalent to (𝐴 + 𝜆𝐹)(𝑢∗ ) = 𝑢∗ . So we obtained the following fixed point theorem. Theorem 9. Let 𝐴 : 𝐻 → 𝐻 be a linear self-adjoint operator and 𝐹 : 𝐻 → 𝐻 nonlinear, satisfying the following conditions: (i) 𝐹 is a Lipschitz operator, that is, there is a constant 𝑀 > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 (28) 󵄩󵄩𝐹 (𝑥) − 𝐹 (𝑦)󵄩󵄩󵄩 ≤ 𝑀 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , for all 𝑥, 𝑦 ∈ 𝐻; (ii) 𝐴 is a strongly positive operator, that is, there is a constant 𝑐 > 0 such that ⟨𝐴𝑥, 𝑥⟩ ≥ 𝑐‖𝑥‖2 , for all 𝑥 ∈ 𝐻.

(29)

4


If 𝑐 > 1 and 𝜆 ∈ (0; (𝑐 − 1)/𝑀), then the operator 𝐴 + 𝜆𝐹 has a unique fixed point.

Then there exists a unique 𝑥∗ ∈ 𝐷 such that 𝐴𝑆𝑥∗ + 𝜆𝐹𝑆𝑥∗ = 𝑥∗ .

The way used in obtaining Theorem 9 can be applied in the study of the following problem: extracting operators which have the unique fixed point property from a family of operators {𝑉𝜆 /𝜆 ∈ R}. Let 𝐸 be a Banach space, and F = {𝑉𝜆 : 𝐸 → 𝐸/𝜆 ∈ R}, 𝑉𝜆 satisfying some constraints for any 𝜆 ∈ R. Our intention is to extract a subfamily G ⊂ F, so that 𝑉𝜆 can have a unique fixed point for all 𝑉𝜆 ∈ G. It is easy to observe, using the same method as in obtaining Theorem 9, that the following result holds.

Proof. By Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 : 𝐻 → 𝐻 is a contraction with contractive constant 𝛼 = 1/(𝑐−𝜆𝑀) < 1. Thus 𝑇 is a shrinking mapping. Now all of the conditions of Corollary 1.19 [7] are satisfied so there exists an 𝑥∗ ∈ 𝐷 such that 𝑇𝑥∗ = 𝑆𝑥∗ = 𝑥∗ which implies 𝐴𝑆𝑥∗ + 𝜆𝐹𝑆𝑥∗ = 𝑥∗ .

Theorem 10. If (i) there exists 𝐽 ⊂ R, 𝐽 ≠𝜙, so that 𝑉𝜆 is invertible and 󵄩󵄩 −1 󵄩 󵄩󵄩𝑉𝜆 𝑥 − 𝑉𝜆−1 𝑦󵄩󵄩󵄩 ≤ 𝑔 (𝜆) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 , 󵄩 󵄩

𝑥, 𝑦 ∈ 𝐸; 𝑔 (𝜆) > 0, (30)

for all 𝜆 ∈ 𝐽; (ii) there exists 𝐼 ⊂ 𝐽, 𝐼 ≠𝜙, so that 𝑔(𝜆) < 1 for all 𝜆 ∈ 𝐼, then 𝑉𝜆 has a unique fixed point for all 𝜆 ∈ 𝐼. As an application of the observations established above, we develop here the Krasnoselskii and Ky Fan type approximation results for certain pair of maps. Theorem 11. Let 𝐴, 𝐹, 𝜆, 𝑀 and 𝑐 be as in Theorem 9, 𝑇 = (𝐴+ 𝜆𝐹)−1 , and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Let 𝑆 : 𝐻 → 𝐻 be a mapping satisfying the following: (i) 𝑇(𝐷) ⊆ 𝐷 and 𝑆(𝐷) ⊆ 𝐷, (ii) 𝑆 is continuous and 𝜓-condensing, (iii) 𝑇 and 𝑆 are weakly isotone. Then there exists a unique 𝑥∗ ∈ 𝐷 such that 𝐴𝑆𝑥∗ + 𝜆𝐹𝑆𝑥∗ = 𝑥∗ . Proof. By Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 : 𝐻 → 𝐻 is a contraction with contractive constant 𝛼 = 1/(𝑐 − 𝜆𝑀) < 1. Thus 𝑇 is a shrinking mapping. Now all of the conditions of Corollary 1.18 [7] are satisfied so there exists an 𝑥∗ ∈ 𝐷 such that 𝑇𝑥∗ = 𝑆𝑥∗ = 𝑥∗ which implies that (𝐴 + 𝜆𝐹)𝑜𝑆𝑥∗ = 𝑥∗ , and hence 𝐴𝑆𝑥∗ + 𝜆𝐹𝑆𝑥∗ = 𝑥∗ . Theorem 12. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 , and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Let 𝑆 : 𝐻 → 𝐻 be a mapping satisfying the following: (i) 𝑇(𝐷) ⊆ 𝐷 and 𝑆(𝐷) ⊆ 𝐷, (ii) 𝑆 is a nonlinear contraction, (iii) 𝑇 and 𝑆 are weakly isotone.

As an application of Corollaries 1.24 or 1.25 and 1.30 or 1.31 [7], we obtain the following results, respectively Theorem 13. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 , and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of weak noncompactness on 𝐻. Assume that 𝑆, 𝑇 : 𝐻 → 𝐻 are sequentially weakly continuous mappings satisfying the following: (i) 𝑇(𝐷) ⊆ 𝐷 and 𝑆(𝐷) ⊆ 𝐷, (ii) 𝑆 is 𝜓-condensing or 𝑆 is a nonlinear contraction, (iii) 𝑇 and 𝑆 are weakly isotone. Then there exists a unique point 𝑥∗ ∈ 𝐷 such that 𝐴𝑆𝑥∗ + 𝜆𝐹𝑆𝑥∗ = 𝑥∗ . Theorem 14. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 , and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of weak noncompactness on 𝐻. Assume that 𝑆, 𝑇 satisfy the following: (i) 𝑇 is a ww-compact mapping, (ii) 𝑆 is continuous ws-compact and 𝜓-condensing or 𝑆 is continuous ws-compact and nonlinear contraction, (iii) 𝑇 and 𝑆 are weakly isotone, (iv) 𝑇(𝐷) ⊆ 𝐷 and 𝑆(𝐷) ⊆ 𝐷. Then there exists a unique 𝑥∗ ∈ 𝐷 such that 𝐴𝑆𝑥∗ + 𝜆𝐹𝑆𝑥∗ = 𝑥∗ . Theorem 15. Let 𝐻, 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9 and 𝑇 = (𝐴 + 𝜆𝐹)−1 sequentially weakly continuous. Assume that 𝐷 is nonempty closed bounded convex subset of 𝐻 and 𝑆 : 𝐷 → 𝐻 is sequentially weakly continuous mapping satisfying the following: (i) 𝑆(𝐷) is relatively weakly compact, (ii) (𝑥 = 𝑇𝑥 + 𝑆𝑦; 𝑦 ∈ 𝐷) ⇒ 𝑥 ∈ 𝐷. Then there exists a unique point 𝑥∗ ∈ 𝐷 such that (𝐴 + 𝜆𝐹)𝑜(𝐼 − 𝑆)𝑥∗ = 𝑥∗ . Proof. By Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 : 𝐻 → 𝐻 is a strict contraction with contractive constant 𝛼 = 1/(𝑐 − 𝜆𝑀) < 1. Now Theorem 2.1 [6] implies that there exists an 𝑥∗ ∈ 𝐷 such that 𝑆𝑥∗ + 𝑇𝑥∗ = 𝑥∗ which implies that (𝐴 + 𝜆𝐹)𝑜(𝐼 − 𝑆)𝑥∗ = 𝑥∗ .


5

Theorem 16. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 , and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Let 𝑆 : 𝐻 → 𝐻 be a mapping satisfying the following: (i) 𝑆(𝐷) ⊆ 𝐷, (ii) 𝑆 is continuous and 𝜓-condensing or (𝑆 is a nonlinear contraction), (iii) 𝑃𝑇 and 𝑆 are weakly isotone, where 𝑃 is the proximity map on 𝐷. Then there exists 𝑥0 ∈ 𝐷 such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑥0 − 𝑇𝑥0 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑆𝑥0 − 𝑇𝑥0 󵄩󵄩󵄩 (31) = 𝑑 (𝑇𝑥0 , 𝐷) = 𝑑 (𝑇𝑥0 , 𝐼𝐷 (𝑥0 )) . More precisely, either (1) 𝑆 and 𝑇 have a common fixed point 𝑥0 ∈ 𝐷, or (2) there exists 𝑥0 ∈ 𝜕𝐷 with 󵄩 󵄩 󵄩 󵄩 0 < 󵄩󵄩󵄩𝑇𝑥0 − 𝑆𝑥0 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑇𝑥0 − 𝑥0 󵄩󵄩󵄩 = 𝑑 (𝑇𝑥0 , 𝐷) = 𝑑 (𝑇𝑥0 , 𝐼𝐷 (𝑥0 )) .

(32)

Proof. Let 𝑃 be the proximity map on 𝐷; that is, for each 𝑥 ∈ 𝐻, we have ‖𝑃𝑥 − 𝑥‖ = 𝑑(𝑥, 𝐷). It is well known that 𝑃 is nonexpansive in 𝐻. As 𝑇 is shrinking map, so 𝑃𝑇 : 𝐷 → 𝐷 is also shrinking mapping. By Theorem 4.1 (or 4.2), there exists 𝑥0 ∈ 𝑀 such that 𝑥0 = 𝑆𝑥0 = 𝑃𝑇𝑥0 . Thus we obtain, as in Theorem 4.1 [8] the desired conclusion. Following the proof of Corollary 4.5 [8] and using Theorem 16, we obtain the following common fixed point theorem. Theorem 17. Let 𝐴, 𝐹, 𝜆, 𝑀, and 𝑐 be as in Theorem 9, 𝑇 = (𝐴 + 𝜆𝐹)−1 , and let 𝐻 be an ordered Hilbert space. Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝜓 a set additive measure of noncompactness on 𝐻. Assume that 𝑆 : 𝐻 → 𝐻 is a mapping satisfying the following: (i) 𝑆(𝐷) ⊆ 𝐷, (ii) 𝑆 is continuous and 𝜓-condensing or (𝑆 is a nonlinear contraction), (iii) 𝑃𝑇 and 𝑆 are weakly isotone, where 𝑃 is the proximity map on 𝐷. Suppose that 𝑇 satisfies one of the following conditions for each 𝑥 ∈ 𝜕𝐷, with 𝑥 ≠𝑇𝑥: (i) ‖𝑇𝑥 − 𝑦‖ < ‖𝑇𝑥 − 𝑥‖ for some 𝑦 in 𝐼𝐷(𝑥); (ii) there is a 𝛾 such that |𝛾| < 1 and 𝛾𝑥+(1−𝛾)𝑇𝑥 ∈ 𝐼𝐷(𝑥); (iii) 𝑇𝑥 ∈ 𝐼𝐷(𝑥); (iv) for each 𝛾 ∈ (0, 1), 𝑥 ≠𝛾𝑇𝑥; (v) there exists an 𝛼 ∈ (1, ∞) such that, ‖𝑇𝑥 − 𝑥‖𝛼 ≥ ‖𝑇𝑥‖𝛼 − 𝑟𝛼 ; (vi) there exists 𝛽 ∈ (0, 1) such that, ‖𝑇𝑥 − 𝑥‖𝛽 ≤ ‖𝑇𝑥‖𝛽 − 𝑟𝛽 . Then 𝐹(𝑆) ∩ 𝐹(𝑇) ≠0.

4. An Application Fixed point theorems for certain operators have found various applications in differential and integral equations (see [7–10] and references therein). In this section, we present an application of our Theorem 7 to establish a solution of a nonlinear integral equation. Let 𝐾 : [0, 1] × [0, 1] → R be a continuous function, and suppose that 𝐾 is symmetric (i.e., 𝐾(𝑥, 𝑦) = 𝐾(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ [0, 1]). We consider the linear operator 𝐵 : 𝐿2R [0, 1] → 1 𝐿2R [0, 1] defined by 𝐵𝑢(𝑡) = ∫0 𝐾(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠. It is easy to observe that 𝐵 is a self-adjoint operator. Now let 𝐴 : 1 𝐿2R [0, 1] → 𝐿2R [0, 1] defined by 𝐴𝑢(𝑡) = 𝑢(𝑡) + ∬0 𝐾(𝑡, 𝑠) 𝐾(𝑠, 𝑧)𝑢(𝑧) 𝑑𝑧 𝑑𝑠; that is, 𝐴 = 𝐼 + 𝐵2 where 𝐼 is the identity of 𝐿2R [0, 1]. 𝐴 is a self-adjoint strongly positive operator satisfying ⟨𝐴𝑢, 𝑢⟩2 = ⟨𝑢, 𝑢⟩2 + ⟨𝐵2 𝑢, 𝑢⟩2 ≥ ‖𝑢‖22 (⟨⋅, ⋅⟩2 and ‖ ⋅ ‖2 signify the inner product and the norm in 𝐿2R [0, 1]). Define 𝑓 : [0, 1] × R → R, (𝑡, 𝑢) 󳨃→ 𝑓(𝑡, 𝑢), having partial derivative of first order in the second variable 𝑢 and |(𝜕𝑓/𝜕𝑢)(𝑡, 𝑢)| ≤ 𝑀 for all (𝑡, 𝑢) ∈ [0, 1] × R(𝑀 > 0). 𝐹 : 𝐿2R [0, 1] → 𝐿2R [0, 1] defined by 𝐹𝑢(𝑡) = 𝑓(𝑡, 𝑢(𝑡)) is a Lipschitz operator of constant 𝑀 from 𝐿2R [0, 1] into 𝐿2R [0, 1]. Define nonlinear integral equation 1

1

0

0

𝑢 (𝑡) + ∫ ∫ 𝐾 (𝑡, 𝑠) 𝐾 (𝑠, 𝑧) 𝑢 (𝑧) 𝑑𝑧 𝑑𝑠+ 𝜆𝑓 (𝑡, 𝑢 (𝑡)) = 𝑔 (𝑡) , 𝑔 ∈ 𝐿2R [0, 1] , (33) where 𝜆 ∈ (0; 1/𝑀). All of the conditions of Theorem 7 are satisfied, so the above-mentioned integral equation has a unique solution for all 𝜆 ∈ (0; 1/𝑀). Moreover, by Theorem 8 the solution depends continuously on the free term 𝑔.

Acknowledgment The work of first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCSUEFISCDI, project number PN-II-ID-PCE-2011-3-0087.

References [1] H. Amann, “On the unique solvability of semilinear operator equations in Hilbert spaces,” Journal de Mathématiques Pures et Appliquées, vol. 61, no. 2, pp. 149–175, 1982. [2] H. Amann, “Saddle points and multiple solutions of differential equations,” Mathematische Zeitschrift, vol. 169, no. 2, pp. 127– 166, 1979. [3] D. Teodorescu, “A contractive method for a semilinear equation in Hilbert spaces,” Analele Universitatii Bucuresti Matematica, vol. 54, no. 2, pp. 289–292, 2005. [4] J. Bana´s and J. Rivero, “On measures of weak noncompactness,” Annali di Matematica Pura ed Applicata, vol. 151, pp. 213–224, 1988. [5] F. S. de Blasi, “On a property of the unit sphere in a Banach space,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 21, no. 3-4, pp. 259–262, 1977.

6 [6] M. A. Taoudi, “Krasnosel’skii type fixed point theorems under weak topology features,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 478–482, 2010. [7] R. P. Agarwal, N. Hussain, and M.-A. Taoudi, “Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations,” Abstract and Applied Analysis, vol. 2012, Article ID 245872, 15 pages, 2012. [8] N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 475– 489, 2010. [9] N. Hussain and M. A. Taoudi, “Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations,” Fixed Point Theory and Applications, vol. 2013, article 196, 2013. [10] K. Latrach, M. A. Taoudi, and A. Zeghal, “Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations,” Journal of Differential Equations, vol. 221, no. 1, pp. 256–271, 2006.



Research Article 0 A Minimax Theorem for 𝐿 -Valued Functions on Random Normed Modules Shien Zhao1 and Yuan Zhao2 1 2

Elementary Educational College, Capital Normal University, Beijing 100048, China Department of Basic Sciences, Hebei Finance University, Baoding 071051, China

Correspondence should be addressed to Shien Zhao; [email protected] Received 9 April 2013; Accepted 25 July 2013 Academic Editor: Pei De Liu Copyright © 2013 S. Zhao and Y. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 0

We generalize the well-known minimax theorems to 𝐿 -valued functions on random normed modules. We first give some basic properties of an 𝐿0 -valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (𝜀, 𝜆)-topology and the locally 𝐿0 -convex topology. Then, we introduce the definition of random saddle points. Conditions for an 𝐿0 -valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, 0 using relations between the two kinds of topologies, establish the minimax theorem of 𝐿 -valued functions in the framework of random normed modules and random conjugate spaces.

1. Introduction The classical minimax theorem, which originated from game theory, is an important content of nonlinear analysis. It has been applied in many fields, such as optimization theory, different equations, and fixed point theory. The first mathematical formulation was established by Neumann in 1928 [1]. Since then, various generalizations of Neumann’s minimax theorem have been given by several scholars; see [2–7]. The classical minimax theorems for extended realvalued functions 𝐿 : 𝐴 × 𝐵 → 𝑅 show that, under some suitable conditions of compactness, convexity, and continuity, the equality inf sup𝑓 (𝑥, 𝑦) = sup inf 𝑓 (𝑥, 𝑦)

𝑦∈𝐵 𝑥∈𝐴

𝑥∈𝐴 𝑦∈𝐵

(1)

holds. In 1980s, to meet the needs of vectorial optimization, minimax problems in this more general setting have been investigated; see [4–7]. In this paper, we generalize the well0 known minimax theorems to 𝐿 -valued functions on random normed modules (briefly RN modules). Random metric theory is based on the idea of randomizing the classical space theory of functional analysis. All the basic notions such as RN modules and random inner

product modules (briefly RIP modules) and random locally convex modules (briefly RLC modules) together with their random conjugate spaces were naturally presented by Guo in the course of the development of random functional analysis, (cf. [8–12]). In the last ten years, random metric theory and its applications in the theory of conditional risk measures have undergone a systematic and deep development. Especially after 2009, in [13] Guo gives the relations between the basic results currently available derived from the two kinds of topologies, namely, the (𝜀, 𝜆)-topology and the locally 𝐿0 -convex topology. In [14], Guo gives some basic results on 𝐿0 -convex analysis together with some applications to conditional risk measures and studies the relations among the three kinds of conditional convex risk measures. Furthermore, in [15] Guo et al. establish a complete random convex analysis over RN modules and RLC modules by simultaneously considering the two kinds of topologies in order to provide a solid analytic foundation for the module approach to conditional risk measures. These results pave the way for further research of the theory of random convex analysis and conditional risk measures. Motivated by the recent applications of random metric theory to conditional risk measures [13, 16, 17], in this paper,

2

Journal of Function Spaces and Applications 0

𝐿0 (F, 𝐾) = the algebra of equivalence classes of 𝐾valued F-measurable random variables on (Ω, F, 𝑃);

we must employ the (𝜀, 𝜆)-topology also, because the (𝜀, 𝜆)topology is very natural from the viewpoint of probability theory, and under this type of topology we can use the relations between random normed modules and classical normed spaces to prove the main result; see the proof of Theorem 1 in Section 4. The remainder of this paper is organized as follows: in Section 2 we will briefly collect some necessary known facts; 0 in Section 3 we will give some basic properties of an 𝐿 -valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, Theorems 26 and 28; in Section 4 we will present the definition of random saddle points and prove our main result.

𝐿0 (F) = 𝐿0 (F, 𝑅);

2. Preliminaries

we establish a minimax theorem for 𝐿 -valued functions on random normed modules. Theorem 1, which is the main result of this paper, can be seen as a natural extension of the classical minimax theorems and has potential applications in the further study of conditional risk measures. To introduce the main result of this paper, let us first recall some notation and terminology as follows: 𝐾: the scalar field R of real numbers or C of complex numbers; (Ω, F, 𝑃): a probability space;

0

𝐿 (F) = the set of equivalence classes of extended real-valued F-measurable random variables on (Ω, F, 𝑃). Theorem 1. Let (𝐸, ‖ ⋅ ‖) be a random strictly convex and random reflexive random normed module over 𝑅 with base (Ω, F, 𝑃), 𝐴 and 𝐵 T𝑐 -closed, 𝐿0 (F)-convex subset with the 0 countable concatenation property of 𝐸 and 𝐿 : 𝐸×𝐸 → 𝐿 (F). If 𝐿 satisfies the following: 0

(1) for any fixed 𝑝 ∈ 𝐵, 𝐿(⋅, 𝑝) : 𝐸 → 𝐿 (F) is proper 𝐿0 (F)-convex, T𝑐 -lower semicontinuous function on 𝐴 and has the local property; 0

(2) for any fixed 𝑢 ∈ 𝐴, −𝐿(𝑢, ⋅) : 𝐸 → 𝐿 (F) is proper 𝐿0 (F)-convex, T𝑐 -lower semicontinuous function on 𝐵 and has the local property; (3) 𝐴 and 𝐵 are a.s. bounded, then there exists a random saddle point (𝑢0 , 𝑝0 ) ∈ 𝐴 × 𝐵 of 𝐿 with respect to 𝐴 × 𝐵, namely, ⋀ ⋁ 𝐿 (𝑢, 𝑝) = 𝐿 (𝑢0 , 𝑝0 ) = ⋁ ⋀ 𝐿 (𝑢, 𝑝) .

𝑢∈𝐴 𝑝∈𝐵

𝑝∈𝐵 𝑢∈𝐴

(2)

Theorem 1 has the same shape as the classical minimax theorems, and its proof follows a known pattern in [18], but it is not trivial since the complicated stratification structure in the random setting needs to be considered. Besides, the most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. In order to overcome this obstacle, we make full use of the respective advantages of the (𝜀, 𝜆)-topology and the locally 𝐿0 -convex topology. In [13], Guo pointed out that these two kinds of topologies can complement each other (see also Propositions 14 and 15 in this paper), and we can consider them simultaneously in some cases. Specifically, on one hand, in Theorem 1 we require the functions to be T𝑐 -lower semicontinuous; namely, the functions are lower semicontinuous under the locally 𝐿0 -convex topology, because we need a very important inequality to establish this theorem; see Definition 21 and Proposition 22 of this paper for details. On the other hand, in the process of the proof of Theorem 1

0

It is well known from [19] that 𝐿 (F) is a complete lattice under the ordering ≤: 𝜉 ≤ 𝜂 if and only if 𝜉0 (𝜔) ≤ 𝜂0 (𝜔), for almost all 𝜔 in Ω (briefly, a.s.), where 𝜉0 and 𝜂0 are arbitrarily chosen representatives of 𝜉 and 𝜂, respectively. 0 Furthermore, every subset 𝐺 of 𝐿 (F) has a supremum, denoted by ⋁ 𝐺, and an infimum, denoted by ⋀ 𝐺. Finally 0 𝐿0 (F), as a sublattice of 𝐿 (F), is also a complete lattice in the sense that every subset with upper bound has a supremum. 0 The pleasant properties of 𝐿 (F) are summarized as follows. 0

Proposition 2 (see [19]). For every subset 𝐺 of 𝐿 (F), there exist countable subsets {𝑎𝑛 | 𝑛 ∈ 𝑁} and {𝑏𝑛 | 𝑛 ∈ 𝑁} of 𝐺 such that ⋁ 𝐺 = ⋁𝑛≥1 𝑎𝑛 and ⋀ 𝐺 = ⋀𝑛≥1 𝑏𝑛 . Further, if 𝐺 is directed (dually directed) with respect to ≤, then the above {𝑎𝑛 | 𝑛 ∈ 𝑁} (accordingly, {𝑏𝑛 | 𝑛 ∈ 𝑁}) can be chosen as nondecreasing (correspondingly, nonincreasing) with respect to ≤. Specially, 𝐿0+ = {𝜉 ∈ 𝐿0 (F) | 𝜉 ≥ 0}, 𝐿0++ = {𝜉 ∈ 𝐿 (F) | 𝜉 > 0 on Ω}, where for 𝐴 ∈ F, “𝜉 > 𝜂” on 𝐴 means 𝜉0 (𝜔) > 𝜂0 (𝜔) a.s. on 𝐴 for any chosen representatives 𝜉0 and 𝜂0 of 𝜉 and 𝜂, respectively. As usual, 𝜉 > 𝜂 means 𝜉 ≥ 𝜂 and 𝜉 ≠𝜂. For any 𝐴 ∈ F, 𝐴𝑐 denotes the complement of 𝐴, ̃ = {𝐵 ∈ F | 𝑃(𝐴Δ𝐵) = 0} denotes the equivalence and 𝐴 class of 𝐴, where Δ is the symmetric difference operation, 𝐼𝐴 is the characteristic function of 𝐴, and 𝐼̃𝐴 is used to denote the equivalence class of 𝐼𝐴; given two 𝜉 and 𝜂 in 𝐿0 (F), and 𝐴 = {𝜔 ∈ Ω | 𝜉0 ≠𝜂0 }, where 𝜉0 and 𝜂0 are arbitrarily chosen representatives of 𝜉 and 𝜂 respectively, then we always write ̃ [𝜉 ≠𝜂] for the equivalence class of 𝐴 and 𝐼[𝜉 ≠ 𝜂] for 𝐼𝐴 ; one can also understand the implication of such notation as 𝐼[𝜉≤𝜂] , 𝐼[𝜉 1 − 𝜆}; then {𝑁𝜃 (𝜀, 𝜆) | 𝜀 > 0, 0 < 𝜆 < 1} is easily verified to be a local base at the null vector 𝜃 of some Hausdorff linear topology. The linear topology is called the (𝜀, 𝜆)-topology for 𝐸 induced by ‖ ⋅ ‖. From now on, the (𝜀, 𝜆)-topology for each RN module is always denoted by T𝜀,𝜆 when no confusion occurs. Proposition 9 (see [12–14]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃). Then one has the following statements. (1) The (𝜀, 𝜆)-topology for 𝐿0 (F, 𝐾) is exactly the topology of convergence in probability 𝑃, and (𝐿0 (F, 𝐾), T𝜀,𝜆 ) is a topological algebra over 𝐾. (2) If (𝐸, ‖ ⋅ ‖) is an RN modules, then (𝐸, T𝜀,𝜆 ) is a topological module over the topological algebra 𝐿0 (𝐹, 𝐾). (3) A net {𝑥𝛿 , 𝛿 ∈ Γ} converges in the (𝜀, 𝜆)-topology to some 𝑥 in 𝐸 if and only if {‖𝑥𝛿 − 𝑥‖, 𝛿 ∈ Γ} converges in probability 𝑃 to 0. The following locally 𝐿0 -convex topology is easily seen to be much stronger than the (𝜀, 𝜆)-topology and was first introduced by Filipović et al. in [16].

4


Definition 10 (see [14, 16]). Let (𝐸, ‖ ⋅ ‖) be an RN module over 𝐾 with base (Ω, F, 𝑃). For any 𝜀 ∈ 𝐿0++ , let 𝑁𝜃 (𝜀) = {𝑥 ∈ 𝐸 | ‖𝑥‖ ≤ 𝜀}. A subset 𝐺 of 𝐸 is called T𝑐 -open if for each 𝑥 ∈ 𝐺 there exists some 𝑁𝜃 (𝜀) such that 𝑥 + 𝑁𝜃 (𝜀) ⊂ 𝐺, and T𝑐 denotes the family of T𝑐 -open subsets of 𝐸. Then it is easy to see that (𝐸, T𝑐 ) is a Hausdorff topological group with respect to the addition on 𝐸. T𝑐 is called the locally 𝐿0 -convex topology for 𝐸 induced by ‖ ⋅ ‖. 0

From now on, the locally 𝐿 -convex topology for each random locally convex space is always denoted by T𝑐 when no confusion occurs.

concatenation Σ𝑛≥1 𝐼̃𝐴 𝑛 𝑥𝑛 is well defined or Σ𝑛≥1 𝐼̃𝐴 𝑛 𝑥𝑛 ∈ 𝐸 if there is 𝑥 ∈ 𝐸 such that 𝐼̃𝐴 𝑛 𝑥 = 𝐼̃𝐴 𝑛 𝑥𝑛 , for all 𝑛 ∈ 𝑁. A subset 𝐺 of 𝐸 is said to have the countable concatenation property if every countable concatenation Σ𝑛≥1 𝐼̃𝐴 𝑛 𝑥𝑛 with 𝑥𝑛 ∈ 𝐺 for each 𝑛 ∈ 𝑁 still belongs to 𝐺; namely, Σ𝑛≥1 𝐼̃𝐴 𝑛 𝑥𝑛 is well defined and there exists 𝑥 ∈ 𝐺 such that 𝑥 = Σ𝑛≥1 𝐼̃𝐴 𝑛 𝑥𝑛 . Proposition 14 (see [13]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃). Then 𝐸 is T𝜀,𝜆 -complete if and only if E is T𝑐 -complete and has the countable concatenation property.

Proposition 11 (see [13, 14, 16]). Let (𝐸, ‖⋅‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃). Then

Proposition 15 (see [13]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃) and 𝐺 ⊂ 𝐸 a subset with the countable concatenation property. Then 𝐺𝜀,𝜆 = 𝐺𝑐 .

(1) 𝐿0 (F, 𝐾) is a topological ring endowed with its locally 𝐿0 -convex topology;

Now, we introduce the definition of random conjugate spaces of RN modules.

(2) 𝐸 is a topological module over the topological ring 𝐿0 (F, 𝐾) when 𝐸 and 𝐿0 (F, 𝐾) are endowed with their respective locally 𝐿0 -convex topologies; (3) a net {𝑥𝛼 | 𝛼 ∈ Γ} in 𝐸 converges in the locally 𝐿0 convex topology to 𝑥 ∈ 𝐸 if and only if {‖𝑥𝛼 −𝑥‖ | 𝛼 ∈ Γ} converges in the locally 𝐿0 -convex topology of 𝐿0 (F, 𝐾) to 0. T𝑐 is called locally 𝐿0 -convex because it has a striking local base U𝜃 = {𝐵(𝜀) | 𝜀 ∈ 𝐿0++ }, each member 𝑈 of which is as follows: (i) 𝐿0 -convex: 𝜉 ⋅ 𝑥 + (1 − 𝜉) ⋅ 𝑦 ∈ 𝑈 for any 𝑥, 𝑦 ∈ 𝑈 and 𝜉 ∈ 𝐿0+ such that 0 ⩽ 𝜉 ⩽ 1; (ii) 𝐿0 -absorbent: there is 𝜉 ∈ 𝐿0++ for each 𝑥 ∈ 𝐸 such that 𝑥 ∈ 𝜉 ⋅ 𝑈; (iii) 𝐿0 -balanced: 𝜉 ⋅ 𝑥 ∈ 𝑈 for any 𝑥 ∈ 𝑈 and any 𝜉 ∈ 𝐿0 (𝐹, 𝐾) such that |𝜉| ⩽ 1. Remark 12. Let (𝐸, ‖ ⋅ ‖) be an RN module over 𝐾 with base (Ω, F, 𝑃) endowed with the locally 𝐿0 -convex topology T𝑐 . Although 𝐸 can be viewed as a linear space over 𝐾 with scalar multiplication 𝛼 ⋅ 𝑥 := (𝛼 ⋅ 1) ⋅ 𝑥 for 𝛼 ∈ 𝐾, 1 ∈ 𝐿0 and 𝑥 ∈ 𝐸, (𝐸, T𝑐 ) is not a topological linear space since the map 𝐾 → (𝐸, T𝑐 ), 𝛼 → 𝛼 ⋅ 𝑥, is not necessarily continuous for 𝑥 ≠𝜃; see [16] for details. In the sequel of this paper, for a subset 𝐺 of an RN module (𝐸, ‖ ⋅ ‖), 𝐺𝜀,𝜆 denotes the T𝜀,𝜆 -closure of 𝐺, and 𝐺𝑐 denotes the T𝑐 -closure of 𝐺. For giving the relations of the two kinds of topologies, which Guo has studied the [13], we need to introduce the definition of the countable concatenation property. Definition 13 (see [13]). Let 𝐸 be a left module over the algebra 𝐿0 (F, 𝐾). A formal sum Σ𝑛≥1 𝐼̃𝐴 𝑛 𝑥𝑛 for some countable partition {𝐴 𝑛 , 𝑛 ∈ 𝑁} of Ω to F and some sequence {𝑥𝑛 | 𝑛 ∈ 𝑁} in 𝐸 is called a countable concatenation of {𝑥𝑛 | 𝑛 ∈ 𝑁} with respect to {𝐴 𝑛 , 𝑛 ∈ 𝑁}. Furthermore a countable

Definition 16 (see [7, 10, 11, 13]). Let (𝐸, ‖ ⋅ ‖) be an RN module over 𝐾 with base (Ω, F, 𝑃). A linear operator 𝑓 from 𝐸 to 𝐿0 (F, 𝐾) is said to be an a.s. bounded random linear functional on 𝐸 if there exists some 𝜉 in 𝐿0+ (F, 𝑅) such that |𝑓(𝑥)| ≤ 𝜉 ⋅ ‖𝑥‖, for all 𝑥 ∈ 𝐸. Denote by 𝐸∗ the linear space of a.s. bounded random linear functionals on 𝐸 with the pointwise addition and scalar multiplication on linear operators; define ‖ ⋅ ‖∗ : 𝐸∗ → 𝐿0+ (F, 𝑅) by ‖𝑓‖∗ = ⋀{𝜉 ∈ 𝐿0+ (F) | |𝑓(𝑥)| ≤ 𝜉‖𝑥‖, for all 𝑥 ∈ 𝐸} for all 𝑓 ∈ 𝐸∗ and define ⋅ : 𝐿0 (F, 𝐾) × 𝐸∗ → 𝐸∗ by (𝜂 ⋅ 𝑓)(𝑥) = 𝜂(𝑓(𝑥)) for all 𝜂 ∈ 𝐿0 (F, 𝐾), 𝑓 ∈ 𝐸∗ , and 𝑥 ∈ 𝐸; then it is easy to check that (𝐸∗ , ‖ ⋅ ‖∗ ) is also an RN module over 𝐾 with base (Ω, F, 𝑃), called the random conjugate space of (𝐸, ‖ ⋅ ‖). Guo et al. gave the topological characterizations of an a.s. bounded random linear functional in [10, 11, 16] as follows: let (𝐸, ‖ ⋅ ‖) be an RN module over 𝐾 with base ∗ the 𝐿0 (F, 𝐾)-module of continuous module (Ω, F, 𝑃), 𝐸𝜀,𝜆 homomorphisms from (𝐸, T𝜀,𝜆 ) to (𝐿0 (F, 𝐾), T𝜀,𝜆 ), and 𝐸𝑐∗ the 𝐿0 (𝐹, 𝐾)-module of continuous module homomorphisms ∗ = from (𝐸, T𝑐 ) to (𝐿0 (𝐹, 𝐾), T𝑐 ), then it was proved that 𝐸𝜀,𝜆 ∗ ∗ 𝐸𝑐 . In fact, Guo et al. also proved in [10, 11, 13] ‖𝑓‖ = ⋁{|𝑓(𝑥)| | 𝑥 ∈ 𝐸 and ‖𝑥‖ ≤ 1} for any 𝑓 ∈ 𝐸. Let (𝐸, ‖ ⋅ ‖) be an RN module, 𝐸∗∗ denotes (𝐸∗ )∗ , and the canonical embedding mapping 𝐽 : 𝐸 → 𝐸∗∗ defined by (𝐽𝑥)(𝑓) = 𝑓(𝑥), for all 𝑥 ∈ 𝐸 and for all 𝑓 ∈ 𝐸∗ , is random-norm preserving. If 𝐽 is subjective, then 𝐸 is called random reflexive. In [13] Guo proved that the random reflexivity is independent of a special choice of T𝜀,𝜆 and T𝑐 . The following propositions are very essential relations, which are established by Guo in [12, 21], between classical reflexive spaces and random reflexive RN modules. Proposition 17 (see [21]). 𝐿0 (F, 𝐵) is random reflexive if and only if 𝐵 is a reflexive Banach space. Proposition 18 (see [12]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃). Then 𝐸 is random reflexive if and only if (𝐿𝑝 (𝐸), ‖ ⋅ ‖𝑝 ) is reflexive, where 1 < 𝑝 < +∞.


5

Proposition 19 (see [12]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃), 1 ≤ 𝑝 < +∞ and 1 < 𝑞 ≤ +∞ a pair of Hölder conjugate numbers. Then (𝐿𝑞 (𝐸∗ ), ‖ ⋅ ‖𝑞 ) is isometrically isomorphic with the classical conjugate space of (𝐿𝑝 (𝐸), ‖ ⋅ ‖), denoted by (𝐿𝑝 (𝐸))󸀠 , under the canonical mapping 𝑇 : 𝐿𝑞 (𝐸∗ ) → (𝐿𝑝 (𝐸))󸀠 defined as follows. For each 𝑓 ∈ 𝐿𝑞 (𝐸∗ ), 𝑇𝑓 (denoting 𝑇(𝑓)): 𝐿𝑝 (𝐸) → 𝐾 is defined by 𝑇𝑓 (𝑔) = ∫Ω 𝑓(𝑔)𝑑𝑃 for all 𝑔 ∈ 𝐿𝑝 (𝐸). 0

3. Some Basic Properties of 𝐿 -Valued Lower Semicontinuous Functions 0

In this section, we give some basic properties of 𝐿 -valued lower semicontinuous functions. First, we recall the defini0 tion of 𝐿 -valued lower semicontinuous functions under two kinds of topologies, which was presented by Guo in [14] for the first time. Let 𝐸 be a left module over the algebra 𝐿0 (F). The 0 effective domain of function 𝑓 : 𝐸 → 𝐿 (F) is denoted by 0 dom(𝑓) := {𝑥 ∈ 𝐸 | 𝑓(𝑥) ∈ 𝐿 (F)}. The epigraph of 𝑓 is denoted by epi(𝑓) := {(𝑥, 𝑦) ∈ 𝐸 × 𝐿0 (F) | 𝑓(𝑥) ≤ 𝑦}. The function 𝑓 is called proper if 𝑓(𝑥) > −∞ on Ω for every 𝑥 ∈ 𝐸 and dom(𝑓) ≠0. Definition 20 (see [14]). Let 𝐸 be a left module over the alge0 bra 𝐿0 (F) and 𝑓 : 𝐸 → 𝐿 (F). (1) 𝑓 is 𝐿0 (F)-convex if 𝑓(𝜉𝑥 + (1 − 𝜉)𝑦) ≤ 𝜉𝑓(𝑥) + (1 − 𝜉)𝑓(𝑦) for all 𝑥 and 𝑦 in 𝐸 and 𝜉 ∈ 𝐿0+ such that 0 ≤ 𝜉 ≤ 1 (here we make the convention that 0 ⋅ (±∞) = 0 and ∞ − ∞ = ∞).

(3) lim𝛼 𝑓(𝑥𝛼 ) ≥ 𝑓(𝑥0 ) for each 𝑥0 ∈ 𝐸 and each net {𝑥𝛼 , 𝛼 ∈ Γ} in 𝐸 such that {𝑥𝛼 , 𝛼 ∈ Γ} is T𝑐 -convergent to 𝑥0 , where lim𝛼 𝑓(𝑥𝛼 ) = ⋁𝛼∈Γ (⋀𝛽≥𝛼 𝑓(𝑥𝛽 )). Remark 23. Proposition 22 first occurred in [16] where the countable concatenation property of 𝐸 was not assumed, but this condition should be added (see [14] for details). For T𝜀,𝜆 -lower semicontinuous functions, we only have the following proposition. Proposition 24 (see [14]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 0 𝑅 with base (Ω, F, 𝑃) and 𝑓 : 𝐸 → 𝐿 (F) a function. Then one has the following statements: (1) 𝑓 is T𝜀,𝜆 -lower semicontinuous if lim𝛼 𝑓(𝑥𝛼 ) ≥ 𝑓(𝑥0 ) for each 𝑥0 ∈ 𝐸 and each net {𝑥𝛼 , 𝑥 ∈ Γ} in 𝐸 such that {𝑥𝛼 , 𝛼 ∈ Γ} is T𝜀,𝜆 -convergent to 𝑥0 ; (2) {𝑥 ∈ 𝐸 | 𝑓(𝑥) ≤ 𝑟} is T𝜀,𝜆 -closed for each 𝑟 ∈ 𝐿0 (F) if 𝑓 is T𝜀,𝜆 -lower semicontinuous. If we define 𝑓 to be lower semicontinuous via “lim𝛼 𝑓(𝑥𝛼 ) ≥ 𝑓(𝑥) for all net {𝑥𝛼 , 𝛼 ∈ Λ} in 𝐸 such that it converges in the (𝜀, 𝜆)-topology to some 𝑥 ∈ 𝐸”, the notion is, however, meaningless in the random setting, since we can construct an RN module 𝐸 and a T𝜀,𝜆 -continuous 𝐿0 -convex function 𝑓 from 𝐸 to 𝐿0 (F), whereas 𝑓 is not a T𝜀,𝜆 -lower semicontinuous function. Hence, we cannot use this inequality for T𝜀,𝜆 -lower semicontinuous functions. Since this inequality is very important for the proof of Theorem 1 (see Section 4 for details), we can only establish 0 𝐿 -valued minimax theorems for T𝑐 -lower semicontinuous functions.

(3) 𝑓 is regular if 𝐼̃𝐴𝑓(𝑥) = 𝑓(𝐼̃𝐴𝑥) for all 𝑥 ∈ 𝐸 and 𝐴 ∈ F.

Proposition 25 (see [14]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝐾 with base (Ω, F, 𝑃) such that 𝐸 has the countable 0 concatenation property and 𝑓 : 𝐸 → 𝐿 (F) a function with the local property. Then 𝑓 is T𝜀,𝜆 -lower semicontinuous if and only if 𝑓 is T𝑐 -lower semicontinuous, specially this is true for an 𝐿0 (F)-convex function 𝑓.

Definition 21 (see [14]). Let (𝐸, ‖ ⋅ ‖) be an RN module 0 over 𝑅 with base (Ω, F, 𝑃). A function 𝑓 : 𝐸 → 𝐿 (F) is called T𝑐 -lower semicontinuous if epi(𝑓) is closed in 0 (𝐸, T𝑐 ) × (𝐿0 (F), T𝑐 ). A function 𝑓 : 𝐸 → 𝐿 (F) is called T𝜀,𝜆 -lower semicontinuous if epi(𝑓) is closed in (𝐸, T𝜀,𝜆 ) × (𝐿0 (F), T𝜀,𝜆 ).

Now, we give some important properties of 𝐿 -valued lower semicontinuous functions on RN modules. To pave the way for Theorem 26, we first introduce some notation: if (𝐸, ‖ ⋅ ‖) is an RN module over 𝐾 with base (Ω, F, 𝑃), Π denotes the set of all probability measures equivalent to 𝑝 𝑃 on (Ω, F), 𝐿 𝑄(𝐸) = {𝑥 ∈ 𝐸 | ∫Ω ‖𝑥‖𝑝 𝑑𝑄 < +∞},

(2) 𝑓 has the local property if 𝐼̃𝐴𝑓(𝑥) = 𝐼̃𝐴𝑓(𝐼̃𝐴𝑥) for all 𝑥 ∈ 𝐸 and 𝐴 ∈ F.

Proposition 22 (see [14]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝑅 with base (Ω, F, 𝑃) such that 𝐸 has the countable 0 concatenation property and 𝑓 : 𝐸 → 𝐿 (F) a function with the local property. Then the following are equivalent to each other: (1) 𝑓 is T𝑐 -lower semicontinuous; (2) {𝑥 ∈ 𝐸 | 𝑓(𝑥) ≤ 𝑟} is T𝑐 -closed for each 𝑟 ∈ 𝐿0 (F);

0

𝑝

where 𝑄 ∈ Π, and ‖ ⋅ ‖𝑄 𝑝 denotes the norm on 𝐿 𝑄(𝐸), namely, 𝑝

𝑝 1/𝑝 for any 𝑥 ∈ 𝐿 𝑄(𝐸). ‖𝑥‖𝑄 𝑝 = (∫Ω ‖𝑥‖ 𝑑𝑄)

Theorem 26. Let (𝐸, ‖ ⋅ ‖) be a random reflexive 𝑅𝑁 module over 𝑅 with base (Ω, F, 𝑃), 𝐺 ⊂ 𝐸 a T𝑐 -closed, 𝐿0 (F)-convex, and a.s. bounded set with the countable concatenation property 0

and 𝑓 : 𝐸 → 𝐿 (F) a T𝑐 -lower semicontinuous function with the local property. If 𝑓|𝐺 is proper, then 𝑓 is bounded from below on 𝐺.

6


Proof. Let 𝜂 = ∨{‖𝑥‖ : 𝑥 ∈ 𝐺}; then it is easy to see that 𝜂 ∈ 𝐿0+ . If 𝑓 is not bounded from below on 𝐺, then there exists 𝐵 ∈ F such that 𝑃(𝐵) > 0 and ⋀ {𝑓 (𝑥) | 𝑥 ∈ 𝐺} = −∞

𝑗

𝑓𝑗 (𝐼̃𝐵𝑗 𝑥) = 𝐼̃𝐵𝑗 𝑓 (𝑥) ,

(6)

on 𝐵. Since 𝐺 is 𝐿0 (F)-convex and 𝑓 has the local property, it is easy to see that {𝑓(𝑥) | 𝑥 ∈ 𝐺} is directed. Hence there exists a sequence {𝑥𝑛 , 𝑛 ∈ 𝑁} such that {𝑓(𝑥𝑛 ), 𝑛 ∈ 𝑁} ↘ ⋀{𝑓(𝑥) | 𝑥 ∈ 𝐺}. Let 𝐺𝑛 = {𝑥 ∈ 𝐸 | 𝑓(𝑥) ≤ 𝑓(𝑥𝑛 )} ⋂ 𝐺; then it is clear that 𝐺𝑛 is T𝑐 -closed by Definition 21. Since 𝑓 has the local property and 𝐺 has the countable concatenation property, for any 𝑛 ∈ 𝑁, we have that 𝐺𝑛 has the countable concatenation property and 𝐺𝑛 is T𝜀,𝜆 closed by Proposition 15. By the fact that 𝐺 is a.s. bounded, we can define a probability measure 𝑄 on (Ω, F) by 𝑑𝑄/𝑑𝑃 = 1/𝑐(1 + 𝜂)2 , where 𝑐 = 𝐸[1/(1 + 𝜂)2 ]. Then 𝑄 is equivalent to 𝑃 and ∫Ω ‖𝑥‖2 𝑑𝑄 ≤ ∫Ω (𝜂2 /𝑐(1 + 𝜂)2 )𝑑𝑃 < +∞, for any 𝑥 ∈ 𝐺, which means that 𝐺 is bounded in (𝐿2𝑄(𝐸), ‖ ⋅ ‖𝑄 2 ). Noting that replacing the probability measure 𝑃 of the base space (Ω, F, 𝑃) with a probability measure 𝑄 does not change the (𝜀, 𝜆)-topology of 𝐸, for any given 𝑛 ∈ 𝑁, we can obtain that 𝐺𝑛 is norm-closed and convex in (𝐿2𝑄(𝐸), ‖ ⋅ ‖𝑄 2 ). Since (𝐸, ‖ ⋅ ‖) is a random reflexive RN module, we have that (𝐿2𝑄(𝐸), ‖ ⋅ ‖𝑄 2 ) is reflexive normed space from Proposition 18. Hence 𝐺𝑛 is compact under the weak topology of the normed space 𝐿2𝑄(𝐸). Let O = {𝐺𝑛 | 𝑛 ∈ 𝑁}, then one can obtain that O has the finite intersection property and ⋂ O ≠0. Let 𝑥∗ ∈ ⋂ O; then 𝑓 (𝑥∗ ) = −∞

partition of Ω to F. For any 𝑗 ∈ 𝑁; define a function 𝑓𝑗 : 0 𝐼̃𝐵 ⋅ 𝐸 → 𝐿 (F) as follows: ∀𝑥 ∈ 𝐸.

(8)

Since 𝑓 has the local property, for any 𝑗 ∈ 𝑁, we have that 𝑓𝑗 (𝐼̃𝐵 𝑥) = 𝐼̃𝐵 𝑓(𝐼̃𝐵 𝑥) and 𝐼̃𝐵 ⋅ ⋀ 𝑓(𝐺) = ⋀ 𝑓𝑗 (𝐼̃𝐵 ⋅ 𝐺). 𝑗

𝑗

𝑗

𝑗

𝑗

Because 𝑓|𝐺 is proper and 𝑓 is T𝑐 -lower semicontinuous and 𝐿0 (F)-convex, it is clear that 𝑓𝑗 |𝐼̃ ⋅𝐺 is proper, T𝑐 -lower 𝐵𝑗

semicontinuous, and 𝐿0 (F)-convex. Next, we prove that 𝑓𝑗 has the local property. We need only to prove that 𝐼̃𝐴𝑓𝑗 (𝐼̃𝐴𝐼̃𝐵𝑗 𝑥) = 𝐼̃𝐴 𝑓𝑗 (𝐼̃𝐵𝑗 𝑥) ,

∀𝐴 ∈ F ⋂ 𝐵𝑗 .

(9)

In fact, since 𝑓 has the local property, for any 𝐴 ∈ F ⋂ 𝐵𝑗 , we have that 𝐼̃𝐴𝑓𝑗 (𝐼̃𝐵𝑗 𝑥) = 𝐼̃𝐴𝐼̃𝐵𝑗 𝑓 (𝑥) = 𝐼̃𝐴 𝐼̃𝐵𝑗 𝑓 (𝐼̃𝐵𝑗 𝑥) = 𝐼̃𝐴𝐼̃𝐵𝑗 𝑓 (𝐼̃𝐴𝐼̃𝐵𝑗 𝑥)

(10)

= 𝐼̃𝐴𝑓𝑗 (𝐼̃𝐴 𝐼̃𝐵𝑗 𝑥) . Let 𝜂𝑗 = 𝐼̃𝐵𝑗 ⋅ 𝜂 for any 𝑗 ∈ 𝑁. It is easy to see that 𝐼̃𝑗 𝐺 ⊆ (𝐿2 (𝐸), ‖ ⋅ ‖2 ) is a bounded and convex set. Since 𝐼̃𝐵𝑗 𝐺 is 𝐿0 (F)-convex, T𝑐 -closed, and a.s. bounded in 𝐸 and has the countable concatenation, we can obtain that 𝐼̃𝐵𝑗 𝐺 is T𝜀,𝜆 closed in 𝐸 by Proposition 15. It is easy to see that 𝐼̃𝐵 𝐺 is 𝑗

(7)

on 𝐵, which it contradicts to the fact that 𝑓|𝐺 is proper. For giving Theorem 28, we need to introduce the following Proposition 27, which was established by Guo and Yang in 0 [22] for studying Ekeland’s variational principle for 𝐿 -valued functions on RN modules. Proposition 27 (see [22]). Let (𝐸, ‖ ⋅ ‖) be an 𝑅𝑁 module over 𝑅 with base (Ω, F, 𝑃), 𝐺 ⊂ 𝐸 a subset with the countable 0 concatenation property and 𝑓 : 𝐸 → 𝐿 (F) have the local property. If 𝑓|𝐺 is proper and bounded from below on 𝐺 (resp., bounded from above on 𝐺), then for each 𝜀 ∈ 𝐿0++ (F), there exists 𝑥𝜀 ∈ 𝐺 such that 𝑓(𝑥𝜀 ) ≤ ⋀ 𝑓(𝐺) + 𝜀 (accordingly, 𝑓(𝑥𝜀 ) ≥ ⋁ 𝑓(𝐺) − 𝜀). Theorem 28. Let (𝐸, ‖ ⋅ ‖) be a random reflexive 𝑅𝑁 module over 𝑅 with base (Ω, F, 𝑃), 𝐺 ⊂ 𝐸 a T𝑐 -closed, 𝐿0 (F)-convex and a.s. bounded set with the countable concatenation property 0 and 𝑓 : 𝐸 → 𝐿 (F) a T𝑐 -lower semicontinuous and 𝐿0 (F)convex function with the local property. If 𝑓|𝐺 is proper, then there exists 𝑥∗ ∈ 𝐺 such that 𝑓(𝑥∗ ) = ⋀ 𝑓(𝐺). Proof. Let 𝜉 = ⋁{‖𝑥‖ | 𝑥 ∈ 𝐺} and 𝜂 = ⋀ 𝑓(𝐺). It is clear that 𝜉 ∈ 𝐿0+ and 𝜂 ∈ 𝐿0 (F) by Theorem 26. Take 𝐵𝑗 = [𝑗 − 1 ≤ 𝜉 < 𝑗], for all 𝑗 ∈ 𝑁, then {𝐵𝑗 | 𝑗 = 1, 2, . . .} is a countable

convex and ‖ ⋅ ‖2 -closed in (𝐿2 (𝐸), ‖ ⋅ ‖2 ) from the fact that the topology induced by ‖ ⋅ ‖2 is stronger than the (𝜀, 𝜆)topology. Since (𝐸, ‖ ⋅ ‖) is random reflexive, (𝐿2 (𝐸), ‖ ⋅ ‖2 ) is reflexive normed space, and 𝐼̃𝐵𝑗 𝐺 is compact in 𝐿2 (𝐸) under the weak topology of 𝐿2 (𝐸). For any 𝜀 ∈ 𝐿0++ , define 𝐺𝑗 (𝜀) = {𝐼̃𝐵𝑗 𝑥 | 𝑓𝑗 (𝐼̃𝐵𝑗 𝑥) ≤ 𝐼̃𝐵𝑗 𝜂 + 𝜀, 𝑥 ∈ 𝐺}. It is clear that 𝐺𝑗 (𝜀) ≠0 by Proposition 27. Since 𝑓𝑗 |𝐼̃ ⋅𝐺 is T𝑐 -lower 𝐵𝑗

semicontinuous, we have that 𝐺𝑗 (𝜀) is T𝑐 -closed. Thus, we have that 𝐺𝑗 (𝜀) is 𝐿0 (F)-convex and ‖ ⋅ ‖2 -closed in 𝐿2 (𝐸) by the fact that 𝑓𝑗 has the local property and Proposition 15. By Hahn-Banach theorem, we have that 𝐺𝑗 (𝜀) is closed under the weak topology of 𝐿2 (𝐸). Take O = {𝐺𝑗 (𝜀) | 𝜀 ∈ 𝐿0++ } ,

(11)

it is easy to prove that O has the finite intersection property. Since 𝐼̃𝐵𝑗 𝐺 is compact under the weak topology of 𝐿2 (𝐸), we have that ⋂ O ≠0. Let 𝑥𝑗 ∈ ⋂ O for any 𝑗 ∈ 𝑁 and ∞

𝑥∗ = ∑𝐼̃𝐵𝑗 ⋅ 𝑥𝑗 . 𝑗=1

(12)

We have that 𝑓𝑗 (𝑥𝑗 ) = 𝜂𝑗 and 𝑓 (𝑥∗ ) = ⋀ 𝑓 (𝐺) . This completes the proof.

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7

Definition 29. Let (𝐸, ‖ ⋅ ‖) be an RN module over 𝑅 with base (Ω, F, 𝑃), and 𝐺 a 𝐿0 (F)-convex subset in 𝐸. 𝑓 : 𝐺 → 0 𝐿 (F) is called strictly 𝐿0 (F)-convex if 𝑓 (𝛼𝑥 + (1 − 𝛼) 𝑦) < 𝛼𝑓 (𝑥) + (1 − 𝛼) 𝑓 (𝑦) ,

(14)

for any 𝑥, 𝑦 ∈ 𝐺, 𝑥 ≠𝑦 and 0 < 𝛼 < 1 on Ω. Corollary 30. Let (𝐸, ‖ ⋅ ‖) be a random reflexive 𝑅𝑁 module over 𝑅 with base (Ω, F, 𝑃), 𝐺 ⊂ 𝐸 a T𝑐 -closed, 𝐿0 (F)-convex and a.s. bounded set with the countable concatenation property, 0 and 𝑓 : 𝐸 → 𝐿 (F) a T𝑐 -lower semicontinuous and strictly 𝐿0 (F)-convex function with the local property. If 𝑓|𝐺 is proper, then there exists an unique 𝑥∗ ∈ 𝐺 such that 𝑓(𝑥∗ ) = ⋀ 𝑓(𝐺).

4. Main Results Now, we give the definition of random saddle points. Definition 31. Let 𝐴 and 𝐵 be any two nonempty sets, 0 (𝑢0 , 𝑝0 ) ∈ 𝐴 × 𝐵 and 𝐿 : 𝐴 × 𝐵 → 𝐿 (F). Then (𝑢0 , 𝑝0 ) is called a random saddle point of 𝑓 with respect to 𝐴 × 𝐵 if 𝐿 (𝑢0 , 𝑝) ≤ 𝐿 (𝑢0 , 𝑝0 ) ≤ 𝐿 (𝑢, 𝑝0 ) ,

∀𝑢 ∈ 𝐴, 𝑝 ∈ 𝐵. (15)

Remark 32. Let 𝐴 and 𝐵 be any two nonempty sets, 𝐿 : 𝐴 × 0 𝐵 → 𝐿 (F). It is easy to see that the following statements are equivalent: (1) (𝑢0 , 𝑝0 ) ∈ 𝐴 × 𝐵 is a random saddle point of 𝑓 with respect to 𝐴 × 𝐵; (2) ⋀𝑢∈𝐴⋁𝑝∈𝐵 𝐿(𝑢, 𝑝) ≤ ⋁𝑝∈𝐵 ⋀𝑢∈𝐴𝐿(𝑢, 𝑝); (3) ⋀𝑢∈𝐴⋁𝑝∈𝐵 𝐿(𝑢, 𝑝) = 𝐿(𝑢0 , 𝑝0 ) = ⋁𝑝∈𝐵 ⋀𝑢∈𝐴𝐿(𝑢, 𝑝). Before giving the proof of main result in this paper, we first recall the definition of random strictly convex RN module, which is presented by Guo and Zeng in [23] for the first time. Definition 33 (see [23]). An RN module (𝐸, ‖ ⋅ ‖) is said to be random strictly convex if for any 𝑥 and 𝑦 ∈ 𝐸 \ {𝜃} such that ‖𝑥 + 𝑦‖ = ‖𝑥‖ + ‖𝑦‖, then there exist 𝐴 ∈ F and 𝜉 ∈ 𝐿0+ such that 𝑃(𝐴) > 0, 𝜉 > 0 on 𝐴 and 𝐼̃𝐴𝑥 = 𝜉(𝐼̃𝐴𝑦). Definition 34 (see [24]). Let (𝐸, ‖ ⋅ ‖) be an RN module over 𝑅 with base (Ω, F, 𝑃), 𝑥, 𝑦 ∈ 𝐸 and 𝐹 ∈ F. Then 𝑥 and 𝑦 are called 𝐿0 -independent on 𝐹 if 𝜉𝐼̃𝐹 = 𝜂𝐼̃𝐹 = 0 whenever 𝜉, 𝜂 ∈ 𝐿0 (F) such that 𝜉𝐼̃𝐹 𝑥 + 𝜂𝐼̃𝐹 𝑦 = 𝜃. By Definitions 33 and 34, we can obtain the following lemma easily. Lemma 35. Let (𝐸, ‖ ⋅ ‖) be a random strictly convex 𝑅𝑁 module over 𝑅 with base (Ω, F, 𝑃). Then the mapping 𝑓 : 𝐸 → 𝐿0 (F) 𝑓 (𝑥) = ‖𝑥‖2 0

is strictly 𝐿 (F)-convex.

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For giving the proof of Theorem 1, we need the following lemma and remark. Lemma 36 (Mazur lemma). Let (𝑋, ‖ ⋅ ‖) be a normed space, {𝑢𝑛 ∈ 𝑋, 𝑛 ∈ 𝑁} converge to 𝑢 under the weak topology on 𝑋. Then there exists a sequence {V𝑛 ∈ 𝑋, 𝑛 ∈ 𝑁} such that it converges to 𝑢 in norm, where 𝑁𝑛

V𝑛 = ∑ 𝜆 𝑘 𝑢𝑘 , 𝑘=𝑛

𝑁𝑛

∑ 𝜆 𝑘 = 1,

𝑘=𝑛

(17) 𝜆 𝑘 ≥ 0.

Remark 37. Let {𝜉𝑛 ∈ 𝐿0 (F), 𝑛 ∈ 𝑁} converge to 𝜉 uniformly. Then we can obtain a net {𝜉𝜀 ∈ 𝐿0 (F) | 𝜀 ∈ 𝐿0++ , 𝜀 ≤ 1} such that it converges to 𝜉 under the locally 𝐿0 -convex topology of (𝐿0 (F, 𝑅), | ⋅ |). In fact, for any 𝜀 ∈ 𝐿0++ let 𝐴 1 = [𝜀 > 1], 𝐴 𝑖 = [1/(𝑖 + 1) < 𝜀 ≤ 1/𝑖], for all 𝑖 ∈ 𝑁. Then {𝐴 𝑛 , 𝑛 ∈ 𝑁} is a countable partition of Ω to F. Since the sequence {𝜉𝑛 , 𝑛 ∈ 𝑁} converges to 𝜉 uniformly, thus for any number 𝑘 > 0, there exists 𝑁(𝑘) ∈ 𝑁 such that 󵄨 1 󵄨󵄨 󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 < 𝑘

(18)

for any 𝑛 > 𝑁(𝑘). Let ∞

𝜉𝜀 = ∑𝐼̃𝐴 𝑖 𝜉𝑁(𝑖)+1 ;

(19)

𝑖=1

then it is easy to see that |𝜉𝜀 − 𝜉| < 𝜀. Set ∧ = {𝜀 ∈ 𝐿0++ | 𝜀 ≤ 1}; then ∧ is directed with respect to ≤, and one can easy to see that the net {𝜉𝜀 , 𝜀 ∈ ∧} converges to 𝜉 under the locally 𝐿0 convex topology of (𝐿0 (F), | ⋅ |). With the above preparations, we now give the proof of Theorem 1. Proof of Theorem 1. First, let us assume that for any 𝑝 ∈ 𝐵, 𝐿(⋅, 𝑝) is strictly 𝐿0 (F)-convex on 𝐴. Set 𝐹(𝑢) = ⋁𝑝∈𝐵 𝐿(𝑢, 𝑝) and 𝐺(𝑝) = ⋀𝑢∈𝐴𝐿(𝑢, 𝑝). We show that the functional 𝐹 has the local property and 𝐹 is 𝐿0 (F)-convex and T𝑐 -lower semicontinuous on 𝐴. By conditions (1) and (2), it is clear that 𝐹 has the local property. For any 𝑥1 , 𝑥2 ∈ 𝐴, 𝛼 ∈ 𝐿0+ , we have that 𝐹 (𝛼𝑥1 + (1 − 𝛼) 𝑥2 ) = ⋁ 𝐿 (𝛼𝑥1 + (1 − 𝛼) 𝑥2 , 𝑝) 𝑝∈𝐵

≤ ⋁ [𝛼𝐿 (𝑥1 , 𝑝) + (1 − 𝛼) 𝐿 (𝑥2 , 𝑝)] 𝑝∈𝐵

≤ 𝛼⋁ 𝐿 (𝑥1 , 𝑝) + (1 − 𝛼) ⋁ 𝐿 (𝑥2 , 𝑝) 𝑝∈𝐵

𝑝∈𝐵

= 𝛼𝐹 (𝑥1 ) + (1 − 𝛼) 𝐹 (𝑥2 ) . (20)

8


Thus, 𝐹 is 𝐿0 (F)-convex on 𝐴. For any 𝑟 ∈ 𝐿0 (F, 𝑅), let 𝐴 𝑟 = {𝑢 ∈ 𝐴 | 𝐹(𝑢) ≤ 𝑟}. Since 𝐹 has the local property, we have that 𝐴 𝑟 has the countable concatenation property. Let {𝑥𝛼 , 𝛼 ∈ ∧} ⊂ 𝐴 𝑟 converge to 𝑥0 under the locally 𝐿0 -convex topology of 𝐸. By 𝐹(𝑥𝛼 ) ≤ 𝑟, we can obtain that 𝐿 (𝑥𝛼 , 𝑝) ≤ 𝑟,

∀𝑝 ∈ 𝐵.

∀𝑝 ∈ 𝐵.

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Hence, 𝐹(𝑥0 ) ≤ 𝑟 and 𝑥0 ∈ 𝐴 𝑟 . So we have that 𝐹 is T𝑐 lower semicontinuous. Similarly, replacing 𝐿 with −𝐿, we see that −𝐺 has the local property and −𝐺 is 𝐿0 (F)-convex and T𝑐 -lower semicontinuous on 𝐵. By Theorems 26 and 28, we have that ⋁𝑝∈𝐵 𝐺(𝑝) ∈ 𝐿0 (F), and there exists 𝑝0 ∈ 𝐵 such that 𝐺 (𝑝0 ) = ⋁ 𝐺 (𝑝) . 𝑝∈𝐵

󵄩 󵄩 {󵄩󵄩󵄩V𝜀 − 𝜔𝑖 󵄩󵄩󵄩 , 𝜀 ∈ ∧}

(21)

Since 𝐿(⋅, 𝑝) is T𝑐 -lower semicontinuous, it is easy to see that 𝐿 (𝑥0 , 𝑝) ≤ 𝑟,

𝑁

𝑛 𝜆 𝑘 = 1, 𝜆 𝑘 ≥ 0. Since ‖V𝑛 − 𝜔𝑖 ‖2 → 0, we have that ∑𝑘=𝑛 {‖V𝑛 − 𝜔𝑖 ‖, 𝑛 ∈ 𝑁} converges in probability 𝑃 to 𝜃. By Egoroff theorem, for any number 𝛿 > 0, there exists 𝐶𝛿 ∈ F such that 𝑃(𝐶𝛿 ) < 𝛿 and {‖V𝑛 − 𝜔𝑖 ‖, 𝑛 ∈ 𝑁} converges uniformly to 0 on 𝐶𝛿𝑐 . Then there is a net

as in Remark 37, which converges 0 on 𝐶𝛿𝑐 under the locally 𝐿0 -convex topology of (𝐿0 (F), | ⋅ |). By the construction of {‖V𝜀 − 𝜔𝑖 ‖, 𝜀 ∈ ∧} as in Remark 37, we have that lim 𝐿 (V𝑛 , 𝑝) ≥ lim𝐿 (V𝜀 , 𝑝) .

𝑛→∞

𝑢∈𝐴

𝑁𝑛

𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 𝐺 (𝑝0 ) ≥ 𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 lim ∑ 𝜆 𝑘 𝐿 (𝑢𝑘 , 𝑝) 𝑛 → ∞𝑘=𝑛

(23)

(24)

for any 𝑝 ∈ 𝐵. Let 𝑢0 = 𝑢𝑝0 , we need only to prove that 𝐺 (𝑝0 ) ≥ 𝐿 (𝑢0 , 𝑝) ,

∀𝑝 ∈ 𝐵.

≥ 𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 lim 𝐿 (V𝑛 , 𝑝)

𝐺 (𝑝0 ) ≥ 𝐺 (𝑝𝑛 ) = 𝐿 (𝑢𝑛 , 𝑝𝑛 ) .

(26)

≥ 𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 lim𝐿 (V𝜀 , 𝑝) ≥ 𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 𝐿 (𝜔𝑖 , 𝑝) . 𝜀∈∧

Hence, one can obtain that 𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 𝐺(𝑝0 ) ≥ 𝐼̃𝐶𝛿𝑐 𝐼̃𝐶𝑖 𝐿(𝜔𝑖 , 𝑝). Because 𝛿 is an arbitrary nonnegative number and 𝐿(⋅, 𝑝) has the local property, we have that 𝐼̃𝐶𝑖 𝐺 (𝑝0 ) ≥ 𝐼̃𝐶𝑖 𝐿 (𝜔𝑖 , 𝑝) ,

−1

−1

𝐺 (𝑝0 ) ≥ (1 − 𝑛 ) 𝐺 (𝑝0 ) + 𝑛 𝐿 (𝑢𝑛 , 𝑝) , (27) namely, 𝐺(𝑝0 ) ≥ 𝐿(𝑢𝑛 , 𝑝). Let 𝜂 = ⋁{‖𝑥‖ | 𝑥 ∈ 𝐴} and 𝐶𝑛 = {𝜔 ∈ Ω | 𝑛 − 1 ≤ 𝜂 < 𝑛}, for all 𝑛 ∈ 𝑁, according to the condition that 𝐴 is a.s. bounded in 𝐸, and it is easy to see that {𝐶𝑛 , 𝑛 ∈ 𝑁} is a countable partition of Ω to F. For any fixed 𝑖 ∈ 𝑁, we have that 𝐼̃𝐶𝑖 ⋅ 𝑢𝑛 ∈ 𝐿2 (𝐸), for all 𝑛 ∈ 𝑁. It is easy to see that 𝐼̃𝐶𝑖 ⋅ 𝐴 is a bounded and closed subset of (𝐿2 (𝐸), ‖ ⋅ ‖2 ). Thus, there exists a subsequence {𝐼̃𝐶𝑖 ⋅ 𝑢𝑛𝑘 , 𝑘 ∈ 𝑁} and 𝜔𝑖 ∈ 𝐼̃𝐶𝑖 ⋅ 𝐴 such that {𝐼̃𝐶𝑖 ⋅ 𝑢𝑛𝑘 , 𝑘 ∈ 𝑁} converges to 𝜔𝑖 under the weak topology of 𝐿2 (𝐸). Without loss of generality, denote this subsequence by {𝐼̃𝐶𝑖 ⋅ 𝑢𝑛 , 𝑛 ∈ 𝑁}. By Lemma 36, there exists a sequence {V𝑛 ∈ 𝐼̃𝐶𝑖 ⋅ 𝐴, 𝑛 ∈ 𝑁} such that it converges to 𝜔𝑖 in norm, where

𝐿 (𝑢𝑛 , 𝑝𝑛 ) ≤ 𝐿 (𝑢, 𝑝𝑛 ) ,

V𝑛 = ∑ 𝜆 𝑘 𝑢𝑘 , 𝑘=𝑛

(28)

∀𝑢 ∈ 𝐴.

(33)

By condition (2), we have that (1 − 𝑛−1 ) 𝐿 (𝑢𝑛 , 𝑝0 ) + 𝑛−1 𝐿 (𝑢𝑛 , 𝑝) ≤ 𝐿 (𝑢, 𝑝𝑛 ) ,

∀𝑝 ∈ 𝐵. (34)

Hence, we can obtainthat lim (1 − 𝑛−1 ) 𝐿 (𝑢𝑛 , 𝑝0 ) ≤ lim 𝐿 (𝑢, 𝑝𝑛 ) ,

𝑛→∞

𝑛→∞

∀𝑝 ∈ 𝐵. (35)

Since 𝐿(𝑢𝑛 , 𝑝) ≥ 𝐺(𝑝), we can obtain that (1 − 𝑛−1 )𝐿(𝑢𝑛 , 𝑝0 ) + 𝑛−1 𝐺(𝑝) ≤ 𝐿(𝑢, 𝑝𝑛 ) and (1 − 𝑛−1 )𝐿(V𝑛 , 𝑝0 ) + 𝑛−1 𝐺(𝑝) ≤ 𝐼̃𝐶𝑖 ⋅ lim𝑛 → ∞ ((1 − 𝑛−1 )𝐿(𝑢𝑛 , 𝑝0 ) + 𝑛−1 𝐿(𝑢𝑛 , 𝑝)). According to lim 𝐿 (V𝑛 , 𝑝0 ) = lim (1 − 𝑛−1 ) 𝐿 (V𝑛 , 𝑝0 )

𝑛→∞

𝑛→∞

𝑁𝑛

≤ lim (1 − 𝑛−1 ) ∑ 𝜆 𝑘 𝐿 (𝑢𝑘 , 𝑝0 ) 𝑛→∞

𝑘=𝑛

≤ lim (1 − 𝑛−1 ) 𝐿 (𝑢𝑛 , 𝑝0 ) 𝑛→∞

𝑁𝑛

(32)

for any 𝑝 ∈ 𝐵. Now, we prove that 𝜔𝑖 = 𝐼̃𝐶𝑖 𝑢0 for any 𝑖 ∈ 𝑁. By the definition of 𝑢𝑛 , it is clear that

By condition (2), we have that 𝐺 (𝑝0 ) ≥ 𝐿 (𝑢𝑛 , 𝑝𝑛 ) ≥ (1 − 𝑛−1 ) 𝐿 (𝑢𝑛 , 𝑝0 ) + 𝑛−1 𝐿 (𝑢𝑛 , 𝑝) ,

(31)

𝑛→∞

(25)

For every 𝑝 ∈ 𝐵, let 𝑝𝑛 = (1 − 𝑛−1 )𝑝0 + 𝑛−1 𝑝 and 𝑢𝑛 = 𝑢𝑝𝑛 , for all 𝑛 ∈ 𝑁. It is clear that

(30)

𝜀∈∧

Since 𝐿(⋅, 𝑝) is 𝐿0 (F)-convex for any 𝑝 ∈ 𝐵, we have that

Since for all 𝑝 ∈ 𝐵, 𝐿(⋅, 𝑝) is strictly 𝐿0 (F)-convex on 𝐴, according to Corollary 30 there exists an unique 𝑢𝑝 ∈ 𝐴 such that 𝐺 (𝑝) = ⋀ 𝐿 (𝑢, 𝑝) = 𝐿 (𝑢𝑝 , 𝑝) ,

(29)

≤ lim 𝐿 (𝑢, 𝑝𝑛 ) , 𝑛→∞

(36)


9

𝑁

𝑛 where ∑𝑘=𝑛 𝜆 𝑘 = 1, it is obvious that

for all 𝑢 ∈ 𝐴, 𝑝 ∈ 𝐵. Similarly, for any 𝑖, 𝑗 ∈ 𝑁, one can have a net {𝐼̃𝐻𝑖𝑗 𝑝𝛽 ∈ 𝐸 | 𝛽 ∈ Γ} such that it converges to ]𝑖𝑗 under the locally 𝐿0 -convex topology of 𝐸 and

𝐼̃𝐶𝑖 𝐿 (𝜔𝑖 , 𝑝0 ) ≤ lim 𝐼̃𝐶𝑖 𝐿 (V𝑛 , 𝑝0 ) ≤ lim 𝐼̃𝐶𝑖 𝐿 (𝑢𝑛 , 𝑝0 ) 𝑛→∞

𝑛→∞

(37)

≤ lim 𝐼̃𝐶𝑖 𝐿 (𝑢, 𝑝𝑛 ) . 𝑛→∞

Since {‖𝑝𝑛 − 𝑝0 ‖, 𝑛 ∈ 𝑁} converges in probability to 0, by Egoroff theorem, we can obtain that for any 𝜎 > 0, there exists 𝐶𝜎 ∈ F such that 𝑃(𝐶𝜎 ) < 𝜎 and {‖𝑝𝑛 −𝑝0 ‖, 𝑛 ∈ 𝑁} converges to 0 uniformly on 𝐶𝜎𝑐 . By Remark 37, we can construct a net {𝑝𝛼 ∈ 𝐵, 𝛼 ∈ ∧} as in Remark 37 such that ‖𝑝𝛼 − 𝑝0 ‖ → 0 under the locally 𝐿0 -convex topology of 𝐸. Hence, by (2) we can obtain that 𝐼̃𝐶𝜎𝑐 𝐼̃𝐶𝑖 𝐿 (𝜔𝑖 , 𝑝0 ) ≤ 𝐼̃𝐶𝜎𝑐 𝐼̃𝐶𝑖 lim 𝐿 (𝑢, 𝑝𝛼 ) ≤ 𝐼̃𝐶𝜎𝑐 𝐼̃𝐶𝑖 𝐿 (𝑢, 𝑝0 ) . 𝛼∈∧

(38) Since 𝜎 is an arbitrary nonnegative number, it is clear that 𝐼̃𝐶𝑖 𝐿 (𝜔𝑖 , 𝑝0 ) ≤ 𝐼̃𝐶𝑖 𝐿 (𝑢, 𝑝0 ) .

∀𝑛 ∈ 𝑁.

(40)

Since 𝐸 is random strictly convex RN module, we can obtain that 𝐿 𝑛 is strictly 𝐿0 (F)-convex from Lemma 35. By the similar method, we have that for any 𝑛 ∈ 𝑁, there exists (𝑢𝑛 , 𝑝𝑛 ) ∈ 𝐴×𝐵 such that it is a saddle point of 𝐿 𝑛 with respect to 𝐴 × 𝐵. Let 𝜁 = ⋁{‖𝑦‖ | 𝑦 ∈ 𝐵} and 𝐷𝑚 be any representation element of [𝑚 − 1 ≤ 𝜂 < 𝑚], for all 𝑚 ∈ 𝑁, according to the condition that 𝐵 is a.s. bounded in 𝐸, and it is easy to see that {𝐷𝑚 , 𝑚 ∈ 𝑁} is a countable partition of Ω to F. It is easy to see that {𝐶𝑖 ⋂ 𝐷𝑗 , 𝑖, 𝑗 ∈ 𝑁} is also a countable partition of Ω to F. For any 𝑖, 𝑗 ∈ 𝑁, let 𝐻𝑖𝑗 = 𝐶𝑖 ⋂ 𝐷𝑗 . We can suppose that, without loss of generality, 𝐼̃𝐻𝑖𝑗 𝑢𝑛 ∈ 𝐿2 (𝐸) converge to 𝜇𝑖𝑗

under the weak topology of 𝐿2 (𝐸). Then we have that there exists a net {𝐼̃𝐻𝑖𝑗 𝑢𝛼 ∈ 𝐸 | 𝛼 ∈ Λ} such that it converges to 𝜇𝑖𝑗 under the locally 𝐿0 -convex topology of 𝐸. Thus, we have that 𝐼̃𝐻𝑖𝑗 𝐿 (𝜇𝑖𝑗 , 𝑝) ≤ lim𝐼̃𝐻𝑖𝑗 𝐿 (𝐼̃𝐻𝑖𝑗 𝑢𝛼 , 𝑝) 𝛼

󵄩 󵄩2 ≤ lim 𝐼̃𝐻𝑖𝑗 𝐿 (𝐼̃𝐻𝑖𝑗 𝑢𝑛 , 𝑝) + 𝑛−1 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 𝑛→∞

≤ lim (𝐼̃𝐻𝑖𝑗 𝐿 (𝑢, 𝐼̃𝐻𝑖𝑗 𝑝𝑛 ) + 𝑛−1 ‖𝑢‖2 ) 𝑛→∞

≤ lim 𝐼̃𝐻𝑖𝑗 𝐿 (𝑢, 𝐼̃𝐻𝑖𝑗 𝑝𝑛 ) + lim 𝑛−1 ‖𝑢‖2 𝑛→∞

𝑛→∞

≤ lim 𝐼̃𝐻𝑖𝑗 𝐿 (𝑢, 𝐼̃𝐻𝑖𝑗 𝑝𝛽 ) + lim 𝑛−1 ‖𝑢‖2 𝛽∈Γ

(42)

𝑛→∞

≤ 𝐼̃𝐻𝑖𝑗 𝐿 (𝑢, ]𝑖𝑗 ) . Let 𝑢0 = ∑𝑖,𝑗∈𝑁 𝐼̃𝐻𝑖𝑗 𝜇𝑖𝑗 and 𝑝0 = ∑𝑖,𝑗∈𝑁 𝐼̃𝐻𝑖𝑗 ]𝑖𝑗 ; it is easy to check that 𝐿 (𝑢0 , 𝑝) ≤ 𝐿 (𝑢0 , 𝑝0 ) ≤ 𝐿 (𝑢, 𝑝0 )

(43)

for all 𝑢 ∈ 𝐴, 𝑝 ∈ 𝐵; namely, (𝑢0 , 𝑝0 ) is a random saddle point of 𝐿 with respect to 𝐴 × 𝐵. This completes the proof.

Acknowledgment (39)

Therefore, for any 𝑖 ∈ 𝑁, 𝜔𝑖 = 𝐼̃𝐶𝑖 𝑢0 and 𝐺(𝑝0 ) ≥ 𝐿(𝑢0 , 𝑝), for all 𝑝 ∈ 𝐵; namely, (𝑢0 , 𝑝0 ) is a random saddle point of 𝐿 with respect to 𝐴 × 𝐵. If there is 𝑝 ∈ 𝐵 such that 𝐿(⋅, 𝑝) is not strictly 𝐿0 (F)convex on 𝐴, define 𝐿 𝑛 (𝑢, 𝑝) = 𝐿 (𝑢, 𝑝) + 𝑛−1 ‖𝑢‖2 ,

lim (𝐼̃𝐻𝑖𝑗 𝐿 (𝑢, 𝐼̃𝐻𝑖𝑗 𝑝𝑛 ) + 𝑛−1 ‖𝑢‖2 )

𝑛→∞

(41)

This work is supported by NNSF no. 11171015.

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Research Article New Sequence Spaces and Function Spaces on Interval [0, 1] Cheng-Zhong Xu1 and Gen-Qi Xu2 1 2

Université de Lyon, LAGEP, Bâtiment CPE, Université Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne, France Department of Mathematics, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Gen-Qi Xu; [email protected] Received 27 April 2013; Accepted 17 August 2013 Academic Editor: Ji Gao Copyright © 2013 C.-Z. Xu and G.-Q. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the sequence spaces and the spaces of functions defined on interval [0, 1] in this paper. By a new summation method of sequences, we find out some new sequence spaces that are interpolating into spaces between ℓ𝑝 and ℓ𝑞 and function spaces that are interpolating into the spaces between the polynomial space 𝑃[0, 1] and 𝐶∞ [0, 1]. We prove that these spaces of sequences and functions are Banach spaces.

1. Introduction With development of sciences and technologies, more and more information are obtaining and need to be reserved and transmitted in the form of data sequence, such as DNA sequence, protein structure [1], brain imaging data, optic spectral analysis, text retrieval, financial data, and climate data. These data have common features: (1) there are at most finite many nonzero elements in the sequence; (2) their dimensions have not bounded from above; (3) the sample size is relatively small. In particular, some elements in the sequence repeat many times, for instance, there are only four different elements in DNA sequence: 𝑇, 𝐴, 𝐶, and 𝐺. When the data have much greater dimension, their record and reserve also become a serious problem. On the other hand, we usually use the data to obtain some information, such as the image reconstruction, sequence comparison in medicine, and plant classification in biology. From application point view, the basic requirement is that one can draw easily information from the reservoir; the is to use this data to handle some things. When the data have lower dimension and the samples have larger size, the statistics method such as the covariance matrix can give a good treatment; for instance, see [2] for the semiparameter estimation, [3] for the sparse data estimation, and [4, 5] for the threshold sparse sample covariance matrix method. However, when the data have higher dimensional

and the sample size is smaller, the statistics method shall lead to great errors. So, we need new methods to treat them. Let us consider a simple example from a classification problem. Set 𝑆 as a set of some class samples and 𝑎 as a given data. Is 𝑎 close to someone of 𝑆 or a new class? A simpler approach is to consider problem inf 𝑠∈𝑆 ‖𝑎 − 𝑠‖𝑝 , where 𝑝 denotes the norm in ℓ𝑝 space. In most cases, there is at least one 𝑠0 ∈ 𝑆 such that ‖𝑎 − 𝑠0 ‖𝑝 = inf 𝑠∈𝑆 ‖𝑎 − 𝑠‖𝑝 . We denote by 𝐹(𝑎) the feasible set. Can we say that 𝑎 is close to some 𝑠0 ∈ 𝐹(𝑎)? To see disadvantage, we divide sequence 𝑠 ∈ 𝑆 into three segments (𝑠1 , 𝑠2 , 𝑠3 ); the first segment 𝑠1 is composed of the first 𝑛1 elements, the second segment 𝑠2 is made of the next 𝑛2 elements, and the third is composed of the others. Similarly, we also divide 𝑎 into corresponding three parts (𝑎1 , 𝑎2 , 𝑎3 ). Now, we reconsider 󵄩󵄩 󵄩 inf 󵄩𝑎 − 𝑠1 󵄩󵄩󵄩𝑝 , 𝑠 󵄩 1 1

󵄩󵄩 󵄩 inf 󵄩𝑎 − 𝑠2 󵄩󵄩󵄩𝑝 , 𝑠 󵄩 2 2

󵄩󵄩 󵄩 inf 󵄩𝑎 − 𝑠3 󵄩󵄩󵄩𝑝 . 𝑠 󵄩 3 3

(1)

Perhaps we would find that 𝐹(𝑎1 ) ∩ 𝐹(𝑎2 ) ∩ 𝐹(𝑎3 ) = 0. Can one say that 𝑎 is a new class? From the above example, we see that we need a new definition of the norm to fit application. Motivated by these questions, we revisit the sequence spaces and function spaces defined on [0, 1] in this paper. We have observed recent studies on the sequence spaces, for instants, [6–8] for different requirements. Here, the sequence spaces we work on are different from the existing spaces, this

2


is because the spaces are aimed to solve our problem. Now, let us introduce our idea and the resulted sequence space and function spaces. Let 𝑎 = (𝑎1 , 𝑎2 , 𝑎3 , . . . , 𝑎𝑛 , . . .) be a DNA sequence. Obviously, there are at most finite many nonzero terms and 𝑎𝑛 ∈ {𝐴, 𝐶, 𝑇, 𝐺}. To shorten the representation, we can embed 𝑎 into a polynomial. In this way, we can write 𝑎 as 𝑎 (𝑥) = 𝐴𝑝1 (𝑥) + 𝑇𝑝2 (𝑥) + 𝐶𝑝3 (𝑥) + 𝐺𝑝4 (𝑥) .

(2)

For a different DNA sequence, we have different polynomial 𝑝𝑗 (𝑥). Obviously, it is a simpler reserve form. How do we extract original sequence from the polynomial? By the classical mathematics, we know that 1 𝑑(𝑘) 𝑎 (𝑥) | , 𝑎𝑘 = 𝑛! 𝑑𝑥𝑘 𝑥=0

𝑘 = 0, 1, 2, . . . ,

(3)

so we recover the sequence. To extend this form into a sequence of infinite many nonzero terms, we usually take 𝑥 ∈ [0, 1]; 𝑎(𝑥) is called the generation function in the classical queuing theory. Note that the generation function is not a continuous function defined on [0, 1]. So, the differential operation is not fitting to such a function, although the formal differential is always feasible. To find out a feasible form of 𝑎(𝑥), let us consider the operations of integral and derivative. We denote by 𝐿 the integral operation. Operating for the constant 1 leads to 𝑥

1

0

𝑥

0

2

𝑥 , 2

𝑥

𝐿3 (1) (𝑥) = ∫ 𝐿2 (1) (𝑥) 𝑑𝑥 = 0

(4)

𝑥3 . 3!

𝑎𝑘 = 𝐷𝑘 𝑝𝑛 (0) , . . . ,

𝐿𝑛 (1) (𝑥) =

𝑥𝑛 , 𝑛!

∀𝑛 ∈ N.

Therefore, the coefficient sequence is given by (𝑎0 , 𝑎1 , . . . , 𝑎𝑛 ) = (𝐷0 , 𝐷1 , . . . , 𝐷𝑛 ) 𝑝𝑛 |𝑥=0 .

For any polynomial of 𝑛-order, 𝑝𝑛 (𝑥), it can be written as 𝑥2 𝑥𝑛 𝑝𝑛 (𝑥) = 𝑎0 1 + 𝑎1 𝑥 + 𝑎2 + ⋅ ⋅ ⋅ + 𝑎𝑛 2! 𝑛!

(6)

𝑘

= [ ∑ 𝑎𝑘 𝐿 ] (1) (𝑥) . 𝑘=0

Next, let us consider the differential operation 𝐷(𝑓) = 𝑓 (𝑥). Taking deferential for function 𝑥𝑛 /𝑛! leads to 󸀠

𝑥𝑛 𝑥𝑛−2 𝐷 ( )= . 𝑛! (𝑛 − 2)! 2

(7)

󵄩 𝑛 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑝󵄩󵄩𝜙 = sup {󵄩󵄩󵄩𝐷 𝑝󵄩󵄩󵄩∞ } ; 𝑛≥0

(11)

where ‖𝑓‖∞ = max0≤𝑥≤1 |𝑓(𝑥)|, then it is a normed linear space. In this space, the integral and differential operations are bounded linear operators. To extend to an infinite sequence, we should choose such a function space in which the integral and differential operations are bounded linear operators. What is such a function space? Consider a subset of 𝐶∞ [0, 1] defined as 󵄩 󵄩 ∞ 𝐶𝑀 [0, 1] = {𝑓 ∈ 𝐶∞ [0, 1] | sup󵄩󵄩󵄩𝐷𝑛 𝑓󵄩󵄩󵄩∞ < ∞} . 𝑛≥0

(12)

The set is in fact a linear space. We define a norm on it by (13)

then it becomes a Banach space. Now, for the function spaces over interval [0, 1], there are the following inclusion relations (14)

But the completion of (𝑃[0, 1], ‖ ⋅ ‖𝜙 ) is not the space ∞ [0, 1], ‖ ⋅ ‖𝜙 ). Let 𝐶𝜙,0 [0, 1] be the completion space of (𝐶𝑀 (𝑃[0, 1], ‖ ⋅ ‖𝜙 ). Clearly, ∞ 𝑃 [0, 1] ⊂ 𝐶𝜙,0 [0, 1] ⊂ 𝐶𝑀 [0, 1] .

(15)

And the integral and differential operations are bounded operators. For space 𝐶𝜙,0 [0, 1], we have the following representation theorem. Theorem 1. The set 𝐶𝜙,0 [0, 1] has the following representation: ∞

𝐶𝜙,0 [0, 1] = {𝑓 (𝑥) = ∑ 𝑎𝑛 𝑛=0

𝑥𝑛 | lim 𝑎 = 0} . 𝑛! 𝑛 → ∞ 𝑛

(16)

Moreover, the space C𝜙,0 [0, 1] is isomorphic to the sequence space 𝑐0 , and the isomorphism mapping is given by

In general, for any 1 ≤ 𝑘 ≤ 𝑛, it holds that 𝑥𝑛 𝑥𝑛−𝑘 . )= 𝑛! (𝑛 − 𝑘)!

(10)

Clearly, we should take functions 𝑥𝑛 /𝑛! as the basis functions. Moreover, we note that in the polynomial space over [0, 1], denoted by 𝑃[0, 1], if the norm is defined as

⊂ 𝐿∞ [0, 1] ⊂ 𝐿𝑝 [0, 1] ⊂ 𝐿1 [0, 1] . (5)

(9)

𝑎𝑛 = 𝐷𝑛 𝑝𝑛 (0) .

∞ 𝑃 [0, 1] ⊂ 𝐶𝑀 [0, 1] ⊂ 𝐶∞ [0, 1] ⊂ 𝐶𝑘 [0, 1] ⊂ 𝐶 [0, 1]

Generally, we have

𝐷𝑘 (

𝑎1 = 𝐷𝑝𝑛 (0) , . . . ,

𝑛≥0

𝐿2 (1) (𝑥) = ∫ 𝐿1 (1) 𝑥 𝑑𝑥 =

𝑥𝑛 𝑥𝑛−1 𝐷( ) = , 𝑛! (𝑛 − 1)!

𝑎0 = 𝑝𝑛 (0) ,

󵄩 𝑛 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝜙 = sup󵄩󵄩󵄩𝐷 𝑓󵄩󵄩󵄩∞ ,

𝐿 (1) (𝑥) = ∫ 1 𝑑𝑥 = 𝑥,

𝑛

Obviously, using 𝐷, we get once again the coefficients in (6):

∞

(8)

𝑇 {𝑎𝑛 } = 𝑓 (𝑥) = ∑ 𝑎𝑛 𝑛=0

𝑥𝑛 , 𝑛!

∀ {𝑎𝑛 } ∈ 𝑐0 .

(17)


3

Motivated by above result, for 𝑝 ≥ 1, we define a new norm on the space 𝑃[0, 1] by 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝜙,𝑝 =

1/𝑝 󵄩 󵄩𝑝 (∑󵄩󵄩󵄩󵄩𝐷𝑘 𝑓󵄩󵄩󵄩󵄩∞ ) . 𝑘

(18)

The completion of space 𝑃[0, 1] under this norm is denoted by 𝐶𝜙,𝑝 . Similarly, we have the following result. Theorem 2. The space 𝐶𝜙,𝑝 [0, 1] has a representation: ∞

𝐶𝜙,𝑝 [0, 1] = {𝑓 (𝑥) = ∑ 𝑎𝑛 𝑛=0

𝑥𝑛 ∞ 󵄨󵄨 󵄨󵄨𝑝 | ∑ 󵄨𝑎 󵄨 < ∞} . 𝑛! 𝑛=0󵄨 𝑛 󵄨

(19)

𝑝

Furthermore, 𝐶𝜙,𝑝 [0, 1] is isomorphic to the space ℓ , and the isomorphism operator is given by ∞

𝑇 {𝑎𝑛 } = 𝑓 (𝑥) = ∑ 𝑎𝑛 𝑛=0

𝑥𝑛 , 𝑛!

∀ {𝑎𝑛 } ∈ ℓ𝑝 .

𝐶𝜙,∞ [0, 1] = {𝑓 (𝑥) = ∑ 𝑎𝑛 𝑛=0

(21)

∞ Moreover, 𝐶𝜙,∞ [0, 1] = 𝐶𝑀 [0, 1] is isomorphic to ℓ∞ , and the isomorphism operator is given by ∞

𝑇 {𝑎𝑛 } = 𝑓 (𝑥) = ∑ 𝑎𝑛 𝑛=0

𝑥𝑛 , 𝑛!

∀ {𝑎𝑛 } ∈ ℓ∞ .

(22)

By now, we have gotten a series spaces in which both the differential operator and integral operator are bounded linear operators. Obviously, these sets have the following inclusion relations: 𝑃 [0, 1] ⊂ 𝐶𝜙,1 [0, 1] ⊂ 𝐶𝜙,𝑝 [0, 1] ⊂ 𝐶𝜙,0 [0, 1] ⊂ 𝐶𝜙,∞ [0, 1] =

∞ 𝐶𝑀

[0, 1] ,

1 ≤ 𝑝 < ∞.

(23)

We observe that for 𝑝 ≥ 1, in definition of these spaces, the term ∞

󵄩 󵄩𝑝 ∑ 󵄩󵄩󵄩𝐷𝑛 𝑓󵄩󵄩󵄩∞ < ∞,

(24)

𝑛=0

𝑝

means that the terms ∑𝑛𝑘=1 ‖𝐷𝑛+𝑘 𝑓‖∞ are small as 𝑛 is large enough. To insert a new space between 𝐶𝜙,𝑝 [0, 1] and 𝐶𝜙,0 [0, 1], let us consider the sequence formed by 𝑓 ∈ 𝐶𝜙,0 [0, 1], 𝑓 (𝑥) , 𝑓󸀠 (𝑥) , 𝑓󸀠󸀠 (𝑥) , . . . , 𝑓(𝑛) (𝑥) , . . . ;

𝜎 {𝑎𝑛 } = {𝑎𝜎(𝑘) ; 𝑘 ≥ 0} ,

(25)

the norm ‖𝑓‖𝜙 = sup𝑛≥0 ‖𝐷𝑛 𝑓‖∞ is maximum of the function sequence under space 𝐶[0, 1]. To define a new normed space, in Section 2, we shall study new summation approach for a sequence. Based on such a new summation approach, we shall define some new sequence spaces. In Section 3, we shall define some function spaces and discuss their completeness. Furthermore, we compare these spaces and give some inclusion relations.

(26)

where |𝑎𝜎(𝑘) | ≥ |𝑎𝜎(𝑘+1) |. It is called the absolutely dominant queuing operator. Removing the first 𝑛 terms, the remainder sequence {𝑎𝑛+𝑘 ; 𝑘 ≥ 0} has a new queuing: 𝜎 {𝑎𝑛+𝑘 } = {𝑎𝑛+𝜎(𝑘) ; 𝑘 ≥ 0} .

(27)

Using the queuing operator, for a sequence {𝑎𝑛 } of zero limit (this is only used to ensure that we can take the maximal value for such a sequence), we define a positive number by 󵄨 󵄨 󵄨 󵄨 𝑠0 = sup 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑎𝜎(0) 󵄨󵄨󵄨 . 𝑛≥0

𝑛

𝑥 󵄨 󵄨 | sup 󵄨󵄨𝑎 󵄨󵄨 < ∞} . 𝑛! 𝑛≥0 󵄨 𝑛 󵄨

2.1. Summation of Absolutely Dominant Operator. Let {𝑎𝑛 } be a sequence (real or complex number) and satisfy lim𝑛 → ∞ 𝑎𝑛 = 0. We define an operator 𝜎 by

(20)

Theorem 3. The space 𝐶𝜙,∞ [0, 1] has a representation: ∞

2. New Summation Method

(28)

By removing the first term 𝑎0 from {𝑎𝑛 }∞ 𝑛=0 , we define the second number 𝑠1 , that is, the absolute summation of the first two terms in 𝜎{𝑎𝑛+1 }, that is, 󵄨 󵄨 󵄨 󵄨 𝑠1 = ∑𝜎 {𝑎1+𝑛 , 𝑛 ≥ 0} = 󵄨󵄨󵄨𝑎1+𝜎(0) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜎1+𝜎(1) 󵄨󵄨󵄨 , 2

(29)

where the number in subscription of summation is the number of the terms 𝜎 denotes the absolutely dominant queuing operator. After removing the first two terms 𝑎0 and 𝑎1 from {𝑎𝑛 }∞ 𝑛=0 , we define the third positive number 𝑠2 , that is, the absolute summation of the first three terms in 𝜎{{𝑎2+𝑛 }}, that is, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑠2 = ∑𝜎 {𝑎2+𝑛 , 𝑛 ≥ 0} = 󵄨󵄨󵄨𝑎2+𝜎(0) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎2+𝜎(1) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎2+𝜎(2) 󵄨󵄨󵄨 . 3

(30) Generally, we define positive number 𝑠𝑘 as 󵄨 󵄨 󵄨 󵄨 𝑠𝑘 = ∑ 𝜎 {𝑎𝑘+𝑛, 𝑛≥0 } = 󵄨󵄨󵄨𝑎𝑘+𝜎(0) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎𝑘+𝜎(1) 󵄨󵄨󵄨 𝑘+1

󵄨 󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝑎𝑘+𝜎(2) 󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨𝑎𝑘+𝜎(𝑘) 󵄨󵄨󵄨 .

(31)

According to this rule, we get a new nonnegative sequence {𝑠𝑘 , ; 𝑘 ≥ 0}. Example 4. Let {𝑎𝑘 } = {1, 2, 3, 4, 5, 6}. Then, we have 𝑠0 = 6, 𝑠2 = 6 + 5 + 4, 𝑠4 = 6 + 5,

𝑠1 = 6 + 5, 𝑠3 = 6 + 5 + 4,

(32)

𝑠5 = 6.

2.2. Relationship between Summation and Order of Sequence. To explain the thing we concerned about, let us see an example.

4


Example 5. Let scalar group be {𝑏𝑘 } = {6, 5, 4, 3, 2, 1}. Then, we have

In this case, the 𝑠-sequence has a 𝑈-type distribution shown as follows:

𝑠̂1 = 5 + 4,

𝑠̂0 = 6,

𝑠̂3 = 3 + 2 + 1,

𝑠̂2 = 4 + 3 + 2,

𝑠0

(33)

𝑠̂5 = 1.

𝑠̂4 = 2 + 1,

Comparing this with Example 4, we see that the summation of a sequence has a relationship with its order. From Examples 4 and 5, we see that the sequences {̂𝑠𝑛 } and {𝑠𝑛 } generated by a new summation method have relation of order of a sequence. In the sequel, we mainly discuss the infinite sequence. If a sequence has only finite many terms, we shall complement zero after the last term so that it becomes an infinite sequence.

𝑠1

𝑠2

(37)

𝑚−1 𝑗=0

𝑚

𝑠𝑚 = ∑ 𝑎𝑁−𝑗 , 𝑗=0

𝑚−1

𝑠𝑚+1 = ∑ 𝑎𝑁−𝑗 , . . . ,

(38)

𝑗=0

(34)

𝑚−𝑘

𝑠𝑚+𝑘 = ∑ 𝑎𝑁−𝑗 , . . . 𝑗=0

𝑠0 = 𝑎𝑁,

𝑠𝑁−1 = 𝑎𝑁 + 𝑎𝑁−1 ,

𝑠1 = 𝑎𝑁 + 𝑎𝑁−1 , 𝑠2 = 𝑎𝑁 + 𝑎𝑁−1 + 𝑎𝑁−2 , .... 𝑘

𝑠𝑁−2

.

𝑠𝑚−1 = ∑ 𝑎𝑁−𝑗 ,

According to the absolutely dominant summation, we have

𝑠𝑘 = ∑𝑎𝑁−𝑗 ,

𝑠𝑚 𝑠𝑚+1 ⋅ ⋅ ⋅

𝑠𝑁

If 𝑁 = 2𝑚, the sequence has odd terms (2𝑚 + 1); then

2.3. Distribution of 𝑠-Sequence. In this subsection, we shall consider the distribution of the sequence {𝑠𝑘 , 𝑘 ≥ 0}. We discuss it according to the different cases. (1) Let {𝑎𝑘 } be a positive and increasing sequence, that is, 0 < 𝑎0 < 𝑎1 < 𝑎2 < 𝑎3 < ⋅ ⋅ ⋅ < 𝑎𝑁−1 < 𝑎𝑁.

d

𝑠𝑁−1

𝑘≤

𝑗=0

𝑠𝑁 = 𝑎𝑁. (35)

𝑁 , 2

In this case, the 𝑠-sequence has 𝑉-type distribution as follows: 𝑠0

when 𝑁 = 2𝑚 + 1, the sequence has even terms 2(𝑚 + 1); then 𝑚−1

𝑠1

𝑠2

d

𝑠𝑚−1

𝑠𝑚+1

𝑠𝑚−1 = ∑ 𝑎𝑁−𝑗 ,

𝑠𝑚+1

⋅⋅⋅

𝑠𝑁−2

𝑠𝑁−1

𝑠𝑁 . (39)

𝑗=0

If {𝑎𝑘 } is an increasing sequence, then {𝑠𝑘 } has a symmetrical form; the 𝑠-data at the medial term is its maximal value. (2) {𝑎𝑛 } is a positive and decreasing sequence

𝑚

𝑠𝑚 = ∑𝑎𝑁−𝑗 , 𝑗=0

𝑚

𝑎𝑁 > 𝑎𝑁−1 > 𝑎𝑁−2 > ⋅ ⋅ ⋅ > 𝑎𝑁/2 > 𝑎(𝑁/2)−1 > ⋅ ⋅ ⋅ > 𝑎1 > 𝑎0 . (40)

𝑠𝑚+1 = ∑𝑎𝑁−𝑗 , 𝑗=0

𝑚−1

𝑠𝑚+2 = ∑ 𝑎𝑁−𝑗 , . . . , 𝑗=0

𝑚+1−𝑘

𝑠𝑚+𝑘 = ∑ 𝑎𝑁−𝑗 𝑗=0

.... 𝑠𝑁−1 = 𝑎𝑁 + 𝑎𝑁−1 , 𝑠𝑁 = 𝑎𝑁.

(36)

According to the absolutely dominant summation, we have 𝑠0 = 𝑎𝑁, 𝑠1 = 𝑎𝑁−1 + 𝑎𝑁−2 , 𝑠2 = 𝑎𝑁−2 + 𝑎𝑁−3 + 𝑎𝑁−4 , 𝑘

𝑠𝑘 = ∑𝑎𝑁−𝑘−𝑗 , 𝑗=0

0≤𝑘≤

(41) 𝑁 . 2


5

If 𝑁 = 2𝑚 + 1, the sequence has an even term 2(𝑚 + 1); then 𝑚−1

𝑠𝑚−1 = ∑ 𝑎𝑁−(𝑚−1)−𝑗 ,

From above, we see that 𝑠-sequence undergoes a great contortion due to different queuing order of a sequence. Denote by 𝐽(𝑎𝑛 ) the maximum change of 𝑠-sequence for {𝑎𝑛 } under different queuing order; then

𝑗=0

𝐽 (𝑎𝑛 ) =

𝑚

𝑠𝑚 = ∑ 𝑎𝑁−𝑚−𝑗 , 𝑚

𝑠𝑚+1 = ∑ 𝑎𝑁−𝑗

(42)

𝑗=0

𝑚−1

𝑠𝑚+2 = ∑ 𝑎𝑁−𝑗 , . . . , 𝑗=0 𝑘

𝑗=0

𝑠𝑚−1 = ∑ 𝑎𝑁−𝑗 , 𝑚

3,𝑝

𝑗=0

𝑚−1

(43)

𝑠𝑚+1 = ∑ 𝑎𝑗 , . . . , 𝑗=0 𝑘

𝑘+1,𝑝

󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 = (󵄨󵄨󵄨𝑎𝑘+𝜎(0) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎𝑘+𝜎(1) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎𝑘+𝜎(2) 󵄨󵄨󵄨

𝑠𝑁 = 𝑎0 .

d

𝑠𝑚

(50)

󵄨𝑝 1/𝑝 󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨𝑎𝑘+𝜎(𝑘) 󵄨󵄨󵄨 ) .

So, 𝑠-sequence has distribution as

. d

(44)

𝑠𝑁

In this case, the 𝑠-sequence has a character that the initial original data is the absolute largest; at the first several steps, 𝑠sequence arrives at its maximum value; after then, 𝑠-sequence decreases until it arrives at its minimum value. The final value is minimal. (3) Let {𝑎𝑛 } be an arbitrary sequence. In this case, it is difficult to give a general distribution. Although so, we have the following relation 𝑘≥0

(49)

𝑠𝑘,𝑝 = ∑ 𝜎 {𝑎𝑘+𝑛 , 𝑛 ≥ 0}

𝑗=0

󵄨 󵄨 󵄨 󵄨 𝑠0 = sup 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑎𝜎(0) 󵄨󵄨󵄨 ,

󵄨𝑝 󵄨 󵄨𝑝 󵄨 󵄨𝑝 1/𝑝 󵄨 = (󵄨󵄨󵄨𝑎2+𝜎(0) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎2+𝜎(1) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎2+𝜎(2) 󵄨󵄨󵄨 ) . In general, we define

𝑠𝑁−𝑘 = ∑𝑎𝑗 ,

𝑠0

󵄨𝑝 󵄨 󵄨𝑝 1/𝑝 󵄨 𝑠1,𝑝 = ∑𝜎 {𝑎1+𝑛 , 𝑛 ≥ 0} = (󵄨󵄨󵄨𝑎1+𝜎(0) 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑎1+𝜎(1) 󵄨󵄨󵄨 ) . (48)

𝑠2,𝑝 = ∑𝜎 {𝑎2+𝑛 , 𝑛 ≥ 0}

𝑠𝑚 = ∑𝑎𝑗 ,

𝑠𝑘+1

2.4. Generalized 𝑝-Summation. From the absolutely dominant summation, we get a new sequence {𝑠𝑘 }. Now, we define a new 𝑝-summation for {𝑎𝑛 } sequence: 󵄨 󵄨 󵄨 󵄨 𝑠0,𝑝 = 𝑠0 = sup 󵄨󵄨󵄨𝑎𝑛 󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑎𝜎(0) 󵄨󵄨󵄨 . (47) 𝑛≥0

Again, removing the second term 𝑎1 , we define the summation of the first three absolute maximum value as 𝑠2,𝑝 , that is,

𝑗=0

⋅⋅⋅

(46)

2,𝑝

𝑚−1

𝑠1

.

Removing the first term 𝑎0 , we define the second value 𝑠1,𝑝 by

𝑠𝑁 = 𝑎0 .

If 𝑁 = 2𝑚, the sequence has odd terms (2𝑚 + 1), then

𝑠𝑘

max {𝑠𝑘𝑚 }

For the sequence {𝑎𝑛 } = {1, 2, 3, 4, 5, 6}, it is easy to see that 𝐽(𝑎𝑛 ) = 15/9.

𝑗=0

𝑠𝑁−𝑘 = ∑𝑎𝑗 ,

max {𝑠𝑘𝑀}

𝑠𝑘𝑚 ≤ 𝑠𝑘 ≤ 𝑠𝑘𝑀, ∀𝑘,

(45)

where 𝑠𝑘𝑚 denotes the value in decreasing queuing, 𝑠𝑘𝑀 denotes the value in increasing queuing.

According to this definition, we get a new nonnegative sequence {𝑠𝑘,𝑝 , ; 𝑘 ≥ 0}. Obviously, when 𝑝 = 1, we recover the above summation sequence 𝑠𝑘 = 𝑠𝑘,1 , for all 𝑘 ≥ 0. For the sequence {𝑎𝑛 }, we define a positive number ‖{𝑎𝑛 }‖𝐺,𝑝 by 󵄩󵄩 󵄩 󵄩󵄩{𝑎𝑛 }󵄩󵄩󵄩𝐺,𝑝 = sup 𝑠𝑘,𝑝 , 𝑘≥0

(51)

where 𝐺 denotes the generalized summation and 𝑝 denotes the 𝑝-norm in finite-dimensional space. In particular, when {𝑎𝑛 } ∈ ℓ𝑝 , 𝑠𝑘,𝑝 → 0 as 𝑘 → ∞, and 𝑝 1/𝑝 < ∞. hence ‖{𝑎𝑛 }‖𝐺,𝑝 < (∑∞ 𝑛≥0 |𝑎𝑛 | ) Now, we define the sequence spaces by 𝑐𝐺,𝑝,𝑀 = {{𝑎𝑛 } | sup 𝑠𝑘,𝑝 < ∞} , 𝑘≥0

𝑐𝐺,𝑝 = {{𝑎𝑛 } | lim 𝑠𝑘,𝑝 = 0} . 𝑘→∞

We can prove the following result.

(52)

6


Theorem 6. (𝑐𝐺,𝑝,𝑀, ‖ ⋅ ‖𝐺,𝑝 ) and (𝑐𝐺,𝑝 , ‖ ⋅ ‖𝐺,𝑝 ) are Banach spaces.

𝑠2 (𝑒𝑛 ) = ∑𝜎 {𝐷2+𝑘 𝑒𝑛 , 𝑘 ≥ 0} = 1 + 1 + 3

....

3. 𝐶𝐺,𝑝 [0, 1]-Type Spaces 3.1. Definition of Spaces 𝐶𝐺,𝑝,𝑀[0, 1]. Let 𝑝 ≥ 1. For each 𝑓 ∈ 𝐶𝜙,0 [0, 1], we define positive number: 𝑠𝑘,𝑝 (𝑓) = ∑𝜎 {𝐷𝑘+𝑛 𝑓, 𝑛 ≥ 0}

𝑠𝑗 (𝑒𝑛 ) = ∑𝜎 {𝐷𝑗+𝑘 𝑒𝑛 , 𝑘 ≥ 0} 𝑘

=1+1+

1 1 + ⋅⋅⋅ + , 2! 𝑗!

𝑛/2

1 , 𝑘! 𝑘=0

𝑠(𝑛/2)+1 = ∑

(𝑛/2)−1

𝑠(𝑛/2)+2 = ∑ Example 7. To show the computation method, let us consider the case that 𝑝 = 1 and discuss the function 𝑒𝑛 (𝑥) = 𝑥𝑛 /𝑛!. By computing various derivative functions

𝑘=0

1 , 𝑘!

𝑠𝑛−2 = 1 + 1 +

1 , 2!

𝑠𝑛−1 = 1 + 1,

1 󵄩󵄩 0 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = , 󵄩 󵄩∞ 𝑛!

𝑠𝑛 = 1, 𝑠𝑛+𝑗 = 0,

1 󵄩󵄩 1 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 󵄩 󵄩∞ (𝑛 − 1)! ,

𝑗 ≥ 1. (55)

According to the definition of 𝑠𝑘,𝑝 (𝑓), it has the following properties:

1 󵄩󵄩 2 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 󵄩∞ (𝑛 − 2)! 󵄩

(1) for any 𝑘 ≥ 0,

....

󵄩 󵄩 𝑠0,𝑝 (𝑓) = sup󵄩󵄩󵄩𝐷𝑛 𝑓󵄩󵄩󵄩∞ , 𝑛≥0

1 󵄩󵄩 𝑘 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 󵄩 󵄩∞ (𝑛 − 𝑘)!

We calculate 𝑠𝑘 (𝑒𝑛 ) as follows:

1 , 𝑘! 𝑘=0

𝑛/2

󵄩 󵄩𝑝 󵄩 󵄩𝑝 1/𝑝 + 󵄩󵄩󵄩󵄩𝐷𝑘+𝜎(2) (𝑓)󵄩󵄩󵄩󵄩∞ + ⋅ ⋅ ⋅ + 󵄩󵄩󵄩󵄩𝐷𝑘+𝜎(𝑘) (𝑓)󵄩󵄩󵄩󵄩∞ ) . (53)

....

𝑛 , 2 𝑛/2

󵄩𝑝 󵄩 󵄩𝑝 󵄩 = (󵄩󵄩󵄩󵄩𝐷𝑘+𝜎(0) (𝑓)󵄩󵄩󵄩󵄩∞ + 󵄩󵄩󵄩󵄩𝐷𝑘+𝜎(1) (𝑓)󵄩󵄩󵄩󵄩∞

1 󵄩󵄩 𝑛−2 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 󵄩 󵄩∞ 2! 󵄩󵄩 𝑛−1 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 1, 󵄩 󵄩∞ 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩∞ = 1 󵄩󵄩 𝑛+1 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 0, 󵄩∞ 󵄩 󵄩󵄩 𝑛+2 󵄩󵄩 󵄩󵄩𝐷 𝑒𝑛 󵄩󵄩 = 0, 󵄩∞ 󵄩

𝑗≤

𝑠(𝑛/2) (𝑒𝑛 ) = ∑𝜎 {𝐷(𝑛/2)+𝑘 𝑒𝑛 , 𝑘 ≥ 0} = ∑

𝑘,𝑝

....

1 , 2!

(54)

󵄩 󵄩 𝑠𝑘,𝑝 (𝑓) ≥ 󵄩󵄩󵄩󵄩𝐷𝑘 𝑓󵄩󵄩󵄩󵄩∞ ≥ 0;

(56)

(2) for each 𝛼 ∈ C, 𝑠𝑘,𝑝 (𝛼𝑓) = |𝛼| 𝑠𝑘,𝑝 (𝑓) ,

∀𝑓 ∈ 𝐶𝜙,0 [0, 1] ;

(57)

(3) for any 𝑔, 𝑓 ∈ 𝐶𝜙,0 [0, 1], 𝑠𝑘,𝑝 (𝑓 + 𝑔) ≤ 𝑠𝑘,𝑝 (𝑓) + 𝑠𝑘,𝑝 (𝑔) .

(58)

This can be obtained from the Minkovwski inequality. These properties show that 𝑠0,𝑝 (𝑓) satisfies the norm axiom, and hence it is a norm on space 𝐶𝜙,0 [0, 1]. But for 𝑘 ≥ 1, 𝑠𝑘,𝑝 (𝑓) only is a seminorm (or prenorm), it is continuous in the topology of 𝐶𝜙,0 [0, 1]. For each 𝑓 ∈ 𝐶𝜙,0 [0, 1], we define the following: 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝐺,𝑝 = sup𝑠𝑛,𝑝 (𝑓) . (59) 𝑛≥0

󵄩 󵄩 𝑠0 (𝑒𝑛 ) = sup󵄩󵄩󵄩󵄩𝐷𝑘 𝑒𝑛 󵄩󵄩󵄩󵄩∞ = 1, 𝑘≥0

We define the following functions spaces: 󵄩 󵄩 𝐶𝐺,𝑝,𝑀 [0, 1] = {𝑓 ∈ 𝐶𝜙,0 [0, 1] | 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐺,𝑝 < ∞} ,

(60)

𝑠1 (𝑒𝑛 ) = ∑𝜎 {𝐷1+𝑘 𝑒𝑛 , 𝑘 ≥ 0} = 1 + 1,

𝐶𝐺,𝑝 [0, 1] = {𝑓 ∈ 𝐶𝜙,0 [0, 1] | lim 𝑠𝑘,𝑝 (𝑓) = 0} .

(61)

2

𝑘→∞


7

Theorem 8. For 𝑓 ∈ 𝐶𝜙,0 [0, 1], the scalar ‖𝑓‖𝐺,𝑝 is defined as above. Then, ‖𝑓‖𝐺,𝑝 is a norm on 𝐶𝐺,𝑝,𝑀[0, 1], and hence (𝐶𝐺,𝑝,𝑀[0, 1], ‖ ⋅ ‖𝐺,𝑝 ) is a normed linear space. Moreover, (𝐶𝐺,𝑝,𝑀[0, 1], ‖ ⋅ ‖𝐺,𝑝 ) is a Banach space, which is isomorphic to the sequence space 𝑐𝐺,𝑝,𝑀. Proof. Here, we only prove the second assertion. Let 𝑓𝑛 ∈ 𝐶𝐺,𝑝,𝑀[0, 1] be a Cauchy sequence, that is, for any 𝜀 > 0, there exists 𝑁(𝜀) such that 󵄩 󵄩󵄩 󵄩󵄩𝑓𝑛 − 𝑓𝑚 󵄩󵄩󵄩𝐺,𝑝 < 𝜀,

∀𝑛, 𝑚 > 𝑁 (𝜀) .

󵄩 󵄩󵄩 𝑘 󵄩󵄩𝐷 (𝑓𝑛 − 𝑓𝑚 )󵄩󵄩󵄩 ≤ 𝑠𝑘,𝑝 (𝑓𝑛 − 𝑓𝑚 ) ≤ 󵄩󵄩󵄩󵄩𝑓𝑛 − 𝑓𝑚 󵄩󵄩󵄩󵄩𝐺,𝑝 . 󵄩∞ 󵄩

󵄩 󵄩󵄩 𝑘 󵄩󵄩𝐷 (𝑓𝑛 − 𝑓0 )󵄩󵄩󵄩 ≤ 𝑠𝑘,𝑝 (𝑓𝑛 − 𝑓0 ) ≤ 𝜀. 󵄩∞ 󵄩

(71)

is a bounded invertible linear operator from 𝐶𝐺,𝑝,𝑀[0, 1] to 𝑐𝐺,𝑝,𝑀. Theorem 9. Let 𝐶𝐺,𝑝 [0, 1] be defined as (61). Then, 𝐶𝐺,𝑝 [0, 1] is the completion of space (𝑃[0, 1], ‖ ⋅ ‖𝐺,𝑝 ), and 𝐶𝜙,𝑝 [0, 1] ⊂ 𝐶𝐺,𝑝 [0, 1] ⊂ 𝐶𝜙,0 [0, 1] .

(72)

Moreover, space 𝐶𝐺,𝑝 [0, 1] is isomorphic to 𝑐𝐺,𝑝 . (63)

This shows that {𝑓𝑛 } also is a Cauchy sequence in 𝐶𝜙,0 [0, 1]. By the completeness of space 𝐶𝜙,0 [0, 1], there is a 𝑓0 ∈ 𝐶𝜙,0 [0, 1] such that ‖𝑓𝑛 − 𝑓0 ‖𝜙 → 0. For each 𝑘 ≥ 0, by the continuity of 𝑠𝑘,𝑝 (𝑓), we can get

𝑘≥0

∞

𝑇 (𝑓) = {𝑓(𝑛) (0)}𝑛=0 ,

(62)

For any 𝑘 ≥ 0, it holds that

Thus, we find that 󵄩󵄩 󵄩 󵄩󵄩𝑓𝑛 − 𝑓0 󵄩󵄩󵄩𝐺,𝑝 = sup 𝑠𝑘,𝑝 (𝑓𝑛 − 𝑓0 ) ≤ 𝜀,

Therefore, the mapping

(64)

The proof of Theorem 9 is similar to that of Theorem 8; we omit the details. 3.2. Comparison of Spaces. So far, we have introduced some Banach spaces. The question is whether there is an inclusion relation 𝐶𝜙,𝑞 [0, 1] ⊂ 𝐶𝐺,𝑝 [0, 1] for 𝑞 > 𝑝. In general, the answer is negative. Example 10. Let 𝑞 > 𝑝 ≥ 1. Then, 𝐶𝜙,𝑞 [0, 1] ⊄𝐶𝐺,𝑝,𝑀[0, 1]

∀𝑛 > 𝑁 (𝜀) . (65)

Take 𝑝󸀠 ∈ (𝑝, 𝑞). Define a function by 󸀠

This means that the sequence converges to 𝑓0 in the sense of norm on 𝐶𝐺,𝑝,𝑀[0, 1]. Since 𝑠𝑘,𝑝 (𝑓) is a seminorm, it holds that 𝑠𝑘,𝑝 (𝑓0 ) ≤ 𝑠𝑘,𝑝 (𝑓𝑛 − 𝑓0 ) + 𝑠𝑘,𝑝 (𝑓𝑛 ) < 𝜀 + 𝑠𝑘,𝑝 (𝑓𝑛 ) ,

for 𝑛 > 𝑁 (𝜀) .

(66)

∞

𝑛=0

Obviously, 𝑓 ∈ 𝐶𝜙,𝑞 [0, 1], but 𝑓 ∉ 𝐶𝜙,𝑝 [0, 1]. Furthermore, we also have 𝑓 ∉ 𝐶𝐺,𝑝,𝑀[0, 1]. In fact, for any 𝑛 ∈ N, 1 1/𝑝 𝐷𝑛 𝑓 (0) = ( ) . 𝑛

∞

∞

1 󵄨 󵄨𝑞 ∑ 󵄨󵄨󵄨𝐷𝑛 𝑓 (0)󵄨󵄨󵄨 = ∑ ( ) 𝑛 𝑛=1 𝑛=1

∞ 󵄨 󵄩 𝑘 󵄩 󵄨 󵄨 1 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 𝑘 󵄨󵄨𝑎𝑘 󵄨󵄨 = 󵄨󵄨𝐷 𝑓 (0)󵄨󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩𝐷 𝑓󵄩󵄩󵄩󵄩∞ ≤ ∑ 󵄨󵄨󵄨𝑎𝑛+𝑘 󵄨󵄨󵄨 ≤ 𝑒 sup 󵄨󵄨󵄨𝑎𝑘+𝑛 󵄨󵄨󵄨 . 𝑛! 𝑛≥0

(67) Clearly, 󵄨󵄨 󵄨 󵄩 𝑘+𝜎(0) 󵄩󵄩󵄩 󵄨 󵄨 𝑓󵄩󵄩∞ ≤ 𝑒 󵄨󵄨󵄨𝑎𝑘+𝜎(0) 󵄨󵄨󵄨 , 󵄨󵄨𝑎𝑘+𝜎(0) 󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩𝐷 󵄨󵄨 󵄨󵄨󵄨𝑎 󵄨󵄨 󵄩󵄩 𝑘+𝜎(𝑗) 𝑓󵄩󵄩󵄩 ≤ 𝑒 󵄨󵄨󵄨𝑎 󵄨󵄨 𝑘+𝜎(𝑗) 󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝐷 󵄩󵄩∞ 󵄨󵄨 𝑘+𝜎(𝑗) 󵄨󵄨󵄨 .

< ∞.

(75)

󸀠

󸀠

∞ ∞ 1 1/𝑝 𝑥𝑘−𝑛 1 1/𝑝 𝑥𝑟 = ∑( 𝐷𝑛 𝑓 (𝑥) = ∑ ( ) ) , 𝑘 𝑟! (𝑘 − 𝑛)! 𝑟=0 𝑛 + 𝑟 𝑘=1

󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝐷 𝑓󵄩󵄩∞ ≤ ∑

𝑘=0 (𝑛

(68)

1 + 𝑘)1/𝑝

󸀠

1 , 𝑘!

(76)

∀𝑛 ≥ 1,

it holds that 󸀠

∞

(69)

This implies that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩{𝑎𝑛 }󵄩󵄩󵄩𝐺,𝑝 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐺,𝑝 ≤ 𝑒󵄩󵄩󵄩{𝑎𝑛 }󵄩󵄩󵄩𝐺,𝑝 .

𝑞/𝑝󸀠

Due to

∞

Thus, we have 𝑠𝑘,𝑝 ≤ 𝑠𝑘,𝑝 (𝑓) ≤ 𝑒 𝑠𝑘,𝑝 .

(74)

Clearly,

𝑥𝑛 , 𝑛!

𝑛=0

(73)

󸀠

Therefore, 𝑓0 ∈ 𝐶𝐺,𝑝,𝑀[0, 1], which implies that 𝐶𝜙,𝑝,𝑀[0, 1] is a Banach space. For each 𝑓 ∈ 𝐶𝐺,𝑝,𝑀[0, 1], 𝑓 (𝑥) = ∑ 𝑎𝑛

∞ 1 1/𝑝 𝑥𝑘 𝑓 (𝑥) = ∑ ( ) . 𝑘 𝑘! 𝑘=1

𝑞

∞ ∞ ∞ 1 1/𝑝 1 󵄨𝑞 󵄨 󵄩 󵄩𝑞 ) ) ∑ 󵄨󵄨󵄨𝐷𝑛 𝑓 (0)󵄨󵄨󵄨 ≤ ∑ 󵄩󵄩󵄩𝐷𝑛 𝑓󵄩󵄩󵄩∞ ≤ ∑ ( ∑ ( 𝑘! 𝑛=1 𝑛=1 𝑛=1 𝑘=0 𝑛 + 𝑘

< ∞. (70)

(77)

8

Journal of Function Spaces and Applications For any 𝑛 ∈ N,

Example 12. It holds that 𝐶𝐺,𝑝 [0, 1] ⊂ 𝐶𝐺,𝑝,𝑀[0, 1]. 󸀠

𝑛 𝑛 1 1 𝑝/𝑝 1 𝑛 󵄨 󵄨𝑝 = 𝑝/𝑝󸀠 ∑ , ) ∑ 󵄨󵄨󵄨󵄨𝐷𝑛+𝑘 𝑓 (0)󵄨󵄨󵄨󵄨 = ∑ ( 𝑝/𝑝󸀠 𝑛 + 𝑘 𝑛 𝑘=1 𝑘=1 𝑘=1 (1 + (𝑘/𝑛)) (78)

while

Let us consider the following function: ∞

𝑥𝑛 . 1/𝑝 𝑛! 𝑛=0 (1 + 𝑛)

(79)

this shows that sup𝑛≥1 ∑𝑛𝑘=1 |𝐷𝑛+𝑘 𝑓(0)|𝑝 = ∞, and hence sup𝑛≥0 𝑠𝑛,𝑝 (𝑓) = ∞. So, 𝑓 ∉ 𝐶𝐺,𝑝,𝑀[0, 1]. Example 11. It holds that 𝐶𝜙,𝑝 [0, 1] ≠ 𝐶𝐺,𝑝 [0, 1].

𝑛=1

, (1 + 𝑛)1/𝑝 󵄨 󵄩 𝑛 󵄩 󵄨 󵄨 𝑛 󵄨󵄨 𝑛 󵄨󵄨𝐷 𝑓 (0)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝐷 𝑓󵄩󵄩󵄩∞ ≤ 𝑒 󵄨󵄨󵄨𝐷 𝑓 (0)󵄨󵄨󵄨 , 𝑛

1 1 𝑑𝑥 =∫ , 𝑛→∞ (1 + 𝑘 + 𝑛) 0 1+𝑥 𝑘=1

(80)

Obviously, 1/𝑝 1 ) , (1 + 𝑘) ln (1 + 𝑘)

(86)

this means that 𝑓 ∈ 𝐶𝐺,𝑝,𝑀[0, 1], but 𝑓 ∉ 𝐶𝐺,𝑝 [0, 1]. A new question is whether there is a inclusion relation between 𝐶𝐺,𝑝,𝑀[0, 1] and 𝐶𝐺,𝑞 [0, 1] for any 𝑞 > 𝑝? The following result gives a positive answer. Theorem 13. For any 𝑞 > 𝑝 ≥ 1, the inclusion 𝐶𝐺,𝑝,𝑀[0, 1] ⊂ 𝐶𝐺,𝑞 [0, 1] holds true.

𝑛 𝑛 󵄨 󵄨𝑝 ∑ 󵄨󵄨󵄨󵄨𝐷𝑘 𝑓 (0)󵄨󵄨󵄨󵄨 = ∑

1 + 𝑘) ln (1 (1 + 𝑘) 𝑘=1

𝑘=1

(85)

we have lim ∑

1/𝑝 𝑛 𝑥 1 ) . 𝑛! (1 + 𝑛) ln (1 + 𝑛)

𝐷𝑘 𝑓 (0) = (

1

𝐷𝑛 𝑓 (0) =

Define a function as ∞

(84)

Since

1 1 𝑛 1 𝑑𝑥 lim ∑ = ∫ 󸀠 󸀠 , 𝑝/𝑝 𝑛→∞𝑛 0 (1 + 𝑥)𝑝/𝑝 𝑘=1 (1 + (𝑘/𝑛))

𝑓 (𝑥) = ∑ (

1

𝑓 (𝑥) = ∑

(81)

Proof. Set 𝑓 ∈ 𝐶𝐺,𝑝,𝑀[0, 1]. Then, ∞

𝑓 (𝑥) = ∑ 𝑎𝑛

𝑛

1 = ∞. 𝑛→∞ + 𝑘) ln (1 (1 + 𝑘) 𝑘=1 lim ∑

𝑛=0

𝑥𝑛 , 𝑛!

(87)

and for any 𝑛 ∈ N,

So, 𝑓 ∉ 𝐶𝜙,𝑝 [0, 1].

∞ 󵄨 󵄩 󵄨 󵄨 1 󵄨 󵄨 󵄩 󵄨󵄨 𝑛 󵄨󵄨𝐷 𝑓 (0)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝐷𝑛 𝑓󵄩󵄩󵄩∞ ≤ ∑ 󵄨󵄨󵄨𝑎𝑛+𝑘 󵄨󵄨󵄨 ≤ 𝑒 sup 󵄨󵄨󵄨𝑎𝑛+𝑘 󵄨󵄨󵄨 𝑘!

Furthermore, 󵄨 󵄩 𝑛 󵄩 󵄨󵄨 𝑛 󵄨󵄨𝐷 𝑓 (0)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝐷 𝑓󵄩󵄩󵄩∞

𝑘=0

𝑘≥0

(88)

󵄨 󵄨 = 𝑒 󵄨󵄨󵄨𝑎𝑛+𝜎(0) 󵄨󵄨󵄨 .

1/𝑝 1 1 ) + 𝑛 + 𝑘) ln + 𝑛 + 𝑘) 𝑘! (1 (1 𝑘=1 󵄨 󵄨 ≤ 𝑒 󵄨󵄨󵄨𝐷𝑛 𝑓 (0)󵄨󵄨󵄨 , ∞

≤ ∑(

From above, we get that 󵄨󵄨 󵄨󵄨󵄨𝐷𝑛+𝜎(0) 𝑓 (0)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝐷𝑛+𝜎(0) 𝑓󵄩󵄩󵄩 ≤ 𝑒 󵄨󵄨󵄨𝑎 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩∞ 󵄨 𝑛+𝜎(0) 󵄨󵄨 . Similarly, for 𝑘 ∈ {1, 2, . . . , 𝑛}, 󵄨󵄨 𝑛+𝜎(𝑘) 󵄨 󵄩 󵄩 󵄨 󵄨 󵄨󵄨𝐷 𝑓 (0)󵄨󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩𝐷𝑛+𝜎(𝑘) 𝑓󵄩󵄩󵄩󵄩∞ ≤ 𝑒 󵄨󵄨󵄨𝑎𝑛+𝜎(𝑘) 󵄨󵄨󵄨 . 󵄨

𝑛 󵄨𝑝 󵄨 ∑ 󵄨󵄨󵄨󵄨𝐷𝑛+𝑘 𝑓 (0)󵄨󵄨󵄨󵄨

𝑘=1

𝑛

1 =∑ (𝑛 + 𝑘) ln (𝑛 + 𝑘) 𝑘=1

(89)

(90)

Since 𝑓 ∈ 𝐶𝐺,𝑝,𝑀[0, 1], it holds that

1 1 𝑛 . = ∑ 𝑛 ln 𝑛 𝑘=1 (1+(𝑘/𝑛)) (1 + (ln (1+(𝑘/𝑛)) / ln 𝑛))

𝑛 󵄩 󵄩𝑝 󵄩 󵄩 ∑ 󵄩󵄩󵄩󵄩𝐷(𝑛+𝜎(𝑘)) 𝑓󵄩󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐺,𝑝 ,

∀𝑛 ∈ N.

(91)

𝑘=1

Let 𝑞 > 𝑝 ≥ 1. Set 𝑞 = 𝑝 + 𝛿, then (82) Notice the identity 𝑛

𝑘=1 1

1 1 𝑑𝑥 =∫ , ∑ 𝑛→∞𝑛 1 +𝑥 + + + / ln 𝑛)) (1 (𝑘/𝑛)) (1 (ln (1 (𝑘/𝑛)) 0 𝑘=1 (83) lim

and so 𝑓 ∈ 𝐶𝐺,𝑝 [0, 1].

𝑛 󵄩 󵄩𝑞 ∑ 󵄩󵄩󵄩󵄩𝐷(𝑛+𝜎(𝑘)) 𝑓󵄩󵄩󵄩󵄩∞ 𝑛 󵄩 󵄩𝛿 󵄩 󵄩𝑝 = ∑ 󵄩󵄩󵄩󵄩𝐷(𝑛+𝜎(𝑘)) 𝑓󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩𝐷(𝑛+𝜎(𝑘)) 𝑓󵄩󵄩󵄩󵄩∞ 𝑘=1

󵄩 󵄩𝛿 󵄩 󵄩𝑝 ≤ 󵄩󵄩󵄩󵄩𝐷𝑛+𝜎(0) 𝑓󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐺,𝑝 ,

∀𝑛 ∈ N.

(92)


9

Since 󵄩 󵄩 lim 󵄩󵄩𝐷𝑛 𝑓󵄩󵄩󵄩∞ = 0, 𝑛 → ∞󵄩

(93)

𝑛 󵄩 󵄩𝑞 lim ∑ 󵄩󵄩󵄩󵄩𝐷(𝑛+𝜎(𝑘)) 𝑓󵄩󵄩󵄩󵄩∞ = 0. 𝑛→∞

(94)

it holds that

𝑘=1

Since 𝑠𝑘 (𝑓) is a seminorm, the above relation shows that ‖𝑓‖𝑔,1 is a norm. In what follows, we shall prove that 𝐶𝜙,𝑔,1 [0, 1] is a Banach space. Let 𝑓𝑛 ∈ 𝐶𝜙,𝑔,1 [0, 1] be a Cauchy sequence, that is, 󵄩󵄩

lim 󵄩𝑓 𝑚,𝑛 → ∞󵄩 𝑛

3.3. 𝐶𝜙,𝑔,1 [0, 1]-Type Space. In the previous discussion, we insert the some spaces between 𝐶𝜙,𝑝 [0, 1] and 𝐶𝜙,𝑞 [0, 1]. The questions is can one insert a Banach space between 𝑃[0, 1] and 𝐶𝜙,1 [0, 1]? For each 𝑓 ∈ 𝐶𝜙,1 [0, 1], whose norm is ∞

󵄩 𝑛 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝜙,1 = ∑ 󵄩󵄩󵄩𝐷 𝑓󵄩󵄩󵄩∞ ,

(95)

𝑛=0

(103)

this implies that {𝑓𝑛 } ⊂ 𝐶𝜙,1 [0, 1] also is a Cauchy sequence, so there exists a 𝑓0 ∈ 𝐶𝜙,1 [0, 1] such that 󵄩󵄩

lim 󵄩𝑓 𝑚 → ∞󵄩 𝑚

󵄩 − 𝑓0 󵄩󵄩󵄩𝜙,1 = 0.

(104)

For any 𝑘 ∈ N, when 𝑁 is large enough, we have 𝑘

∀𝑛, 𝑚 > 𝑁.

(105)

𝑗=0

𝑘 󵄩 󵄩 𝑠𝑘 (𝑓) = ∑𝜎 {𝐷𝑘+𝑛 𝑓, 𝑛 ≥ 0} = ∑󵄩󵄩󵄩󵄩𝐷𝑘+𝜎(𝑗) 𝑓󵄩󵄩󵄩󵄩∞ ,

∀𝑘 ≥ 0.

𝑗=0

𝑘

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑓𝑛 − 𝑓𝑛 󵄩󵄩󵄩𝜙,1 ≤ 󵄩󵄩󵄩𝑓𝑛 − 𝑓𝑚 󵄩󵄩󵄩𝑔,1 ,

󵄩 󵄩 ∑𝑠𝑗 (𝑓𝑛 − 𝑓𝑚 ) ≤ 󵄩󵄩󵄩𝑓𝑛 − 𝑓𝑚 󵄩󵄩󵄩𝑔,1 < 𝜀,

the corresponding 𝑠-sequence is

(96)

Using the continuity of 𝑠𝑘 (𝑓) with respect to ‖𝑓|𝜙,1 , and taking 𝑚 → ∞, we get that 𝑘

∑𝑠𝑗 (𝑓𝑛 − 𝑓0 ) ≤ 𝜀,

Define a positive number ∞

(97)

𝑘=0

where (𝑔, 1) denotes the ℓ1 sum after the generalized 1summation. Define the function space by 󵄩 󵄩 𝐶𝜙,𝑔,1 [0, 1] = {𝑓 ∈ 𝐶𝜙,1 [0, 1] | 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑔,1 < ∞} .

Theorem 14. Let ‖𝑓‖𝑔,1 be defined as before. Then, for any 𝑓 ∈ 𝐶𝜙,𝑔,1 [0, 1], (1) the following inequality holds 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝑔,1 ≥ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝜙,1 , ∀𝑓 ∈ 𝐶𝜙,𝑔,1 [0, 1] ; (2) ‖𝑓‖𝑔,1 is a norm on 𝐶𝜙,𝑔,1 [0, 1]. (𝐶𝜙,𝑔,1 [0, 1], ‖ ⋅ ‖𝑔,1 ) is a Banach space;

󵄩󵄩 󵄩 󵄩󵄩𝑓𝑛 − 𝑓0 󵄩󵄩󵄩𝑔,1 ≤ 𝜀,

(99)

∀𝑛 > 𝑁.

(107)

So, 𝑓0 ∈ 𝐶𝜙,𝑔,1 [0, 1], and 󵄩󵄩

lim 󵄩𝑓 𝑛 → ∞󵄩 𝑛

󵄩 − 𝑓0 󵄩󵄩󵄩𝑔,1 = 0.

(108)

The completeness of the space follows. Let 𝑝(𝑥) be a 𝑛-order polynomial. Then, 󵄨 󵄨 𝑠0 (𝑝) ≥ sup 󵄨󵄨󵄨𝑝 (𝑥)󵄨󵄨󵄨 , 𝑘≥0

󵄩 󵄩 𝑠𝑛 (𝑝) = 󵄩󵄩󵄩𝐷𝑛 𝑝󵄩󵄩󵄩∞ ,

󵄩 󵄩 𝑠𝑘 (𝑝) ≤ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩𝜙,1 , (109) 𝑠𝑛+𝑘 (𝑝) = 0,

𝑘 ≥ 1.

Therefore, Further,

𝑛

󵄩󵄩 󵄩󵄩 󵄩󵄩𝑝󵄩󵄩𝑔,1 = ∑ 𝑠𝑘 (𝑝) < ∞.

(3) 𝑃[0, 1] ⊂ 𝐶𝜙,𝑔,1 [0, 1] is a dense subspace.

(110)

𝑘=0

For any 𝑓 ∈ 𝐶𝜙,𝑔,1 [0, 1],

Proof. For any 𝑓 ∈ 𝐶𝜙,1 [0, 1], we have ∀𝑘 ≥ 1.

(100)

∞

𝑓 (𝑥) = ∑ 𝑎𝑗 𝑗=0

This leads to 𝑛 𝑛 󵄩 󵄩 ∑ 𝑠𝑘 (𝑓) ≥ ∑ 󵄩󵄩󵄩󵄩𝐷𝑘 𝑓󵄩󵄩󵄩󵄩∞ .

(106)

Again taking 𝑘 → ∞, we get that

(98)

Then, we have the following result.

󵄩 󵄩 𝑠𝑘 (𝑓) ≥ 󵄩󵄩󵄩󵄩𝐷𝑘 𝑓󵄩󵄩󵄩󵄩∞ ,

∀𝑛 ≥ 𝑁.

𝑗=0

󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝑔,1 = ∑ 𝑠𝑘 (𝑓) ,

𝑘=0

(102)

Since

This gives that 𝑓 ∈ 𝐶𝐺,𝑞 [0, 1].

󵄩 󵄩 𝑠0 (𝑓) ≥ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩∞ ,

󵄩 − 𝑓𝑚 󵄩󵄩󵄩𝑔,1 = 0.

(101)

𝑘=0

Obviously, when 𝑓 ∈ 𝐶𝜙,𝑔,1 [0, 1], it holds that ‖𝑓‖𝜙,1 ≤ ‖𝑓‖𝑔,1 .

𝑥𝑗 , 𝑗!

(111)

for 𝜀 > 0, there exists a 𝑁 such that 𝑒 󵄨 󵄨 ∞ sup 󵄨󵄨󵄨󵄨𝑎𝑗+𝑘 󵄨󵄨󵄨󵄨 + ∑ 𝑠𝑘 (𝑓) < 𝜀, 2 𝑗≥𝑛+1 𝑘=𝑛

∀𝑛 > 𝑁.

(112)

10


References

Taking a polynomial 𝑛

𝑝𝑛 (𝑥) = ∑ 𝑎𝑗 𝑗=0

𝑥𝑗 𝑗!

(113)

and noticing ∞ 󵄩󵄩 󵄩 1 󵄩󵄩(𝑓 − 𝑝𝑛 )(𝑘) 󵄩󵄩󵄩 ≤ ∑ 󵄨󵄨󵄨󵄨𝑎𝑗 󵄨󵄨󵄨󵄨 󵄩󵄩 󵄩󵄩∞ 󵄨 󵄨 (𝑗 − 𝑘)! 𝑗=𝑛

1 󵄨 󵄨∞ , ≤ sup 󵄨󵄨󵄨󵄨𝑎𝑗 󵄨󵄨󵄨󵄨 ∑ (𝑗 − 𝑘)! 𝑗≥𝑛

(114) 𝑘 ≤ 𝑛,

𝑗=𝑛

and (𝑓 − 𝑝𝑛 )(𝑘) = 𝑓(𝑘) for 𝑘 > 𝑛, we have 𝑒 󵄨 󵄨 ∞ 󵄩󵄩 󵄩 󵄩󵄩𝑓 − 𝑝𝑛 󵄩󵄩󵄩𝑔,1 ≤ sup 󵄨󵄨󵄨󵄨𝑎𝑗+𝑘 󵄨󵄨󵄨󵄨 + ∑ 𝑠𝑘 (𝑓) < 𝜀. 2 𝑗≥𝑛+1 𝑘=𝑛

(115)

This means that 𝑃[0, 1] is dense in 𝐶𝜙,𝑔,1 [0, 1]. 3.4. Some Open Questions. From the previous discussion we see that 𝐶𝜙,1 [0, 1] ⊂ 𝐶𝜙,𝑝 [0, 1] ⊂ 𝐶𝜙,𝑞 [0, 1] ⊂ 𝐶𝜙,𝑀 [0, 1] ,

∀1 < 𝑝 < 𝑞,

𝐶𝜙,𝑝 [0, 1] ⊂ 𝐶𝐺,𝑝 [0, 1] ⊂ 𝐶𝐺,𝑝,𝑀 [0, 1] ⊂ 𝐶𝐺,𝑞 [0, 1] ⊂ 𝐶𝐺,𝑞,𝑀 [0, 1] ,

(116)

∀1 < 𝑝 < 𝑞.

Note that the first relation 𝐶𝜙,𝑞 [0, 1] ⊂ 𝐶𝐺,𝑞 [0, 1] ⊂ 𝐶𝐺,𝑞,𝑀 [0, 1] ,

(117)

and the relation 𝐶𝜙,𝑞 [0, 1] ⊄𝐶𝐺,𝑝,𝑀[0, 1] (see, Example 10). These spaces are based on the new summation approach. Now, we propose some open questions in mathematics aspect. Question 1. Does the inclusion 𝐶𝐺,𝑝,𝑀[0, 1] ⊂ 𝐶𝜙,𝑞 [0, 1] hold? yes or no. Question 2. What is the dual space of 𝐶𝐺,𝑝 [0, 1]? Is it a reflexive space? ̃ 1], ‖ ⋅ ‖∗ ), Question 3. Can one find out a Banach space (𝑃[0, ̃ 1] is the comthat is, the least in the following sense: 𝑃[0, pletion of 𝑃[0, 1] under the norm ‖ ⋅ ‖∗ such that, for every ̃ 1] embeds densely and Banach space X of functions, 𝑃[0, continuously into X?

Acknowledgment The research is supported by the Natural Science Foundation of China (NSFC-61174080). The work has been completed during the visit of Gen-Qi Xu in the University of Lyon, April 2013.

[1] E. Bornberg-Bauer, “How are model protein structures distributed in sequence space?” Biophysical Journal, vol. 73, no. 5, pp. 2393–2403, 1997. [2] J. Fan, T. Huang, and R. Li, “Analysis of longitudinal data with semiparametric estimation of convariance function,” Journal of the American Statistical Association, vol. 102, no. 478, pp. 632– 641, 2007. [3] E. Levina, A. J. Rothman, and J. Zhu, “Sparse estimation of large covariance matrices via a nested Lasso penalty,” Annals of Applied Statistics, vol. 2, no. 1, pp. 245–263, 2008. [4] N. El Karoui, “Operator norm consistent estimation of largedimensional sparse covariance matrices,” The Annals of Statistics, vol. 36, no. 6, pp. 2717–2756, 2008. [5] A. J. Rothman, E. Levina, and J. Zhu, “A new approach to Cholesky-based covariance regularization in high dimensions,” Biometrika, vol. 97, no. 3, pp. 539–550, 2010. [6] A. Delcroix, M. F. Hasler, S. Pilipović, and V. Valmorin, “Generalized function algebras as sequence space algebras,” Proceedings of the American Mathematical Society, vol. 132, no. 7, pp. 2031–2038, 2004. [7] N. Subramanian and U. K. Misra, “The semi normed space defined by a double gai sequence of modulus function,” Fasciculi Mathematici, no. 45, pp. 111–120, 2010. [8] M. Kiris¸çi and F. Bas¸ar, “Some new sequence spaces derived by the domain of generalized difference matrix,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1299–1309, 2010.


Research Article Semicommutators and Zero Product of Block Toeplitz Operators with Harmonic Symbols Puyu Cui,1 Yufeng Lu,1 and Yanyue Shi2 1 2

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Correspondence should be addressed to Yanyue Shi; [email protected] Received 15 February 2013; Accepted 17 August 2013 Academic Editor: Gen-Qi Xu Copyright © 2013 Puyu Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We completely characterize the finite rank semicommutators, commutators, and zero product of block Toeplitz operators 𝑇𝐹 and 𝑇𝐺 with 𝐹, 𝐺 ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 on the vector valued Bergman space 𝐿2𝑎 (D, C𝑛 ).

1. Introduction Let D be the open unit disk in the complex plane C, and let 𝑑𝐴 be the normalized Lebesgue area measure on D. Denote by 𝐿∞ (D, 𝑑𝐴) and 𝐿2 (D, 𝑑𝐴) the space of essential bounded measurable functions and the space of square integral functions with respect to 𝑑𝐴, respectively. The Bergman space 𝐿2𝑎 consists of all analytic functions in 𝐿2 (D, 𝑑𝐴). Let 𝐿2 (D, C𝑛 ) = 𝐿2 (D, 𝑑𝐴) ⊗ C𝑛 and 𝐿2𝑎 (D, C𝑛 ) = 𝐿2𝑎 ⊗ C𝑛 , where ⊗ means the Hilbert space tensor product. Let 𝑀𝑛×𝑛 be the set of all 𝑛 × 𝑛 complex matrixes in the complex plane C, and let 𝐿∞ (D, 𝑑𝐴) ⊗ 𝑀𝑛×𝑛 be the space of matrix valued essential bounded Lebesgue measurable functions on D. Given a matrix valued function Φ(𝑧) = [𝜑𝑖𝑗 (𝑧)]𝑛×𝑛 ∈ 𝐿∞ (D, 𝑑𝐴) ⊗ 𝑀𝑛×𝑛 , the Toeplitz operator 𝑇Φ and the Hankel operator 𝐻Φ on 𝐿2𝑎 (D, C𝑛 ) with symbol Φ are defined by 𝑇Φ ℎ = 𝑃(Φℎ) and 𝐻Φ ℎ = (𝐼 − 𝑃)(Φℎ), respectively, where 𝑃 is the orthogonal projection from 𝐿2 (D, C𝑛 ) onto 𝐿2𝑎 (D, C𝑛 ). Then the operators 𝑇Φ and 𝐻Φ have the following matrix representations: 𝑇𝜑11 ⋅ ⋅ ⋅ 𝑇𝜑1𝑛 𝐻𝜑11 ⋅ ⋅ ⋅ 𝐻𝜑1𝑛 [ .. [ ] . .. ] . (1) .. ] , 𝐻Φ = [ ... 𝑇Φ = [ . . ] [𝑇𝜑𝑛1 ⋅ ⋅ ⋅ 𝑇𝜑𝑛𝑛 ] [𝐻𝜑𝑛1 ⋅ ⋅ ⋅ 𝐻𝜑𝑛𝑛 ] The problem of the finite rank semicommutator of two Toeplitz operators on the Hardy space has been completely solved in [1, 2]. The analogous problems on the Bergman

space have been characterized in [3–5]. In the case of vector valued Hardy space, Gu and Zheng [6] have studied the problems of compact semicommutators and commutators of two block Toeplitz operators. In this paper, we study the related problems of the two block Toeplitz operators 𝑇𝐹 and 𝑇𝐺 with 𝐹, 𝐺 ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 on the vector valued Bergman space 𝐿2𝑎 (D, C𝑛 ), where ℎ∞ denotes the space of all bounded harmonic functions on D. Our main idea here is to reduce the problems on 𝐿2𝑎 (D, C𝑛 ) to the problems of the finite sum of some Toeplitz operator products or some Hankel operator products on the Bergman space 𝐿2𝑎 . In the following, we recall two useful theorems. For every 𝑓 ∈ ℎ∞ , denote by 𝑓+ (or 𝑓− = 𝑓 − 𝑓+ ) the analytic part (or the coanalytic part) of 𝑓. Theorem 1 (see [7]). Suppose that 𝑓𝑗 and 𝑔𝑗 are bounded harmonic functions on D for 𝑗 = 1, . . . , 𝑘. Then the following are equivalent: (1) ∑𝑘𝑗=1 𝐻𝑔∗𝑗 𝐻𝑓𝑗 has finite rank, (2) ∑𝑘𝑗=1 𝐻𝑔∗𝑗 𝐻𝑓𝑗 = 0, (3) 𝜎(𝑓1 , . . . , 𝑓𝑘 ; 𝑔1 , . . . , 𝑔𝑘 ) ≡ 0, ̃ 𝑘 (𝑓𝑖 )− (𝑔𝑖 )+ ] = (1 − where 𝜎(𝑓1 , . . . , 𝑓𝑘 ; 𝑔1 , . . . , 𝑔𝑘 ) = Δ[∑ 𝑖=1 |𝑧|2 )2 ∑𝑘𝑖=1 (𝑓𝑖 )󸀠− (𝑔𝑖 )󸀠+ .

2


Theorem 2 (see [8]). Let 𝑢1 , . . . 𝑢𝑛 , V1 , . . . , V𝑛 be bounded harmonic functions on D, and 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝐿2𝑎 . Then ∑𝑛𝑗=1 𝑇𝑢𝑗 𝑇V𝑗 = ∑𝑛𝑗=1 𝑥𝑗 ⊗ 𝑦𝑗 if and only if the following two conditions hold: (a) ∑𝑛𝑗=1 𝑢𝑗 V𝑗 = (1 − |𝑧|2 )2 ∑𝑛𝑗=1 𝑥𝑗 𝑦𝑗 ; (b) ∑𝑛𝑗=1 [𝑃𝑢𝑗 − 𝑢𝑗 (0)][𝑃V𝑗 − V𝑗 (0)] = 0.

2. Semicommutators of Block Toeplitz Operators In this section, we discuss the finite rank semicommutators and commutators of the block Toeplitz operators 𝑇𝐹 , 𝑇𝐺 on 𝐿2𝑎 (D, C𝑛 ) with 𝐹, 𝐺 ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 . In the following part of this paper, 𝐸𝑗 denote the matrix unit with (𝑗, 𝑗) th entry equal to one and all others equal to zero. Theorem 3. Let 𝐹(𝑧) = 𝐹+ (𝑧) + 𝐹− (𝑧), and let 𝐺(𝑧) = 𝐺+ (𝑧) + 𝐺− (𝑧) ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 such that 𝐹+ (0) = 𝐺− (0) = 0𝑛×𝑛 . Then the following statements are equivalent:

Note that det 𝐹+ (𝑧) is the analytic function of 𝑧 and det 𝐺− (𝑤) is the coanalytic function of 𝑤. If det 𝐹+ (𝑧) is not identically zero, then 𝐺− = 0𝑛×𝑛 ; that is, 𝐺 is analytic. If det 𝐺− (𝑤) is not identically zero, then 𝐹+ = 0𝑛×𝑛 ; that is, 𝐹 is coanalytic. In the following, we deal with the case that det 𝐹+ ≡ 0 𝑗 and det 𝐺− ≡ 0. Suppose 𝐺− 𝐸𝑗 = (𝑔1 , . . . , 𝑔𝑛𝑗 )𝑇 for 1 ≤ 𝑗 ≤ 𝑛. Since det 𝐺− ≡ 0, 𝐺− 𝐸𝑗 is linearly dependent. Let 𝑄𝑗 be a permutation matrix such that

𝑗

𝑇

𝑗

𝑗

(3)

𝑗

where {𝑔𝜎(1) , . . . , 𝑔𝜎(𝑘 ) } is maximally linearly independent of 𝑗

0

𝑗

{𝑔𝜎(1) , . . . , 𝑔𝜎(𝑛) }. And 𝑗

𝑗

𝑗

𝑔𝜎 𝑘 +1 = 𝑎11 𝑔𝜎(1) + ⋅ ⋅ ⋅ + 𝑎𝑘10 𝑔𝜎 𝑘 , (0 ) ( 0) .. .

(1) 𝑇𝐹𝐺 − 𝑇𝐹 𝑇𝐺 has finite rank; (2) 𝑇𝐹𝐺 = 𝑇𝐹 𝑇𝐺; (3) for each 1 ≤ 𝑗 ≤ 𝑛, there exist a matrix 𝑀𝑗 ∈ 𝑀𝑛×𝑛 and a permutation matrix 𝑄𝑗 , such that 𝑀𝑗 𝑄𝑗 𝐺− 𝐸𝑗 = 0𝑛×𝑛 and 𝐹+ 𝑄𝑗 (𝐼 + 𝑀𝑗 ) = 0𝑛×𝑛 . Proof. (1) ⇒ (2) Note that 𝑇𝐹𝐺 − 𝑇𝐹 𝑇𝐺 = 𝐻𝐹∗∗ 𝐻𝐺. If 𝐻𝐹∗∗ 𝐻𝐺 has a finite rank, then ∑𝑛𝑝=1 𝐻𝑓∗ 𝐻𝑔𝑝𝑗 has a finite rank for 1 ≤

𝑗

𝑄𝑗 𝐺− 𝐸𝑗 = (𝑔𝜎(1) , . . . , 𝑔𝜎 𝑘 , . . . , 𝑔𝜎(𝑛) ) , ( 0)

𝑗

𝑛−𝑘0 𝑗 𝑔𝜎(1)

𝑔𝜎(𝑛) = 𝑎1

(4)

𝑛−𝑘0 𝑗 𝑔𝜎 𝑘 , 0 ( 0)

+ ⋅ ⋅ ⋅ + 𝑎𝑘

with the matrix representation

𝑖𝑝

𝑖, 𝑗 ≤ 𝑛. By Theorem 1, we have that ∑𝑛𝑝=1 𝐻𝑓∗ 𝐻𝑔𝑝𝑗 = 0 for 𝑖𝑝

1 ≤ 𝑖, 𝑗 ≤ 𝑛. Hence, 𝑇𝐹𝐺 = 𝑇𝐹 𝑇𝐺. (2) ⇒ (3) If 𝑇𝐹𝐺 − 𝑇𝐹 𝑇𝐺 = 𝐻𝐹∗∗ 𝐻𝐺 = 0, then [𝐻𝐹∗∗ 𝐻𝐺]𝑖,𝑗 = 𝑛 ∑𝑝=1 𝐻𝑓∗ 𝐻𝑔𝑝𝑗 = 0, for 1 ≤ 𝑖, 𝑗 ≤ 𝑛. By Theorem 1, 𝑖𝑝

2 2

∑𝑛𝑝=1 [(𝑓𝑖𝑝 )󸀠+ (𝑔𝑝𝑗 )󸀠− ]

= 0 on D. Therefore, we get (1 − |𝑧| ) ∑𝑛𝑝=1 [(𝑓𝑖𝑝 )󸀠+ (𝑔𝑝𝑗 )󸀠− ] = 0 on D for 1 ≤ 𝑖, 𝑗 ≤ 𝑛. It is known that if 𝐻(𝑧, 𝑤) is holomorphic in D × D and 𝐻(𝑧, 𝑧) = 0, then 𝐻(𝑧, 𝑤) ≡ 0 for (𝑧, 𝑤) ∈ D × D in several complex variables. So we can “complexificate” the above identity and have ∑𝑛𝑝=1 [(𝑓𝑖𝑝 )󸀠+ (𝑧)(𝑔𝑝𝑗 )󸀠− (𝑤)] = 0 for 1 ≤ 𝑖, 𝑗 ≤ 𝑛, (𝑧, 𝑤) ∈ D × D. Therefore, 𝑛

󸀠

󸀠

0 = ∑ [(𝑓𝑖𝑝 )+ (𝑢) (𝑔𝑝𝑗 )− (V)]

0 [ . [ . [ . [ [ 0 [ 1 [ 𝑎1 [ [ . [ .. [ 𝑛−𝑘 𝑎 0 [ 1

𝑗

𝑔𝜎(1) ][ .. ][ . ][ [ ][ 𝑗 ] [ 𝑔𝜎 𝑘 ( 0) ][ ][ .. ][ . ][ ][ 𝑗 ] [ 𝑔𝜎(𝑛−1) 𝑗 0 ⋅ ⋅ ⋅ −1 ] [ 𝑔𝜎(𝑛)

0 .. . 0 0 ⋅⋅⋅ −1 ⋅ ⋅ ⋅ 0 .. .. . . 0 ⋅⋅⋅ .. .

⋅⋅⋅ 0 .. .

⋅⋅⋅ 0 ⋅ ⋅ ⋅ 𝑎𝑘10 .. . 𝑛−𝑘 ⋅ ⋅ ⋅ 𝑎𝑘 0 0

] ] ] ] ] ] ] = 0𝑛×𝑛 , (5) ] ] ] ] ] ]

that is, 𝑀𝑗 𝑄𝑗 𝐺− 𝐸𝑗 = 0𝑛×𝑛 . Since 𝐹+ (𝑧)[𝐺− (𝑤)𝐸𝑗 ] = 0, we have [𝐹+ (𝑧)𝑄𝑗 ][𝑄𝑗 𝐺− ×(𝑤)𝐸𝑗 ] = 0𝑛×𝑛 . Denote 𝐹+ = [𝑓𝑖𝑗 ]𝑛×𝑛 . It follows that

𝑝=1

𝑧 𝑛

󸀠

𝑗

󸀠

0 𝑝=1

𝑛

󸀠

= ∑ [(𝑓𝑖𝑝 )+ (𝑧) (𝑔𝑝𝑗 )− (V)] 𝑝=1

𝑤 𝑛

󸀠

= ∫ ∑ [(𝑓𝑖𝑝 )+ (𝑧) (𝑔𝑝𝑗 )− (V)] 𝑑V 0 𝑝=1

𝑛

= ∑ [(𝑓𝑖𝑝 )+ (𝑧) (𝑔𝑝𝑗 )− (𝑤)] , 𝑝=1

which means that 𝐹+ (𝑧)𝐺− (𝑤) = 0𝑛×𝑛 on D × D.

𝑗

𝑗

𝑓𝑘𝜎(1) 𝑔𝜎(1) + ⋅ ⋅ ⋅ + 𝑓𝑘𝜎(𝑘0 ) 𝑔𝜎 𝑘 + ⋅ ⋅ ⋅ + 𝑓𝑘𝜎(𝑛) 𝑔𝜎(𝑛) = 0 ( 0)

= ∫ ∑ [(𝑓𝑖𝑝 )+ (𝑢) (𝑔𝑝𝑗 )− (V)] 𝑑𝑢 (2)

𝑗

𝑗

(6)

𝑗

for 1 ≤ 𝑘 ≤ 𝑛. Note that 𝑔𝜎(𝑘 +𝑙) = 𝑎1𝑙 𝑔𝜎(1) + ⋅ ⋅ ⋅ + 𝑎𝑘𝑙 0 𝑔𝜎(𝑘 ) , 0 0 1 ≤ 𝑙 ≤ 𝑛 − 𝑘0 . Therefore 𝑗

𝑗

𝑓𝑘𝜎(1) 𝑔𝜎(1) + ⋅ ⋅ ⋅ + 𝑓𝑘𝜎(𝑘0 ) 𝑔𝜎 𝑘 ( 0) 𝑛−𝑘0

𝑗

𝑗

+ ∑ (𝑎1𝑙 𝑔𝜎(1) + ⋅ ⋅ ⋅ + 𝑎𝑘𝑙 0 𝑔𝜎 𝑘 ) 𝑓𝑘𝜎(𝑘0 +𝑙) = 0, ( 0) 𝑙=1

(7)


3

𝑗

𝑛−𝑘

𝑛−𝑘

and hence, (𝑓𝑘𝜎(1) + ∑𝑙=1 0 𝑎1𝑙 𝑓𝑘𝜎(𝑘0 +𝑙) )𝑔𝜎(1) + (𝑓𝑘𝜎(2) + ∑𝑙=1 0 𝑎2𝑙 𝑗 𝑗 𝑛−𝑘 ×𝑓𝑘𝜎(𝑘0 +𝑙) )𝑔𝜎(2) + ⋅ ⋅ ⋅ + (𝑓𝑘𝜎(𝑘0 ) + ∑𝑙=1 0 𝑎𝑘𝑙 0 𝑓𝑘𝜎(𝑘0 +𝑙) )𝑔𝜎(𝑘 ) 0 𝑗 𝑗 Since {𝑔𝜎(1) , . . . , 𝑔𝜎(𝑘 ) } is linearly independent, we have 0

= 0.

Corollary 5. Let 𝐹(𝑧) = 𝐹+ (𝑧) + 𝐹− (𝑧), and let 𝐺(𝑧) = 𝐺+ (𝑧) + 𝐺− (𝑧) ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 such that 𝐹+ (0) = 𝐺− (0) = 0𝑛×𝑛 . Suppose that | det 𝐹+ | + | det 𝐺− | is not identically zero. Then the following are equivalent:

𝑛−𝑘0

𝑓𝑘𝜎(1) + ∑ 𝑎1𝑙 𝑓𝑘𝜎(𝑘0 +𝑙) = 0, 𝑙=1

.. . 𝑛−𝑘0

𝑓𝑘𝜎(𝑘0 ) + ∑ 𝑙=1

𝑎𝑘𝑙 0 𝑓𝑘𝜎(𝑘0 +𝑙)

(8)

= 0,

⋅ ⋅ ⋅ 𝑓1𝜎(𝑘0 ) ⋅ ⋅ ⋅ 𝑓2𝜎(𝑘0 ) .. . .. .

𝑛−1𝜎(1)

[ 𝑓𝑛𝜎(1)

⋅ ⋅ ⋅ 𝑓1𝜎(𝑛) ⋅ ⋅ ⋅ 𝑓2𝜎(𝑛) .. . .. .

(3) if det 𝐹+ is not identically zero, then 𝐺 is analytic; if det 𝐺− is not identically zero, then 𝐹 is coanalytic.

... ⋅⋅⋅ ⋅⋅⋅

0 .. .

1 𝑎𝑘10 .. . 𝑛−𝑘 ⋅ ⋅ ⋅ 𝑎𝑘 0 0

Proof. The proof is obvious.

] ] ] ] ] ] ] ] ]

⋅ ⋅ ⋅ 𝑓𝑛−1𝜎(𝑘0 ) ⋅ ⋅ ⋅ 𝑓𝑛−1𝜎(𝑛) ⋅ ⋅ ⋅ 𝑓𝑛𝜎(𝑘0 ) ⋅ ⋅ ⋅ 𝑓𝑛𝜎(𝑛) ]

1 [ .. [ . [ [ 0 [ × [ 𝑎1 [ 1 [ . [ . [ . 𝑛−𝑘0 [𝑎1

0 ⋅⋅⋅ .. .

0 .. ] .] ] 0 ⋅ ⋅ ⋅ 0] ] = 0𝑛×𝑛 , 0 ⋅ ⋅ ⋅ 0] ] ] .. .. ] . .] 0 ⋅ ⋅ ⋅ 0]

Let

(9)

− 𝑇(𝐹+ 𝐺+ +𝐹− 𝐺− +𝐹− 𝐺+ ) − 𝑇𝐹+ 𝐺−

Ψ=[

𝐺 0 ]. 𝐹 0

(12)

𝑇ΦΨ − 𝑇Φ 𝑇Ψ

𝑇𝐹 𝑇𝐺 − 𝑇𝐹𝐺 = 𝑇(𝐹+ +𝐹− ) 𝑇(𝐺+ +𝐺− ) − 𝑇(𝐹+ +𝐹− )(𝐺+ +𝐺− ) = 𝑇(𝐹+ 𝐺+ +𝐹− 𝐺− +𝐹− 𝐺+ ) + 𝑇𝐹+ 𝐺−

𝐹 −𝐺 Φ=[ ], 0 0

As in [6], the commutator 𝑇𝐹 𝑇𝐺 − 𝑇𝐺𝑇𝐹 can be reduced to the semicommutator 𝑇ΦΨ − 𝑇Φ 𝑇Ψ . It is easy to check that

that is 𝐹+ 𝑄𝑗 (𝐼 + 𝑀𝑗 ) = 0𝑛×𝑛 . In the following, we prove (3) ⇒ (1). Note that

(10)

=[

𝑇𝐹𝐺−𝐺𝐹 0 𝑇 −𝑇𝐺 𝑇𝐺 0 ][ ] ]−[ 𝐹 0 0 0 0 𝑇𝐹 0

=[

𝑇𝐹𝐺−𝐺𝐹 0 𝑇 𝑇 − 𝑇𝐺𝑇𝐹 0 ]−[ 𝐹 𝐺 ]. 0 0 0 0

(13)

If 𝑇𝐹 𝑇𝐺 = 𝑇𝐺𝑇𝐹 , Theorem 2 implies that 𝐹𝐺 = 𝐺𝐹. Therefore, 𝑇Φ 𝑇Ψ = 𝑇ΨΦ . Combining Theorem 3, we get the following explicit theorem. Theorem 6. Suppose that 𝐹 = 𝐹+ + 𝐹− ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 and 𝐺 = 𝐺+ + 𝐺− ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 . Then 𝑇𝐹 𝑇𝐺 = 𝑇𝐺𝑇𝐹 if and only if 𝐹 and 𝐺 satisfy the following two conditions:

= 𝑇𝐹+ 𝑇𝐺− − 𝑇𝐹+ 𝐺− . By the hypothesis, we have 𝑇𝐹+ 𝑇𝐺− 𝐸𝑗 = 𝑇𝐹+ 𝑄𝑗 (𝐼+𝑀𝑗 −𝑀𝑗 )𝑄𝑗 𝑇𝐺− 𝐸𝑗

(1) 𝐹𝐺 = 𝐺𝐹;

= 𝑇𝐹+ 𝑄𝑗 (𝐼+𝑀𝑗 )𝑄𝑗 𝑇𝐺− 𝐸𝑗 − 𝑇𝐹+ 𝑄𝑗 𝑀𝑗 𝑄𝑗 𝑇𝐺− 𝐸𝑗 = 𝑇𝐹+ 𝑄𝑗 (𝐼+𝑀𝑗 )𝑄𝑗 𝑇𝐺− 𝐸𝑗 − 𝑇𝐹+ 𝑄𝑗 𝑇𝑀𝑗 𝑄𝑗 𝐺− 𝐸𝑗 = 0𝑛×𝑛 , 𝐹+ 𝐺− 𝐸𝑗 = 𝐹+ 𝑄𝑗 [𝐼 + 𝑀𝑗 − 𝑀𝑗 ] 𝑄𝑗 𝐺− 𝐸𝑗 = [𝐹+ 𝑄𝑗 (𝐼 + 𝑀𝑗 )] 𝑄𝑗 𝐺− 𝐸𝑗

(2) for each 1 ≤ 𝑗 ≤ 𝑛, there exist permutation matrixes 󸀠 ̃𝑗 [ 𝐺−󸀠 ] 𝐸𝑗 = 0𝑛×𝑛 ̃𝑗 and M𝑗 ∈ 𝑀2𝑛×2𝑛 such that M𝑗 𝑄 𝑄 ̃𝑗 (𝐼 + M𝑗 ) = 0𝑛×𝑛 . and [𝐹+󸀠 − 𝐺+󸀠 ]𝑄

𝐹−

3. Zero Product of Two Block Toeplitz Operators

− 𝐹+ 𝑄𝑗 [𝑀𝑗 𝑄𝑗 𝐺− 𝐸𝑗 ] = 0𝑛×𝑛 . (11) Hence, we obtain the desired result.

(1) 𝑇𝐹𝐺 − 𝑇𝐹 𝑇𝐺 has finite rank; (2) 𝑇𝐹𝐺 = 𝑇𝐹 𝑇𝐺;

with the matrix representation 𝑓1𝜎(1) [ 𝑓2𝜎(1) [ [ .. [ . [ [ . [ .. [ [𝑓

Remark 4. If the assumption 𝐹+ (0) = 𝐺− (0) = 0𝑛×𝑛 is removed and 𝐹+ , 𝐺− are replaced by 𝐹+󸀠 , 𝐺−󸀠 in statement (3), respectively, then the above theorem still holds.

In this section, we discuss the zero product of two block Toeplitz operators 𝑇𝐹 and 𝑇𝐺 with 𝐹, 𝐺 ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 on 𝐿2𝑎 (D, C𝑛 ).

4


Theorem 7. Let 𝐹(𝑧) = 𝐹+ (𝑧) + 𝐹− (𝑧) and 𝐺(𝑧) = 𝐺+ (𝑧) + 𝐺− (𝑧) ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 such that 𝐹+ (0) = 𝐺− (0) = 𝐹− (0) = 𝐺+ (0) = 0𝑛×𝑛 . Then 𝑇𝐹 𝑇𝐺 = 0𝑛×𝑛 if and only if (1) 𝐹𝐺 = 0𝑛×𝑛 ; (2) for each 1 ≤ 𝑗 ≤ 𝑛, there exist a matrix 𝑁𝑗 ∈ 𝑀𝑛×𝑛 and a permutation matrix 𝑄𝑗1 , such that 𝑁𝑗 𝑄𝑗1 𝐺− 𝐸𝑗 = 0𝑛×𝑛 and 𝐹+ 𝑄𝑗1 (𝐼 + 𝑁𝑗 ) = 0𝑛×𝑛 . Proof. By Theorem 2, it is easy to know that ∑𝑛𝑝=1 𝑓𝑖𝑝 (𝑧)𝑔𝑝𝑗 (𝑧) = 0 for 1 ≤ 𝑖, 𝑗 ≤ 𝑛; that is, 𝐹𝐺 = 0𝑛×𝑛 . Since 𝑇𝐹 𝑇𝐺 = 𝑇𝐹 𝑇𝐺 − 𝑇𝐹𝐺 = 0𝑛×𝑛 , we obtain the desired result by Theorem 3. Corollary 8. Let 𝐹(𝑧) = 𝐹+ (𝑧) + 𝐹− (𝑧) and let 𝐺(𝑧) = 𝐺+ (𝑧) + 𝐺− (𝑧) ∈ ℎ∞ ⊗ 𝑀𝑛×𝑛 such that 𝐹+ (0) = 𝐺− (0) = 𝐹− (0) = 𝐺+ (0) = 0𝑛×𝑛 . (1) If neither det 𝐺+ nor det 𝐺− is identically zero, then 𝑇𝐹 𝑇𝐺 = 0 if and only if 𝐹 = 0𝑛×𝑛 . (2) If neither det 𝐹+ nor det 𝐹− is identically zero, then 𝑇𝐹 𝑇𝐺 = 0 if and only if 𝐺 = 0𝑛×𝑛 . Proof. (1) We only need to prove the necessity. By the proof of Theorem 3, we have 𝐹+ (𝑧)𝐺− (𝑤) = 0𝑛×𝑛 on D × D. By the theorem given above, we know that 𝐹𝐺 = [𝐹+ +𝐹− ][𝐺+ +𝐺− ] = 0𝑛×𝑛 . Taking Laplace transform, it follows that 𝐹−󸀠 𝐺+󸀠 = 0𝑛×𝑛 . As in Theorem 3, we can prove that 𝐹− (𝑤)𝐺+ (𝑧) = 0𝑛×𝑛 on D × D. “Complexificate” the identity 𝐹𝐺 = 0𝑛×𝑛 , and then [𝐹+ (𝑧) + 𝐹− (𝑤)][𝐺+ (𝑧) + 𝐺− (𝑤)] = 0𝑛×𝑛 . Hence, 𝐹+ (𝑧)𝐺+ (𝑧) = −𝐹− (𝑤)𝐺− (𝑤) on D × D. Since 𝐹+ (0) = 𝐺− (0) = 0𝑛×𝑛 , we have 𝐹+ 𝐺+ = 𝐹− 𝐺− = 0𝑛×𝑛 . It is clear that det 𝐺+ (𝑧) is the analytic function of 𝑧 and det 𝐺− (𝑤) is the coanalytic function of 𝑤. If det 𝐺+ (𝑧) is not identically zero, then 𝐹− ≡ 0𝑛×𝑛 and 𝐹+ ≡ 0𝑛×𝑛 . Therefore, 𝐹 = 0𝑛×𝑛 . If det 𝐺− (𝑤) is not identically zero, then 𝐹− ≡ 0𝑛×𝑛 and 𝐹+ ≡ 0𝑛×𝑛 . Therefore, 𝐹 = 0𝑛×𝑛 . (2) The proof is similar to the proof of (1).

Acknowledgments This research is supported by NSFC (11271059, 11201438), Research Fund for the Doctoral Program of Higher Education of China, Shandong Province Young Scientist Research Award Fund (BS2012SF031), and the Fundamental Research Funds for the Central Universities (201213011).

References [1] S. Axler, S. Y. A. Chang, and D. Sarason, “Products of Toeplitz operators,” Integral Equations and Operator Theory, vol. 1, no. 3, pp. 285–309, 1978. [2] X. Ding and D. Zheng, “Finite rank commutator of Toeplitz operators or Hankel operators,” Houston Journal of Mathematics, vol. 34, no. 4, pp. 1099–1119, 2008. ˇ Cuˇ ˇ cković, “A theorem of Brown-Halmos type [3] P. Ahern and Z. for Bergman space Toeplitz operators,” Journal of Functional Analysis, vol. 187, no. 1, pp. 200–210, 2001.

ˇ Cuˇ ˇ cković, “Commuting Toeplitz operators with [4] S. Axler and Z. harmonic symbols,” Integral Equations and Operator Theory, vol. 14, no. 1, pp. 1–12, 1991. [5] D. C. Zheng, “Hankel operators and Toeplitz operators on the Bergman space,” Journal of Functional Analysis, vol. 83, no. 1, pp. 98–120, 1989. [6] C. Gu and D. Zheng, “Products of block Toeplitz operators,” Pacific Journal of Mathematics, vol. 185, no. 1, pp. 115–148, 1998. [7] K. Guo, S. Sun, and D. Zheng, “Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols,” Illinois Journal of Mathematics, vol. 51, no. 2, pp. 583– 596, 2007. [8] B. R. Choe, H. Koo, and Y. J. Lee, “Sums of Toeplitz products with harmonic symbols,” Revista Mathemática Iberoamericana, vol. 24, no. 1, pp. 43–70, 2008.


Research Article On Supra-Additive and Supra-Multiplicative Maps Jin Xi Chen1,2 and Zi Li Chen2 1 2

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

Correspondence should be addressed to Jin Xi Chen; [email protected] Received 9 May 2013; Revised 28 July 2013; Accepted 14 August 2013 Academic Editor: Gen-Qi Xu Copyright © 2013 J. X. Chen and Z. L. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 𝑏 ∈ 𝐵. Let 𝐴 and 𝐵 be ordered algebras over R, where 𝐴 has a generating positive cone and 𝐵 satisfies the property that 𝑏2 > 0 if 0 ≠ We give some conditions for a map 𝑇 : 𝐴 → 𝐵 which is supra-additive and supra-multiplicative for all positive and negative elements to be linear and multiplicative; that is, 𝑇 is a homomorphism of algebras. Our results generalize some known results on supra-additive and supra-multiplicative maps between spaces of real functions.

Let 𝑋 be a compact Hausdorff space. Rˇadulescu [1] proved that whenever a map 𝑇 : 𝐶(𝑋) → 𝐶(𝑋) satisfies the conditions 𝑇(𝑓 + 𝑔) ≥ 𝑇(𝑓) + 𝑇(𝑔) (supra-additive) and 𝑇(𝑓𝑔) ≥ 𝑇(𝑓)𝑇(𝑔) (supra-multiplicative) for all 𝑓, 𝑔 ∈ 𝐶(𝑋), then 𝑇 is linear and multiplicative. Ercan [2] generalized this result to arbitrary topological spaces. For arbitrary topological spaces 𝑋, 𝑌, a supra-additive and supra-multiplicative map 𝑇 : 𝐶(𝑋) → 𝐶(𝑌) is both linear and multiplicative if and only if lim𝑛 𝑇(𝑓+ ∧ 𝑛 − 𝑓− ∧ 𝑛)(𝑦) = 𝑇(𝑓)(𝑦) for each 𝑓 ∈ 𝐶(𝑋) and 𝑦 ∈ 𝑌 [2, Theorem 2]. Similar results were obtained for rings (see [3, 4]). In [3] Dhombres showed that if 𝐴 is a ring and 𝑅 is an ordered ring in which nonzero elements have nonzero positive squares, the supra-additive and supra-multiplicative map 𝑇 : 𝐴 → 𝑅 is both additive and multiplicative, that is, a homomorphism of rings. Recently, Gusić [5] considered supra-additive and supra-multiplicative maps between ordered fields. It was proved that on ordered fields every supra-additive and supra-multiplicative nonzero map is an injective homomorphism of fields. Let 𝐴 and 𝐵 be ordered algebras over R, where 𝐴 has a generating positive cone and 𝐵 satisfies the property that 𝑏2 > 0 if 0 ≠ 𝑏 ∈ 𝐵. In this short paper, we prove that if a positive map 𝑇 : 𝐴 → 𝐵 is supra-additive and supra-multiplicative for all positive and negative elements in 𝐴, then 𝑇 is indeed linear and multiplicative. In particular, if 𝐴 is squareroot closed, then every map which is supra-additive and supramultiplicative for all positive and negative elements is both

linear and multiplicative. From our result, it follows that for arbitrary topological spaces 𝑋, 𝑌, a supra-additive and supramultiplicative map 𝑇 : 𝐶(𝑋) → 𝐶(𝑌) is indeed both linear and multiplicative. This generalizes the results of Rˇadulescu [1] and Ercan [2]. As a special case we consider the supraadditive and supra-multiplicative maps on 𝐶0 (𝑋) and obtain a Banach-Stone type result. Recall that an ordered real vector space 𝐴 under a multiplication is said to be an ordered algebra whenever the multiplication makes 𝐴 an algebra, and in addition it satisfies the following property: if 𝑎1 , 𝑎 2 ≥ 0, then 𝑎1 𝑎 2 ≥ 0. 𝐴 is called Archimedean if 𝑥, 𝑦 ∈ 𝐴 and 𝑛𝑥 ≤ 𝑦 for all 𝑛 ∈ N implies that 𝑥 ≤ 0. The set 𝐴+ = {𝑎 ∈ 𝐴 : 𝑎 ≥ 0} is called the positive cone of 𝐴. The positive cone of 𝐴 is said to be generating (or 𝐴 is positively generated) if 𝐴 = 𝐴+ − 𝐴+ . A map 𝑇 : 𝐴 → 𝐵 between two ordered algebras is called positive whenever 𝑇(𝐴+ ) ⊆ 𝐵+ . Let 𝐴− := −𝐴+ and 𝐴± := 𝐴+ ∪ 𝐴− . Theorem 1. Let 𝐴 be an ordered algebra which has a generating positive cone. Let 𝐵 be an Archimedean ordered algebra satisfying the property that 𝑏2 > 0 if 0 ≠ 𝑏 ∈ 𝐵. If a positive map 𝑇 : 𝐴 → 𝐵 satisfies the following inequalities: (1) 𝑇(𝑎1 + 𝑎2 ) ≥ 𝑇(𝑎1 ) + 𝑇(𝑎2 ) (supra-additive), (2) 𝑇(𝑎1 𝑎2 ) ≥ 𝑇(𝑎1 )𝑇(𝑎2 ) (supra-multiplicative), for all 𝑎1 , 𝑎2 ∈ 𝐴± , then 𝑇 is both linear and multiplicative.

2


Proof. From the inequality 𝑇(0) = 𝑇(0 + 0) ≥ 𝑇(0) + 𝑇(0), it follows that 𝑇(0) ≤ 0. On the other hand, 𝑇(0) ≥ 𝑇(0 ⋅ 0) ≥ 𝑇(0)𝑇(0) ≥ 0. Therefore, 𝑇(0) = 0. It should be noted that 𝑎2 ≥ 0 for every 𝑎 ∈ 𝐴± . By the supra-additivity of 𝑇, we have 0 = 𝑇 (0) = 𝑇 (𝑎2 − 𝑎2 ) ≥ 𝑇 (𝑎2 ) + 𝑇 (−𝑎2 ) , ±

(1)

±

for every 𝑎 ∈ 𝐴 . Thus, for every 𝑎 ∈ 𝐴 , from the following inequalities: 2

0 ≤ [𝑇 (𝑎) + 𝑇 (−𝑎)] = 𝑇(𝑎) + 𝑇 (𝑎) 𝑇 (−𝑎)

= 𝑇 (𝑢1 𝑢2 ) − 𝑇 (𝑢1 V2 ) − 𝑇 (V1 𝑢2 ) + 𝑇 (V1 V2 ) = 𝑇 (𝑢1 ) 𝑇 (𝑢2 ) − 𝑇 (𝑢1 ) 𝑇 (V2 ) − 𝑇 (V1 ) 𝑇 (𝑢2 ) + 𝑇 (V1 ) 𝑇 (V2 )

(6)

= 𝑇 (𝑎1 ) 𝑇 (𝑎2 ) .

2

(2)

≤ 𝑇 (𝑎2 ) + 𝑇 (−𝑎2 ) + 𝑇 (−𝑎2 ) + 𝑇 (𝑎2 ) 2

𝑇 (𝑎1 𝑎2 ) = 𝑇 [(𝑢1 − V1 ) (𝑢2 − V2 )]

= [𝑇 (𝑢1 ) − 𝑇 (V1 )] [𝑇 (𝑢2 ) − 𝑇 (V2 )]

2

+ 𝑇 (−𝑎) 𝑇 (𝑎) + 𝑇(−𝑎)

is positively generated, there exist 𝑢𝑖 , V𝑖 ∈ 𝐴+ (𝑖 = 1, 2), such that 𝑎1 = 𝑢1 − V1 and 𝑎2 = 𝑢2 − V2 . We have

2

= 2 [𝑇 (𝑎 ) + 𝑇 (−𝑎 )] ≤ 0, it follows that [𝑇(𝑎) + 𝑇(−𝑎)]2 = 0 for every 𝑎 ∈ 𝐴± . By our hypothesis on 𝐵, we have 𝑇(−𝑎) = −𝑇(𝑎) for all 𝑎 ∈ 𝐴± . Now, for all 𝑎1 , 𝑎2 ∈ 𝐴± , we have

That is, 𝑇 is multiplicative, as desired. Remark 2. Let 𝐴 and 𝐵 be as in the above theorem. It may be asked whether the positive map 𝑇 : 𝐴 → 𝐵 is linear and multiplicative whenever 𝑇 : 𝐴 → 𝐵 is supra-additive and supramultiplicative only on 𝐴+ or only on 𝐴− . Indeed, this is not the case. For instance, the positive nonlinear map 𝑓 : R → R defined by 𝑓(𝑥) = |𝑥| is additive and multiplicative on R+ (or R− , resp.). Recall that an ordered algebra 𝐴 is said to be squareroot closed and that whenever for any 𝑎 ∈ 𝐴+ there exists 𝑥 ∈ 𝐴+ , such that 𝑎 = 𝑥 2 . When 𝐴 is square-root closed, we have the following corollary.

𝑇 (𝑎1 + 𝑎2 ) ≥ 𝑇 (𝑎1 ) + 𝑇 (𝑎2 ) = −𝑇 (−𝑎1 ) − 𝑇 (−𝑎2 ) = − [𝑇 (−𝑎1 ) + 𝑇 (−𝑎2 )]

(3)

≥ −𝑇 (−𝑎1 − 𝑎2 ) = 𝑇 (𝑎1 + 𝑎2 ) . That is, 𝑇(𝑎1 +𝑎2 ) = 𝑇(𝑎1 )+𝑇(𝑎2 ) for all 𝑎1 , 𝑎2 ∈ 𝐴± . Similarly, for all 𝑎1 , 𝑎2 ∈ 𝐴± , from 𝑇 (𝑎1 𝑎2 ) ≥ 𝑇 (𝑎1 ) 𝑇 (𝑎2 ) = −𝑇 (𝑎1 ) 𝑇 (−𝑎2 )

(4)

≥ −𝑇 (−𝑎1 𝑎2 ) = 𝑇 (𝑎1 𝑎2 ) ,

Corollary 3. Let 𝐴 be a square-root closed ordered algebra with a generating positive cone. Let 𝐵 be an Archimedean 𝑏 ∈ ordered algebra satisfying the property that 𝑏2 > 0 if 0 ≠ 𝐵. If a map 𝑇 : 𝐴 → 𝐵 is supra-additive and supra-multiplicative on 𝐴± , then 𝑇 is both linear and multiplicative on 𝐴. Proof. By the above Theorem, to complete the proof, we need only to verify that 𝑇 is positive. Since 𝐴 is square-root closed for each 𝑎 ≥ 0 in 𝐴 there exists 𝑥 ∈ 𝐴+ , such that 𝑎 = 𝑥 2 . Hence, by our hypothesis on 𝐵, we have 𝑇 (𝑎) = 𝑇 (𝑥2 ) ≥ 𝑇(𝑥)2 ≥ 0.

±

it follows that 𝑇(𝑎1 𝑎2 ) = 𝑇(𝑎1 )𝑇(𝑎2 ) for all 𝑎1 , 𝑎2 ∈ 𝐴 . Next, because 𝐴 is positively generated and 𝐵 is Archimedean, by the positive additivity of 𝑇 on 𝐴+ and the Kantorovich theorem (cf. [6, Proposition 1, page 150] or [7, Theorem 1.7]) there exists a unique positive linear map 𝑆 : 𝐴 → 𝐵 such that 𝑆 = 𝑇 on 𝐴+ , where 𝑆 is defined by 𝑆(𝑎1 − 𝑎2 ) = 𝑇(𝑎1 ) − 𝑇(𝑎2 ) for all 𝑎1 , 𝑎2 ∈ 𝐴+ . For every 𝑎 ∈ 𝐴, there exist 𝑢, V ∈ 𝐴+ satisfying 𝑎 = 𝑢 − V since 𝐴 is positively generated. From

(7)

This implies that 𝑇 is positive.

(5)

Remark 4. It should be noted that the space of all real functions (all real continuous functions) on a nonempty set (a topological space, resp.), with the pointwise algebraic operations and the pointwise ordering, is a square-root closed Archimedean lattice-ordered algebra with the property mentioned in Corollary 3. Thus, the results on supra-additive and supra-multiplicative maps between spaces of real functions obtained by Rˇadulescu [1], Volkmann [4], and Ercan [2] can now follow from Corollary 3. In their earlier proofs, the constant function or the multiplicative unit element plays an essential role.

it follows that 𝑇 = 𝑆 on 𝐴. This implies that 𝑇 is linear. On the other hand, let 𝑎1 and 𝑎2 be arbitrary elements in 𝐴. Since 𝐴+

Let 𝑋 be a locally compact Hausdorff space, and let 𝐶0 (𝑋) be the Banach lattice of all continuous real functions defined on 𝑋 and vanishing at infinity. Note that 𝐶0 (𝑋) does not necessarily contain the constant function or a unit element

𝑇 (𝑎) = 𝑇 (𝑢 − V) = 𝑇 (𝑢) + 𝑇 (−V) = 𝑇 (𝑢) − 𝑇 (V) = 𝑆 (𝑢 − V) = 𝑆 (𝑎) ,

Journal of Function Spaces and Applications unless 𝑋 is compact. The following result is an immediate consequence of Corollary 3. Corollary 5. Let 𝑋 and 𝑌 be locally compact Hausdorff spaces. If 𝑇 : 𝐶0 (𝑋) → 𝐶0 (𝑌) is supra-additive and supramultiplicative for all elements in 𝐶0 (𝑋)± , then 𝑇 is an algebra and lattice homomorphism. Recall that 𝑇 : 𝐶0 (𝑋) → 𝐶0 (𝑌) for any 𝑓, 𝑔 is said to be separating or disjointness preserving if, for any 𝑓, 𝑔 ∈ 𝐶0 (𝑋), 𝑓𝑔 = 0 implies that 𝑇𝑓𝑇𝑔 = 0. Clearly, every multiplicative map is disjointness preserving. Combining Corollary 5 with Theorem 2 of [8] or Theorem 8 in [9], we obtain the following Banach-Stone type result. Corollary 6. Let 𝑋, 𝑌 be locally compact Hausdorff spaces. If there exists a bijection from 𝐶0 (𝑋) onto 𝐶0 (𝑌) which is supra-additive and supra-multiplicative on 𝐶0 (𝑋)± , then 𝑋 is topologically homeomorphic to 𝑌.

Acknowledgment The authors would like to thank the reviewer for his/her kind comments and valuable suggestions which have improved this paper. The authors were supported in part by the Fundamental Research Funds for the Central Universities (SWJTU11CX154, SWJTU12ZT13) and NSFC (No. 11301285).

References [1] M. R˘adulescu, “On a supra-additive and supra-multiplicative operator of 𝐶(𝑋),” Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, vol. 24, no. 3, pp. 303–305, 1980. [2] Z. Ercan, “A remark on supra-additive and supra-multiplicative operators on 𝐶(𝑋),” Mathematica Bohemica, vol. 132, no. 1, pp. 55–58, 2007. [3] J. Dhombres, “Sur les fonctions simultanément suradditives et surmultiplicatives,” Comptes Rendus Mathématiques. La Société Royale du Canada. L’Academie des Sciences, vol. 5, no. 5, pp. 207– 210, 1983. [4] P. Volkmann, “Sur les fonctions simultanément suradditives et surmultiplicatives,” Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, vol. 28, no. 2, pp. 181–184, 1984. [5] I. Gusić, “A note on certain maps between ordered fields,” Journal of Mathematical Inequalities, vol. 3, no. 4, pp. 657–661, 2009. [6] R. Cristescu, Topological Vector Spaces, Editura Academiei, Bucharest, Romania, 1977. [7] C. D. Aliprantis and O. Burkinshaw, Positive Operators, vol. 119 of Pure and Applied Mathematics, Academic Press, Orlando, Fla, USA, 1985. [8] J. J. Font and S. Hernández, “On separating maps between locally compact spaces,” Archiv der Mathematik, vol. 63, no. 2, pp. 158–165, 1994. [9] J.-S. Jeang and N.-C. Wong, “Weighted composition operators of 𝐶0 (𝑋)’s,” Journal of Mathematical Analysis and Applications, vol. 201, no. 3, pp. 981–993, 1996.

3


Research Article On the Aleksandrov-Rassias Problems on Linear 𝑛-Normed Spaces Yumei Ma Department of Mathematics, Dalian Nationality University, Dalian 116600, China Correspondence should be addressed to Yumei Ma; [email protected] Received 11 May 2013; Accepted 17 July 2013 Academic Editor: Ji Gao Copyright © 2013 Yumei Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper generalizes T. M. Rassias’ results in 1993 to n-normed spaces. If X and Y are two real n-normed spaces and Y is n-strictly convex, a surjective mapping 𝑓 : 𝑋 → 𝑌 preserving unit distance in both directions and preserving any integer distance is an n-isometry.

1. Introduction Let 𝑋 and 𝑌 be two metric spaces. A mapping 𝑓 : 𝑋 → 𝑌 is called an isometry if 𝑓 satisfies 𝑑𝑌 (𝑓(𝑥), 𝑓(𝑦)) = 𝑑𝑋 (𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋, where 𝑑𝑋 (⋅, ⋅) and 𝑑𝑌 (⋅, ⋅) denote the metrics in the spaces 𝑋 and 𝑌, respectively. For some fixed number 𝑟 > 0, suppose that 𝑓 preserves distance 𝑟, that is, for all 𝑥, 𝑦 ∈ 𝑋 with 𝑑𝑋 (𝑥, 𝑦) = 𝑟, we have 𝑑𝑌 (𝑓(𝑥), 𝑓(𝑦)) = 𝑟, then 𝑟 is called a conservative (or preserved) distance for the mapping 𝑓. In particular, we denote DOPP as 𝑓 preserving the one distance property and SDOPP as 𝑓 preserving the strong one distance property and also for 𝑓−1 . In 1970 [1], Aleksandrov posed the following problem. Examine whether the existence of a single conservative distance for some mapping 𝑇 implies that 𝑇 is an isometry. This question is of great significance for the Mazur-Ulam Theorem [2]. ˇ In 1993, T. M. Rassias and P. Semrl proved the following. Theorem 1 (see [3]). Let 𝑋 and 𝑌 be two real normed linear spaces such that one of them has a dimension greater than one. Assume also that one of them is strictly convex. Suppose that 𝑓 : 𝑋 → 𝑌 is a surjective mapping that satisfies SDOPP. Then, 𝑓 is an affine isometry (a linear isometry up to translation). Theorem 2 (see [3]). Let 𝑋 and 𝑌 be two real normed linear spaces such that one of them has a dimension greater than one. Suppose that 𝑓 : 𝑋 → 𝑌 is a Lipschitz mapping. Assume also

that 𝑓 is a surjective mapping satisfying (SDOPP). Then, 𝑓 is an isometry. Since 2004, the Aleksandrov problem in 𝑛-normed spaces (𝑛 ≥ 2) has been discussed, and some results are obtained [4– 8]. Definition 3 (see [7]). Let 𝑋 be a real linear space with dim 𝑋 ≥ 𝑛 and ‖⋅, . . . , ⋅‖ : 𝑋𝑛 → 𝑅, a function, then (𝑋, ‖⋅, . . . , ⋅‖) is called a linear 𝑛-normed space if for any 𝛼 ∈ 𝑅 and all 𝑥, 𝑦, 𝑥1 , . . . , 𝑥𝑛 ∈ 𝑋 𝑛𝑁1 : ‖𝑥1 , . . . , 𝑥𝑛 ‖ = 0 ⇔ 𝑥1 , . . . , 𝑥𝑛 are linearly dependent, 𝑛𝑁2 : ‖𝑥1 , . . . , 𝑥𝑛 ‖ = ‖𝑥𝑗1 , . . . , 𝑥𝑗𝑛 ‖ for every permutation (𝑗1 , . . . , 𝑗𝑛 ) of (1, . . . , 𝑛), 𝑛𝑁3 : ‖𝛼𝑥1 , . . . , 𝑥𝑛 ‖ = |𝛼|‖𝑥1 , . . . , 𝑥𝑛 ‖, 𝑛𝑁4 : ‖𝑥 + 𝑦, 𝑥2 , . . . , 𝑥𝑛 ‖ ≤ ‖𝑥, 𝑥2 , . . . , 𝑥𝑛 ‖ + ‖𝑦, 𝑥2 , . . . , 𝑥𝑛 ‖. The function ‖⋅, . . . , ⋅‖ is called the 𝑛-norm on 𝑋. Definition 4 (see [8]). Let 𝑋 and 𝑌 be two real linear 𝑛normed spaces. (i) A mapping 𝑓 : 𝑋 → 𝑌 is defined to be an 𝑛-isometry if for all 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋, 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩 (1) 󵄩 󵄩 = 󵄩󵄩󵄩𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .

2

Journal of Function Spaces and Applications (ii) A mapping 𝑓 : 𝑋 → 𝑌 is called the 𝑛-distance one preserving property (𝑛-DOPP) if for 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋, ‖𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 1, it follows that ‖𝑓(𝑥1 ) − 𝑓(𝑦1 ), . . . , 𝑓(𝑥𝑛 ) − 𝑓(𝑦𝑛 )‖ = 1. (iii) A mapping 𝑓 : 𝑋 → 𝑌 is called the 𝑛-strong distance one preserving property (𝑛-SDOPP) if for 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋, ‖𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 1, it follows that ‖𝑓(𝑥1 ) − 𝑓(𝑦1 ), . . . , 𝑓(𝑥𝑛 ) − 𝑓(𝑦𝑛 )‖ = 1 and conversely. (iv) A mapping 𝑓 : 𝑋 → 𝑌 is called an 𝑛-Lipschitz if for all 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋, 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 (2) 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .

Set 𝑦0 = 𝑦1 − 𝑥1 . For any 𝛼 ∈ 𝑅, we have 󵄩 󵄩󵄩 󵄩󵄩𝑧0 + 𝛼𝑦0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≠0. Let us define ℎ(𝛼) by 𝑟 (𝑧0 + 𝛼𝑦0 ) ℎ (𝛼) = 󵄩󵄩 󵄩, 󵄩󵄩𝑧0 + 𝛼𝑦0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

C. Park and T. M. Rassias obtained the following. Theorem 7 (see [8]). Let 𝑋 and 𝑌 be real linear 𝑛-normed spaces. If a mapping 𝑓 : 𝑋 → 𝑌 satisfies the following conditions: (i) 𝑓 has the 𝑛-DOPP,

󵄩󵄩 󵄩 󵄩󵄩ℎ (𝛼) , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟.

(iv) 𝑓 preserves the 𝑛-collinearity, then 𝑓 is an 𝑛-isometry. In 2009, Gao [6] researched another 𝑛-isometry and gave the 2-strictly convex concept [6]. In this paper, we generalize T. M. Rassias Theorems 1 and 7 on 𝑛-strictly convex normed spaces (𝑛 > 1).

2. Main Results The proof of the following lemma was presented in [9], to be published; the proof is given again for the convenience of readers. Lemma 8. Let 𝑋 be an 𝑛-normed space such that 𝑋 has dimension greater than 𝑛 and 𝑟 > 0. Suppose that 0 < ‖𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ ≤ 2𝑟 for 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋. Then, there exists 𝜔 ∈ 𝑋 such that 󵄩󵄩 󵄩 󵄩󵄩𝑥1 − 𝜔, 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟, (3) 󵄩󵄩 󵄩 󵄩󵄩𝜔 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟. Proof. Since 𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 are linearly independent and dim 𝑋 > 𝑛, then there exists 𝑧0 ∈ 𝑋 \ span{𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 } with ‖𝑧0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 𝑟.

(6)

Set −𝑟 (𝑦1 − 𝑥1 ) 𝑧1 = 󵄩󵄩 󵄩, 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 𝑟 (𝑦1 − 𝑥1 ) 𝑧2 = 󵄩󵄩 󵄩. 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

(7)

Clearly, 𝑧0 ≠𝑧1 , 𝑧2 . And we have 󵄩 󵄩󵄩 󵄩󵄩𝑧1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟, 󵄩󵄩 󵄩 󵄩󵄩𝑧2 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟.

(8)

On the other hand, lim ℎ (𝛼) = 𝑧1 ,

𝛼 → −∞

lim ℎ (𝛼) = 𝑧2 .

𝛼→∞

(9)

Thus, ℎ (−∞) = 𝑧1 ,

(ii) 𝑓 is 𝑛-Lipschitz, (iii) 𝑓 preserves the 2-collinearity,

(5)

then, we obtain

Definition 5 (see [7]). The points 𝑥0 , 𝑥1 , . . . , 𝑥𝑛 of 𝑋 are called 𝑛-collinear if for every 𝑖, {𝑥𝑗 − 𝑥𝑖 : 0 ≤ 𝑗 ≠𝑖 ≤ 𝑛} is linearly dependent. Definition 6. 𝑋 is said to be 𝑛-strictly convex normed spaces if for any 𝑥0 , 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ∈ 𝑋, 𝑥2 , . . . , 𝑥𝑛 ∉ span{𝑥0 , 𝑥1 }, and ‖𝑥0 + 𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ‖ = ‖𝑥0 𝑥2 , . . . , 𝑥𝑛 ‖ + ‖𝑥1 𝑥2 , . . . , 𝑥𝑛 ‖ imply that 𝑥0 and 𝑥1 are linearly dependent.

(4)

ℎ (+∞) = 𝑧2 .

(10)

Define 𝑔 : ℎ(𝑅) → 𝑅 by 󵄩 󵄩 𝑔 (𝑧) = 󵄩󵄩󵄩𝑧 − 𝑦0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .

(11)

It follows that 󵄩 󵄩 𝑔 (𝑧1 ) = 󵄩󵄩󵄩𝑧1 − 𝑦0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 𝑟 = (1 + 󵄩󵄩 󵄩) 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 × 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≥ 𝑟, 𝑔 (𝑧2 ) 𝑟 󵄩 󵄩 { { (1− 󵄩󵄩 { 󵄩󵄩 )󵄩󵄩󵄩𝑥1 −𝑦1 , . . . , 𝑥𝑛 −𝑦𝑛 󵄩󵄩󵄩 , { 𝑥 −𝑦 , . . . , 𝑥 −𝑦 󵄩󵄩 1 1 { 𝑛 𝑛󵄩 󵄩 { { { 󵄩 󵄩 { { if 󵄩󵄩󵄩𝑥1 −𝑦1 , . . . , 𝑥𝑛 −𝑦𝑛 󵄩󵄩󵄩 > 𝑟 ={ 𝑟 { 󵄩 󵄩 {( { 󵄩󵄩 󵄩󵄩 −1)󵄩󵄩󵄩𝑥1 −𝑦1 , . . . , 𝑥𝑛 −𝑦𝑛 󵄩󵄩󵄩 , { { 𝑥 −𝑦 , . . . , 𝑥 −𝑦 󵄩 󵄩 { 𝑛 𝑛󵄩 󵄩 1 1 { { { 󵄩 󵄩 if 󵄩󵄩󵄩𝑥1 −𝑦1 , . . . , 𝑥𝑛 −𝑦𝑛 󵄩󵄩󵄩 < 𝑟. { (12) Thus, 𝑔(𝑧2 ) ≤ 𝑟. Obviously, 𝑔(ℎ(𝛼)) is continuous on 𝑅. Using the mean value theorem, there exists 𝛼0 ∈ 𝑅 such that 𝑔(ℎ(𝛼0 )) = 𝑟.

Journal of Function Spaces and Applications Set 𝜔0 = ℎ(𝛼0 ), 𝜔 = 𝜔0 + 𝑥1 , we have 󵄩󵄩 󵄩 󵄩󵄩𝜔0 − 𝑦0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟.

3

(13)

And from ‖ℎ(𝛼), 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 𝑟, we have 󵄩󵄩 󵄩 󵄩󵄩𝜔 − 𝑥1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟, 󵄩󵄩 󵄩 󵄩󵄩𝜔 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

󵄩 󵄩 = 󵄩󵄩󵄩𝜔0 + 𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

(14)

It follows from the hypothesis of 𝑓 preserving any integer 𝑘; then, 󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑤1 ) , 𝑓 (𝑥2 − 𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 1, 󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 )−𝑓 (V1 ) , 𝑓 (𝑥2 )−𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 )−𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 𝑘 − 1, 󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑤1 ) − 𝑓 (V1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 𝑘. (22) Clearly, we have

󵄩 󵄩 = 󵄩󵄩󵄩𝜔0 − 𝑦0 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑟.

Lemma 9. Let 𝑋 and 𝑌 be two real linear 𝑛-normed spaces whose dimensions are greater than 𝑛, and let 𝑌 be 𝑛-strictly convex normed space. Suppose that 𝑓 : 𝑋 → 𝑌 is a surjective mapping satisfying (𝑛-SDOPP) with preserving distance 𝑘 for any 𝑘 ∈ 𝑁. Then, 𝑓 preserves distance 1/𝑘 for any 𝑘 ∈ 𝑁.

󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑤1 ) − 𝑓 (V1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (V1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑤1 ) , 𝑓 (𝑥2 − 𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 . (23) We conclude that 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )

Proof. Firstly, 𝑓 is injective. Suppose, on the contrary, that there are 𝑥0 , 𝑥1 ∈ 𝑋, 𝑥0 ≠𝑥1 , such that 𝑓(𝑥0 ) = 𝑓(𝑥1 ). As dim 𝑋 > 𝑛, it follows that there exist vectors 𝑥2 , . . . , 𝑥𝑛 ∈ 𝑋 such that 𝑥1 − 𝑥0 , . . . , 𝑥𝑛 − 𝑥0 are linearly independent. Then, ‖𝑥1 − 𝑥0 , . . . , 𝑥𝑛 − 𝑥0 ‖ ≠0. Set 𝑥2 − 𝑥0 𝑧2 := 𝑥0 + 󵄩󵄩 (15) 󵄩󵄩 . 𝑥 − 𝑥 󵄩 0 , . . . , 𝑥𝑛 − 𝑥0 󵄩 󵄩󵄩 1

Otherwise, if for some 𝑓(𝑥𝑖 ) − 𝑓(𝑦𝑖 ), we have 𝜇𝑖 , 𝜆 𝑖 ∈ 𝑅 with 𝜇𝑖 ≠0 or 𝜆 𝑖 ≠0 such that

Clearly,

Suppose that 𝜆 𝑖 ≠0. Then,

󵄩󵄩 󵄩 󵄩󵄩𝑥1 − 𝑥0 , 𝑧2 − 𝑥0 , 𝑥3 − 𝑥0 , . . . , 𝑥𝑛 − 𝑥0 󵄩󵄩󵄩 = 1. Then

󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑥0 ) , 𝑓 (𝑧2 ) − 𝑓 (𝑥0 ) ,

󵄩 𝑓 (𝑥3 ) − 𝑓 (𝑥0 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑥0 )󵄩󵄩󵄩 = 1.

By Lemma 8, we can find 𝑤1 ∈ 𝑋 with 󵄩󵄩 󵄩 󵄩󵄩𝑥1 − 𝑤1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 1, 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑤1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩 = 1. V1 = 𝑤1 + 𝑘 (𝑥1 − 𝑤1 ) .

Clearly, we have 󵄩 󵄩󵄩 󵄩󵄩𝑥1 − V1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩(𝑘 − 1) (𝑥1 − 𝑤1 ) , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 =𝑘−1 󵄩󵄩 󵄩 󵄩󵄩𝑤1 − V1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑘.

󵄨 󵄨󵄩 = 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 󵄩󵄩󵄩(𝑥1 ) − 𝑓 (V1 ) , . . . , 𝑓 (𝑥1 )

(17)

(26)

󵄩 −𝑓 (𝑤1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 .

Assume that 󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (V1 ) , . . . , 𝑓 (𝑥1 ) − 𝑓 (𝑤1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 ≠0. (27)

(19)

Set 𝑢1 = 𝑤1 + 𝑘 (𝑦1 − 𝑤1 ) ,

󵄩 𝑘 − 1 = 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (V1 ) , . . . , 𝑓 (𝑥𝑖 )

󵄩 −𝑓 (𝑦𝑖 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩

(18)

(24)

𝑓 (𝑥𝑖 ) − 𝑓 (𝑦𝑖 ) = 𝜇𝑖 (𝑓 (𝑥1 ) − 𝑓 (V1 ))+𝜆 𝑖 (𝑓 (𝑥1 ) − 𝑓 (𝑤1 )) . (25)

(16)

This implies that 𝑓(𝑥0 ) ≠𝑓(𝑥1 ), which is a contradiction. Therefore, 𝑓 is a bijective mapping. Let 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋 and (𝑘 ∈ 𝑁 \ {1}) satisfying 󵄩󵄩 󵄩 1 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = . 𝑘

∉ span {𝑓 (𝑥1 ) − 𝑓 (V1 ) , 𝑓 (𝑥1 ) − 𝑓 (𝑤1 )} .

Set 𝑠𝑗 = 𝑓 (𝑥𝑗 ) + (𝑓 (𝑥𝑗 ) − 𝑓 (𝑦𝑗 ))

(20)

󵄩 × (󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (V1 ) , . . . , 𝑓 (𝑥1 ) − 𝑓 (𝑤1 ) , . . . , 󵄩 −1 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩) ,

(28)

(𝑗 ≥ 2) .

Then, for 𝑗 ≠𝑖, (21)

󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (V1 ) , . . . , 𝑠𝑗 − 𝑓 (𝑥𝑗 ) , . . . , 𝑓 (𝑥1 ) 󵄩 󵄩 −𝑓 (𝑤1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 1.

(29)

4


Since 𝑓 is bijective and preserves 𝑛-SDOPP on both directions. Then, there exists 𝑡𝑗 ∈ 𝑋 with 𝑓(𝑡𝑗 ) = 𝑠𝑗 which satisfies that 󵄩󵄩 󵄩 󵄩󵄩𝑥1 − V1 , 𝑡𝑗 − 𝑥𝑗 , . . . , 𝑥1 − 𝑤1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 1. (30) 󵄩 󵄩

𝑥𝑚 − 𝜔𝑚 , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 are linearly independent. Hence, it holds that 󵄩 󵄩󵄩 󵄩󵄩𝑥1 − 𝜔1 , . . . , 𝑥𝑚 − 𝜔𝑚 , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≠0. (39)

However, by (20), 𝑥1 − V1 = (1 − 𝑘)(𝑥1 − 𝑤1 ), and thus 𝑥1 − V1 , 𝑥1 − 𝑤1 are linear dependent. Then,

󵄩󵄩 󵄩 1 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 ≤ , 𝑘 (40)

󵄩 󵄩󵄩 󵄩󵄩𝑥1 − V1 , 𝑡𝑗 − 𝑥𝑗 , . . . , 𝑥1 − 𝑤1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 0. 󵄩 󵄩

(31)

This contradiction implies that

We will prove that

for every 𝑘 ∈ N. Let 𝑚 = 1. We can find a vector 𝜔1 ∈ 𝑋 such that 𝑥1 − 𝜔1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 are linearly independent. Set

󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (V1 ) , . . . , 𝑓 (𝑥1 ) − 𝑓 (𝑤1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 0. (32) This also contradicts with (26). Since 𝑌 is 𝑛-strictly convex, then there exists 𝛼 > 0 such that 𝑓 (𝑥1 ) − 𝑓 (V1 ) = 𝛼 (𝑓 (𝑥1 ) − 𝑓 (𝑤1 )) .

(33)

Then, 𝑓 (𝑥1 ) =

1 𝛼 𝑓 (V1 ) + 𝑓 (𝑤1 ) . 1+𝛼 1+𝛼

(34)

Since 󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 )−𝑓 (V1 ) , 𝑓 (𝑥2 )−𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 )−𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 𝑘 − 1,

]1 = 𝑥1 +

𝑓 (𝑥1 ) =

1 𝑘−1 𝑓 (V1 ) + 𝑓 (𝑤1 ) , 𝑘 𝑘

(36)

𝑓 (𝑦1 ) =

1 𝑘−1 𝑓 (𝑢1 ) + 𝑓 (𝑤1 ) . 𝑘 𝑘

(37)

Similarly,

Hence, 󵄩󵄩 󵄩 1 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = . 𝑘 (38)

Lemma 10. Let 𝑋 and 𝑌 be real 𝑛-normed spaces such that dim 𝑋 ≥ 𝑛. If a mapping 𝑓 : 𝑋 → 𝑌 preserves the distance 1/𝑘 for each 𝑘 ∈ N, then 𝑓 preserves the distance zero. Proof. Choose 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋 such that ‖𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 0; that is, 𝑥1 − 𝑦1 , . . . , 𝑥𝑛 − 𝑦𝑛 are linearly dependent. Assume that {𝑥𝑚+1 −𝑦𝑚+1 , . . . , 𝑥𝑛 −𝑦𝑛 } is a maximum linearly independent group of {𝑥1 −𝑦1 , . . . , 𝑥𝑛 −𝑦𝑛 } (𝑚 < 𝑛). As dim 𝑋 ≥ 𝑛, we can find a finite sequence of vectors 𝜔1 , 𝜔2 , . . . , 𝜔𝑚 ∈ 𝑋 such that 𝑥1 − 𝜔1 , . . . ,

(41)

for arbitrarily fixed 𝑘 ∈ N. Then, 1 󵄩󵄩 󵄩 , 󵄩󵄩𝑥1 − ]1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 2𝑘 󵄩 󵄩󵄩 󵄩󵄩]1 − 𝑥1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 − 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(]1 − 𝑥1 ) + (𝑥1 − 𝑦1 ) , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩]1 − 𝑥1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .

(42)

Since ‖𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 0, we get

󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑤1 ) , 𝑓 (𝑥2 − 𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 1, (35)

then 𝛼 = 𝑘 − 1. Thus,

𝑥1 − 𝜔1 󵄩, 󵄩󵄩 2𝑘 󵄩󵄩𝑥1 − 𝜔1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

1 󵄩 󵄩󵄩 . 󵄩󵄩]1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 2𝑘

(43)

Since 𝑓 preserves the distance 1/(2𝑘), we see that 󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (]1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑓 (]1 )−𝑓 (𝑦1 ) , 𝑓 (𝑥2 )− 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 )− 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 =

1 1 ⋅2= . 2𝑘 𝑘

(44)

For 𝑚 ≥ 2, we set ]1 = 𝑥1 + (𝑥1 − 𝜔1 ) 󵄩 × (2𝑚 𝑘 󵄩󵄩󵄩𝑥1 − 𝜔1 , . . . , 𝑥𝑚 − 𝜔𝑚 , 𝑥𝑚+1

(45)

󵄩 −1 −𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩) , ]𝑖 = 2𝑥𝑖 − 𝜔𝑖 ,

(46)

for any 𝑖 ∈ {2, 3, . . . , 𝑚}. Then, we have 𝑥𝑖 − ]𝑖 = 𝜔𝑖 − 𝑥𝑖 ,

]𝑖 − 𝑦𝑖 = (𝑥𝑖 − 𝜔𝑖 ) + (𝑥𝑖 − 𝑦𝑖 ) , (47)

for each 𝑖 ∈ {2, 3, . . . , 𝑚}. Since 𝑥𝑖 −𝑦𝑖 , 𝑥𝑚+1 −𝑦𝑚+1 , . . . , 𝑥𝑛 −𝑦𝑛 are linearly dependent, we get 󵄩 󵄩󵄩 (48) 󵄩󵄩. . . , 𝑥𝑖 − 𝑦𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 0,


5 󵄩 + 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (]1 ) ,

and hence, 󵄩󵄩 󵄩 󵄩󵄩. . . , 𝑥𝑖 − 𝜔𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 − 󵄩󵄩󵄩. . . , 𝑥𝑖 − 𝑦𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩. . . , (𝑥𝑖 − 𝜔𝑖 ) + (𝑥𝑖 − 𝑦𝑖 ) , . . . , 𝑥𝑚+1 󵄩 −𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩. . . , 𝑥𝑖 − 𝜔𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩. . . , 𝑥𝑖 − 𝑦𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ,

𝑓 (𝑥2 ) − 𝑓 (]2 ) , . . . , 𝑓 (𝑥𝑚−1 ) − 𝑓 (]𝑚−1 ) , 𝑓 (]𝑚 ) − 𝑓 (𝑦𝑚 ) , 󵄩 𝑓 (𝑥𝑚+1 ) − 𝑓 (𝑦𝑚+1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩

󵄩 + 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (]1 ) ,

𝑓 (𝑥2 ) − 𝑓 (]2 ) , . . . , 𝑓 (]𝑚−1 ) − 𝑓 (𝑦𝑚−1 ) , 𝑓 (𝑥𝑚 ) − 𝑓 (]𝑚 ) , 󵄩 𝑓 (𝑥𝑚+1 ) − 𝑓 (𝑦𝑚+1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩

(49)

󵄩 + 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (]1 ) ,

𝑓 (𝑥2 ) − 𝑓 (]2 ) , . . . , 𝑓 (]𝑚−1 ) − 𝑓 (𝑦𝑚−1 ) ,

which together with (48) implies that 󵄩󵄩 󵄩 󵄩󵄩. . . , ]𝑖 − 𝑦𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩. . . , 𝑥𝑖 − 𝜔𝑖 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ,

𝑓 (]𝑚 ) − 𝑓 (𝑦𝑚 ) , 󵄩 𝑓 (𝑥𝑚+1 ) − 𝑓 (𝑦𝑚+1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩

(50)

+ ⋅⋅⋅+ 󵄩 + 󵄩󵄩󵄩𝑓 (]1 ) − 𝑓 (𝑦1 ) ,

for all 𝑖 ∈ {2, 3, . . . , 𝑚}. By a similar argument, we further obtain that 󵄩󵄩 󵄩 󵄩󵄩]1 − 𝑦1 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩]1 − 𝑥1 , . . . , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .

𝑓 (]2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (]𝑚−1 ) − 𝑓 (𝑦𝑚−1 ) , 𝑓 (]𝑚 ) − 𝑓 (𝑦𝑚 ) , 󵄩 𝑓 (𝑥𝑚+1 ) − 𝑓 (𝑦𝑚+1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩

(51) =

1 2𝑚 𝑘

⋅ 2𝑚 =

1 , 𝑘 (53)

In view of (45), (50), and (51), we conclude that 󵄩󵄩 󵄩 󵄩󵄩]1 − 𝑦1 , 𝜇2 , . . . , 𝜇𝑚 , 𝑥𝑚+1 − 𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 = 󵄩󵄩󵄩𝑥1 − ]1 , 𝑥2 − 𝜔2 , . . . , 𝑥𝑚 − 𝜔𝑚 , 𝑥𝑚+1 󵄩 −𝑦𝑚+1 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 =

1 2𝑚 𝑘

where 𝑘 is an arbitrary positive integer. Hence, we conclude that 󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = 0, (54) (52)

,

where 𝜇𝑖 denotes either ]𝑖 − 𝑦𝑖 or 𝑥𝑖 − ]𝑖 for 𝑖 ∈ {2, 3, . . . , 𝑚}. Since 𝑓 preserves the distance 1/(2𝑚 𝑘) for any 𝑘 ∈ N, it follows from (52) that 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , 𝑓 (𝑥3 ) 󵄩 −𝑓 (𝑦3 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (]1 ) , 𝑓 (𝑥2 ) − 𝑓 (]2 ) , . . . , 𝑓 (𝑥𝑚−1 ) − 𝑓 (]𝑚−1 ) , 𝑓 (𝑥𝑚 ) − 𝑓 (]𝑚 ) , 󵄩 𝑓 (𝑥𝑚+1 ) − 𝑓 (𝑦𝑚+1 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩

which implies that 𝑓 preserves the distance zero. Remark 11. In ([9], Lemma 2.2 to be published), we give the same method under the condition of 𝑓 preserving 2-colinear. Theorem 12. Let 𝑋 and 𝑌 be real 𝑛-normed spaces such that dim 𝑋 > 𝑛 and 𝑌 is 𝑛-strictly convex. If a surjective mapping 𝑓 : 𝑋 → 𝑌 has the 𝑛-SDOPP and preserves the distance 𝑘 for any 𝑘 ∈ N, then 𝑓 is an affine 𝑛-isometry. Proof. Assume that ‖𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ > 0 for 𝑥1 , . . . , 𝑥𝑛 , 𝑦1 , . . . , 𝑦𝑛 ∈ 𝑋. Take positive integers 𝑘, 𝑚 such that 𝑚 − 1 󵄩󵄩 󵄩 𝑚 ≤ 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≤ . 𝑘 𝑘

(55)

𝑦1 − 𝑥1 ⋅ 󵄩󵄩 󵄩, 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

(56)

Set 𝑝𝑖 = 𝑥1 +

𝑖 𝑘

6


for 𝑖 = 0, 1, . . . , 𝑚 − 2, and

For any 𝑥1 , 𝑦1 , 𝑥2 , 𝑦2 , . . . , 𝑥𝑛 , 𝑦𝑛 ∈ 𝑋, with 𝑝𝑚 = 𝑦1 .

(57)

Clearly, for 𝑖 = 1, . . . , 𝑚 − 2, 󵄩 1 󵄩󵄩 󵄩󵄩𝑝𝑖 − 𝑝𝑖−1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = , 𝑘 󵄩󵄩 󵄩󵄩 0 < 󵄩󵄩𝑝𝑚 − 𝑝𝑚−2 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩 󵄩󵄩 𝑦1 − 𝑥1 𝑚−2 󵄩 = 󵄩󵄩󵄩󵄩𝑦1 − 𝑥1 − ⋅ 󵄩󵄩 󵄩󵄩 , 𝑘 󵄩󵄩 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩 󵄩󵄩 󵄩 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩󵄩 󵄩󵄩 𝑚−2 𝑘

1 ⋅ 󵄩󵄩 󵄩) 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩

= (1 −

󵄩 󵄩 𝑚−2 = 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 − 𝑘 ≤

𝑚 𝑚−2 2 − = . 𝑘 𝑘 𝑘

󵄩 1 󵄩󵄩 󵄩󵄩𝑝𝑚−1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = . 𝑘 It follows from Lemma 9 that we have

(64)

find a positive integer 𝑘0 satisfying ‖𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ < 𝑘0 . Set 𝑧1 = 𝑥1 + 𝑘0 (𝑦1 − 𝑥1 )/‖𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖. Clearly, ||𝑧1 − 𝑥1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 || = 𝑘0 , and ‖𝑧1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖ = 𝑘0 − ‖𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 ‖. It follows that ‖𝑓(𝑧1 ) − 𝑓(𝑥1 ), 𝑓(𝑥2 ) − 𝑓(𝑦2 ), . . . , 𝑓(𝑥𝑛 ) − 𝑓(𝑦𝑛 )‖ = 𝑘0 and 󵄩 󵄩 𝑘0 = 󵄩󵄩󵄩𝑓 (𝑧1 ) − 𝑓 (𝑥1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑓 (𝑧1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑓 (𝑥1 )−𝑓 (𝑦1 ) , 𝑓 (𝑥2 )−𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 )−𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 < 𝑘0 − 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 𝑘0 . (65) Then (63) is not valid. Hence,

(58)

According to Lemma 8, there exists 𝑝𝑚−1 ∈ 𝑋 such that 󵄩 1 󵄩󵄩 󵄩󵄩𝑝𝑚−1 − 𝑝𝑚−2 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = , 𝑘

󵄩󵄩 󵄩 󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 ≠0,

󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 . (66)

(59)

󵄩 1 󵄩󵄩 󵄩󵄩𝑓 (𝑝𝑖 )−𝑓 (𝑝𝑖−1 ) , 𝑓 (𝑥2 )−𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 )−𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = , 𝑘 (60) for 𝑖 = 0, 1, 2, . . . , 𝑚. On the other hand, 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩

Corollary 13. Let 𝑋 and 𝑌 be two real linear 𝑛-normed spaces. Suppose that mapping 𝑓 : 𝑋 → 𝑌 preserves any positive integer 𝑘-distance and Lipschitz condition. Then, 𝑓 is an 𝑛isometry.

Acknowledgments This work is supported by the Fundamental Research Funds for the Central Universities in China and Education Department of Liaoning province in china.

𝑚

󵄩 ≤ ∑ 󵄩󵄩󵄩𝑓 (𝑝𝑖 ) − 𝑓 (𝑝𝑖−1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 )

References

𝑖=1

󵄩 𝑚 −𝑓 (𝑦𝑛 )󵄩󵄩󵄩 = . 𝑘

(61)

Hence 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 . (62) Suppose that 󵄩󵄩 󵄩 󵄩󵄩𝑓 (𝑥1 ) − 𝑓 (𝑦1 ) , 𝑓 (𝑥2 ) − 𝑓 (𝑦2 ) , . . . , 𝑓 (𝑥𝑛 ) − 𝑓 (𝑦𝑛 )󵄩󵄩󵄩 󵄩 󵄩 < 󵄩󵄩󵄩𝑥1 − 𝑦1 , 𝑥2 − 𝑦2 , . . . , 𝑥𝑛 − 𝑦𝑛 󵄩󵄩󵄩 .

(63)

[1] A. D. Aleksandrov, “Mappings of families of sets,” Soviet Mathematics Doklady, vol. 11, pp. 116–120, 1970. [2] S. Mazur and S. Ulam, “Sur les transformationes isométriques d’espaces vectoriels normés,” Comptes Rendus de L’académie des Sciences, vol. 194, pp. 946–948, 1932. ˇ [3] T. M. Rassias and P. Semrl, “On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings,” Proceedings of the American Mathematical Society, vol. 118, no. 3, pp. 919–925, 1993. [4] H.-Y. Chu, C.-G. Park, and W.-G. Park, “The Aleksandrov problem in linear 2-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 666–672, 2004. [5] H.-Y. Chu, S. K. Choi, and D. S. Kang, “Mappings of conservative distances in linear 𝑛-normed spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 3, pp. 1168–1174, 2009.

Journal of Function Spaces and Applications [6] J. Gao, “On the Alexandrov problem of distance preserving mapping,” Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 583–590, 2009. [7] H.-Y. Chu, K. Lee, and C.-G. Park, “On the Aleksandrov problem in linear 𝑛-normed spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 59, no. 7, pp. 1001–1011, 2004. [8] C.-G. Park and T. M. Rassias, “Isometries on linear 𝑛-normed spaces,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, pp. 1–17, 2006. [9] Y. Ma, “The Aleksandrov problem on linear n-normed spaces,” Acta Mathematica Scientia. In press.

7


Research Article Construction of Frames for Shift-Invariant Spaces Stevan PilipoviT1 and Suzana SimiT2 1

Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia 2 Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovića 12, 34000 Kragujevac, Serbia Correspondence should be addressed to Suzana Simić; [email protected] Received 9 May 2013; Accepted 25 July 2013 Academic Editor: Satit Saejung Copyright © 2013 S. Pilipović and S. Simić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We construct a sequence {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z, 𝑖 = 1, . . . , 𝑟} which constitutes a 𝑝-frame for the weighted shift-invariant space 𝑉𝜇𝑝 (Φ) = 𝑟 {∑𝑖=1 ∑𝑗∈Z 𝑐𝑖 (𝑗)𝜙𝑖 (⋅ − 𝑗) | {𝑐𝑖 (𝑗)}𝑗∈Z ∈ ℓ𝜇𝑝 , 𝑖 = 1, . . . , 𝑟}, 𝑝 ∈ [1, ∞], and generates a closed shift-invariant subspace of 𝐿𝑝𝜇 (R). The first construction is obtained by choosing functions 𝜙𝑖 , 𝑖 = 1, . . . , 𝑟, with compactly supported Fourier transforms 𝜙̂𝑖 , 𝑖 = 1, . . . , 𝑟. The second construction, with compactly supported 𝜙𝑖 , 𝑖 = 1, . . . , 𝑟, gives the Riesz basis.

1. Introduction and Preliminaries The shift-invariant spaces 𝑉𝜇𝑝 (Φ), 𝑝 ∈ [1, ∞], quoted in the abstract, are used in the wavelet analysis, approximation theory, sampling theory, and so forth. They have been extensively studied by many authors [1–18]. The aim of this paper is to construct 𝑉𝜇𝑝 (Φ), 𝑝 ∈ [1, ∞], spaces with specially chosen functions 𝜙𝑖 , 𝑖 = 1, . . . , 𝑟, which generate its 𝑝-frame. These results extend and correct the construction obtained in [19]. For the first construction, we take functions 𝜙𝑖 , 𝑖 = 1, . . . , 𝑟, so that the Fourier transforms are compactly supported smooth functions. Also, we derive conditions for the collection {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z, 𝑖 = 1, . . . , 𝑟} to form a Riesz basis for 𝑉𝜇𝑝 (Φ). We note that the properties of the constructed frame guarantee the feasibility of a stable and continuous reconstruction algorithm in 𝑉𝜇𝑝 (Φ) [20]. We generalize these results for a shift-invariant subspace of 𝐿𝑝𝜇 (R𝑑 ). The second construction is obtained by choosing compactly supported functions 𝜙𝑖 , 𝑖 = 1, . . . , 𝑟. In this way, we obtain the Riesz basis. This paper is organized as follows. In Section 2 we quote some basic properties of certain subspaces of the weighted 𝐿𝑝 and ℓ𝑝 spaces. In Section 3 we derive conditions for functions

of the form 𝜙̂𝑖 (⋅) = 𝜃(⋅ + 𝑘𝑖 𝜋), 𝑘𝑖 ∈ Z, 𝑖 = 1, 2, . . . , 𝑟, 𝑟 ∈ N, to form a Riesz basis for 𝑉𝜇𝑝 (Φ). We also show that using functions of the form 𝜙̂𝑖 (⋅) = 𝜃(⋅ + 𝑖𝜋), 𝑖 = 1, . . . , 𝑟, where 𝜃 is compactly supported smooth function whose length of support is less than or equal to 2𝜋, we cannot construct a 𝑝-frame for the shift-invariant space 𝑉𝜇𝑝 (Φ). In Section 4 we construct a sequence {𝜙𝑖 (⋅−𝑗) | 𝑗 ∈ Z, 𝑖 = 0, . . . , 𝑟}, where 𝑟 ∈ 2N or 𝑟 ∈ 3N, which constitutes a 𝑝-frame for the weighted shift-invariant space 𝑉𝜇𝑝 (Φ). Our construction shows that the sampling and reconstruction problem in the shift-invariant spaces is robust in the sense of [1]. In Section 5 we construct 𝑝-Riesz basis by using compactly supported functions 𝜙𝑖 , 𝑖 = 1, . . . , 𝑟.

2. Basic Spaces Let a function 𝜔 be nonnegative, continuous, symmetric, and submultiplicative; that is, 𝜔(𝑥 + 𝑦) ≤ 𝜔(𝑥)𝜔(𝑦), 𝑥, 𝑦 ∈ R𝑑 ; let a function 𝜇 be 𝜔-moderate; that is, 𝜇(𝑥 + 𝑦) ≤ 𝐶𝜔(𝑥)𝜇(𝑦), 𝑥, 𝑦 ∈ R𝑑 . Functions 𝜇 and 𝜔 are called weights. We consider the weighted function spaces 𝐿𝑝𝜇 and the weighted sequence

2


spaces ℓ𝜇𝑝 (Z𝑑 ) with 𝜔-moderate weights 𝜇 (see [19]). Let 𝑝 ∈ [1, ∞). Then (with obvious modification for 𝑝 = ∞) L𝑝𝜔

{ { { 󵄩 󵄩 = {𝑓 | 󵄩󵄩󵄩𝑓󵄩󵄩󵄩L𝑝𝜔 { { { = (∫

[0,1]

Let Φ = (𝜙1 , . . . , 𝜙𝑟 )𝑇 . Let 𝑝

󵄨 󵄨 ( ∑ 󵄨󵄨󵄨𝑓 (𝑥 + 𝑗)󵄨󵄨󵄨 𝜔 (𝑥 + 𝑗)) 𝑑𝑥) 𝑑

1/𝑝

𝑗∈Z𝑑

} } } < +∞} , } } } 𝑊𝜔𝑝

Remark 1 (see [22]). Let Φ ∈ 𝑊𝜔1 and let 𝜇 be 𝜔-moderate. Then 𝑉𝜇𝑝 (Φ) is a subspace (not necessarily closed) of 𝐿𝑝𝜇 and 𝑊𝜇𝑝 for any 𝑝 ∈ [1, ∞]. Clearly (2) implies that ℓ𝜇𝑝 and 𝑉𝜇𝑝 (Φ) are isomorphic Banach spaces.

̂ Φ] ̂ (𝜉) = [ ∑ 𝜙̂𝑖 (𝜉 + 2𝑘𝜋) 𝜙̂𝑗 (𝜉 + 2𝑘𝜋)] , [Φ, 𝑑 𝑘∈Z [ ]1≤𝑖≤𝑟,1≤𝑗≤𝑟 (4) ̂𝑖 (𝜉)𝜙̂𝑗 (𝜉) is integrable for any 1 ≤ 𝑖, where we assume that 𝜙 𝑗 ≤ 𝑟. Let 𝐴 = [𝑎(𝑗)]𝑗∈Z𝑑 be an 𝑟 × ∞ matrix and 𝐴𝐴𝑇 = [∑𝑗∈Z𝑑 𝑎𝑖 (𝑗)𝑎𝑖󸀠 (𝑗)]1≤𝑖,𝑖󸀠 ≤𝑟 . Then rank 𝐴 = rank 𝐴𝐴𝑇 . We recall results from [1, 19] which are needed in the sequel.

{ { { { 󵄩 󵄩 := {𝑓 | 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑊𝜔𝑝 { { { {

Lemma 2 (see [1]). The following statements are equivalent. ̂ + 2𝑗𝜋)]𝑗∈Z𝑑 is a constant function on R𝑑 . (1) rank[Φ(𝜉 1/𝑝

𝑝 󵄨𝑝 󵄨 = ( ∑ sup 󵄨󵄨󵄨𝑓 (𝑥 + 𝑗)󵄨󵄨󵄨 𝜔(𝑗) ) 𝑑 𝑗∈Z𝑑 𝑥∈[0,1]

} } } } < +∞} . } } } } (1) 𝑇

In what follows, we use the notation Φ = (𝜙1 , . . . , 𝜙𝑟 ) . Define ‖Φ‖H = ∑𝑟𝑖=1 ‖𝜙𝑖 ‖H , where H = 𝐿𝑝𝜔 , L𝑝𝜔 or 𝑊𝜔𝑝 , 𝑝 ∈ [1, ∞]. With F𝜙 = 𝜙̂ we denote the Fourier transform of the ̂ = ∫ 𝑑 𝜙(𝑥)𝑒−𝑖𝜋𝑥⋅𝜉 d𝑥, 𝜉 ∈ R𝑑 . function 𝜙; that is, 𝜙(𝜉) R Let 𝑐 = {𝑐𝑖 }𝑖∈N ∈ ℓ𝜇𝑝 and 𝑓, 𝑔 ∈ 𝐿𝑝𝜔 , 𝑝 ∈ [1, ∞]. We define, as in [1], the semiconvolution 𝑓∗󸀠 𝑐 as (𝑓∗󸀠 𝑐)(𝑥) = ∑𝑗∈Z𝑑 𝑐𝑗 𝑓(𝑥 − 𝑗), 𝑥 ∈ R𝑑 , and ⟨𝑓, 𝑔⟩ = ∫R𝑑 𝑓(𝑥)𝑔(𝑥) d𝑥. The concept of a 𝑝-frame is introduced in [1]. It is said that a collection {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z𝑑 , 𝑖 = 1, . . . , 𝑟} is a 𝑝-frame for 𝑉𝜇𝑝 (Φ) if there exists a positive constant 𝐶 (dependent upon Φ, 𝑝, and 𝜔) such that 󵄩 󵄩 𝐶−1 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝𝜇

̂ Φ](𝜉) ̂ (2) rank[Φ, is a constant function on R𝑑 . (3) There exists a positive constant 𝐶 independent of 𝜉 such that ̂ Φ] ̂ (𝜉) ≤ [Φ, ̂ Φ] ̂ (𝜉) [Φ, ̂ Φ] ̂ (𝜉)𝑇 𝐶−1 [Φ, ̂ Φ] ̂ (𝜉) , ≤ 𝐶 [Φ,

𝜉 ∈ [−𝜋, 𝜋]𝑑 .

(5)

The next theorem [19] derives necessary and sufficient conditions for an indexed family {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z𝑑 , 𝑖 = 1, . . . , 𝑟} to constitute a 𝑝-frame for 𝑉𝜇𝑝 (Φ), which is equivalent with the closedness of this space in 𝐿𝑝𝜇 . Thus, it is shown that under appropriate conditions on the frame vectors, there is an equivalence between the concept of 𝑝-frames, Banach frames, and the closedness of the space they generate. Theorem 3 (see [19]). Let Φ = (𝜙1 , . . . , 𝜙𝑟 )𝑇 ∈ (𝑊𝜔1 )𝑟 , 𝑝0 ∈ [1, ∞], and let 𝜇 be 𝜔-moderate. The following statements are equivalent. (i) 𝑉𝜇𝑝0 (Φ) is closed in 𝐿𝑝𝜇0 .

󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 ≤ ∑󵄩󵄩󵄩󵄩{∫ 𝑓 (𝑥) 𝜙𝑖 (𝑥 − 𝑗) d𝑥} 󵄩𝑝 𝑑 𝑑 󵄩 𝑗∈Z 󵄩 󵄩ℓ𝜇 𝑖=1 󵄩 R 𝑟

󵄩 󵄩 ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝𝜇 ,

(2)

(iii) There exists a positive constant 𝐶 such that

𝑓 ∈ 𝑉𝜇𝑝 (Φ) .

̂ Φ] ̂ (𝜉) ≤ [Φ, ̂ Φ] ̂ (𝜉) [Φ, ̂ Φ] ̂ (𝜉)𝑇 𝐶−1 [Φ,

Recall [21] that the shift-invariant spaces are defined by 𝑟 { 𝑉𝜇𝑝 (Φ) := {𝑓 ∈ 𝐿𝑝𝜇 | 𝑓 (⋅) = ∑ ∑ 𝑐𝑗𝑖 𝜙𝑖 (⋅ − 𝑗) , 𝑖=1 𝑗∈Z𝑑 {

{𝑐𝑗𝑖 }

𝑗∈Z𝑑

} ∈ ℓ𝜇𝑝 , 𝑖 = 1, . . . , 𝑟} . }

(ii) {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z𝑑 , 𝑖 = 1, . . . , 𝑟} is a 𝑝0 -frame for 𝑉𝜇𝑝0 (Φ).

̂ Φ] ̂ (𝜉) , ≤ 𝐶 [Φ,

𝜉 ∈ [−𝜋, 𝜋]𝑑 .

(6)

(iv) There exist positive constants 𝐶1 and 𝐶2 (depending on Φ and 𝜔) such that (3)

󵄩 󵄩 𝐶1 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝𝜇0 ≤

𝑟 󵄩 󵄩󵄩 󵄩󵄩 𝑖 󵄩󵄩{𝑐𝑗 } 𝑑 󵄩󵄩󵄩 𝑝 ∑ 󵄩 󵄩󵄩ℓ𝜇 0 󸀠 𝑖 𝑗∈Z 𝑓=∑𝑖=1 𝜙𝑖 ∗ 𝑐 𝑖=1 󵄩

inf 𝑟

󵄩 󵄩 ≤ 𝐶2 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝𝜇0 ,

𝑓∈

𝑉𝜇𝑝0

(Φ) .

(7)


3

(v) There exists Ψ = (𝜓1 , . . . , 𝜓𝑟 )𝑇 ∈ (𝑊𝜔1 )𝑟 , such that

Proof. By the Paley-Wiener theorem, 𝜙𝑖 ∈ S(R) ⊂ 𝑊𝜇1 (R), 𝑖 = 1, . . . , 𝑟. All possible cases are described in the following lemmas.

𝑟

𝑓 = ∑ ∑ ⟨𝑓, 𝜓𝑖 (⋅ − 𝑗)⟩ 𝜙𝑖 (⋅ − 𝑗) 𝑖=1 𝑗∈Z𝑑 𝑟

Lemma 6. Let Φ = (𝜙𝑘1 , 𝜙𝑘2 )𝑇 , 𝑘2 − 𝑘1 = 2, 𝑘1 , 𝑘2 ∈ Z. The ̂ rank of matrix [Φ(𝜉+2𝑗𝜋)] 𝑗∈Z is a constant function on R and equals 2.

𝑓 ∈ 𝑉𝜇𝑝0 (Φ) .

= ∑ ∑ ⟨𝑓, 𝜙𝑖 (⋅ − 𝑗)⟩ 𝜓𝑖 (⋅ − 𝑗) , 𝑖=1 𝑗∈Z𝑑

(8) Corollary 4 (see [19]). Let Φ = (𝜙1 , . . . , 𝜙𝑟 )𝑇 ∈ (𝑊𝜔1 )𝑟 , 𝑝0 ∈ [1, ∞], and let 𝜇 be 𝜔-moderate. (i) If {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z𝑑 , 𝑖 = 1, . . . , 𝑟} is a 𝑝0 -frame for 𝑉𝜇𝑝0 (Φ), then the collection {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z𝑑 , 𝑖 = 1, . . . , 𝑟} is a 𝑝-frame for 𝑉𝜇𝑝 (Φ) for any 𝑝 ∈ [1, ∞]. (ii) If 𝑉𝜇𝑝0 (Φ) is closed in 𝐿𝑝𝜇0 and 𝑊𝜇𝑝0 , then 𝑉𝜇𝑝 (Φ) is closed in 𝐿𝑝𝜇 and 𝑊𝜇𝑝 for any 𝑝 ∈ [1, ∞]. (iii) If (7) holds for 𝑝0 , then it holds for any 𝑝 ∈ [1, ∞].

3. Construction of Frames Using a Band-Limited Function

𝑘 ∈ Z,

(9)

and Φ = (𝜙𝑖 , 𝜙𝑖+1 , . . . , 𝜙𝑖+𝑟 )𝑇 , 𝑖 ∈ Z, 𝑟 ∈ N. ̂ + 2𝑗𝜋)]𝑗∈Z is not a constant Then the rank of matrix [Φ(𝜉 function on R and it depends on 𝜉 ∈ R. As a matter of fact, by the Paley-Wiener theorem, 𝜙𝑖 ∈ S(R) ⊂ 𝑊𝜇1 (R), 𝑖 ∈ Z. For any 𝑖 ∈ Z, matrix [𝜙̂𝑖 , 𝜙̂𝑖 ](𝜉) = ∑𝑗∈Z |𝜃(𝜉 + 𝑖𝜋 + 2𝑗𝜋)|2 , 𝜉 ∈ R, has the rank 0 or 1, depending on 𝜉. Moreover, we have [𝜙̂2𝑖 , 𝜙̂2𝑖 ](𝜋) = 0 and [𝜙̂2𝑖 , 𝜙̂2𝑖 ](0) > 0. ̂ + 2𝑗𝜋)]𝑗∈Z is not Because of that, the rank of the matrix [Φ(𝜉 a constant function on R and it depends on 𝜉 ∈ R. 𝐶0∞ (R)

Theorem 5. Let 𝜃 ∈ be a non-negative function such that 𝜃(𝑥) > 0, 𝑥 ∈ (−𝜋 − 𝜀, 𝜋 + 𝜀), and supp 𝜃 = [−𝜋 − 𝜀, 𝜋 + 𝜀], where 0 < 𝜀 < 1/4. Moreover, let 𝜙̂𝑖 (𝜉) = 𝜃 (𝜉 + 𝑘𝑖 𝜋) ,

𝑘𝑖 ∈ Z, 𝑖 = 1, 2, . . . , 𝑟,

(10)

and Φ = (𝜙1 , 𝜙2 , . . . , 𝜙𝑟 )𝑇 . (1) If |𝑘2 − 𝑘1 | = 2 and |𝑘𝑖 − 𝑘𝑗 | ≥ 2 for different 𝑖, 𝑗 ≤ 𝑟, then ̂ (𝜉 + 2𝑗𝜋)] rank [Φ = 𝑟, 𝑗∈Z

⋅ ⋅ ⋅ 0 0 𝑎1 𝑏1 0 ⋅ ⋅ ⋅ ], [ ⋅ ⋅ ⋅ 0 𝑎2 𝑏2 0 0 ⋅ ⋅ ⋅

𝜉 ∈ R.

(11)

(2) If |𝑘2 − 𝑘1 | = 2 and, at least for 𝑘𝑖1 and 𝑘𝑖2 , it holds that ̂ + |𝑘𝑖1 − 𝑘𝑖2 | = 1, where 1 ≤ 𝑖1 , 𝑖2 ≤ 𝑟, then rank[Φ(𝜉 2𝑗𝜋)]𝑗∈Z is not a constant function on R.

(12)

̂ + for some 𝑎𝑖 , 𝑏𝑖 > 0, 𝑖 = 1, 2. It is obvious that rank[Φ(𝜉 2𝑗𝜋)]𝑗∈Z = 2, 𝜉 ∈ (−𝜋 − 𝜀 − 𝑘1 𝜋 + 2ℓ𝜋, −𝜋 + 𝜀 − 𝑘1 𝜋 + 2ℓ𝜋), ℓ ∈ Z. (2∘ ) For 𝜉 ∈ [−𝜋 + 𝜀 − 𝑘1 𝜋 + 2ℓ𝜋, 𝜋 − 𝜀 − 𝑘1 𝜋 + 2ℓ𝜋], ℓ ∈ Z, there are only two nonzero values 𝑎1 and 𝑎2 which are ̂ + 2𝑗𝜋)]𝑗∈Z . Since in different columns of matrix [Φ(𝜉 ̂ (𝜉 + 2𝑗𝜋)] [Φ =[ 𝑗∈Z

Considering the length of the support of a function 𝜃, we have ̂ + 2𝑗𝜋)]𝑗∈Z . different cases for the rank of matrix [Φ(𝜉 First, we consider the next claim. Let 𝜃 ∈ 𝐶0∞ (R) be a nonnegative function such that 𝜃(𝑥) > 0, 𝑥 ∈ (−𝜋, 𝜋), and supp 𝜃 ⊆ [−𝜋, 𝜋]. Moreover, let 𝜙̂𝑘 (𝜉) = 𝜃 (𝜉 + 𝑘𝜋) ,

Proof. We have the next two cases. (1∘ ) If 𝜉 ∈ (−𝜋 − 𝜀 − 𝑘1 𝜋 + 2ℓ𝜋, −𝜋 + 𝜀 − 𝑘1 𝜋 + 2ℓ𝜋), ℓ ∈ Z, ̂ + 2𝑗𝜋)]𝑗∈Z we obtain 2 × ∞ matrix for matrix [Φ(𝜉

⋅ ⋅ ⋅ 0 0 𝑎1 0 ⋅ ⋅ ⋅ ] , ⋅ ⋅ ⋅ 0 𝑎2 0 0 ⋅ ⋅ ⋅ 2×∞

(13)

it has the rank 2 for all 𝜉 ∈ [−𝜋+𝜀−𝑘1 𝜋+2ℓ𝜋, 𝜋−𝜀−𝑘1 𝜋+2ℓ𝜋], ℓ ∈ Z. ̂ + 2𝑗𝜋)]𝑗∈Z , Φ = We conclude that the rank of matrix [Φ(𝜉 𝑇 (𝜙𝑘1 , 𝜙𝑘2 ) , 𝑘2 − 𝑘1 = 2, 𝑘1 , 𝑘2 ∈ Z, is a constant function on R and equals 2. ̂ Lemma 7. The rank of matrix [Φ(𝜉+2𝑗𝜋)] 𝑗∈Z is not a constant 𝑇 function on R if Φ = (𝜙𝑘1 , 𝜙𝑘2 ) , 𝑘2 − 𝑘1 = 1, 𝑘1 , 𝑘2 ∈ Z. ̂ Proof. We have four different cases for matrix [Φ(𝜉+2𝑗𝜋)] 𝑗∈Z . Suppose, without losing generality, that 𝑘1 ∈ 2Z. (1∘ ) If 𝜉 ∈ (−𝜋 − 𝜀 + 2ℓ𝜋, −𝜋 + 𝜀 + 2ℓ𝜋), ℓ ∈ Z, then ̂ (𝜉 + 2𝑗𝜋)] [Φ =[ 𝑗∈Z

⋅ ⋅ ⋅ 0 𝑎1 𝑏1 0 ⋅ ⋅ ⋅ ], ⋅ ⋅ ⋅ 0 𝑎2 0 0 ⋅ ⋅ ⋅ 1

2

(14)

1

𝑎 , 𝑎 , 𝑏 > 0, ̂ + 2𝑗𝜋)]𝑗∈Z = 2, for all 𝜉 ∈ (−𝜋 − 𝜀 + 2ℓ𝜋, −𝜋 + and rank[Φ(𝜉 𝜀 + 2ℓ𝜋), ℓ ∈ Z. (2∘ ) For 𝜉 ∈ [−𝜋+𝜀+2ℓ𝜋, −𝜀+2ℓ𝜋], ℓ ∈ Z, nonzero values 1 ̂ + 2𝑗𝜋)]𝑗∈Z . 𝑎 and 𝑎2 are in the same column of matrix [Φ(𝜉 For any choice of a 2 × 2 matrix, we get that the determinant equals 0. So we obtain rank [

⋅ ⋅ ⋅ 0 𝑎1 0 ⋅ ⋅ ⋅ ] = 1, ⋅ ⋅ ⋅ 0 𝑎2 0 ⋅ ⋅ ⋅

(15)

for all 𝜉 ∈ [−𝜋 + 𝜀 + 2ℓ𝜋, −𝜀 + 2ℓ𝜋], ℓ ∈ Z. (3∘ ) If 𝜉 ∈ (−𝜀 + 2ℓ𝜋, 𝜀 + 2ℓ𝜋), ℓ ∈ Z, then ⋅ ⋅ ⋅ 0 0 𝑎1 0 ⋅ ⋅ ⋅ ̂ (𝜉 + 2𝑗𝜋)] [Φ ], =[ 𝑗∈Z ⋅ ⋅ ⋅ 0 𝑏2 𝑎2 0 ⋅ ⋅ ⋅

(16)

4


for some 𝑎1 , 𝑎2 , 𝑏2 > 0, has the rank 2, for all 𝜉 ∈ (−𝜀+2ℓ𝜋, 𝜀+ 2ℓ𝜋), ℓ ∈ Z. (4∘ ) For 𝜉 ∈ [𝜀+2ℓ𝜋, 𝜋−𝜀+2ℓ𝜋], ℓ ∈ Z, there are two non̂ + zero values 𝑎1 and 𝑏2 in different columns of matrix [Φ(𝜉 2𝑗𝜋)]𝑗∈Z and the block with these elements determines the rank 2 for all 𝜉 ∈ [−𝜀 + 2ℓ𝜋, 𝜋 − 𝜀 + 2ℓ𝜋], ℓ ∈ Z. ̂ + Considering possible cases, we conclude that rank[Φ(𝜉 2𝑗𝜋)]𝑗∈Z , Φ = (𝜙𝑘1 , 𝜙𝑘2 )𝑇 , 𝑘2 − 𝑘1 = 1, 𝑘1 , 𝑘2 ∈ Z, depends on 𝜉 ∈ R and equals 1 or 2. This rank is a nonconstant function on R. Proof of Theorem 5. (1) Using Lemmas 6 and 7, it is obvious that if |𝑘2 − 𝑘1 | = 2 and |𝑘𝑖 − 𝑘𝑗 | ≥ 2 for different 𝑖, 𝑗 ≤ 𝑟, then the position of the first non-zero element in each row of ̂ + 2𝑗𝜋)]𝑗∈Z is unique for each row. So the rank of matrix [Φ(𝜉 ̂ matrix [Φ(𝜉+2𝑗𝜋)] 𝑗∈Z is a constant function on R and equals 𝑟 for all 𝜉 ∈ R. (2) If |𝑘2 − 𝑘1 | = 2 and, at least for 𝑘𝑖1 and 𝑘𝑖2 , it holds that |𝑘𝑖1 − 𝑘𝑖2 | = 1, 1 ≤ 𝑖1 , 𝑖2 ≤ 𝑟, then, in the row with the index 𝑖2 (suppose, without losing generality, that 𝑖2 ∈ 2Z + 1), we will have a new column with a non-zero element for 𝜉 ∈ (−𝜋 − 𝜀 + 2ℓ𝜋, −𝜋 + 𝜀 + 2ℓ𝜋), ℓ ∈ Z, but for 𝜉 ∈ [𝜀 + 2ℓ𝜋, 𝜋 − 𝜀 + 2ℓ𝜋], ℓ ∈ Z, the positions of all non-zero elements in that row will appear in the previous columns. It is obvious that the rank of ̂ + 2𝑗𝜋)]𝑗∈Z depends on 𝜉 ∈ R and is not the the matrix [Φ(𝜉 same for all 𝜉 ∈ R. As a consequence of Theorems 3 and 5(1), we have the following result. Theorem 8. Let the functions 𝜃 and Φ satisfy all the conditions of Theorem 5(1). Then space 𝑉𝜇𝑝 (Φ) is closed in 𝐿𝑝𝜇 for any 𝑝 ∈ [1, ∞], and the family {𝜙𝑖 (⋅−𝑗) | 𝑗 ∈ Z, 1 ≤ 𝑖 ≤ 𝑟} is a 𝑝-Riesz basis for 𝑉𝜇𝑝 (Φ) for any 𝑝 ∈ [1, ∞]. The following theorem is a generalisation of Theorem 5 and can be proved in the same way, so we omit the proof. Theorem 9. Let 𝜃 ∈ 𝐶0∞ (R) be a positive function such that 𝜃(𝑥) > 0, 𝑥 ∈ (𝑎, 𝑏), 𝑏 > 𝑎, and supported by [𝑎, 𝑏] where 𝑏 − 𝑎 > 2𝜋. Moreover, let ̂𝑖 (𝜉) = 𝜃 (𝜉 + 𝑘𝑖 𝜋) , 𝑘𝑖 ∈ Z, 𝑖 = 1, 2, . . . , 𝑟, 𝑟 ∈ N, 𝜙 (17)

Lemma 10. Let 𝜃 ∈ 𝐶0∞ (R), 𝜓 ∈ 𝐶0∞ (R) be positive functions such that 𝜃 (𝑥) > 0,

𝑥 ∈ (−𝜀, 2𝜋 + 𝜀) ,

𝜓 (𝑥) > 0,

𝑥 ∈ (𝜀, 2𝜋 − 𝜀) , (19)

supp 𝜃 = [−𝜀, 2𝜋 + 𝜀] , supp 𝜓 = [𝜀, 2𝜋 − 𝜀] ,

1 0 0,

has a constant rank equal to 1. (2∘ ) For 𝜉 ∈ (𝜀 + 2ℓ𝜋, 2𝜋 − 𝜀 + 2ℓ𝜋), ℓ ∈ Z, the rank of matrix ̂ (𝜉 + 2𝑗𝜋)] =[ [Φ 𝑗∈Z

⋅⋅⋅ 0 𝑐 0 ⋅⋅⋅ ], ⋅⋅⋅ 0 𝑑 0 ⋅⋅⋅

(2) If |𝑘2 − 𝑘1 | = 2 and, at least for 𝑘𝑖1 and 𝑘𝑖2 , it holds that ̂ + |𝑘𝑖1 − 𝑘𝑖2 | = 1, where 1 ≤ 𝑖1 , 𝑖2 ≤ 𝑟, then rank[Φ(𝜉 2𝑗𝜋)]𝑗∈Z is not a constant function on R.

4. Construction of Frames Using Several Band-Limited Functions Firstly, we consider two smooth functions with proper compact supports.

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where 𝑐, 𝑑 are non-zero values, is equal to 1. An equivalent matrix is obtained for 𝜉 = 𝜀 + 2ℓ𝜋 and 𝜉 = −𝜀 + 2ℓ𝜋, so we ̂ + 2𝑗𝜋)]𝑗∈Z = 1, for 𝜉 ∈ [𝜀 + 2ℓ𝜋, 2𝜋 − conclude that rank[Φ(𝜉 𝜀 + 2ℓ𝜋], ℓ ∈ Z. ̂ + Considering these two cases, the rank of matrix [Φ(𝜉 ̂ 2𝑗𝜋)]𝑗∈Z is a constant function on R, rank[Φ(𝜉+2𝑗𝜋)]𝑗∈Z = 1, 𝜉 ∈ R. Using functions 𝜃 and 𝜓 from Lemma 10, in the next lemma we construct the 𝑝-frame with four functions. Lemma 11. Let the functions 𝜃 and 𝜓 satisfy all the conditions of Lemma 10. Moreover, let 𝜙̂𝑘 (𝜉) = 𝜃 (𝜉 + 2𝑘𝜋) ,

𝜙̂𝑘+2 (𝜉) = 𝜓 (𝜉 + 2𝑘𝜋) ,

and Φ = (𝜙1 , 𝜙2 , . . . , 𝜙𝑟 )𝑇 . (1) If |𝑘2 − 𝑘1 | = 2 and |𝑘𝑖 − 𝑘𝑗 | ≥ 2 for different 𝑖, 𝑗 ≤ 𝑟, then ̂ (𝜉 + 2𝑗𝜋)] = 𝑟, 𝜉 ∈ R. rank [Φ (18) 𝑗∈Z

(20)

𝑘 = 0, 1,

(22)

̂ + 2𝑗𝜋)]𝑗∈Z = 2, and Φ = (𝜙0 , 𝜙1 , 𝜙2 , 𝜙3 )𝑇 . Then rank[Φ(𝜉 𝜉 ∈ R. Proof. The proof is similar to the proof of Lemma 10. (1∘ ) If 𝜉 ∈ (−𝜀 + 2ℓ𝜋, 𝜀 + 2ℓ𝜋), ℓ ∈ Z, then matrix

̂ (𝜉 + 2𝑗𝜋)] [Φ 𝑗∈Z

⋅⋅⋅ [⋅ ⋅ ⋅ =[ [⋅ ⋅ ⋅ [⋅ ⋅ ⋅

0 0 0 0

0 𝑎2 0 0

𝑎1 𝑏2 0 0

𝑏1 0 0 0

0 0 0 0

⋅⋅⋅ ⋅ ⋅ ⋅] ], ⋅ ⋅ ⋅] ⋅ ⋅ ⋅]

where 𝑎𝑖 , 𝑏𝑖 > 0, 𝑖 = 1, 2, has a constant rank equal to 2.

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5

(2∘ ) For 𝜉 ∈ [𝜀 + 2ℓ𝜋, 2𝜋 − 𝜀 + 2ℓ𝜋], ℓ ∈ Z, we have ̂ (𝜉 + 2𝑗𝜋)] rank [Φ 𝑗∈Z

⋅⋅⋅ [⋅ ⋅ ⋅ =[ [⋅ ⋅ ⋅ [⋅ ⋅ ⋅

0 0 0 0

0 0 𝑐2 𝑑2

𝑐1 𝑑1 0 0

0 0 0 0

(2∘ ) Consider

⋅⋅⋅ ⋅ ⋅ ⋅] ] = 2, ⋅ ⋅ ⋅] ⋅ ⋅ ⋅] (24)

where 𝑐𝑖 , 𝑑𝑖 > 0, 𝑖 = 1, 2. ̂ + 2𝑗𝜋)]𝑗∈Z is a We conclude that the rank of matrix [Φ(𝜉 constant function on R equal to 2. Lemma 10 can be easily generalised for an even number of functions 𝜙𝑖 , 𝑖 = 0, . . . , 2𝑟 − 1, with compactly supported 𝜙̂𝑖 , 𝑖 = 0, . . . , 2𝑟 − 1. The proof of the next theorem is similar to the previous proofs. Theorem 12. Let the functions 𝜃 and 𝜓 satisfy all the conditions of Lemma 10. Moreover, let 𝜙̂𝑘 (𝜉) = 𝜃 (𝜉 + 2𝑘𝜋) ,

𝜙̂𝑘+𝑟 (𝜉) = 𝜓 (𝜉 + 2𝑘𝜋) ,

𝑘 = 0, . . . , 𝑟 − 1, 𝑟 ∈ N,

(25)

and Φ = (𝜙0 , 𝜙1 , . . . , 𝜙2𝑟−1 )𝑇 . The following statements hold. ̂ + 2𝑗𝜋)]𝑗∈Z = 𝑟 for all 𝜉 ∈ R. (1 ) rank[Φ(𝜉 ∘

(2∘ ) 𝑉𝜇𝑝 (Φ) is closed in 𝐿𝑝𝜇 for any 𝑝 ∈ [1, ∞]. (3∘ ) {𝜙𝑖 (⋅ − 𝑗) | 𝑗 ∈ Z, 0 ≤ 𝑖 ≤ 2𝑟 − 1} is a 𝑝-frame for 𝑉𝜇𝑝 (Φ) for any 𝑝 ∈ [1, ∞]. Now we consider three functions with compact supports. Lemma 13. Let the function 𝜃 satisfies all the conditions of Lemma 10, and let 𝜏 ∈ 𝐶0∞ (R) and 𝜔 ∈ 𝐶0∞ (R) be positive functions such that 𝜏 (𝑥) > 0,

𝑥 ∈ (𝜀, 𝜋 − 𝜀) ∪ (𝜋 + 𝜀, 2𝜋 − 𝜀) ,

𝜔 (𝑥) > 0,

supp 𝜔 = [−3𝜋 − 𝜀, −𝜋 + 𝜀] ,

(26)

Moreover, let 𝜙̂1 (𝜉) = 𝜃(𝜉), 𝜙̂2 (𝜉) = 𝜏(𝜉), 𝜙̂3 (𝜉) = 𝜔(𝜉), 𝜉 ∈ R, ̂ + 2𝑗𝜋)]𝑗∈Z = 2, 𝜉 ∈ R. and Φ = (𝜙1 , 𝜙2 , 𝜙3 )𝑇 . Then rank[Φ(𝜉 ̂ Proof. We have four different forms for matrix [Φ(𝜉+2𝑗𝜋)] 𝑗∈Z and in all we show that the rank of matrix is equal to 2. Now we will show all possible cases. Denote with 𝑎𝑖 , 𝑖 = 1, 2, 3, and 𝑏𝑖 , 𝑖 = 1, 2, some positive values. (1∘ ) Consider ̂ (𝜉 + 2𝑗𝜋)] [Φ 𝑗∈Z

𝜉 ∈ (−𝜀 + 2ℓ𝜋, 𝜀 + 2ℓ𝜋) .

𝜉 ∈ [𝜀 + 2ℓ𝜋, 𝜋 − 𝜀 + 2ℓ𝜋] . (3∘ ) Consider ⋅ ⋅ ⋅ 0 𝑏1 0 0 0 ⋅ ⋅ ⋅ [ ̂ [Φ (𝜉 + 2𝑗𝜋)]𝑗∈Z = ⋅ ⋅ ⋅ 0 0 𝑎2 𝑏2 0 ⋅ ⋅ ⋅] , [⋅ ⋅ ⋅ 0 0 0 0 0 ⋅ ⋅ ⋅]

(29)

𝜉 ∈ (𝜋 − 𝜀 + 2ℓ𝜋, 𝜋 + 𝜀 + 2ℓ𝜋) . (4∘ ) Consider ⋅ ⋅ ⋅ 0 𝑏1 0 0 0 ⋅ ⋅ ⋅ [ ̂ [Φ (𝜉 + 2𝑗𝜋)]𝑗∈Z = ⋅ ⋅ ⋅ 0 0 0 𝑏2 0 ⋅ ⋅ ⋅] , 3 [⋅ ⋅ ⋅ 0 𝑎 0 0 0 ⋅ ⋅ ⋅]

(30)

𝜉 ∈ [𝜋 + 𝜀 + 2ℓ𝜋, 2𝜋 − 𝜀 + 2ℓ𝜋] . Remark 14. In Lemma 13 the support of the function 𝜔 must have an empty intersection with the supports of 𝜃 and 𝜏. In the opposite case, that is, supp 𝜃 ∩ supp 𝜏 ∩ supp 𝜔 ≠0, the ̂ + 2𝑗𝜋)]𝑗∈Z is a non-constant function rank of the matrix [Φ(𝜉 on R. Lemma 13 can be easily generalised for functions 𝜙𝑖 , 𝑖 = 0, . . . , 3𝑟 − 1, with compactly supported 𝜙̂𝑖 , 𝑖 = 0, . . . , 3𝑟 − 1. The proof of the next theorem is similar to the previous proofs. Theorem 15. Let the functions 𝜃, 𝜏, and 𝜔 satisfy all the conditions of Lemma 13. Moreover, let

𝜙̂𝑘+𝑟 (𝜉) = 𝜏 (𝜉 + 2𝑘𝜋) ,

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𝜙̂𝑘+2𝑟 (𝜉) = 𝜔 (𝜉 + 2𝑘𝜋) ,

1 0 0, 𝜙2 := 𝜙1 ∗ 𝜙1 , 𝜙3 := 𝜙1 ∗ 𝜙1 ∗ 𝜙1 , . . .; that is, 𝜙𝑛 := ⏟⏟ 𝜙⏟1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∗ 𝜙1 ⏟∗ ⋅ ⋅ ∗ 𝜙⏟1⏟, ⏟⏟⏟⋅⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑛 ∈ N,

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𝑛−1 times

where ∗ denotes the convolution of the functions. We obtain 1 𝜙2 (𝑥) = 2 (𝑥𝐻 (𝑥) − 2 (𝑥 − 𝑎) 𝐻 (𝑥 − 𝑎) 𝑎 + (𝑥 − 2𝑎) 𝐻 (𝑥 − 2𝑎)) , 𝜙3 (𝑥) =

1 (𝑥2 𝐻 (𝑥) − 3(𝑥 − 𝑎)2 𝐻 (𝑥 − 𝑎) + 3(𝑥 − 2𝑎)2 2!𝑎3 ×𝐻 (𝑥 − 2𝑎) − (𝑥 − 3𝑎)2 𝐻 (𝑥 − 3𝑎)) ,

𝜙4 (𝑥) =

1 (𝑥3 𝐻 (𝑥) − 4(𝑥 − 𝑎)3 𝐻 (𝑥 − 𝑎) 3!𝑎4 3

+ 6(𝑥 − 2𝑎) 𝐻 (𝑥 − 2𝑎) − 4(𝑥 − 3𝑎)

(33) Continuing in this manner, for all 𝑛 ∈ N, we have 1 𝑎𝑛 (𝑛 − 1)! 𝑛 𝑛 × ( ( ) 𝑥𝑛−1 𝐻 (𝑥) − ( ) (𝑥 − 𝑎)𝑛−1 1 0 𝑛 × 𝐻 (𝑥 − 𝑎) + ( ) (𝑥 − 2𝑎)𝑛−1 2 𝑛 × 𝐻 (𝑥 − 2𝑎) − ( ) (𝑥 − 3𝑎)𝑛−1 𝐻 (𝑥 − 3𝑎) 3 𝑛−1

+ ⋅ ⋅ ⋅ + (−1)

𝑛 ( ) (𝑥 − (𝑛 − 1) 𝑎)𝑛−1 𝑛−1

𝑛 × 𝐻 (𝑥 − (𝑛 − 1) 𝑎) + (−1)𝑛 ( ) 𝑛 × (𝑥 − 𝑛𝑎)𝑛−1 𝐻 (𝑥 − 𝑛𝑎) ) . (34) Calculating the Fourier transform of functions 𝜙𝑛 , 𝑛 ∈ N, we get −𝑖 1 𝜙̂1 (𝜉) = V ⋅ 𝑝 ⋅ ( ) (𝑒𝑖𝑎𝜉 − 1) , 𝑎 𝜉 2 1 (−𝑖)2 𝜙̂2 (𝜉) = 2 V ⋅ 𝑝 ⋅ ( 2 ) (𝑒𝑖𝑎𝜉 − 1) , 𝑎 𝜉 3 1 (−𝑖)3 𝜙̂3 (𝜉) = 3 V ⋅ 𝑝 ⋅ ( 3 ) (𝑒𝑖𝑎𝜉 − 1) . 𝑎 𝜉

𝑅 (𝜉) ⋅⋅⋅ [ [ [⋅ ⋅ ⋅ [ [ [⋅ ⋅ ⋅ [ =[ [⋅ ⋅ ⋅ [ [ [ [ [ [⋅ ⋅ ⋅

𝛼−4𝜋 𝛽−4𝜋 𝛼−2𝜋 𝛽−2𝜋 𝛼0 𝛽0 𝛼2𝜋 𝛽2𝜋 𝛼4𝜋 𝛽4𝜋 ⋅ ⋅ ⋅ ] ] 2 2 2 2 2 2 2 2 𝛽−4𝜋 𝛼−2𝜋 𝛽−2𝜋 𝛼02 𝛽02 𝛼2𝜋 𝛽2𝜋 𝛼4𝜋 𝛽4𝜋 ⋅ ⋅ ⋅] 𝛼−4𝜋 ] ] 3 3 3 3 3 3 3 3 𝛼−4𝜋 𝛽−4𝜋 𝛼−2𝜋 𝛽−2𝜋 𝛼03 𝛽03 𝛼2𝜋 𝛽2𝜋 𝛼4𝜋 𝛽4𝜋 ⋅ ⋅ ⋅] ] ], 4 4 4 4 4 4 4 4 4 4 𝛼−4𝜋 𝛽−4𝜋 𝛼−2𝜋 𝛽−2𝜋 𝛼0 𝛽0 𝛼2𝜋 𝛽2𝜋 𝛼4𝜋 𝛽4𝜋 ⋅ ⋅ ⋅] ] ] ] .. .. .. .. .. ] . . . . . ] 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝛼−4𝜋 𝛽−4𝜋 𝛼−2𝜋 𝛽−2𝜋 𝛼0 𝛽0 𝛼2𝜋 𝛽2𝜋 𝛼4𝜋 𝛽4𝜋 ⋅ ⋅ ⋅] (36)

where 𝛼𝑘𝑚 = V ⋅ 𝑝 ⋅ (1/(𝜉 − 𝑘))𝑚 and 𝛽𝑘𝑚 = (𝑒𝑖𝑎(𝜉−𝑘) − 1)𝑚 . Since rank 𝑅(𝜉) = 𝑟, 𝜉 ∈ R, we have the next result.

3

× 𝐻 (𝑥 − 3𝑎) + (𝑥 − 4𝑎)3 𝐻 (𝑥 − 4𝑎)) .

𝜙𝑛 (𝑥) =

Continuing in this manner, we obtain 𝜙̂𝑛 (𝜉) = (−𝑖)𝑛 /𝑎𝑛 V ⋅ 𝑝 ⋅ (1/𝜉𝑛 )(𝑒𝑖𝑎𝜉 − 1)𝑛 , 𝑛 ∈ N, where V ⋅ 𝑝⋅ denotes the principal value. ̂ + 2𝑗𝜋)]𝑗∈Z Let Φ = (𝜙1 , 𝜙2 , . . . , 𝜙𝑟 )𝑇 , 𝑟 ∈ N. matrix [Φ(𝜉 has for all 𝜉 ∈ R the same rank as matrix

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Theorem 16. Let Φ = (𝜙𝑘 , 𝜙𝑘+1 , . . . , 𝜙𝑘+(𝑟−1) )𝑇 , for 𝑘 ∈ Z, 𝑟 ∈ N. Then 𝑉𝜇𝑝 (Φ) is closed in 𝐿𝑝𝜇 for any 𝑝 ∈ [1, ∞] and {𝜙𝑘+𝑠 (⋅ − 𝑗) | 𝑗 ∈ Z, 0 ≤ 𝑠 ≤ 𝑟 − 1} is a 𝑝-Riesz basis for 𝑉𝜇𝑝 (Φ) for any 𝑝 ∈ [1, ∞]. Remark 17. Let 𝑘 be a positive integer. We refer to [23] for the 𝐿𝑝 -approximation order 𝑘. Shift-invariant spaces generated by a finite number of compactly supported functions in 𝐿𝑝 (R), 1 ≤ 𝑝 ≤ ∞, were studied in [23] by Jia, who gave a characterization of the approximation order providing such shift-invariant spaces. Theorem 3 in [23] shows that the shiftinvariant space generated with the family of splines, which we constructed in Section 5, provides 𝐿𝑝 -approximation order 𝑘. Remark 18. (1) We refer to [3, 20] for the 𝛾-dense set 𝑋 = {𝑥𝑗 | 𝑗 ∈ 𝐽}. Let 𝜙𝑘 (𝑥) = F−1 (𝜃(⋅ − 𝑘𝜋))(𝑥), 𝑥 ∈ R. Following the notation of [20], we put 𝜓𝑥𝑗 = 𝜙𝑥𝑗 , where

{𝑥𝑗 | 𝑗 ∈ 𝐽} is 𝛾-dense set determined by 𝑓 ∈ 𝑉2 (𝜙) = 𝑉2 (F−1 (𝜃)). Theorems 3.1, 3.2, and 4.1 in [20] give conditions and explicit form of 𝐶𝑝 > 0 and 𝑐𝑝 > 0 such that inequality 𝑐𝑝 ‖𝑓‖𝐿𝑝𝜇 ≤ (∑𝑗∈𝐽 |⟨𝑓, 𝜓𝑥𝑗 ⟩𝜇(𝑥𝑗 )|𝑝 )1/𝑝 ≤ 𝐶𝑝 ‖𝑓‖𝐿𝑝𝜇 holds. This inequality guarantees the feasibility of a stable and continuous reconstruction algorithm in the signal spaces 𝑉𝜇𝑝 (Φ) [20]. ̂ Φ](𝜉), ̂ (2) Since the spectrum of the Gram matrix [Φ, where Φ is defined in Theorem 16, is bounded and bounded away from zero (see [7]), it follows that the family {Φ(⋅ − 𝑗) | 𝑗 ∈ Z} forms a 𝑝-Riesz basis for 𝑉𝜇𝑝 (Φ). (3) Frames of the above sections may be useful in applications since they satisfy assumptions of Theorems 3.1 and 3.2 in [4]. They show that error analysis for sampling and reconstruction can be tolerated or that the sampling and reconstruction problem in shift-invariant space is robust with respect to appropriate set of functions 𝜙𝑘1 , . . . , 𝜙𝑘𝑟 .


7

Acknowledgments The authors are indebted to the referee for the valuable suggestions, which have contributed to the improvement of the presentation of the paper. The authors were supported in part by the Serbian Ministry of Science and Technological Developments (Project no. 174024).

References [1] A. Aldroubi, Q. Sun, and W.-S. Tang, “𝑝-frames and shift invariant subspaces of 𝐿𝑝 ,” The Journal of Fourier Analysis and Applications, vol. 7, no. 1, pp. 1–21, 2001. [2] A. Aldroubi, Q. Sun, and W.-S. Tang, “Non-uniform sampling in multiply generated shift-invariant subspaces of 𝐿𝑝 (R𝑑 ),” in Wavelet Analysis and Applications (Guangzhou, 1999), vol. 25 of AMS/IP Stud. Adv. Math., pp. 1–8, American Mathematical Society, Providence, RI, USA, 2002. [3] A. Aldroubi, “Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces,” Applied and Computational Harmonic Analysis, vol. 13, no. 2, pp. 151– 161, 2002. [4] A. Aldroubi and I. Krishtal, “Robustness of sampling and reconstruction and Beurling-Landau-type theorems for shiftinvariant spaces,” Applied and Computational Harmonic Analysis, vol. 20, no. 2, pp. 250–260, 2006. [5] A. Aldroubi and M. Unser, “Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory,” Numerical Functional Analysis and Optimization, vol. 15, no. 1-2, pp. 1–21, 1994. [6] A. Aldroubi, A. Baskakov, and I. Krishtal, “Slanted matrices, Banach frames, and sampling,” Journal of Functional Analysis, vol. 255, no. 7, pp. 1667–1691, 2008. [7] C. de Boor, R. A. DeVore, and A. Ron, “The structure of finitely generated shift-invariant spaces in 𝐿2 (R𝑑 ),” Journal of Functional Analysis, vol. 119, no. 1, pp. 37–78, 1994. [8] R. A. DeVore, B. Jawerth, and B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Transactions on Information Theory, vol. 38, no. 2, pp. 719–746, 1992. [9] H. G. Feichtinger, “Banach convolution algebras of Wiener type,” in Functions, Series, Operators, Vol. I, II (Budapest, 1980), vol. 35 of Colloq. Math. Soc. János Bolyai, pp. 509–524, NorthHolland, Amsterdam, The Netherlands, 1983. [10] H. G. Feichtinger and K. Gröchenig, “A unified approach to atomic decompositions via integrable group representations,” in Function Spaces and Applications (Lund, 1986), vol. 1302 of Lecture Notes in Math., pp. 52–73, Springer, Berlin, Germany, 1988. [11] H. G. Feichtinger and K. H. Gröchenig, “Banach spaces related to integrable group representations and their atomic decompositions. I,” Journal of Functional Analysis, vol. 86, no. 2, pp. 307– 340, 1989. [12] H. G. Feichtinger, “Generalized amalgams, with applications to Fourier transform,” Canadian Journal of Mathematics, vol. 42, no. 3, pp. 395–409, 1990. [13] H. G. Feichtinger and K. Gröchenig, “Iterative reconstruction of multivariate band-limited functions from irregular sampling values,” SIAM Journal on Mathematical Analysis, vol. 23, no. 1, pp. 244–261, 1992. [14] H. G. Feichtinger and K. Gröchenig, “Theory and practice of irregular sampling,” in Wavelets: Mathematics and Applications,

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Research Article Suzuki-Type Fixed Point Results in Metric-Like Spaces Nabiollah Shobkolaei,1 Shaban Sedghi,2 Jamal Rezaei Roshan,2 and Nawab Hussain3 1

Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2

Correspondence should be addressed to Nawab Hussain; [email protected] Received 16 May 2013; Accepted 17 July 2013 Academic Editor: Pei De Liu Copyright © 2013 Nabiollah Shobkolaei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We demonstrate a fundamental lemma for the convergence of sequences in metric-like spaces, and by using it we prove some Suzuki-type fixed point results in the setup of metric-like spaces. As an immediate consequence of our results we obtain certain recent results in partial metric spaces as corollaries. Finally, three examples are presented to verify the effectiveness and applicability of our main results.

1. Introduction There are a lot of generalizations of Banach fixed-point principle in the literature. So far several authors have studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions (e.g., [1– 20]). In 2008, Suzuki introduced an interesting generalization of Banach fixed-point principle. This interesting fixed-point result is as follows. Theorem 1 (see [19]). Let (𝑋, 𝑑) be a complete metric space, and let 𝑇 be a mapping on 𝑋. Define a nonincreasing function 𝜃 from [0, 1] into [1/2, 1] by √5 − 1 { { 1, 0≤𝑟≤ { { 2 { { { { { 1 − 𝑟 √5 − 1 1 𝜃 (𝑟) = { , ≤𝑟≤ 2 { √ 𝑟 2 { 2 { { { { 1 1 { { ≤ 𝑟 < 1. , { 1 + 𝑟 √2

(1)

Assume that there exists 𝑟 ∈ [0, 1], such that 𝜃 (𝑟) 𝑑 (𝑥, 𝑇𝑥) ≤ 𝑑 (𝑥, 𝑦) 󳨐⇒ 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟𝑑 (𝑥, 𝑦) ,

(2)

for all 𝑥, 𝑦 ∈ 𝑋, then there exists a unique fixed-point 𝑧 of 𝑇. Moreover, lim𝑛 → ∞ 𝑇𝑛 𝑥 = 𝑧 for all 𝑥 ∈ 𝑋.

Suzuki proved also the following version of Edelstein’s fixed point theorem. Theorem 2. Let (𝑋, 𝑑) be a compact metric space. Let 𝑇 : 𝑋 → 𝑋 be a self-map, satisfying for all 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≠𝑦 the condition 1 𝑑 (𝑥, 𝑇𝑥) ≤ 𝑑 (𝑥, 𝑦) 󳨐⇒ 𝑑 (𝑇𝑥, 𝑇𝑦) < 𝑑 (𝑥, 𝑦) . 2

(3)

Then 𝑇 has a unique fixed point in 𝑋. This theorem was generalized in [3]. In addition to the above results, Kikkawa and Suzuki [8] provided a Kannan type version of the theorems mentioned before. In [14], Chatterjea type version is provided. Popescu [15] presented a Cirić type version. Recently, Kikkawa and Suzuki also provided multivalued versions which can be found in [9, 10]. Very recently Hussain et al. [4] have extended Suzuki’s Theorems 1 and 2, as well as Popescu’s results from [15] to the case of metric type spaces and cone metric type spaces (see also [5–7, 11]). The aim of this paper is to generalize the above-mentioned results. Indeed we prove a fixed point theorem in the set up of metric-like spaces and derive certain new results as corollaries. Finally, three examples are presented to verify the effectiveness and applicability of our main results.

2


In the rest of this section, we recall some definitions and facts which will be used throughout the paper. First, we present some known definitions and propositions in partial metric and metric-like spaces. A partial metric on a nonempty set 𝑋 is a mapping 𝑝 : 𝑋 × 𝑋 → R+ such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (p1 ) 𝑥 = 𝑦 if and only if 𝑝(𝑥, 𝑥) = 𝑝(𝑥, 𝑦) = 𝑝(𝑦, 𝑦), (p2 ) 𝑝(𝑥, 𝑥) ≤ 𝑝(𝑥, 𝑦), (p3 ) 𝑝(𝑥, 𝑦) = 𝑝(𝑦, 𝑥), (p4 ) 𝑝(𝑥, 𝑦) ≤ 𝑝(𝑥, 𝑧) + 𝑝(𝑧, 𝑦) − 𝑝(𝑧, 𝑧). A partial metric space is a pair (𝑋, 𝑝) such that 𝑋 is a nonempty set and 𝑝 is a partial metric on 𝑋. It is clear that if 𝑝(𝑥, 𝑦) = 0, then from (p1 ) and (p2 ) 𝑥 = 𝑦. But if 𝑥 = 𝑦, 𝑝(𝑥, 𝑦) may not be 0. A basic example of a partial metric space is the pair (R+ , 𝑝), where 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ R+ . Lemma 3 (see [17]). Let (𝑋, 𝑑) and (𝑋, 𝑝) be a metric space and partial metric space, respectively. Then (i) the function 𝜌 : 𝑋 × 𝑋 → R+ defined by 𝜌(𝑥, 𝑦) = 𝑑(𝑥, 𝑦) + 𝑝(𝑥, 𝑦) is a partial metric; (ii) let 𝜌 : 𝑋 × 𝑋 → R+ be defined by 𝜌(𝑥, 𝑦) = 𝑑(𝑥, 𝑦) + max{𝜔(𝑥), 𝜔(𝑦)}; then 𝜌 is a partial metric on 𝑋, where 𝜔 : 𝑋 → R+ is an arbitrary function; (iii) Let 𝜌 : R × R → R be defined by 𝜌(𝑥, 𝑦) = max{2𝑥 , 2𝑦 }; then 𝜌 is a partial metric on R; (iv) Let 𝜌 : 𝑋×𝑋 → R+ be defined by 𝜌(𝑥, 𝑦) = 𝑑(𝑥, 𝑦)+𝑎; then 𝜌 is a partial metric on 𝑋, where 𝑎 ≥ 0. Other examples of the partial metric spaces which are interesting from a computational point of view may be found in [7, 11, 12, 18]. Each partial metric 𝑝 on 𝑋 generates a 𝑇0 topology 𝜏𝑝 on 𝑋 which has as a base the family of open 𝑝-balls {𝐵𝑝 (𝑥, 𝜀) : 𝑥 ∈ 𝑋, 𝜀 > 0}, where 𝐵𝑝 (𝑥, 𝜀) = {𝑦 ∈ 𝑋 : 𝑝(𝑥, 𝑦) < 𝑝(𝑥, 𝑥) + 𝜀} for all 𝑥 ∈ 𝑋 and 𝜀 > 0. Let (𝑋, 𝑝) be a partial metric. A sequence {𝑥𝑛 } in a partial metric space (𝑋, 𝑝) converges to a point 𝑥 ∈ 𝑋 if and only if 𝑝(𝑥, 𝑥) = lim𝑛 → ∞ 𝑝(𝑥, 𝑥𝑛 ). A sequence {𝑥𝑛 } in a partial metric space (𝑋, 𝑝) is called a Cauchy sequence if there exists (and is finite) lim𝑛,𝑚 → ∞ 𝑝(𝑥𝑛 , 𝑥𝑚 ). A partial metric space (𝑋, 𝑝) is said to be complete if every Cauchy sequence {𝑥𝑛 } in 𝑋 converges, with respect to 𝜏𝑝 , to a point 𝑥 ∈ 𝑋 such that 𝑝(𝑥, 𝑥) = lim𝑛,𝑚 → ∞ 𝑝(𝑥𝑛 , 𝑥𝑚 ). Suppose that {𝑥𝑛 } is a sequence in partial metric space (𝑋, 𝑝); then we define 𝐿(𝑥𝑛 ) = {𝑥 | 𝑥𝑛 → 𝑥}. The following example shows that every convergent sequence {𝑥𝑛 } in a partial metric space (𝑋, 𝑝) may not be a Cauchy sequence. In particular, it shows that the limit is not unique.

Example 4 (see [17]). Let 𝑋 = [0, ∞) and 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦}. Let 𝑥𝑛 = {

0, 𝑛 = 2𝑘, 1, 𝑛 = 2𝑘 + 1.

(4)

Then clearly it is a convergent sequence, and for every 𝑥 ≥ 1 we have lim𝑛 → ∞ 𝑝(𝑥𝑛 , 𝑥) = 𝑝(𝑥, 𝑥), hence 𝐿(𝑥𝑛 ) = [1, ∞). But lim𝑛,𝑚 → ∞ 𝑝(𝑥𝑛 , 𝑥𝑚 ) does not exist; that is, it is not a Cauchy sequence. Definition 5 (see [2]). A metric-like on a nonempty set 𝑋 is a mapping 𝜎 : 𝑋 × 𝑋 → R+ such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (𝜎1) 𝜎(𝑥, 𝑦) = 0 ⇒ 𝑥 = 𝑦, (𝜎2) 𝜎(𝑥, 𝑦) = 𝜎(𝑦, 𝑥), (𝜎3) 𝜎(𝑥, 𝑦) ≤ 𝜎(𝑥, 𝑧) + 𝜎(𝑧, 𝑦). The pair (𝑋, 𝜎) is called a metric-like space. Then a metric-like on 𝑋 satisfies all of the conditions of a metric except that 𝜎(𝑥, 𝑥) may be positive for 𝑥 ∈ 𝑋. Each metriclike 𝜎 on 𝑋 generates a topology 𝜏𝜎 on 𝑋 whose base is the family of open 𝜎-ball, {𝐵𝜎 (𝑥, 𝜀) : 𝑥 ∈ 𝑋, 𝜀 > 0}, where 𝐵𝜎 (𝑥, 𝜀) = {𝑦 ∈ 𝑋 : |𝜎(𝑥, 𝑦) − 𝜎(𝑥, 𝑥)| < 𝜀} for all 𝑥 ∈ 𝑋 and 𝜀 > 0. A sequence {𝑥𝑛 } in a metric-like space (𝑋, 𝜎) converges to a point 𝑥 ∈ 𝑋 if and only if lim𝑛 → ∞ 𝜎(𝑥, 𝑥𝑛 ) = 𝜎(𝑥, 𝑥). A sequence {𝑥𝑛 } in a metric-like space (𝑋, 𝜎) is called a 𝜎-Cauchy sequence if there exists (and is finite) lim𝑛,𝑚 → ∞ 𝜎(𝑥𝑛 , 𝑥𝑚 ). A metric-like space (𝑋, 𝜎) is said to be complete if every 𝜎-Cauchy sequence {𝑥𝑛 } in 𝑋 converges, with respect to 𝜏𝜎 , to a point 𝑥 ∈ 𝑋 such that lim 𝜎 (𝑥𝑛 , 𝑥) = 𝜎 (𝑥, 𝑥) = lim 𝜎 (𝑥𝑛 , 𝑥𝑚 ) .

𝑛→∞

𝑛,𝑚 → ∞

(5)

Every partial metric space is a metric-like space. Below we give some examples of a metric-like space. Example 6. Let 𝑋 = [0, 1]; then mapping 𝜎1 : 𝑋 × 𝑋 → R+ defined by 𝜎1 (𝑥, 𝑦) = 𝑥 + 𝑦 − 𝑥𝑦 is a metric-like on 𝑋. Example 7. Let 𝑋 = R; then mappings 𝜎𝑖 : 𝑋 × 𝑋 → R+ (𝑖 ∈ {2, 3, 4}) defined by 󵄨 󵄨 𝜎2 (𝑥, 𝑦) = |𝑥| + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 + 𝑎, 󵄨 󵄨 𝜎3 (𝑥, 𝑦) = |𝑥 − 𝑏| + 󵄨󵄨󵄨𝑦 − 𝑏󵄨󵄨󵄨 , (6) 𝜎4 (𝑥, 𝑦) = 𝑥2 + 𝑦2 are metric-like space on 𝑋, where 𝑎 ≥ 0 and 𝑏 ∈ R.

2. Main Results We start our work by proving the following crucial lemma. Lemma 8. Let (𝑋, 𝜎) be a metric-like space, and suppose that {𝑥𝑛 } is 𝜎-convergent to 𝑥. Then for every 𝑦 ∈ 𝑋, one has 𝜎 (𝑥, 𝑦) − 𝜎 (𝑥, 𝑥) ≤ lim inf 𝜎 (𝑥𝑛 , 𝑦) 𝑛→∞


3

≤ lim sup𝜎 (𝑥𝑛 , 𝑦)

Also, by the condition 𝜎3 of the definition of metric-like space, for all 𝑚 ≥ 𝑛, we have

𝑛→∞

≤ 𝜎 (𝑥, 𝑦) + 𝜎 (𝑥, 𝑥) . (7)

𝜎 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) + 𝜎 (𝑥𝑛+1 , 𝑥𝑛+2 )

In particular, if 𝜎(𝑥, 𝑥) = 0, then one has lim𝑛 → ∞ 𝜎(𝑥𝑛 , 𝑦) = 𝜎(𝑥, 𝑦).

+ ⋅ ⋅ ⋅ + 𝜎 (𝑥𝑚−1 , 𝑥𝑚 ) ≤ 𝑟𝑛 𝜎 (𝑥0 , 𝑥1 ) + 𝑟𝑛+1 𝜎 (𝑥0 , 𝑥1 )

Proof. Using the triangle inequality in a metric-like space, it is easy to see that 𝜎 (𝑥𝑛 , 𝑦) ≤ 𝜎 (𝑥𝑛 , 𝑥) + 𝜎 (𝑥, 𝑦) ,

+ ⋅ ⋅ ⋅ + 𝑟𝑚−1 𝜎 (𝑥0 , 𝑥1 )

(8)

=

𝑟𝑛 − 𝑟𝑚 𝜎 (𝑥0 , 𝑥1 ) 1−𝑟

Taking the upper limit as 𝑛 → ∞ in the first inequality and the lower limit as 𝑛 → ∞ in the second inequality, we obtain the desired result.

𝜎 (𝑥𝑛 , 𝑧) 1+𝑟

(31)

𝜃 (𝑟) 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝜎 (𝑥𝑛 , 𝑧) .

(32)

𝜃 (𝑟) 𝜎 (𝑥2𝑛−1 , 𝑥2𝑛 ) ≤ 𝜎 (𝑥2𝑛−1 , 𝑧) ,

(33)

𝜃 (𝑟) 𝜎 (𝑥2𝑛 , 𝑥2𝑛+1 ) ≤ 𝜎 (𝑥2𝑛 , 𝑧) .

(34)

In particular,

or

So relation (23) is proved by induction. Now 𝑇𝑧 ≠𝑧 and (23) implies that 𝑇𝑛 𝑧 ≠𝑧 for each 𝑛 ∈ N. Hence, (18) imply that

≤ 𝑟2 𝜎 (𝑧, 𝑇𝑛−1 𝑧)

𝜃 (𝑟) 𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) ≤ 𝜎 (𝑥𝑛−1 , 𝑧) , or

Assumption (10) and relation (21) imply that 𝜎 (𝑇𝑧, 𝑇𝑛+1 𝑧) ≤ 𝑟𝜎 (𝑧, 𝑇𝑛 𝑧)

(30)

(29)

In other words, there is a subsequence {𝑥𝑛𝑘 } of {𝑥𝑛 } such that (28) holds for each 𝑘 ∈ N. But then assumption (10) implies that 𝜎 (𝑇𝑥𝑛𝑘 , 𝑇𝑧) ≤ 𝑟𝜎 (𝑥𝑛𝑘 , 𝑧) ,

(35)

𝜎 (𝑇𝑥𝑛𝑘−1 , 𝑇𝑧) ≤ 𝑟𝜎 (𝑥𝑛𝑘−1 , 𝑧) .

(36)

or

Passing to the limit when 𝑘 → ∞, we get that 𝜎(𝑧, 𝑇𝑧) ≤ 0, which is possible only if 𝑇𝑧 = 𝑧. Thus, we have proved that 𝑧 is a fixed point of 𝑇. The uniqueness of the fixed point follows easily from (10). Indeed, if 𝑦 and 𝑧 are two fixed points of 𝑇 such that 𝑦 ≠𝑧, then from (18) we have 𝜎 (𝑦, 𝑧) = 𝜎 (𝑦, 𝑇𝑧) ≤ 𝑟𝜎 (𝑦, 𝑧) ,

(37)

which is a contradiction. According to Theorem 9, we get the following result. Corollary 10 (see [19]). Let (𝑋, 𝑑) be a complete metric space, and let 𝑇 be a mapping on 𝑋. Define a nonincreasing function 𝜃 from [0, 1] into [1/2, 1] by √5 − 1 { { , 1, 0≤𝑟≤ { { 2 { { { { { √5 − 1 1 𝜃 (𝑟) = { 1 − 𝑟 , , ≤𝑟≤ 2 { √ 𝑟 2 { 2 { { { { 1 { { 1 , ≤ 𝑟 < 1. √ 1 + 𝑟 2 {

(38)


5

Assume that there exists 𝑟 ∈ [0, 1], such that

Proof. Let 𝑥0 ∈ 𝑋 be arbitrary, and define a sequence {𝑥𝑛 } by

𝜃 (𝑟) 𝑑 (𝑥, 𝑇𝑥) ≤ 𝑑 (𝑥, 𝑦) 󳨐⇒ 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟𝑑 (𝑥, 𝑦) , (39)

𝑥𝑛 = 𝑆 (𝑥𝑛−1 ) ,

for all 𝑥, 𝑦 ∈ 𝑋; then there exists a unique fixed-point 𝑧 of 𝑇. Moreover, lim𝑛 → ∞ 𝑇𝑛 𝑥 = 𝑧 for all 𝑥 ∈ 𝑋.

= 𝑇 (𝑥𝑛−1 ) ,

Proof. Using a similar argument given in Theorem 9 for 𝜎(𝑥, 𝑦) = 𝑑(𝑥, 𝑦), the desired result is obtained. Now, in order to support the useability of our results, let us introduce the following example. +

Example 11. Let 𝑋 = [0, ∞). Define 𝜎 : 𝑋 × 𝑋 → R by 𝜎 (𝑥, 𝑦) = 𝑥 + 𝑦

1 𝑥) √2

(41)

= 𝜎 (𝑆 (𝑥𝑛−1 ) , 𝑇 (𝑥𝑛 )) = 𝜎 (𝑆 (𝑥𝑛−1 ) , 𝑇𝑆 (𝑥𝑛−1 ))

𝜎 (𝑇 (𝑥𝑛−1 ) , 𝑆𝑇 (𝑥𝑛−1 ))} ≤ 𝑟 min {𝜎 (𝑥𝑛−1 , 𝑆 (𝑥𝑛−1 )) , 𝜎 (𝑥𝑛−1 , 𝑇 (𝑥𝑛−1 ))} ≤ 𝑟𝜎 (𝑥𝑛−1 , 𝑆 (𝑥𝑛−1 )) If 𝑛 is even, then by (44), we have 𝜎 (𝑥𝑛 , 𝑥𝑛+1 )

(42)

= 𝜎 (𝑇 (𝑥𝑛−1 ) , 𝑆 (𝑥𝑛 )) = 𝜎 (𝑇 (𝑥𝑛−1 ) , 𝑆𝑇 (𝑥𝑛−1 ))

≤ 𝑥 + 𝑦 = 𝜎 (𝑥, 𝑦) .

≤ max {𝜎 (𝑇 (𝑥𝑛−1 ) , 𝑆𝑇 (𝑥𝑛−1 )) ,

On the other hand, we have 𝜎 (𝑇𝑥, 𝑇𝑦) = ln (1 +

𝜎 (𝑆 (𝑥𝑛−1 ) , 𝑇𝑆 (𝑥𝑛−1 ))}

1 1 𝑥) + ln (1 + 𝑦) √2 √2

≤

1 1 𝑥+ 𝑦 √2 √2

=

1 𝜎 (𝑥, 𝑦) . √2

(43)

≤ 𝑟𝜎 (𝑥𝑛−1 , 𝑇 (𝑥𝑛−1 )) = 𝑟𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) . Thus for any positive integer 𝑛, it must be the case that

Theorem 12. Let (𝑋, 𝜎) be a complete metric-like space. Let 𝑆, 𝑇 : 𝑋 → 𝑋 be two self-mappings. Suppose that there exists 𝑟 ∈ [0, 1) such that ≤ 𝑟 min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))}

(48)

≤ 𝑟 min {𝜎 (𝑥𝑛−1 , 𝑇 (𝑥𝑛−1 )) , 𝜎 (𝑥𝑛−1 , 𝑆 (𝑥𝑛−1 ))}

Thus 𝑇 satisfies all the hypotheses of Theorem 9, and hence 𝑇 has a unique fixed point. Indeed, 𝑟 = 1/√2, 𝜃(𝑟) = 1/(1 + 𝑟), and 0 is the unique fixed point of 𝑇.

max {𝜎 (𝑆 (𝑥) , 𝑇𝑆 (𝑥)) , 𝜎 (𝑇 (𝑥) , 𝑆𝑇 (𝑥))}

(47)

= 𝑟𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) .

√2 1 1 𝜎 (𝑥, 𝑇𝑥) = (𝑥 + ln (1 + 𝑥)) √ √ √ 1 + 1/ 2 2+1 2 √2 1 (𝑥 + 𝑥) = 𝑥 √2 + 1 √2

(46)

𝜎 (𝑥𝑛 , 𝑥𝑛+1 )

≤ max {𝜎 (𝑆 (𝑥𝑛−1 ) , 𝑇𝑆 (𝑥𝑛−1 )) ,

for 𝑥 ∈ 𝑋. Then for each 𝑥, 𝑦 ∈ 𝑋, we have

≤

if 𝑛 is even.

Then if 𝑛 ∈ N is odd, we have

(40)

for all 𝑥, 𝑦 ∈ 𝑋. Then (𝑋, 𝜎) is a complete metric-like space. Define a map 𝑇 : 𝑋 → 𝑋 by 𝑇 (𝑥) = ln (1 +

if 𝑛 is odd

(44)

for every 𝑥 ∈ 𝑋 and that 𝛼 (𝑦) = inf {𝜎 (𝑥, 𝑦) + min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} : 𝑥 ∈ 𝑋} > 0 (45) for every 𝑦 ∈ 𝑋 with 𝑦 that is not a common fixed point of 𝑆 and 𝑇. Then there exists 𝑧 ∈ 𝑋 such that 𝑧 = 𝑆(𝑧) = 𝑇(𝑧). Moreover, if V = 𝑆(V) = 𝑇(V), then 𝜎(V, V) = 0.

𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝑟𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) .

(49)

Repeating (49), we obtain 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝑟𝑛 𝜎 (𝑥0 , 𝑥1 ) .

(50)

So, if 𝑚 > 𝑛, then 𝜎 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) + 𝜎 (𝑥𝑛+1 , 𝑥𝑛+2 ) + ⋅ ⋅ ⋅ + 𝜎 (𝑥𝑚−1 , 𝑥𝑚 ) ≤ [𝑟𝑛 + 𝑟𝑛+1 + ⋅ ⋅ ⋅ + 𝑟𝑚−1 ] 𝜎 (𝑥0 , 𝑥1 ) ≤

(51)

𝑟𝑛 𝜎 (𝑥0 , 𝑥1 ) . 1−𝑟

Thus lim𝑛,𝑚 → ∞ 𝜎(𝑥𝑛 , 𝑥𝑚 ) = 0. That is, {𝑥𝑛 } is a 𝜎-Cauchy sequence in the metric-like space (𝑋, 𝜎). Since (𝑋, 𝜎) is 𝜎-complete, there exist 𝑧 ∈ 𝑋 such that 𝜎 (𝑧, 𝑧) = lim 𝜎 (𝑥𝑛 , 𝑧) = lim 𝜎 (𝑥𝑛 , 𝑥𝑚 ) = 0. 𝑛→∞

𝑛,𝑚 → ∞

(52)

6


Assume that 𝑧 is not a common fixed point of 𝑆 and 𝑇. Then by hypothesis 0 < inf {𝜎 (𝑥, 𝑧) + min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} : 𝑥 ∈ 𝑋}

Corollary 14. Let (𝑋, 𝜎) be a complete metric-like space, and let 𝑇 : 𝑋 → 𝑋 be a mapping. Suppose that there exists 𝑟 ∈ [0, 1) such that 𝜎 (𝑇 (𝑥) , 𝑇2 (𝑥)) ≤ 𝑟𝜎 (𝑥, 𝑇 (𝑥))

≤ inf {𝜎 (𝑥𝑛 , 𝑧) + min {𝜎 (𝑥𝑛 , 𝑆 (𝑥𝑛 )) , 𝜎 (𝑥𝑛 , 𝑇 (𝑥𝑛 ))} :

for every 𝑥 ∈ 𝑋 and that

𝑛 ∈ N} ≤ inf {

𝛼 (𝑦) = inf {𝜎 (𝑥, 𝑦) + 𝜎 (𝑥, 𝑇 (𝑥)) : 𝑥 ∈ 𝑋} > 0

𝑟𝑛 𝜎 (𝑥0 , 𝑥1 ) + 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) : 𝑛 ∈ N} 1−𝑟

𝑟𝑛 ≤ inf { 𝜎 (𝑥0 , 𝑥1 ) + 𝑟𝑛 𝜎 (𝑥0 , 𝑥1 ) : 𝑛 ∈ N} = 0 1−𝑟

(53)

= 𝑟 min {𝜎 (V, V) , 𝜎 (V, V)}

min {𝜎 (𝑇𝑆 (𝑥) , 𝑆 (𝑥)) , 𝜎 (𝑆𝑇 (𝑥) , 𝑇 (𝑥))} (54)

= 𝑟𝜎 (V, V)

≥ 𝑟 max {𝜎 (𝑆𝑥, 𝑥) , 𝜎 (𝑇𝑥, 𝑥)}

(59)


which gives that 𝜎(V, V) = 0.

𝛼 (𝑦) = inf {𝜎 (𝑥, 𝑦) + min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} :

Example 13. Let (𝑋, 𝜎) be a metric-like space where 𝑋 = {1/𝑛}∞ 𝑛=1 ∪ {0} and 𝜎(𝑥, 𝑦) = 𝑥 + 𝑦. Define 𝑆 : 𝑋 → 𝑋 by 𝑆(0) = 0, 𝑆(1/2𝑛) = 1/(4𝑛 + 3), and 𝑆(1/(2𝑛 − 1)) = 0 and 𝑇(0) = 0, 𝑇(1/(2𝑛 − 1)) = 1/(4𝑛 + 4), and 𝑇(1/2𝑛) = 0. Then for 𝑥 = 1/2𝑛, we have max {𝜎 (𝑆 (𝑥) , 𝑇𝑆 (𝑥)) , 𝜎 (𝑇 (𝑥) , 𝑆𝑇 (𝑥))} 1 1 ) , 𝑇 (𝑆 ( ))) , 2𝑛 2𝑛

𝜎 (𝑇 ( = max {

Proof. Taking 𝑆 = 𝑇 in Theorem 12, the conclusion of the corollary follows. Theorem 15. Let (𝑋, 𝜎) be a complete metric-like space. Let 𝑆, 𝑇 be mappings from 𝑋 onto itself. Suppose that there exists 𝑟 > 1 such that

𝜎 (V, V) = max {𝜎 (𝑆 (V) , 𝑇𝑆 (V)) , 𝜎 (𝑇 (V) , 𝑆𝑇 (V))} ≤ 𝑟 min {𝜎 (V, 𝑆 (V)) , 𝜎 (V, 𝑇 (V))}

(58)

for every 𝑦 ∈ 𝑋 with 𝑦 ≠𝑇(𝑦). Then there exists 𝑧 ∈ 𝑋 such that 𝑧 = 𝑇(𝑧). Moreover, if V = 𝑇(V), then 𝜎(V, V) = 0.

which is a contradiction. Therefore, 𝑧 = 𝑆(𝑧) = 𝑇(𝑧). If V = 𝑆(V) = 𝑇(V) for some V ∈ 𝑋, then

= max {𝜎 (𝑆 (

(57)

1 1 ) , 𝑆 (𝑇 ( )))} 2𝑛 2𝑛

1 1 1 1 + , 0} = + 4𝑛 + 3 8𝑛 + 12 4𝑛 + 3 8𝑛 + 12

≤ 𝑟 min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} 1 1 1 1 = 𝑟 min { + , + 0} = 𝑟 . 2𝑛 4𝑛 + 3 2𝑛 2𝑛

𝑥 ∈ 𝑋} > 0 (60) for every 𝑦 ∈ 𝑋 with 𝑦 that is not a common fixed point of 𝑆 and 𝑇. Then there exists 𝑧 ∈ 𝑋 such that 𝑧 = 𝑆(𝑧) = 𝑇(𝑧). Moreover, if V = 𝑆(V) = 𝑇(V), then 𝜎(V, V) = 0. Proof. Let 𝑥0 ∈ 𝑋 be arbitrary. Since 𝑆 is onto, there is an element 𝑥1 satisfying 𝑥1 ∈ 𝑆−1 (𝑥0 ). Since 𝑇 is also onto, there is an element 𝑥2 satisfying 𝑥2 ∈ 𝑇−1 (𝑥1 ). Proceeding in the same way, we can find that 𝑥2𝑛+1 ∈ 𝑆−1 (𝑥2𝑛 ) and 𝑥2𝑛+2 ∈ 𝑇−1 (𝑥2𝑛+1 ) for 𝑛 = 1, 2, 3, . . . . Therefore, 𝑥2𝑛 = 𝑆𝑥2𝑛+1 and 𝑥2𝑛+1 = 𝑇𝑥2𝑛+2 for 𝑛 = 0, 1, 2, . . . . If 𝑛 = 2𝑚, then using (59) 𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) = 𝜎 (𝑥2𝑚−1 , 𝑥2𝑚 )

(55)

It is easy to see that the above inequality is true for 𝑥 = 1/(2𝑛− 1) and for 3/4 ≤ 𝑟 < 1. Also, 𝛼 (𝑦) = inf {𝜎 (𝑥, 𝑦) + min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} :

= 𝜎 (𝑇𝑥2𝑚 , 𝑆𝑥2𝑚+1 ) = 𝜎 (𝑇𝑆𝑥2𝑚+1 , 𝑆𝑥2𝑚+1 ) ≥ min {𝜎 (𝑇𝑆 (𝑥2𝑚+1 ) , 𝑆 (𝑥2𝑚+1 )) , 𝜎 (𝑆𝑇 (𝑥2𝑚+1 ) , 𝑇 (𝑥2𝑚+1 ))} ≥ 𝑟 max {𝜎 (𝑆𝑥2𝑚+1 , 𝑥2𝑚+1 ) , 𝜎 (𝑇𝑥2𝑚+1 , 𝑥2𝑚+1 )}

𝑥 ∈ 𝑋} > 0 (56) for every 𝑦 ∈ 𝑋 with y is not a common fixed point of 𝑆 and 𝑇. This shows that all conditions of Theorem 12 are satisfied and 0 is a common fixed point for 𝑆 and 𝑇.

≥ 𝑟𝜎 (𝑆𝑥2𝑚+1 , 𝑥2𝑚+1 ) = 𝑟𝜎 (𝑥2𝑚 , 𝑥2𝑚+1 ) = 𝑟𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) .

(61)


7

If 𝑛 = 2𝑚 + 1, then using (59)

If V = 𝑆(V) = 𝑇(V) for some V ∈ 𝑋, then

𝜎 (𝑥𝑛−1 , 𝑥𝑛 )

𝜎 (V, V) = min {𝜎 (𝑇𝑆 (V) , 𝑆 (V)) , 𝜎 (𝑆𝑇 (V) , 𝑇 (V))}

= 𝜎 (𝑥2𝑚 , 𝑥2𝑚+1 )

≥ 𝑟 max {𝜎 (𝑆 (V) , V) , 𝜎 (𝑇 (V) , V)}

= 𝜎 (𝑆𝑥2𝑚+1 , 𝑇𝑥2𝑚+2 )

= 𝑟 max {𝜎 (V, V) , 𝜎 (V, V)}

= 𝜎 (𝑆𝑇𝑥2𝑚+2 , 𝑇𝑥2𝑚+2 )

= 𝑟𝜎 (V, V)

≥ min {𝜎 (𝑇𝑆 (𝑥2𝑚+2 ) , 𝑆 (𝑥2𝑚+2 )) , 𝜎 (𝑆𝑇 (𝑥2𝑚+2 ) , 𝑇 (𝑥2𝑚+2 ))}

(62)

≥ 𝑟𝜎 (𝑇𝑥2𝑚+2 , 𝑥2𝑚+2 )

𝜎 (𝑇2 (𝑥) , 𝑇 (𝑥)) ≥ 𝑟𝜎 (𝑇 (𝑥) , 𝑥)

= 𝑟𝜎 (𝑥2𝑚+1 , 𝑥2𝑚+2 )

(70)


= 𝑟𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) .

𝛼 (𝑦) = inf {𝜎 (𝑥, 𝑦) + 𝜎 (𝑇 (𝑥) , 𝑥) : 𝑥 ∈ 𝑋} > 0

Thus for any positive integer 𝑛, it must be the case that 𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) ≥ 𝑟𝜎 (𝑥𝑛 , 𝑥𝑛+1 )

(63)

which implies that 1 1 𝑛 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) ≤ ⋅ ⋅ ⋅ ≤ ( ) 𝜎 (𝑥0 , 𝑥1 ) . (64) 𝑟 𝑟 Let 𝛼 = 1/𝑟; then 0 < 𝛼 < 1 since 𝑟 > 1. Now, (64) becomes 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝛼𝑛 𝜎 (𝑥0 , 𝑥1 ) .

(65)

So, if 𝑚 > 𝑛, then 𝜎 (𝑥𝑛 , 𝑥𝑚 ) ≤ 𝜎 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ [𝛼𝑛 + 𝛼𝑛+1 + ⋅ ⋅ ⋅ + 𝛼𝑚−1 ] 𝜎 (𝑥0 , 𝑥1 )

(66)

𝛼𝑛 𝜎 (𝑥0 , 𝑥1 ) . 1−𝛼

𝑛,𝑚 → ∞

Proof. Taking 𝑆 = 𝑇 in Theorem 15, we have the desired result. Definition 17. Let (𝑋, 𝜎) and (𝑌, 𝜏) be metric-like spaces. Then 𝑓 : 𝑋 → 𝑌 is said to be a continuous mapping, if lim𝑛 → ∞ 𝑥𝑛 = 𝑥 implies that lim𝑛 → ∞ 𝑓(𝑥𝑛 ) = 𝑓(𝑥). Corollary 18. Let (𝑋, 𝜎) be a complete metric-like space, and let 𝑇 be a mapping of 𝑋 into itself. If there is a real number 𝑟 with 𝑟 > 1 satisfying

(67)

Assume that 𝑧 is not a common fixed point of 𝑆 and 𝑇. Then by hypothesis 0 < inf {𝜎 (𝑥, 𝑧) + min {𝜎 (𝑥, 𝑆 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} : 𝑥 ∈ 𝑋}

for every 𝑥 ∈ 𝑋 and 𝑇 is onto and continuous, then 𝑇 has a fixed point.

inf {𝜎 (𝑥, 𝑦) + 𝜎 (𝑇 (𝑥) , 𝑥) : 𝑥 ∈ 𝑋} = 0.

𝑛 ∈ N}

lim {𝜎 (𝑥𝑛 , 𝑦) + 𝜎 (𝑇 (𝑥𝑛 ) , 𝑥𝑛 )} = 0.

𝑛→∞

≤ inf {

𝛼𝑛 𝜎 (𝑥0 , 𝑥1 ) + 𝛼𝑛−1 𝜎 (𝑥0 , 𝑥1 ) : 𝑛 ∈ N} = 0 1−𝛼

which is a contradiction. Therefore, 𝑧 = 𝑆(𝑧) = 𝑇(𝑧).

(74)

So, we have 𝜎(𝑥𝑛 , 𝑦) → 0 and 𝜎(𝑇(𝑥𝑛 ), 𝑥𝑛 ) → 0 as 𝑛 → ∞. Since, 𝜎(𝑦, 𝑦) ≤ 𝜎(𝑦, 𝑥𝑛 ) + 𝜎(𝑥𝑛 , 𝑦), hence 𝜎(𝑦, 𝑦) → 0 as 𝑛 → ∞. Now, 𝜎 (𝑇 (𝑥𝑛 ) , 𝑦) ≤ 𝜎 (𝑇 (𝑥𝑛 ) , 𝑥𝑛 ) + 𝜎 (𝑥𝑛 , 𝑦) 󳨀→ 0 as 𝑛 󳨀→ ∞.

(75)

Since 𝑇 is continuous, we have

𝑛

𝛼 𝜎 (𝑥0 , 𝑥1 ) + 𝜎 (𝑥𝑛−1 , 𝑥𝑛 ) : 𝑛 ∈ N} 1−𝛼

(73)

Then there exists a sequence {𝑥𝑛 } such that

≤ inf {𝜎 (𝑥𝑛 , 𝑧) + min {𝜎 (𝑥𝑛 , 𝑆 (𝑥𝑛 )) , 𝜎 (𝑥𝑛 , 𝑇 (𝑥𝑛 ))} :

≤ inf {

(72)

Proof. Assume that there exists 𝑦 ∈ 𝑋 with 𝑦 ≠𝑇(𝑦) and

Thus lim𝑛,𝑚 → ∞ 𝜎(𝑥𝑛 , 𝑥𝑚 ) = 0. That is, {𝑥𝑛 } is a 𝜎-Cauchy sequence in the metric-like space (𝑋, 𝜎). Since (𝑋, 𝜎) is 𝜎complete, there exists 𝑧 ∈ 𝑋 such that 𝜎 (𝑧, 𝑧) = lim 𝜎 (𝑥𝑛 , 𝑧) = lim 𝜎 (𝑥𝑛 , 𝑥𝑚 ) = 0.

(71)

for every 𝑦 ∈ 𝑋 with 𝑦 ≠𝑇(𝑦). Then there exists 𝑧 ∈ 𝑋 such that 𝑧 = 𝑇(𝑧). Moreover, if V = 𝑇(V), then 𝜎(V, V) = 0.

𝜎 (𝑇2 (𝑥) , 𝑇 (𝑥)) ≥ 𝑟𝜎 (𝑇 (𝑥) , 𝑥)

+ 𝜎 (𝑥𝑛+1 , 𝑥𝑛+2 ) + ⋅ ⋅ ⋅ + 𝜎 (𝑥𝑚−1 , 𝑥𝑚 )

𝑛→∞

which gives that 𝜎(V, V) = 0. Corollary 16. Let (𝑋, 𝜎) be a complete metric-like space, and let 𝑇 : 𝑋 → 𝑋 be an onto mapping. Suppose that there exists 𝑟 ∈ [0, 1) such that

≥ 𝑟 max {𝜎 (𝑆𝑥2𝑚+2 , 𝑥2𝑚+2 ) , 𝜎 (𝑇𝑥2𝑚+2 , 𝑥2𝑚+2 )}

≤

(69)

𝑇 (𝑦) = 𝑇 ( lim 𝑥𝑛 ) = lim 𝑇 (𝑥𝑛 ) = 𝑦. 𝑛→∞

𝑛→∞

(76)

This is a contradiction. Hence if 𝑦 ≠𝑇(𝑦), then inf {𝜎 (𝑥, 𝑦) + 𝜎 (𝑇 (𝑥) , 𝑥) : 𝑥 ∈ 𝑋} > 0, (68)

(77)

which is condition (71) of Corollary 16. By Corollary 16, there exists 𝑧 ∈ 𝑋 such that 𝑧 = 𝑇(𝑧).

8

Journal of Function Spaces and Applications Now we give an example to support our result.

Example 19. Let 𝑋 = [0, ∞) and 𝜎(𝑥, 𝑦) = 𝑥 + 𝑦. Define 𝑇 : 𝑋 → 𝑋 by 𝑇(𝑥) = 2𝑥. Obviously 𝑇 is onto and continuous. Also for each 𝑥, 𝑦 ∈ 𝑋, we have 𝜎 (𝑇2 𝑥, 𝑇𝑥) = 4𝑥 + 2𝑥 = 6𝑥 ≥ 𝑟3𝑥 = 𝑟𝜎 (𝑇𝑥, 𝑥) ,

(78)

where 𝑟 = 2. Thus 𝑇 satisfies the conditions given in Corollary 18, and 0 is the fixed point of 𝑇. Corollary 20. Let (𝑋, 𝜎) be a complete metric-like space, and 𝑇 be a mapping of 𝑋 into itself. If there is a real number 𝑟 with 𝑟 > 1 satisfying 𝜎 (𝑇 (𝑥) , 𝑇 (𝑦)) ≥ 𝑟 min {𝜎 (𝑥, 𝑇 (𝑥)) , 𝜎 (𝑇 (𝑦) , 𝑦) , 𝜎 (𝑥, 𝑦)} (79) for every 𝑥, 𝑦 ∈ 𝑋 and 𝑇 is onto and continuous, then 𝑇 has a fixed point. Proof. Replacing 𝑦 by 𝑇(𝑥) in (79), we obtain 𝜎 (𝑇 (𝑥) , 𝑇2 (𝑥)) ≥ 𝑟 min {𝜎 (𝑥, 𝑇 (𝑥)) , 𝜎 (𝑇2 (𝑥) , 𝑇 (𝑥)) , 𝜎 (𝑥, 𝑇 (𝑥))} (80) for all 𝑥 ∈ 𝑋. Without loss of generality, we may assume that 𝑇(𝑥) ≠ 𝑇2 (𝑥). Otherwise 𝑇 has a fixed point. Since 𝑟 > 1, it follows from (80) that 𝜎 (𝑇2 (𝑥) , 𝑇 (𝑥)) ≥ 𝑟𝜎 (𝑇 (𝑥) , 𝑥)

(81)

for every 𝑥 ∈ 𝑋. By the argument similar to that used in Corollary 18, we can prove that if 𝑦 ≠𝑇(𝑦), then inf {𝜎 (𝑥, 𝑦) + 𝜎 (𝑇 (𝑥) , 𝑥) : 𝑥 ∈ 𝑋} > 0,

(82)

which is condition (71) of Corollary 16. So, Corollary 16 applies to obtain a fixed point of 𝑇. According to Theorem 12, we get the following result. Corollary 21 (see [17, Theorem 1]). Let (𝑋, 𝑝) be a complete partial metric space. Let 𝑆, 𝑇 : 𝑋 → 𝑋 be two self-mappings. Suppose that there exists 𝑟 ∈ [0, 1) such that max {𝑝 (𝑆 (𝑥) , 𝑇𝑆 (𝑥)) , 𝑝 (𝑇 (𝑥) , 𝑆𝑇 (𝑥))} ≤ 𝑟 min {𝑝 (𝑥, 𝑆 (𝑥)) , 𝑝 (𝑥, 𝑇 (𝑥))}

(83)

for every 𝑥 ∈ 𝑋 and that 𝛼 (𝑦) = inf {𝑝 (𝑥, 𝑦) + min {𝑝 (𝑥, 𝑆 (𝑥)) , 𝑝 (𝑥, 𝑇 (𝑥))} : 𝑥 ∈ 𝑋} > 0 (84) for every 𝑦 ∈ 𝑋 with 𝑦 that is not a common fixed point of 𝑆 and 𝑇. Then there exists 𝑧 ∈ 𝑋 such that 𝑧 = 𝑆(𝑧) = 𝑇(𝑧). Moreover, if V = 𝑆(V) = 𝑇(V), then 𝑝(V, V) = 0.

Proof. Using a similar argument given in the Theorem 12 for 𝜎(𝑥, 𝑦) = 𝑝(𝑥, 𝑦), the desired result is obtained, where 𝑝 is a partial metric on 𝑋. Also, according to Theorem 15, we get Theorem 2 from [17].

References [1] A. Aghajani, S. Radenović, and J. R. Roshan, “Common fixed point results for four mappings satisfying almost generalized (𝑆, 𝑇)-contractive condition in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5665– 5670, 2012. [2] A. Amini-Harandi, “Metric-like spaces, partial metric spaces and fixed points,” Fixed Point Theory and Applications, vol. 2012, article 204, 10 pages, 2012. [3] D. Ðorić, Z. Kadelburg, and S. Radenović, “Edelstein-Suzukitype fixed point results in metric and abstract metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 1927–1932, 2012. [4] N. Hussain, D. Ðorić, Z. Kadelburg, and S. Radenović, “Suzukitype fixed point results in metric type spaces,” Fixed Point Theory and Applications, vol. 2012, article 126, 12 pages, 2012. [5] N. Hussain and M. H. Shah, “KKM mappings in cone 𝑏-metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1677–1684, 2011. [6] N. Hussain, M. H. Shah, A. Amini-Harandi, and Z. Akhtar, “Common fixed point theorems for generalized contractive mappings with applications,” Fixed Point Theory and Applications, vol. 2013, article 169, 17 pages, 2013. [7] N. Hussain, Z. Kadelburg, S. Radenović, and F. Al-Solamy, “Comparison functions and fixed point results in partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 605781, 15 pages, 2012. [8] M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 649749, 8 pages, 2008. [9] M. Kikkawa and T. Suzuki, “Some notes on fixed point theorems with constants,” Bulletin of the Kyushu Institute of Technology, no. 56, pp. 11–18, 2009. [10] M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constants in complete metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2942–2949, 2008. [11] M. A. Kutbi, J. Ahmad, N. Hussain, and M. Arshad, “Common fixed point results for mappings with rational expressions,” Abstract and Applied Analysis, vol. 2013, Article ID 549518, 11 pages, 2013. [12] S. G. Matthews, “Partial metric topology,” in Papers on General Topology and Applications, vol. 728 of Annals of the New York Academy of Sciences, pp. 183–197, New York Academy of Sciences, New York, NY, USA, 1994. [13] V. Parvaneh, J. R. Roshan, and S. Radenović, “Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations,” Fixed Point Theory and Applications, vol. 2013, article 130, 19 pages, 2013. [14] O. Popescu, “Fixed point theorem in metric spaces,” Bulletin of the Transilvania University of Bras¸ov, vol. 150, pp. 479–482, 2008.

Journal of Function Spaces and Applications [15] O. Popescu, “Two fixed point theorems for generalized contractions with constants in complete metric space,” Central European Journal of Mathematics, vol. 7, no. 3, pp. 529–538, 2009. [16] J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, and W. Shatanawi, “Common fixed points of almost generalized (𝜓, 𝜑)𝑠 -contractive mappings in ordered b-metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 159, 23 pages, 2013. [17] S. Sedghi and N. Shobkolaei, “Common fixed point of maps in complete partial metric spaces,” East Asian Mathematical Journal, vol. 29, no. 1, pp. 1–12, 2013. [18] N. Shobkolaei, S. Sedghi, J. R. Roshan, and I. Altun, “Common fixed point of mappings satisfying almost generalized (𝑆, 𝑇)contractive condition in partially ordered partial metric spaces,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 443– 452, 2012. [19] T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008. [20] T. Suzuki, “A new type of fixed point theorem in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5313–5317, 2009.

9


Research Article Maximal and Area Integral Characterizations of Bergman Spaces in the Unit Ball of C𝑛 Zeqian Chen1 and Wei Ouyang1,2 1 2

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, 30 West District, Xiao-Hong-Shan, Wuhan 430071, China Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Correspondence should be addressed to Zeqian Chen; [email protected] Received 15 May 2013; Accepted 17 July 2013 Academic Editor: Pei De Liu Copyright © 2013 Z. Chen and W. Ouyang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present maximal and area integral characterizations of Bergman spaces in the unit ball of C𝑛 . The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.

Also, we denote by B𝑛 the closed unit ball, that is,

1. Introduction and Main Results Let C denote the set of complex numbers. Throughout the paper we fix a positive integer 𝑛, and let

B𝑛 = {𝑧 ∈ C𝑛 : |𝑧| ≤ 1} = B𝑛 ∪ S𝑛 .

C𝑛 = C × ⋅ ⋅ ⋅ × C

The automorphism group of B𝑛 , denoted by Aut(B𝑛 ), consists of all biholomorphic mappings of B𝑛 . Traditionally, biholomorphic mappings are also called automorphisms. For 𝛼 ∈ R, the weighted Lebesgue measure 𝑑V𝛼 on B𝑛 is defined by

(1)

denote the Euclidean space of complex dimension 𝑛. Addition, scalar multiplication, and conjugation are defined on C𝑛 component wise. For 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) and 𝑤 = (𝑤1 , . . . , 𝑤𝑛 ) in C𝑛 , we write ⟨𝑧, 𝑤⟩ = 𝑧1 𝑤1 + ⋅ ⋅ ⋅ + 𝑧𝑛 𝑤𝑛 ,

(2)

𝑑V𝛼 (𝑧) = 𝑐𝛼 (1 − |𝑧|2 ) 𝑑V (𝑧) ,

(3)

where 𝑐𝛼 = 1 for 𝛼 ≤ −1 and 𝑐𝛼 = Γ(𝑛 + 𝛼 + 1)/[𝑛!Γ(𝛼 + 1)] if 𝛼 > −1, which is a normalizing constant so that 𝑑V𝛼 is a probability measure on B𝑛 . In the case of 𝛼 = −(𝑛 + 1), we denote the resulting measure by

where 𝑤𝑘 is the complex conjugate of 𝑤𝑘 . We also write 󵄨 󵄨2 󵄨 󵄨2 |𝑧| = √󵄨󵄨󵄨𝑧1 󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨𝑧𝑛 󵄨󵄨󵄨 . The open unit ball in C𝑛 is the set B𝑛 = {𝑧 ∈ C𝑛 : |𝑧| < 1} .

(4)

The boundary of B𝑛 will be denoted by S𝑛 and is called the unit sphere in C𝑛 , that is, S𝑛 = {𝑧 ∈ C𝑛 : |𝑧| = 1} .

𝛼

(6)

(5)

𝑑𝜏 (𝑧) =

𝑑V 𝑛+1

(1 − |𝑧|2 )

(7)

(8)

and call it the invariant measure on B𝑛 , since 𝑑𝜏 = 𝑑𝜏 ∘ 𝜑 for any automorphism 𝜑 of B𝑛 .

2


For 𝛼 > −1 and 𝑝 > 0, the (weighted) Bergman space A𝑝𝛼 consists of holomorphic functions 𝑓 in B𝑛 with 1/𝑝

󵄨𝑝 󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝑝,𝛼 = (∫ 󵄨󵄨󵄨𝑓(𝑧)󵄨󵄨󵄨 𝑑V𝛼 (𝑧)) B

< ∞,

In order to state the area integral characterizations of the Bergman spaces, we require some more notation. For any 𝑓 ∈ H(B𝑛 ) and 𝑧 = (𝑧1 , . . . , 𝑧𝑛 ) ∈ B𝑛 , we define 𝑛

(9)

R𝑓 (𝑧) = ∑ 𝑧𝑘

𝑛

where the weighted Lebesgue measure 𝑑V𝛼 on B𝑛 is defined by 𝛼

𝑑V𝛼 (𝑧) = 𝑐𝛼 (1 − |𝑧|2 ) 𝑑V (𝑧) ,

𝑘=1

(10)

(11)

where H(B𝑛 ) is the space of all holomorphic functions in B𝑛 . 𝑝 When 𝛼 = 0, we simply write A𝑝 for A0 . These are the usual Bergman spaces. Note that for 1 ≤ 𝑝 < ∞, A𝑝𝛼 is a Banach space under the norm ‖ ‖𝑝,𝛼 . If 0 < 𝑝 < 1, the space A𝑝𝛼 is a quasi-Banach space with 𝑝-norm ‖𝑓‖𝑝𝑝,𝛼 . Recall that 𝐷(𝑧, 𝛾) denotes the Bergman metric ball at 𝑧 𝐷 (𝑧, 𝛾) = {𝑤 ∈ B𝑛 : 𝛽 (𝑧, 𝑤) < 𝛾} ,

(12)

with 𝛾 > 0, where 𝛽 is the Bergman metric on B𝑛 . It is known that 󵄨 󵄨 1 + 󵄨󵄨󵄨𝜑𝑧 (𝑤)󵄨󵄨󵄨 1 𝛽 (𝑧, 𝑤) = log 󵄨, 󵄨 2 1 − 󵄨󵄨󵄨𝜑𝑧 (𝑤)󵄨󵄨󵄨

∇𝑓 (𝑧) = (

𝜕𝑓 (𝑧) 𝜕𝑓 (𝑧) ,..., ), 𝜕𝑧1 𝜕𝑧𝑛

(17)

̃ (𝑧) = ∇ (𝑓 ∘ 𝜑𝑧 ) (0) . ∇𝑓 Now, for fixed 1 < 𝑞 < ∞ and 𝛾 > 0, we define for each 𝑓 ∈ H(B𝑛 ) and 𝑧 ∈ B𝑛 : (1) the radial area integral function 1/𝑞

𝛾,𝑞

𝐴 R (𝑓) (𝑧) = (∫

𝐷(𝑧,𝛾)

|(1 − |𝑤|2 )R𝑓(𝑤)|𝑞 𝑑𝜏(𝑤))

, (18)

(2) the complex gradient area integral function 1/𝑞

𝛾,𝑞

𝐴 ∇ (𝑓) (𝑧) = (∫

𝐷(𝑧,𝛾)

|(1 − |𝑤|2 )∇𝑓(𝑤)|𝑞 𝑑𝜏(𝑤))

, (19)

(3) the invariant gradient area integral function 𝑧, 𝑤 ∈ B𝑛 ,

(13)

󵄨 󵄨 sup 󵄨󵄨󵄨𝑓 (𝑤)󵄨󵄨󵄨 ,

𝑤∈𝐷(𝑧,𝛾)

∀𝑧 ∈ B𝑛 .

𝐷(𝑧,𝛾)

1/𝑞

𝑞 ̃ |∇𝑓(𝑤)| 𝑑𝜏(𝑤))

.

(20)

We state the second main result of this paper as follows. Theorem 2. Suppose 1 < 𝑞 < ∞, 𝛾 > 0, and 𝛼 > −1. Let 0 < 𝑝 < ∞. Then, for any 𝑓 ∈ H(B𝑛 ), the following conditions are equivalent: (a) 𝑓 ∈ A𝑝𝛼 , 𝛾,𝑞

(14)

(b) 𝐴 R (𝑓) is in 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ), 𝛾,𝑞

(c) 𝐴 ∇ (𝑓) is in 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ), 𝛾,𝑞

(d) 𝐴 ∇̃ (𝑓) is in 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ).

Then we have the following result. Theorem 1. Suppose 𝛾 > 0 and 𝛼 > −1. Let 0 < 𝑝 < ∞. Then for any 𝑓 ∈ H(B𝑛 ), 𝑓 ∈ A𝑝𝛼 if and only if 𝑀𝛾 𝑓 ∈ 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ). Moreover, 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩𝑝,𝛼 ≈ 󵄩󵄩󵄩󵄩M𝛾 𝑓󵄩󵄩󵄩󵄩𝑝,𝛼 ,

𝛾,𝑞

𝐴 ∇̃ (𝑓) (𝑧) = (∫

whereafter 𝜑𝑧 is the bijective holomorphic mapping in B𝑛 , which satisfies 𝜑𝑧 (0) = 𝑧, 𝜑𝑧 (𝑧) = 0, and 𝜑𝑧 ∘ 𝜑𝑧 = id. As is well known, maximal functions play a crucial role in the real-variable theory of Hardy spaces (cf. [1]). In this paper, we first establish a maximal-function characterization for the Bergman spaces. To this end, we define for each 𝛾 > 0 and 𝑓 ∈ H(B𝑛 ): (𝑀𝛾 𝑓) (𝑧) =

(16)

and call it the radial derivative of 𝑓 at 𝑧. The complex and invariant gradients of 𝑓 at 𝑧 are, respectively, defined as

and 𝑐𝛼 = Γ(𝑛 + 𝛼 + 1)/[𝑛!Γ(𝛼 + 1)] is a normalizing constant so that 𝑑V𝛼 is a probability measure on B𝑛 . Thus, A𝑝𝛼 = H (B𝑛 ) ∩ 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ) ,

𝜕𝑓 (𝑧) 𝜕𝑧𝑘

(15)

where “≈” depends only on 𝛾, 𝛼, 𝑝, and 𝑛. The norm appearing on the right-hand side of (15) can be viewed as an analogue of the so-called nontangential maximal function in Hardy spaces. The proof of Theorem 1 is fairly elementary (see Section 2), using some basic facts and estimates on the Bergman balls.

Moreover, the quantities 󵄩󵄩 𝛾,𝑞 󵄩󵄩 󵄩󵄩𝐴 R (𝑓)󵄩󵄩𝑝,𝛼 ,

󵄩󵄩 𝛾,𝑞 󵄩󵄩 󵄩󵄩𝐴 ∇ (𝑓)󵄩󵄩𝑝,𝛼 ,

󵄩󵄩 𝛾,𝑞 󵄩󵄩 󵄩󵄩𝐴 ̃ (𝑓)󵄩󵄩 󵄩 ∇ 󵄩𝑝,𝛼

(21)

are all comparable to ‖𝑓 − 𝑓(0)‖𝑝,𝛼 , where the comparable constants depend only on 𝑞, 𝛾, 𝛼, 𝑝, and 𝑛. For 0 < 𝑝 < ∞ and −∞ < 𝛼 < ∞, we fix a nonnegative integer 𝑘 with 𝑝𝑘 + 𝛼 > −1 and define the generalized Bergman space A𝑝𝛼 as introduced in [2] to be the space of all 𝑓 ∈ H(B𝑛 ) such that (1−|𝑧|2 )𝑘 R𝑘 𝑓 ∈ 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ). One then easily observes that A𝑝𝛼 is independent of the choice of 𝑘 and consistent with the traditional definition when 𝛼 > −1. Let 𝑁


3

be the smallest nonnegative integer such that 𝑝𝑁 + 𝛼 > −1. Put 1/𝑝 󵄨𝑝 󵄨 󵄨 󵄩󵄩 󵄩󵄩 2 𝑝𝑁 󵄨 𝑁 󵄩󵄩𝑓󵄩󵄩𝑝,𝛼 = 󵄨󵄨󵄨𝑓 (0)󵄨󵄨󵄨 + (∫ (1 − |𝑧| ) 󵄨󵄨󵄨󵄨R 𝑓(𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 (𝑧)) , B 𝑛

𝑓 ∈ A𝑝𝛼 . (22) Equipped with (22), A𝑝𝛼 becomes a Banach space when 𝑝 ≥ 1 and a quasi-Banach space for 0 < 𝑝 < 1. It is known that the family of the generalized Bergman spaces A𝑝𝛼 covers most of the spaces of holomorphic functions in the unit ball of C𝑛 , such as the classical diagonal 𝑝 Besov space 𝐵𝑝𝑠 and the Sobolev space 𝑊𝑘,𝛽 , which has been extensively studied before in the literature under different names (see e.g., [2] for an overview). 𝑝 There are various characterizations for 𝐵𝑝𝑠 or 𝑊𝑘,𝛽 involving complex-variable quantities in terms of radical derivatives, complex and invariant gradients, and fractional differential operators (for a review and details see [2] and references therein). However, as an application of Theorems 1 and 2, we obtain new maximal and area integral characterizations of the Besov spaces as follows, which can be considered as a unified characterization for such spaces involving realvariable quantities. Corollary 3. Suppose 𝛾 > 0 and 𝛼 ∈ R. Let 0 < 𝑝 < ∞ and 𝑘 a positive integer such that 𝑝𝑘 + 𝛼 > −1. Then for any 𝑓 ∈ H(B𝑛 ), 𝑓 ∈ A𝑝𝛼 if and only if 𝑀𝛾(𝑘) (𝑓) ∈ 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ), where 󵄨󵄨 󵄨󵄨 𝑘 sup 󵄨󵄨󵄨󵄨(1 − |𝑤|2 ) R𝑘 𝑓 (𝑤)󵄨󵄨󵄨󵄨 , 𝑧 ∈ B𝑛 . M(𝑘) 𝛾 (𝑓) (𝑧) = 󵄨 𝑤∈𝐷(𝑧,𝛾) 󵄨

The paper is organized as follows. In Section 2 we will prove Theorems 1 and 2. An atomic decomposition of realvariable type with respect to Carleson tubes for Bergman spaces will be presented in Section 3 via duality method. Finally, in Section 4, we will prove Theorem 2 through using the real-variable atomic decomposition of Bergman spaces established in the preceding section. In what follows, 𝐶 always denotes a constant depending (possibly) on 𝑛, 𝑞, 𝑝, 𝛾, or 𝛼 but not on 𝑓, which may be different in different places. For two nonnegative (possibly infinite) quantities 𝑋 and 𝑌, by 𝑋 ≲ 𝑌 we mean that there exists a constant 𝐶 > 0 such that 𝑋 ≤ 𝐶𝑌 and by 𝑋 ≈ 𝑌 that 𝑋 ≲ 𝑌 and 𝑌 ≲ 𝑋. Any notation and terminology not otherwise explained are as used in [3] for spaces of holomorphic functions in the unit ball of C𝑛 .

2. Proofs of Theorems 1 and 2 For the sake of convenience, we collect some elementary facts on the Bergman metric and holomorphic functions in the unit ball of C𝑛 as follows. Lemma 5 (cf. [3, Lemma 2.20]). For each 𝛾 > 0, 1 − |𝑎|2 ≈ 1 − |𝑧|2 ≈ |1 − ⟨𝑎, 𝑧⟩| ,

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for all 𝑎 and 𝑧 in B𝑛 with 𝛽(𝑎, 𝑧) < 𝛾. Lemma 6 (cf. [3, Lemma 2.24]). Suppose 𝛾 > 0, 𝑝 > 0, and 𝛼 > −1. Then there exists a constant 𝐶 > 0 such that for any 𝑓 ∈ H(B𝑛 ),

(23) 󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓(𝑧)󵄨󵄨󵄨 ≤

Moreover, 󵄩 󵄩 󵄩󵄩 𝑘 󵄩 󵄩󵄩𝑓 − 𝑓(0)󵄩󵄩󵄩𝑝,𝛼 ≈ 󵄩󵄩󵄩󵄩M𝛾 (R 𝑓)󵄩󵄩󵄩󵄩𝑝,𝛼 ,

(24)

𝐶 2 𝑛+1+𝛼

(1 − |𝑧| )

∫

𝐷(𝑧,𝛾)

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓(𝑤)󵄨󵄨󵄨 𝑑V𝛼 (𝑤) ,

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where “≈” depends only on 𝛾, 𝛼, 𝑝, 𝑘, and 𝑛. Corollary 4. Suppose 1 < 𝑞 < ∞, 𝛾 > 0, and 𝛼 ∈ R. Let 0 < 𝑝 < ∞ and 𝑘 a nonnegative integer such that 𝑝𝑘 + 𝛼 > −1. 𝛾,𝑞 Then for any 𝑓 ∈ H(B𝑛 ), 𝑓 ∈ A𝑝𝛼 if and only if 𝐴 R𝑘+1 (𝑓) is 𝑝 in 𝐿 (B𝑛 , 𝑑V𝛼 ), where

∀𝑧 ∈ B𝑛 .

Lemma 7 (cf. [3, Lemma 2.27]). For each 𝛾 > 0, |1 − ⟨𝑧, 𝑢⟩| ≈ |1 − ⟨𝑧, V⟩| ,

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𝛾,𝑞

𝐴 R𝑘+1 (𝑓) (𝑧) 1/𝑞 󵄨󵄨 󵄨𝑞 󵄨󵄨(1 − |𝑤|2 )𝑘+1 R𝑘+1 𝑓 (𝑤)󵄨󵄨󵄨 𝑑𝜏 (𝑤)) . = (∫ 󵄨󵄨 󵄨󵄨 󵄨 𝐷(𝑧,𝛾) 󵄨

Moreover,

󵄩 𝛾,𝑞 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑓 − 𝑓(0)󵄩󵄩󵄩𝑝,𝛼 ≈ 󵄩󵄩󵄩󵄩𝐴 R𝑘+1 (𝑓)󵄩󵄩󵄩󵄩𝑝,𝛼 ,

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for all 𝑧 in B𝑛 and 𝑢, V in B𝑛 with 𝛽(𝑢, V) < 𝛾. 2.1. Proof of Theorem 1. One needs the following result (cf. [4, Lemma 5]).

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where “≈” depends only on 𝑞, 𝛾, 𝛼, 𝑝, 𝑘, and 𝑛. To prove Corollaries 3 and 4, one merely notices that 𝑓 ∈ A𝑝𝛼 if and only if R𝑘 𝑓 ∈ 𝐿𝑝 (B𝑛 , 𝑑V𝛼+𝑝𝑘 ) and applies Theorems 1 and 2, respectively, to R𝑘 𝑓 with the help of Lemma 5 below.

Lemma 8. For fixed 𝛾 > 0, there exist a positive integer 𝑁 and a sequence {𝑎𝑘 } in B𝑛 such that (1) B𝑛 = ⋃𝑘 𝐷(𝑎𝑘 , 𝛾), and (2) each 𝑧 ∈ B𝑛 belongs to at most 𝑁 of the sets 𝐷(𝑎𝑘 , 3𝛾).

4


Proof of Theorem 1. Let 𝑝 > 0. By Lemmas 8, 6, and 5, one has 󵄨 󵄨𝑝 ∫ 󵄨󵄨󵄨󵄨𝑀𝛾 (𝑓)(𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 (𝑧) B

Proposition 10. Suppose 𝑝 > 0, 𝛼 > −1, and 𝑏 > 𝑛 max{1, 1/𝑝} + (𝛼 + 1)/𝑝. Then there exists a sequence {𝑎𝑘 } in B𝑛 such that A𝑝𝛼 consists exactly of functions of the form

𝑛

󵄨󵄨 󵄨𝑝 󵄨󵄨𝑀𝛾 (𝑓)(𝑧)󵄨󵄨󵄨 𝑑V𝛼 (𝑧) 󵄨 󵄨 𝐷(𝑎 ,𝛾)

≤ ∑∫ 𝑘

𝑘

󵄨𝑝 󵄨 sup 󵄨󵄨󵄨𝑓(𝑤)󵄨󵄨󵄨 𝑑V𝛼 (𝑧)

𝐷(𝑎𝑘 ,𝛾) 𝑤∈𝐷(𝑧,𝛾)

1 sup 𝑛+1+𝛼 𝐷(𝑎𝑘 ,𝛾) 𝑤∈𝐷(𝑧,𝛾) (1 − |𝑤|2 )

𝑘

𝐷(𝑎𝑘 ,𝛾)

(

𝑘

𝑘

𝑧 ∈ B𝑛 ,

(34)

(35)

where the infimum runs over all the above decompositions.

1 󵄨󵄨 󵄨󵄨2 𝑛+1+𝛼 (1 − 󵄨󵄨𝑎𝑘 󵄨󵄨 ) 󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑢)󵄨󵄨󵄨 𝑑V𝛼 (𝑢) ) 𝑑V𝛼 (𝑧) 𝐷(𝑎 ,3𝛾)

𝐷(𝑎𝑘 ,3𝛾)

,

𝑘

𝑛

×∫ ≲ ∑∫

𝑏

󵄨𝑝 󵄨 󵄨 󵄨𝑝 ∫ 󵄨󵄨󵄨𝑓(𝑧)󵄨󵄨󵄨 𝑑V𝛼 (𝑧) ≈ inf {∑󵄨󵄨󵄨𝑐𝑘 󵄨󵄨󵄨 } , B

󵄨𝑝 󵄨󵄨 ×∫ 󵄨󵄨𝑓 (𝑢)󵄨󵄨󵄨 𝑑V𝛼 (𝑢) 𝑑V𝛼 (𝑧) 𝐷(𝑤,𝛾) ≲ ∑∫

𝑘=1

(1 − ⟨𝑧, 𝑎𝑘 ⟩)

where {𝑐𝑘 } belongs to the sequence space ℓ𝑝 and the series converges in the norm topology of A𝑝𝛼 . Moreover,

≲ ∑∫ 𝑘

󵄨 󵄨2 (𝑝𝑏−𝑛−1−𝛼)/𝑝 (1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 )

𝑓 (𝑧) = ∑ 𝑐𝑘

𝑘

= ∑∫

∞

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑓 (𝑢)󵄨󵄨󵄨 𝑑V𝛼 (𝑢)

Also, we need a characterization of Carleson type measures for Bergman spaces as follows, which can be found in [2, Theorem 45]. Proposition 11. Suppose 𝑛 + 1 + 𝛼 > 0 and 𝜇 is a positive Borel measure on B𝑛 . Then, there exists a constant 𝐶 > 0 such that 𝜇 (𝑄𝑟 (𝜁)) ≤ 𝐶𝑟2(𝑛+1+𝛼) ,

󵄨𝑝 󵄨 ≲ 𝑁 ∫ 󵄨󵄨󵄨𝑓 (𝑢)󵄨󵄨󵄨 𝑑V𝛼 (𝑢) , B 𝑛

(30) where 𝑁 is the constant in Lemma 8 depending only on 𝛾 and 𝑛. 2.2. Proof of Theorem 2. Recall that B(B𝑛 ) is defined as the space of all 𝑓 ∈ H(B𝑛 ) so that 󵄨 󵄨̃ 󵄩󵄩 󵄩󵄩 (𝑧)󵄨󵄨󵄨󵄨 < ∞. 󵄩󵄩𝑓󵄩󵄩B = sup 󵄨󵄨󵄨󵄨∇𝑓 𝑧∈B

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𝑛

B(B𝑛 ) with the norm ‖𝑓‖ = |𝑓(0)| + ‖𝑓‖B is a Banach space and called the Bloch space. Then, the following interpolation result holds.

∫

(1 − |𝑧|2 )

B𝑛

󸀠

𝜃

𝑠

|1 − ⟨𝑧, 𝑤⟩|𝑛+1+𝛼+𝑠

𝑑𝜇 (𝑤) ≤ 𝐶,

(37)

for all 𝑧 ∈ B𝑛 . We are now ready to prove Theorem 2. Note that for any 𝑓 ∈ H(B𝑛 ), 󵄨 󵄨 󵄨 󵄨 (1 − |𝑧|2 ) 󵄨󵄨󵄨R𝑓 (𝑧)󵄨󵄨󵄨 ≤ (1 − |𝑧|2 ) 󵄨󵄨󵄨∇𝑓 (𝑧)󵄨󵄨󵄨 󵄨 󵄨̃ ≤ 󵄨󵄨󵄨󵄨∇𝑓 (𝑧)󵄨󵄨󵄨󵄨 ,

∀𝑧 ∈ B𝑛

(38)

(32)

(cf. [3, Lemma 2.14]). We have that (d) implies (c) and (c) implies (b) in Theorem 2. Then, it remains to prove that (b) implies (a) and (a) implies (d).

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Proof of (𝑏) ⇒ (𝑎). Since Lemma 6 we have

for 0 < 𝜃 < 1 and 1 ≤ 𝑝󸀠 < ∞. Then A𝑝𝛼 (B𝑛 ) = [A𝑝𝛼 (B𝑛 ) , B (B𝑛 )] ,

(36)

if and only if for each 𝑠 > 0, there exists a constant 𝐶 > 0 such that

Lemma 9 (cf. [3, Theorem 3.25]). Let 1 < 𝑝 < ∞. Suppose 𝛼 > −1 and 1 1−𝜃 , = 𝑝 𝑝󸀠

∀𝜁 ∈ S𝑛 , 𝑟 > 0,

with equivalent norms. Moreover, to prove Theorem 2 for the case 0 < 𝑝 ≤ 1, one will use atom decomposition for Bergman spaces due to Coifman and Rochberg [5] (see also [3, Theorem 2.30]) as follows.

󵄨𝑞 󵄨󵄨 󵄨󵄨R𝑓 (𝑧)󵄨󵄨󵄨 ≤

R𝑓(𝑧)

𝐶 2 𝑛+1

(1 − |𝑧| )

∫

𝐷(𝑧,𝛾)

is

holomorphic,

by

󵄨𝑞 󵄨󵄨 󵄨󵄨R𝑓 (𝑤)󵄨󵄨󵄨 𝑑V (𝑤)

󵄨𝑞 󵄨󵄨 ≤ 𝐶𝛾 ∫ 󵄨󵄨R𝑓(𝑤)󵄨󵄨󵄨 𝑑𝜏 (𝑤) . 𝐷(𝑧,𝛾)

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5 where we have used a the fact V(𝐷(𝑧, 𝛾)) ≈ (1 − |𝑧|2 )𝑛+1 . Note that V𝛼 (𝑄𝑟 ) ≈ 𝑟2(𝑛+1+𝛼) (cf. [3, Corollary 5.24]); by Proposition 11 we have

Then, 󵄨 󵄨 (1 − |𝑧|2 ) 󵄨󵄨󵄨R𝑓 (𝑧)󵄨󵄨󵄨 1/𝑞

󵄨𝑞 󵄨󵄨 󵄨󵄨R𝑓 (𝑤)󵄨󵄨󵄨 𝑑𝜏 (𝑤)) 𝐷(𝑧,𝛾)

≤ 𝐶 (1 − |𝑧|2 ) (∫

󵄨󵄨 󵄨𝑞 󵄨󵄨(1 − |𝑤|2 )R𝑓(𝑤)󵄨󵄨󵄨 𝑑𝜏(𝑤)) 󵄨 󵄨 𝐷(𝑧,𝛾)

≤ 𝐶𝛾 (∫ =

𝛾,𝑞 𝐶𝛾 𝐴 R

󵄨𝑝 󵄨 𝛾,𝑞 ∫ 󵄨󵄨󵄨󵄨𝐴 ∇̃ (𝑓𝑘 ) (𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 (𝑧) B 𝑛

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1/𝑞

󵄨 󵄨2 (𝑝𝑏−𝑛−1−𝛼) (1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 ) ≤ 𝐶𝑏 ∫ 󵄨󵄨 󵄨𝑝𝑏 𝑑V𝛼 (𝑧) B𝑛 󵄨󵄨1 − ⟨𝑧, 𝑎𝑘 ⟩󵄨󵄨󵄨 𝑝

(𝑓) (𝑧) . 𝛾,𝑞

Hence, for any 𝑝 > 0, if 𝐴 R (𝑓) ∈ 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ) then (1 − |𝑧|2 )|R𝑓(𝑧)| is in 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ), which implies that 𝑓 ∈ A𝑝𝛼 (cf. [3, Theorem 2.16]). The proof of (a) ⇒ (d) is divided into two steps. We first prove the case 0 < 𝑝 ≤ 1 using the atomic decomposition and then the remaining case via complex interpolation.

≤ 𝐶𝑝,𝛼 . Hence, for 0 < 𝑝 ≤ 1, we have for 𝑓 = ∑∞ 𝑘=1 𝑐𝑘 𝑓𝑘 with ∑𝑘 |𝑐𝑘 |𝑝 < ∞, 󵄨𝑝 󵄨 𝛾,𝑞 ∫ 󵄨󵄨󵄨󵄨𝐴 ∇̃ (𝑓) (𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 B 𝑛

Proof of (𝑎) ⇒ (𝑑) for 0 < 𝑝 ≤ 1. To this end, we write 𝑓𝑘 (𝑧) =

󵄨 󵄨2 (𝑝𝑏−𝑛−1−𝛼)/𝑝 (1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 ) 𝑏

(1 − ⟨𝑧, 𝑎𝑘 ⟩)

.

∞

󵄨 󵄨𝑝 ≤ ∑ 󵄨󵄨󵄨𝑐𝑘 󵄨󵄨󵄨 ∫ B 𝑘=1

(41)

R𝑓𝑘 (𝑧) =

(46)

∞

𝑘=1

󵄨󵄨2 (𝑝𝑏−𝑛−1−𝛼)/𝑝

󵄨 𝑏𝑎𝑘 (1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨 )

𝑛

󵄨𝑝 󵄨󵄨 𝛾,𝑞 󵄨󵄨𝐴 ̃ (𝑓𝑘 ) (𝑧)󵄨󵄨󵄨 𝑑V𝛼 󵄨 󵄨 ∇

󵄨 󵄨𝑝 ≤ 𝐶𝑝,𝛼 ∑ 󵄨󵄨󵄨𝑐𝑘 󵄨󵄨󵄨 .

An immediate computation yields that ∇𝑓𝑘 (𝑧) =

(45)

𝑏+1

(1 − ⟨𝑧, 𝑎𝑘 ⟩)

󵄨 󵄨2 (𝑝𝑏−𝑛−1−𝛼)/𝑝 𝑏⟨𝑧, 𝑎𝑘 ⟩(1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 ) 𝑏+1

(1 − ⟨𝑧, 𝑎𝑘 ⟩)

This concludes that

,

󵄨𝑝 󵄨 𝛾,𝑞 ∫ 󵄨󵄨󵄨󵄨𝐴 ∇̃ (𝑓) (𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 B

(42)

𝑛

.

∞

󵄨 󵄨𝑝 ≤ 𝐶𝑝,𝛼 inf { ∑ 󵄨󵄨󵄨𝑐𝑘 󵄨󵄨󵄨 }

(47)

𝑘=1

Then we have 󵄨󵄨 ̃ 󵄨2 󵄨󵄨∇𝑓𝑘 (𝑧)󵄨󵄨󵄨 󵄨 󵄨

󵄨𝑝 󵄨 ≤ 𝐶𝑝,𝛼 ∫ 󵄨󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨 𝑑V𝛼 (𝑧) . B 𝑛

󵄨󵄨2

󵄨󵄨2

󵄨 󵄨 = (1 − |𝑧|2 ) (󵄨󵄨󵄨∇𝑓𝑘 (𝑧)󵄨󵄨 − 󵄨󵄨󵄨R𝑓𝑘 (𝑧)󵄨󵄨 )

The proof is complete.

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 2(𝑝𝑏−𝑛−1−𝛼)/𝑝 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 − 󵄨󵄨󵄨⟨𝑧, 𝑎𝑘 ⟩󵄨󵄨󵄨 . = 𝑏2 (1 − |𝑧|2 ) (1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 ) 󵄨󵄨󵄨1 − ⟨𝑧, 𝑎𝑘 ⟩󵄨󵄨󵄨2(𝑏+1) 󵄨 󵄨 (43)

Proof of (𝑎) ⇒ (𝑑) for 𝑝 > 1. Set 𝐸 = 𝐿𝑞 (B𝑛 , 𝜒𝐷(0,𝛾) 𝑑𝜏; C𝑛 ). Consider the operator

By Lemmas 5 and 7, one has 𝛾,𝑞

󵄨 󵄨2 (𝑝𝑏−𝑛−1−𝛼)/𝑝 ≤ 𝑏(1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 ) × (∫

𝐷(𝑧,𝛾)

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󵄩 󵄩󵄩 󵄩󵄩𝑇 (𝑓) (𝑧)󵄩󵄩󵄩𝐸 1/𝑞

1/𝑞 󵄨 ̃ 󵄨𝑞 (𝜑𝑧 (𝑤))󵄨󵄨󵄨󵄨 𝜒𝐷(0,𝛾) (𝑤) 𝑑𝜏 (𝑤)) = (∫ 󵄨󵄨󵄨󵄨(∇𝑓) B

󵄨󵄨 󵄨𝑞𝑏 𝑑𝜏 (𝑤)) 󵄨󵄨1 − ⟨𝑤, 𝑎𝑘 ⟩󵄨󵄨󵄨

󵄨 󵄨2 (𝑝𝑏−𝑛−1−𝛼)/𝑝 𝐶𝛾 𝑏(1 − 󵄨󵄨󵄨𝑎𝑘 󵄨󵄨󵄨 ) ≤ , 󵄨󵄨 󵄨𝑏 󵄨󵄨1 − ⟨𝑧, 𝑎𝑘 ⟩󵄨󵄨󵄨

𝑓 ∈ H (B𝑛 ) .

Note that 𝜑𝑧 (𝐷(0, 𝛾)) = 𝐷(𝑧, 𝛾) and the measure 𝑑𝜏 is invariant under any automorphism of B𝑛 (cf. [3, Proposition 1.13]); we have

1/𝑞 󵄨󵄨 ̃ 󵄨𝑞 󵄨󵄨∇𝑓𝑘 (𝑤)󵄨󵄨󵄨 𝑑𝜏(𝑤)) 󵄨 󵄨 𝐷(𝑧,𝛾)

𝐴 ∇̃ (𝑓𝑘 ) (𝑧) = (∫

1

̃ (𝜑𝑧 (𝑤)) , 𝑇 (𝑓) (𝑧, 𝑤) = (∇𝑓)

𝑛

1/𝑞 󵄨󵄨𝑞 󵄨̃ 󵄨󵄨 𝜒𝐷(𝑧,𝛾) (𝑤)𝑑𝜏(𝑤)) = (∫ 󵄨󵄨󵄨󵄨∇𝑓(𝑤) 󵄨 B 𝑛

(44)

=

𝛾,𝑞 𝐴 ∇̃

(𝑓) (𝑧) .

(49)

6

Journal of Function Spaces and Applications The constant function (1) is also considered to be a (1, 𝑞)𝛼 atom. Note that for any (1, 𝑞)𝛼 -atom 𝑎,

On the other hand, 𝛾,𝑞 𝐴 ∇̃

(𝑓) (𝑧) −𝑛−1

≤ [𝐶𝛾 (1 − |𝑧|2 )

1/2

V(𝐷(𝑧, 𝛾))]

󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑓󵄩󵄩B ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩B . (50)

This follows that 𝑇 is bounded from B into 𝐿∞ 𝛼 (B𝑛 , 𝐸). Notice that we have proved that 𝑇 is bounded from A1𝛼 to 𝐿1𝛼 (B𝑛 , 𝐸).

‖𝑎‖1,𝛼 = ∫ |𝑎| 𝑑V𝛼 ≤ V𝛼 (𝑄)1−1/𝑞 ‖𝑎‖𝑞,𝛼 ≤ 1. 𝑄

Recall that 𝑃𝛼 is the orthogonal projection from 𝐿2 (B𝑛 , 𝑑V𝛼 ) onto A2𝛼 , which can be expressed as 𝑃𝛼 𝑓 (𝑧) = ∫ 𝐾𝛼 (𝑧, 𝑤) 𝑓 (𝑤) 𝑑V𝛼 (𝑤) ,

Thus, by Lemma 9 and the well known fact that 𝐿 𝑝 (B𝑛 , 𝐸) =

(𝐿1𝛼 (B𝑛 , 𝐸), 𝐿∞ 𝛼

1 with 𝜃 = 1 − , 𝑝 (51)

(B𝑛 , 𝐸))𝜃

we conclude that 𝑇 is bounded from any 1 < 𝑝 < ∞; that is, 󵄩󵄩 𝛾,𝑞 󵄩󵄩 󵄩󵄩𝐴 ̃ (𝑓)󵄩󵄩 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝,𝛼 , 󵄩 ∇ 󵄩𝑝,𝛼

A𝑝𝛼

into

𝐿𝑝𝛼 (B𝑛 , 𝐸)

∀𝑓 ∈ A𝑝𝛼 ,

where 𝐶 depends only on 𝑞, 𝛾, 𝑛, 𝑝, and 𝛼. The proof is complete. Remark 12. From the proofs of that (b) ⇒ (a) and that (a) ⇒ (d) for 𝑝 > 1, we find that Theorem 2 still holds true for the Bloch space. That is, for any 𝑓 ∈ H(B𝑛 ), 𝑓 ∈ B if and only 𝛾,𝑞 𝛾,𝑞 𝛾,𝑞 if one (or equivalently, all) of 𝐴 R (𝑓), 𝐴 ∇ (𝑓), and 𝐴 ∇̃ (𝑓) is (or, are) in 𝐿∞ (B𝑛 ). Moreover, 󵄩󵄩 󵄩󵄩 󵄩 𝛾,𝑞 󵄩 󵄩󵄩𝑓󵄩󵄩B ≈ 󵄩󵄩󵄩𝐴 R (𝑓)󵄩󵄩󵄩𝐿∞ (B𝑛 ) 󵄩 𝛾,𝑞 󵄩 ≈ 󵄩󵄩󵄩𝐴 ∇ (𝑓)󵄩󵄩󵄩𝐿∞ (B ) 𝑛 󵄩󵄩 𝛾,𝑞 󵄩󵄩 ≈ 󵄩󵄩󵄩𝐴 ∇̃ (𝑓)󵄩󵄩󵄩 ∞ , 𝐿 (B𝑛 )

(53)

3. Atomic Decomposition for Bergman Spaces We let 𝑧, 𝑤 ∈ B𝑛 .

(54)

It is known that 𝑑 satisfies the triangle inequality and the restriction of 𝑑 to S𝑛 is a metric. As usual, 𝑑 is called the nonisotropic metric. For any 𝜁 ∈ S𝑛 and 𝑟 > 0, the set 𝑄𝑟 (𝜁) = {𝑧 ∈ B𝑛 : 𝑑 (𝑧, 𝜁) < 𝑟}

(55)

is called a Carleson tube with respect to the nonisotropic metric 𝑑. We usually write 𝑄 = 𝑄𝑟 (𝜁) in short. As usual, we define the atoms with respect to the Carleson tube as follows: for 1 < 𝑞 < ∞, 𝑎 ∈ 𝐿𝑞 (B𝑛 , 𝑑V𝛼 ) is said to be a (1, 𝑞)𝛼 -atom if there is a Carleson tube 𝑄 such that (1) 𝑎 is supported in 𝑄; (1/𝑞)−1

(2) ‖𝑎‖𝐿𝑞 (B𝑛 ,𝑑V𝛼 ) ≤ V𝛼 (𝑄) (3) ∫B 𝑎(𝑧) 𝑑V𝛼 (𝑧) = 0. 𝑛

;

(57)

1

∀𝑓 ∈ 𝐿 (B𝑛 , 𝑑V𝛼 ) , 𝛼 > −1, where 𝐾𝛼 (𝑧, 𝑤) =

1 , (1 − ⟨𝑧, 𝑤⟩)𝑛+1+𝛼

𝑧, 𝑤 ∈ B𝑛 .

(58)

𝑃𝛼 extends to a bounded projection from 𝐿𝑝 (B𝑛 , 𝑑V𝛼 ) onto A𝑝𝛼 (1 < 𝑝 < ∞). We have the following useful estimates. Lemma 13. For 𝛼 > −1 and 1 < 𝑞 < ∞, there exists a constant 𝐶𝑞,𝛼,𝑛 > 0 such that 󵄩󵄩 󵄩 󵄩󵄩𝑃𝛼 (𝑎)󵄩󵄩󵄩1,𝛼 ≤ 𝐶𝑞,𝛼,𝑛 ,

(59)

for any (1, 𝑞)𝛼 -atom 𝑎. To prove Lemma 13, we need first to show an inequality for reproducing kernel 𝐾𝛼 associated with 𝑑, which is essentially borrowed from [6, Proposition 2.13]. Lemma 14. For 𝛼 > −1, there exists a constant 𝛿 > 0 such that for all 𝑧, 𝑤 ∈ B𝑛 , 𝜁 ∈ S𝑛 satisfying 𝑑(𝑧, 𝜁) > 𝛿𝑑(𝑤, 𝜁), we have

where “≈” depends only on 𝑞, 𝛾, and 𝑛.

𝑑 (𝑧, 𝑤) = |1 − ⟨𝑧, 𝑤⟩|1/2 ,

B𝑛

for

(52)

(56)

󵄨 󵄨󵄨 𝛼 𝛼 󵄨󵄨𝐾 (𝑧, 𝑤) − 𝐾 (𝑧, 𝜁)󵄨󵄨󵄨 ≤ 𝐶𝛼,𝑛

𝑑 (𝑤, 𝜁) 𝑑(𝑧, 𝜁)2(𝑛+1+𝛼)+1

.

(60)

Proof. Note that 𝐾𝛼 (𝑧, 𝑤) − 𝐾𝛼 (𝑧, 𝜁) =∫

1

0

𝑑 1 ) 𝑑𝑡. ( 𝑑𝑡 (1 − ⟨𝑧, 𝜁⟩ − 𝑡⟨𝑧, 𝑤 − 𝜁⟩)𝑛+1+𝛼

(61)

We have 󵄨󵄨 𝛼 󵄨 𝛼 󵄨󵄨𝐾 (𝑧, 𝑤) − 𝐾 (𝑧, 𝜁)󵄨󵄨󵄨

󵄨 󵄨 1 (𝑛 + 1 + 𝛼) 󵄨󵄨󵄨⟨𝑧, 𝑤 − 𝜁⟩󵄨󵄨󵄨 ≤∫ 󵄨 󵄨𝑛+2+𝛼 𝑑𝑡. 0 󵄨󵄨1 − ⟨𝑧, 𝜁⟩ − 𝑡 ⟨𝑧, 𝑤 − 𝜁⟩󵄨󵄨 󵄨 󵄨

(62)

Write 𝑧 = 𝑧1 + 𝑧2 and 𝑤 = 𝑤1 + 𝑤2 , where 𝑧1 and 𝑤1 are parallel to 𝜁, while 𝑧2 and 𝑤2 are perpendicular to 𝜁. Then ⟨𝑧, 𝑤⟩ − ⟨𝑧, 𝜁⟩ = ⟨𝑧2 , 𝑤2 ⟩ − ⟨𝑧1 , 𝑤1 − 𝜁⟩

(63)


7

and so 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨⟨𝑧, 𝑤⟩ − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑧2 󵄨󵄨󵄨 󵄨󵄨󵄨𝑤2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑤1 − 𝜁󵄨󵄨󵄨 .

(64)

Since |𝑤1 − 𝜁| = |1 − ⟨𝑤, 𝜁⟩|, 󵄨 󵄨2 󵄨󵄨 󵄨󵄨2 2 󵄨󵄨𝑧2 󵄨󵄨 = |𝑧| − 󵄨󵄨󵄨𝑧1 󵄨󵄨󵄨 󵄨 󵄨2 < 1 − 󵄨󵄨󵄨𝑧1 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 < (1 + 󵄨󵄨󵄨𝑧1 󵄨󵄨󵄨) (1 + 󵄨󵄨󵄨𝑧1 󵄨󵄨󵄨) 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨1 − ⟨𝑧1 , 𝜁⟩󵄨󵄨󵄨 = 2 󵄨󵄨󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 ,

Proof of Lemma 13. When 𝑎 is the constant function 1, the result is clear. Thus we may suppose that 𝑎 is a (1, 𝑞)𝛼 -atom. Let 𝑎 be supported in a Carleson tuber 𝑄𝑟 (𝜁) and 𝛿𝑟 ≤ √2, where 𝛿 is the constant in Lemma 14. Since 𝑃𝛼 is a bounded operator on 𝐿𝑞 (B𝑛 , 𝑑V𝛼 ), we have ∫

𝑄𝛿𝑟

󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑃𝛼 󵄩󵄩󵄩𝐿𝑞 . (66)

Next, if 𝑑(𝑧, 𝜁) > 𝛿𝑟 then 󵄨󵄨 󵄨󵄨 𝑎 (𝑤) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨∫ 𝑑V (𝑤) 𝛼 󵄨󵄨 󵄨󵄨 B (1 − ⟨𝑧, 𝑤⟩)𝑛+1+𝛼 󵄨 󵄨 𝑛 󵄨󵄨 1 󵄨 = 󵄨󵄨󵄨󵄨∫ 𝑎 (𝑤) [ 󵄨󵄨 𝑄𝑟 (𝜁) (1 − ⟨𝑧, 𝑤⟩)𝑛+1+𝛼

󵄨 󵄨󵄨 󵄨󵄨⟨𝑧, 𝑤⟩ − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 󵄨1/2 󵄨 󵄨 󵄨1/2 󵄨 󵄨 ≤ 2󵄨󵄨󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 (67)

−

1 1 ≤ 2 (1 + ) 𝑑2 (𝑧, 𝜁) . 𝛿 𝛿

≤ 𝐶∫

𝑄𝑟 (𝜁)

This concludes that there is 𝛿 > 1 such that 󵄨 󵄨 1󵄨 󵄨󵄨 󵄨󵄨⟨𝑧, 𝑤 − 𝜁⟩󵄨󵄨󵄨 < 󵄨󵄨󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 , 2

≤ 𝐶𝑟 ∫ ≤

whenever 𝑑(𝑧, 𝜁) > 𝛿𝑑(𝑤, 𝜁). Then, we have 󵄨 󵄨󵄨 󵄨󵄨1 − ⟨𝑧, 𝜁⟩ − 𝑡 ⟨𝑧, 𝑤 − 𝜁⟩󵄨󵄨󵄨

>

|𝑎 (𝑤)|

𝑄𝑟 (𝜁)

∀𝑧, 𝑤 ∈ B𝑛 , 𝜁 ∈ S𝑛 , (68)

󵄨 󵄨 󵄨 󵄨 > 󵄨󵄨󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 − 𝑡 󵄨󵄨󵄨⟨𝑧, 𝜁 − 𝑤⟩󵄨󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 ] 𝑑V (𝑤) 𝛼 󵄨󵄨 𝑛+1+𝛼 󵄨󵄨 (1 − ⟨𝑧, 𝜁⟩) 1

𝑑 (𝑤, 𝜁)

|𝑎 (𝑤)| 𝑑V𝛼 (𝑤)

𝐶𝑟 𝑑(𝑧, 𝜁)2(𝑛+1+𝛼)+1

1 2(𝑛+1+𝛼)+1

𝑑(𝑧, 𝜁)

.

Then (69)

1 󵄨󵄨 󵄨 󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 . 2󵄨

∫

𝑑(𝑧,𝜁)>𝛿𝑟

󵄨 󵄨󵄨 󵄨󵄨𝑃𝛼 (𝑎)󵄨󵄨󵄨 𝑑V𝛼 (𝑧) 1

≤ 𝐶𝑟 ∫

𝑑(𝑧,𝜁)>𝛿𝑟

𝑑(𝑧, 𝜁)

2(𝑛+1+𝛼)+1

𝑑V𝛼 (𝑧)

2

≤ 𝐶𝛼,𝑛

(𝑛 + 1 + 𝛼) (1 + 1/𝛿) 𝑑 (𝑤, 𝜁) 𝑑 (𝑧, 𝜁) 󵄨𝑛+2+𝛼 󵄨󵄨 󵄨󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 𝑑 (𝑤, 𝜁)

𝑑(𝑧, 𝜁)

, 2(𝑛+1+𝛼)+1

and the lemma is proved.

1

= 𝐶𝑟∑ ∫

󵄨󵄨 𝛼 󵄨 𝛼 󵄨󵄨𝐾 (𝑧, 𝑤) − 𝐾 (𝑧, 𝜁)󵄨󵄨󵄨 ≤

(72)

𝑑V𝛼 (𝑤)

𝑑(𝑧, 𝜁)2(𝑛+1+𝛼)+1

Therefore,

𝑛+3+𝛼

(71)

1−(1/𝑞) 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑃𝛼 󵄩󵄩󵄩𝐿𝑞 V𝛼 (𝑄𝛿𝑟 ) ‖𝑎‖𝑞,𝛼

we have

= 2𝑑 (𝑤, 𝜁) [𝑑 (𝑧, 𝜁) + 𝑑 (𝑤, 𝜁)]

󵄩 󵄩󵄩𝑃𝛼 (𝑎)󵄩󵄩󵄩𝑞,𝛼

1−(1/𝑞) 󵄩 󵄩

≤ V𝛼 (𝑄𝛿𝑟 )

(65)

and similarly 󵄨󵄨 󵄨󵄨2 󵄨 󵄨 󵄨󵄨𝑤2 󵄨󵄨 ≤ 2 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 ,

󵄨 󵄨󵄨 󵄨󵄨𝑃𝛼 (𝑎)󵄨󵄨󵄨 𝑑V𝛼 (𝑧)

𝑘≥0

≤ 𝐶𝑟∑

2𝑘 𝛿𝑟 −1 and 1 ≤ 𝑝 < ∞. Then, for any 𝑓 ∈ H(B𝑛 ), 𝑓 is in B if and only if there exists a constant 𝐶 > 0 depending only on 𝛼 and 𝑝 such that 1 󵄨󵄨 󵄨𝑝 󵄨󵄨𝑓 − 𝑓𝛼,𝑄𝑟 (𝜁) 󵄨󵄨󵄨 𝑑V𝛼 ≤ 𝐶, ∫ 󵄨 V𝛼 (𝑄𝑟 (𝜁)) 𝑄𝑟 (𝜁) 󵄨

𝑛

=∫

𝑄𝛿𝑟

󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑃𝛼 (𝑎)󵄨󵄨󵄨 𝑑V𝛼 (𝑧) + ∫ 󵄨󵄨𝑃𝛼 (ℎ)󵄨󵄨󵄨 𝑑V𝛼 (𝑧) 𝑑(𝑧,𝜁)>𝛿𝑟

(74)

for all 𝑟 > 0 and all 𝜁 ∈ S𝑛 , where

≤ 𝐶,

𝑓𝛼,𝑄𝑟 (𝜁) =

where 𝐶 depends only on 𝑞, 𝑛, and 𝛼. Now we turn to the atomic decomposition of A1𝛼 (𝛼 > −1) with respect to the Carleson tubes. Recall that ‖𝑎‖1,𝛼 ≤ 1 for any (1, 𝑞)𝛼 -atom 𝑎. Then, we define A1,𝑞 𝛼 as the space of all 𝑓 ∈ A1𝛼 which admits a decomposition 󵄨 󵄨 󵄩 󵄩 ∑ 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶𝑞 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1,𝛼 ,

𝑓 = ∑𝜆 𝑖 𝑃𝛼 𝑎𝑖 , 𝑖

𝑖

(80)

(75)

1 𝑓 (𝑧) 𝑑V𝛼 (𝑧) . ∫ 𝑄𝑟 (𝜁) 𝑄𝑟 (𝜁)

(81)

Moreover, 1/𝑝 1 󵄨󵄨 󵄨𝑝 󵄩󵄩 󵄩󵄩 󵄨󵄨𝑓 − 𝑓𝛼,𝑄𝑟 (𝜁) 󵄨󵄨󵄨 𝑑V𝛼 ) , ∫ 󵄩󵄩𝑓󵄩󵄩B ≈ sup ( 󵄨 󵄨 𝑟>0, 𝜁∈S V𝛼 (𝑄𝑟 (𝜁)) 𝑄𝑟 (𝜁) (82)

where “≈” depends only on 𝛼, 𝑝, and 𝑛.

where for each 𝑖, 𝑎𝑖 is an (1, 𝑞)𝛼 -atom and 𝜆 𝑖 ∈ C so that ∑𝑖 |𝜆 𝑖 | < ∞. We equip this space with the norm

As noted above, we will prove Theorem 15 via duality. To this end, we first prove the following duality theorem.

󵄨 󵄨 󵄩󵄩 󵄩󵄩 1,𝑞 󵄩󵄩𝑓󵄩󵄩A𝛼 = inf {∑ 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 : 𝑓 = ∑𝜆 𝑖 𝑃𝛼 𝑎𝑖 } ,

Proposition 17. For any 1 < 𝑞 < ∞ and 𝛼 > −1, one has ∗ (A1,𝑞 𝛼 ) = B isometrically. More precisely,

𝑖

(76)

𝑖

where the infimum is taken over all decompositions of 𝑓 described above. It is easy to see that A1,𝑞 𝛼 is a Banach space. By Lemma 13, 1 we have the contractive inclusion A1,𝑞 𝛼 ⊂ A𝛼 . We will prove in what follows that these two spaces coincide. That establishes the “real-variable” atomic decomposition of the Bergman space A1𝛼 . In fact, we will show the remaining inclusion A1𝛼 ⊂ A1,𝑞 𝛼 by duality. Theorem 15. Let 1 < 𝑞 < ∞ and 𝛼 > −1. For every 𝑓 ∈ A1𝛼 there exist a sequence {𝑎𝑖 } of (1, 𝑞)𝛼 -atoms and a sequence {𝜆 𝑖 } of complex numbers such that 𝑓 = ∑𝜆 𝑖 𝑃𝛼 𝑎𝑖 , 𝑖

󵄨 󵄨 󵄩 󵄩 ∑ 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 ≤ 𝐶𝑞 󵄩󵄩󵄩𝑓󵄩󵄩󵄩1,𝛼 . 𝑖

(77)

Moreover, 󵄨 󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓󵄩󵄩1,𝛼 ≈ inf ∑ 󵄨󵄨󵄨𝜆 𝑖 󵄨󵄨󵄨 ,

(78)

𝑖

where the infimum is taken over all decompositions of 𝑓 described above and “≈” depends only on 𝛼, 𝑛, and 𝑞. Recall that the dual space of A1𝛼 is the Bloch space B (we refer to [3] for details). The Banach dual of A1𝛼 can be identified with B (with equivalent norms) under the integral pairing ⟨𝑓, 𝑔⟩𝛼 = lim− ∫ 𝑓 (𝑟𝑧) 𝑔(𝑧)𝑑V𝛼 (𝑧) , 𝑟→1

B𝑛

𝑓∈

A1𝛼 ,

𝑔 ∈ B. (79)

(cf. [3, Theorem 3.17].) In order to prove Theorem 15, we need the following result, which can be found in [4] (see also [3, Theorem 5.25]).

(i) every 𝑔 ∈ B defines a continuous linear functional 𝜑𝑔 on A1,𝑞 𝛼 by 𝜑𝑔 (𝑓) = lim− ∫ 𝑓 (𝑟𝑧) 𝑔 (𝑧)𝑑V𝛼 (𝑧) , 𝑟→1

B𝑛

∀𝑓 ∈ A1,𝑞 𝛼 , (83)

∗ (ii) conversely, each 𝜑 ∈ (A1,𝑞 𝛼 ) is given as (83) by some 𝑔 ∈ B.

Moreover, we have 󵄩󵄩󵄩𝜑 󵄩󵄩󵄩 ≈ 󵄨󵄨󵄨𝑔 (0)󵄨󵄨󵄨 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩 , 󵄩󵄩 𝑔 󵄩󵄩 󵄨 󵄨 󵄩 󵄩B

∀𝑔 ∈ B.

(84)

Proof. Let 𝑝 be the conjugate index of 𝑞, that is, 1/𝑝+1/𝑞 = 1. ∗ We first show B ⊂ (A1,𝑞 𝛼 ) . Let 𝑔 ∈ B. For any (1, 𝑞)𝛼 -atom 𝑎, by Lemma 16 one has 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ 𝑃𝛼 𝑎 (𝑧) 𝑔 (𝑧)𝑑V𝛼 (𝑧)󵄨󵄨󵄨 󵄨󵄨 B 󵄨󵄨 󵄨 𝑛 󵄨 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨⟨𝑃𝛼 (𝑎𝑗 ) , 𝑔⟩𝛼 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨󵄨∫ 𝑎𝑔𝑑V𝛼 󵄨󵄨󵄨󵄨 󵄨󵄨 B𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨󵄨∫ 𝑎(𝑔 − 𝑔𝑄)𝑑V𝛼 󵄨󵄨󵄨󵄨 󵄨󵄨 B𝑛 󵄨󵄨 1/𝑞

≤ (∫ |𝑎|𝑞 𝑑V𝛼 ) 𝑄

1/𝑝

󵄨 󵄨𝑝 (∫ 󵄨󵄨󵄨𝑔 − 𝑔𝑄󵄨󵄨󵄨 𝑑V𝛼 ) 𝑄

1/𝑝 1 󵄨 󵄨𝑝 ∫ 󵄨󵄨󵄨𝑔 − 𝑔𝑄󵄨󵄨󵄨 𝑑V𝛼 ) V𝛼 (𝑄) 𝑄 󵄩󵄩 󵄩󵄩 ≤ 𝐶󵄩󵄩𝑔󵄩󵄩B .

≤(

(85)


9 󵄩 󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 󵄩󵄩󵄩(𝑓 − 𝑓𝑄) 𝜒𝑄󵄩󵄩󵄩𝐿𝑞 (B

On the other hand, for the constant function (1), we have 𝑃𝛼 1 = 1 and so 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨∫ 𝑃𝛼 1 (𝑧) 𝑔(𝑧)𝑑V𝛼 (𝑧)󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ 𝑔 (𝑧) 𝑑V𝛼 (𝑧)󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨𝑔 (0)󵄨󵄨󵄨󵄨 . 󵄨󵄨 B 󵄨󵄨 󵄨󵄨 B 󵄨󵄨 󵄨 𝑛 󵄨 󵄨 𝑛 󵄨 (86)

(93) This concludes that (

(87)

for any finite linear combination 𝑓 of (1, 𝑞)𝛼 -atoms. Hence, 𝑔 defines a continuous linear functional 𝜑𝑔 on a dense subspace of A1,𝑞 𝛼 , and 𝜑𝑔 extends to a continuous linear functional on 1,𝑞 A𝛼 such that 󵄨󵄨 󵄨 󵄨󵄨𝜑𝑔 (𝑓)󵄨󵄨󵄨 ≤ 𝐶 (󵄨󵄨󵄨󵄨𝑔 (0)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩B ) 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩A1,𝑞 , 󵄨 󵄨 𝛼

for all 𝑓 ∈ Next let 𝜑 be a bounded linear functional on A1,𝑞 𝛼 . Note that (89)

Then, 𝜑 is a bounded linear functional on H𝑞 (B𝑛 , 𝑑V𝛼 ). By duality there exists 𝑔 ∈ H𝑝 (B𝑛 , 𝑑V𝛼 ) such that 𝜑 (𝑓) = ∫ 𝑓𝑔𝑑V𝛼 , B𝑛

∀𝑓 ∈ H𝑞 (B𝑛 , 𝑑V𝛼 ) .

(90)

Let 𝑄 = 𝑄𝑟 (𝜁) be a Carleson tube. For any 𝑓 ∈ 𝐿𝑞 (B𝑛 , 𝑑V𝛼 ) supported in 𝑄, it is easy to check that (𝑓 − 𝑓𝑄) 𝜒𝑄 𝑎𝑓 = 󵄩 󵄩 [󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑞 V𝛼 (𝑄)1/𝑝 ]

(91)

Hence, for any 𝑓 ∈ 𝐿𝑞 (B𝑛 , 𝑑V𝛼 ), we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ 𝑓(𝑔 − 𝑔𝑄)𝑑V𝛼 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑄 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨∫ (𝑓 − 𝑓𝑄) 𝑔𝑑V𝛼 󵄨󵄨󵄨 󵄨󵄨 𝑄 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨󵄨∫ (𝑓 − 𝑓𝑄) 𝜒𝑄𝑔𝑑V𝛼 󵄨󵄨󵄨󵄨 󵄨󵄨 B𝑛 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨󵄨∫ 𝑃𝛼 [(𝑓 − 𝑓𝑄) 𝜒𝑄] 𝑔𝑑V𝛼 󵄨󵄨󵄨󵄨 󵄨󵄨 B𝑛 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨𝜑 (𝑃𝛼 [(𝑓 − 𝑓𝑄) 𝜒𝑄])󵄨󵄨󵄨

(94)

By Lemma 16, we have that 𝑔 ∈ B and ‖𝑔‖B ≤ 𝐶‖𝜑‖. Therefore, 𝜑 is given as (83) by 𝑔 with |𝑔(0)| + ‖𝑔‖B ≤ 𝐶‖𝜑‖. Now we are ready to prove Theorem 15. 1 Proof of Theorem 15. By Lemma 13 we know that A1,𝑞 𝛼 ⊂ A𝛼 . 1 ∗ On the other hand, by Proposition 17 we have (A𝛼 ) = ∗ . (A1,𝑞 𝛼 ) . Hence, by duality we have ‖𝑓‖1,𝑞 ≈ ‖𝑓‖A1,𝑞 𝛼

Remark 18. (1) One would like to expect that when 0 < 𝑝 < 1, A𝑝𝛼 also admits an atomic decomposition in terms of atoms with respect to Carleson tubes. However, the proof of Theorem 15 via duality cannot be extended to the case 0 < 𝑝 < 1. At the time of this writing, this problem is entirely open. (2) The real-variable atomic decomposition of Bergman spaces should be known to specialists in the case 𝑝 = 1. Indeed, based on their theory of harmonic analysis on homogeneous spaces, Coifman and Weiss [7] claimed that the Bergman space A1 admits an atomic decomposition in terms of atoms with respect to (B𝑛 , 󰜚, 𝑑V), where 󰜚 (𝑧, 𝑤)

is a (1, 𝑞)-atom. Then, |𝜑(𝑃𝛼 𝑎𝑓 )| ≤ ‖𝜑‖ and so 󵄨 󵄩 󵄩󵄩 󵄩 󵄨󵄨 1/𝑝 󵄨󵄨𝜑 (𝑃𝛼 [(𝑓 − 𝑓𝑄) 𝜒𝑄])󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑞 V𝛼 (𝑄) .

1/𝑝 1 󵄨 󵄨𝑝 󵄩 󵄩 ∫ 󵄨󵄨󵄨𝑔 − 𝑔𝑄󵄨󵄨󵄨 𝑑V𝛼 ) ≤ 2 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 . V𝛼 (𝑄) 𝑄

(88)

A1,𝑞 𝛼 .

H𝑞 (B𝑛 , 𝑑V𝛼 ) = H (B𝑛 ) ∩ 𝐿𝑞 (B𝑛 , 𝑑V𝛼 ) ⊂ A1,𝑞 𝛼 .

V𝛼 (𝑄)1/𝑝

󵄩 󵄩󵄩 󵄩 ≤ 2 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑞 (𝑄,𝑑V𝛼 ) V𝛼 (𝑄)1/𝑝 .

Thus, we deduce that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ 𝑓𝑔𝑑V𝛼 󵄨󵄨󵄨 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩A1,𝑞 (󵄨󵄨󵄨󵄨𝑔 (0)󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩B ) , 󵄨󵄨 B 󵄨󵄨 𝛼 󵄨 𝑛 󵄨

𝑛 ,𝑑V𝛼 )

(92)

󵄨󵄨 󵄨󵄨 1 󵄨 {||𝑧| − |𝑤|| + 󵄨󵄨󵄨󵄨1 − ⟨𝑧, 𝑤⟩󵄨󵄨󵄨 , if 𝑧, 𝑤 ∈ B𝑛 \ {0} , 󵄨󵄨 󵄨󵄨 ={ |𝑧| |𝑤| + , otherwise. |𝑧| |𝑤| { (95) This also applies to A1𝛼 because (B𝑛 , 󰜚, 𝑑V𝛼 ) is a homogeneous space for 𝛼 > −1 (see e.g. [6]). However, the approach of Coifman and Weiss is again based on duality and therefore not constructive and cannot be applied to the case 0 < 𝑝 < 1. Recently, the present authors [8] extend this result to the case 0 < 𝑝 < 1 through using a constructive method.

4. Area Integral Inequalities: Real-Variable Methods In this section, we will prove the area integral inequality for the Bergman space A1𝛼 via atomic decomposition established in Section 3.

10


Theorem 19. Suppose 1 < 𝑞 < ∞, 𝛾 > 0, and 𝛼 > −1. Then,

󵄨𝑞 󵄨̃ ≈ ∫ 󵄨󵄨󵄨󵄨∇𝑓 (𝑤)󵄨󵄨󵄨󵄨 𝑑V𝛼 (𝑤) B 𝑛

󵄩󵄩 𝛾,𝑞 󵄩󵄩 󵄩󵄩𝐴 ̃ (𝑓)󵄩󵄩 ≲ 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩1,𝛼 , 󵄩 ∇ 󵄩1,𝛼

∀𝑓 ∈ H (B𝑛 ) .

(96)

󵄨𝑞 󵄨 ≈ ∫ 󵄨󵄨󵄨𝑓(𝑤) − 𝑓(0)󵄨󵄨󵄨 𝑑V𝛼 (𝑤) . B 𝑛

This is the assertion (a) ⇒ (d) of Theorem 2 in the case 𝑝 = 1. The novelty of the proof here is to involve a realvariable method. The following lemma is elementary.

(98) In the last step, we have used [3, Theorem 2.16(b)]. Proof of (96). By Theorem 15, it suffices to show that for 1 < 𝑞 < ∞, 𝛾 > 0, and 𝛼 > −1, there exists 𝐶 > 0 such that

Lemma 20. Suppose 1 < 𝑞 < ∞, 𝛾 > 0 and 𝛼 > −1. If 𝑓 ∈ A𝑞𝛼 , then 󵄨 𝛾,𝑞 󵄨𝑞 󵄨𝑞 󵄨 ∫ 󵄨󵄨󵄨󵄨𝐴 ∇̃ (𝑓)(𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 ≈ ∫ 󵄨󵄨󵄨𝑓(𝑧) − 𝑓(0)󵄨󵄨󵄨 𝑑V𝛼 , B B 𝑛

(97)

𝑛

𝛾,𝑞

1/𝑞

󵄨 󵄨𝑞 ≤ 𝐶V𝛼 (𝑄)1−(1/𝑞) (∫ 󵄨󵄨󵄨𝑃𝛼 𝑎(𝑧) − 𝑃𝛼 𝑎(0)󵄨󵄨󵄨 𝑑V𝛼 ) B 󵄨󵄨𝑞 ̃ 󵄨 󵄨󵄨∇𝑓 󵄨 (𝑤)󵄨󵄨 𝑑V (𝑤) 𝑑V𝛼 (𝑧)

−1−𝑛 󵄨󵄨

≤ 𝐶V𝛼 (𝑄)1−(1/𝑞) ‖𝑎‖𝑞,𝛼 ≤ 𝐶,

󵄨𝑞 󵄨󵄨∇𝑓(𝑤)󵄨󵄨󵄨 𝑑V (𝑤) 󵄨 󵄨

2 −1−𝑛 󵄨󵄨 ̃

= ∫ V𝛼 (𝐷 (𝑤, 𝛾)) (1 − |𝑤| ) B𝑛

where 2𝑄 = 𝑄2𝑟 (𝜁). On the other hand,

𝛾,𝑞

(2𝑄)𝑐

𝐴 ∇̃ (𝑃𝛼 𝑎) 𝑑V𝛼 1/𝑞 󵄨󵄨 ̃ 󵄨𝑞 󵄨󵄨∇𝑃𝛼 𝑎(𝑤)󵄨󵄨󵄨 𝑑𝜏(𝑤)) 𝑑V𝛼 (𝑧) 󵄨 󵄨 𝐷(𝑧,𝛾)

=∫

(∫ 𝑐

=∫

(∫ 𝑐

(2𝑄)

(2𝑄)

1/𝑞 󵄨󵄨𝑞 󵄨󵄨 󵄨 󵄨󵄨 ̃ 𝛼 𝛼 󵄨󵄨∫ ∇𝑤 [𝐾 (𝑤, 𝑢) − 𝐾 (𝑤, 𝜁)]𝑎(𝑢)𝑑V𝛼 (𝑢)󵄨󵄨󵄨 𝑑𝜏(𝑤)) 𝑑V𝛼 (𝑧) 󵄨󵄨 𝐷(𝑧,𝛾) 󵄨󵄨 𝑄

𝑞−1 󵄨󵄨𝑞/(𝑞−1) 󵄨̃ 𝛼 𝛼 (∫ 󵄨󵄨󵄨󵄨∇ 𝑑V𝛼 (𝑢)) 𝑑𝜏(𝑤)) ≤ ‖𝑎‖𝑞,𝛼 ∫ 𝑐 (∫ 𝑤 [𝐾 (𝑤, 𝑢) − 𝐾 (𝑤, 𝜁)]󵄨󵄨󵄨 (2𝑄) 𝐷(𝑧,𝛾) 𝑄

≤∫

(2𝑄)

󵄨󵄨𝑞 󵄨̃ 𝛼 𝛼 sup󵄨󵄨󵄨󵄨∇ 𝑤 [𝐾 (𝑤, 𝑢) − 𝐾 (𝑤, 𝜁)]󵄨󵄨󵄨 𝑑𝜏 (𝑤))

(∫ 𝑐

𝐷(𝑧,𝛾) 𝑢∈𝑄

𝑢 𝜁 − ] 𝑛+2+𝛼 𝑛+2+𝛼 − ⟨𝑤, 𝑢⟩) (1 (1 − ⟨𝑤, 𝜁⟩)

𝑢(1 − ⟨𝑤, 𝜁⟩)

1/𝑞

𝑑V𝛼 (𝑧)

1/𝑞

𝑑V𝛼 (𝑧) ,

= (𝑛 + 1 + 𝛼) [

∇𝑤 [𝐾𝛼 (𝑤, 𝑢) − 𝐾𝛼 (𝑤, 𝜁)]

𝑛+2+𝛼

(101)

R𝑤 [𝐾𝛼 (𝑤, 𝑢) − 𝐾𝛼 (𝑤, 𝜁)]

where (2𝑄)𝑐 = B𝑛 \ 2𝑄. An immediate computation yields that

= (𝑛 + 1 + 𝛼)

(100)

𝑛

(1 − |𝑤|2 )

= (𝑛 + 1 + 𝛼) [

1/𝑞

2𝑄

𝑛

∫

𝑞

𝛾,𝑞

≤ V𝛼 (2𝑄)1−(1/𝑞) (∫ [𝐴 ∇̃ (𝑃𝛼 𝑎)] 𝑑V𝛼 )

󵄨𝑞 󵄨 𝛾,𝑞 ∫ 󵄨󵄨󵄨󵄨𝐴 ∇̃ (𝑓) (𝑧)󵄨󵄨󵄨󵄨 𝑑V𝛼 B 𝐷(𝑧,𝛾)

for all (1, 𝑞)𝛼 -atoms 𝑎. Given an (1, 𝑞)𝛼 -atom 𝑎 supported in 𝑄 = 𝑄𝑟 (𝜁), by Lemma 20, we have 2𝑄

Proof. Note that V𝛼 (𝐷(𝑧, 𝛾)) ≈ (1 − |𝑧|2 )𝑛+1+𝛼 . Then

B𝑛

(99)

∫ 𝐴 ∇̃ (𝑃𝛼 𝑎) 𝑑V𝛼

where “≈” depends only on 𝑞, 𝛾, 𝛼, and 𝑛.

=∫ ∫

󵄩 󵄩󵄩 𝛾,𝑞 󵄩󵄩𝐴 ̃ (𝑃𝛼 𝑎)󵄩󵄩󵄩 ≤ 𝐶, 󵄩1,𝛼 󵄩 ∇

𝑛+2+𝛼

− 𝜁(1 − ⟨𝑤, 𝑢⟩)

𝑛+2+𝛼

(1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 (1 − ⟨𝑤, 𝜁⟩)

,

⟨𝑤, 𝜁⟩ ⟨𝑤, 𝑢⟩ − ] 𝑛+2+𝛼 𝑛+2+𝛼 (1 − ⟨𝑤, 𝑢⟩) (1 − ⟨𝑤, 𝜁⟩)

= (𝑛 + 1 + 𝛼) ×

⟨𝑤, 𝑢⟩(1 − ⟨𝑤, 𝜁⟩)𝑛+2+𝛼 − ⟨𝑤, 𝜁⟩(1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 (1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 (1 − ⟨𝑤, 𝜁⟩)

𝑛+2+𝛼

.

(102)


11

Moreover,

(cf. [3, Lemma 2.13]). It is concluded that 󵄨 󵄨󵄨 ̃ 󵄨󵄨∇𝑤 [𝐾𝛼 (𝑤, 𝑢) − 𝐾𝛼 (𝑤, 𝜁)]󵄨󵄨󵄨 󵄨 󵄨

󵄨󵄨 󵄨2 󵄨󵄨∇𝑤 [𝐾𝛼 (𝑤, 𝑢) − 𝐾𝛼 (𝑤, 𝜁)]󵄨󵄨󵄨 = (𝑛 + 1 + 𝛼)2

1/2

(𝑛 + 1 + 𝛼) (1 − |𝑤|2 ) ≤ 󵄨𝑛+2+𝛼 󵄨 |1 − ⟨𝑤, 𝑢⟩|𝑛+2+𝛼 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

󵄨2(𝑛+2+𝛼) + |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) |𝑢| 󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 ×{ 󵄨2(𝑛+2+𝛼) 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 2 󵄨󵄨

−

󵄨2(𝑛+2+𝛼) 󵄨 × {(1 − |⟨𝑤, 𝑢⟩|2 ) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

𝑛+2+𝛼

(1 − ⟨𝑢, 𝑤⟩)𝑛+2+𝛼 ⟨𝜁, 𝑢⟩ 󵄨2(𝑛+2+𝛼) 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

(1 − ⟨𝑤, 𝜁⟩)

𝑛+2+𝛼

󵄨2 󵄨 + (1 − 󵄨󵄨󵄨⟨𝑤, 𝜁⟩󵄨󵄨󵄨 ) |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼)

𝑛+2+𝛼

⟨𝑢, 𝜁⟩ (1 − ⟨𝜁, 𝑤⟩) 󵄨2(𝑛+2+𝛼) } , 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 󵄨2 󵄨󵄨 󵄨󵄨R𝑤 [𝐾𝛼 (𝑤, 𝑢) − 𝐾𝛼 (𝑤, 𝜁)]󵄨󵄨󵄨 −

(1 − ⟨𝑤, 𝑢⟩)

󵄨2 󵄨 + [(⟨𝑤, 𝑢 − 𝜁⟩ ⟨𝜁, 𝑤⟩ + (󵄨󵄨󵄨⟨𝑤, 𝜁⟩󵄨󵄨󵄨 − 1) + (1 − ⟨𝜁, 𝑢⟩) ] × (1 − ⟨𝑤, 𝜁⟩)𝑛+2+𝛼 (1 − ⟨𝑢, 𝑤⟩)𝑛+2+𝛼

= (𝑛 + 1 + 𝛼)2 ×{

󵄨2(𝑛+2+𝛼) 󵄨󵄨 󵄨2 󵄨 + 󵄨󵄨⟨𝑤, 𝜁⟩󵄨󵄨󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) |⟨𝑤, 𝑢⟩|2 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 󵄨2(𝑛+2+𝛼) 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 −

−

𝑛+2+𝛼

+ [(⟨𝑤, 𝜁 − 𝑢⟩⟨𝑢, 𝑤⟩ + (|⟨𝑤, 𝑢⟩|2 − 1) + (1 − ⟨𝑢, 𝜁⟩) ] ×(1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 (1 − ⟨𝜁, 𝑤⟩)𝑛+2+𝛼 }

𝑛+2+𝛼

⟨𝑤, 𝑢⟩⟨𝜁, 𝑤⟩(1 − ⟨𝑤, 𝜁⟩) (1 − ⟨𝑢, 𝑤⟩) 󵄨2(𝑛+2+𝛼) 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

1/2

1/2

𝑛+2+𝛼

⟨𝑤, 𝜁⟩⟨𝑢, 𝑤⟩(1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 (1 − ⟨𝜁, 𝑤⟩) 󵄨2(𝑛+2+𝛼) 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

}.

≤

(𝑛 + 1 + 𝛼) (1 − |𝑤|2 )

1/2

(𝑀1 + 𝑀2 + 𝑀3 + 𝑀4 ) 󵄨𝑛+2+𝛼 󵄨 |1 − ⟨𝑤, 𝑢⟩|𝑛+2+𝛼 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

, (106)

(103) where

Then, we have 󵄨󵄨 󵄨2 󵄨 󵄨2 𝛼 𝛼 𝛼 𝛼 󵄨󵄨∇𝑤 [𝐾 (𝑤, 𝑢) − 𝐾 (𝑤, 𝜁)]󵄨󵄨󵄨 − 󵄨󵄨󵄨R𝑤 [𝐾 (𝑤, 𝑢) − 𝐾 (𝑤, 𝜁)]󵄨󵄨󵄨 =

(𝑛 + 1 + 𝛼)2 󵄨2(𝑛+2+𝛼) 󵄨 |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨

󵄨𝑛+2+𝛼 󵄨 𝑀1 = 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 |1 − ⟨𝑢, 𝑤⟩|𝑛+2+𝛼 󵄨 󵄨 × 󵄨󵄨󵄨⟨𝑤, 𝑢 − 𝜁⟩⟨𝜁, 𝑤⟩ + (1 − ⟨𝜁, 𝑢⟩)󵄨󵄨󵄨 , 󵄨𝑛+2+𝛼 󵄨 𝑀2 = |1 − ⟨𝑤, 𝑢⟩|𝑛+2+𝛼 󵄨󵄨󵄨1 − ⟨𝜁, 𝑤⟩󵄨󵄨󵄨 󵄨 󵄨 × 󵄨󵄨󵄨⟨𝑤, 𝜁 − 𝑢⟩⟨𝑢, 𝑤⟩ + (1 − ⟨𝑢, 𝜁⟩)󵄨󵄨󵄨 ,

󵄨2(𝑛+2+𝛼) 󵄨 × {(|𝑢|2 − |⟨𝑤, 𝑢⟩|2 ) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 󵄨2 󵄨 + (1 − 󵄨󵄨󵄨⟨𝑤, 𝜁⟩󵄨󵄨󵄨 ) |1 − ⟨𝑤, 𝑢⟩|2(𝑛+2+𝛼)

󵄨 󵄨 𝑛+2+𝛼 × 󵄨󵄨󵄨󵄨(1 − ⟨𝑤, 𝜁⟩) − (1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 󵄨󵄨󵄨󵄨 ,

+ (⟨𝑤, 𝑢⟩⟨𝜁, 𝑤⟩ − ⟨𝜁, 𝑢⟩)

󵄨2 󵄨 𝑀4 = (1 − 󵄨󵄨󵄨⟨𝑤, 𝜁⟩󵄨󵄨󵄨 ) |1 − ⟨𝑢, 𝑤⟩|𝑛+2+𝛼

× (1 − ⟨𝑤, 𝜁⟩)𝑛+2+𝛼 (1 − ⟨𝑢, 𝑤⟩)𝑛+2+𝛼

󵄨 󵄨 × 󵄨󵄨󵄨󵄨(1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 − (1 − ⟨𝑤, 𝜁⟩)𝑛+2+𝛼 󵄨󵄨󵄨󵄨 ,

+ (⟨𝑤, 𝜁⟩⟨𝑢, 𝑤⟩ − ⟨𝑢, 𝜁⟩) ×(1 − ⟨𝑤, 𝑢⟩)𝑛+2+𝛼 (1 − ⟨𝜁, 𝑤⟩)𝑛+2+𝛼 } . (104) Note that for any 𝑓 ∈ H(B𝑛 ), 󵄨2 󵄨󵄨 ̃ 󵄨󵄨∇𝑓(𝑧)󵄨󵄨󵄨 = (1 − |𝑧|2 ) (󵄨󵄨󵄨󵄨∇𝑓 (𝑧)󵄨󵄨󵄨󵄨2 − 󵄨󵄨󵄨󵄨R𝑓 (𝑧)󵄨󵄨󵄨󵄨2 ) , 󵄨 󵄨

(107)

󵄨𝑛+2+𝛼 󵄨 𝑀3 = (1 − |⟨𝑤, 𝑢⟩|2 ) 󵄨󵄨󵄨1 − ⟨𝜁, 𝑤⟩󵄨󵄨󵄨

for 𝑤 ∈ 𝐷(𝑧, 𝛾), 𝑢 ∈ 𝑄𝑟 (𝜁), and 𝑧 ∈ B𝑛 , 𝜁 ∈ S𝑛 . Hence, ∫

(2𝑄)𝑐

≤∫

𝑧 ∈ B𝑛 (105)

𝛾,𝑞 ∇

𝐴 ̃ (𝑃𝛼 𝑎) 𝑑V𝛼

(2𝑄)

󵄨󵄨𝑞 󵄨̃ 𝛼 𝛼 sup󵄨󵄨󵄨∇ 𝑤 [𝐾 (𝑤, 𝑢) − 𝐾 (𝑤, 𝜁)]󵄨󵄨󵄨 𝑑𝜏(𝑤)) 𝐷(𝑧,𝛾) 𝑢∈𝑄󵄨

(∫ 𝑐

≤ (𝑛 + 1 + 𝛼) ∫(2𝑄)𝑐 (𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 ) 𝑑V𝛼 (𝑧) ,

1/𝑞

𝑑V𝛼 (𝑧)

(108)

12


where

by Lemmas 5 and 7, one has 𝑞/2

𝑞/2

(1 − |𝑤|2 ) 𝑀1 𝑑𝜏(𝑤)) 𝑞(𝑛+2+𝛼) 𝐷(𝑧,𝛾) 𝑢∈𝑄 |1 − ⟨𝑤, 𝑢⟩|𝑞(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 󵄨 󵄨

𝐼1 = (∫

,

sup

2 𝑞/2

𝐼2 = (∫

1/𝑞

sup

𝐷(𝑧,𝛾) 𝑢∈𝑄 |1

𝑞/2 𝑀2

(1 − |𝑤| ) 󵄨 󵄨𝑞(𝑛+2+𝛼) 𝑑𝜏 (𝑤)) − ⟨𝑤, 𝑢⟩|𝑞(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 2 𝑞/2

𝑞/2

(1 − |𝑤| ) 𝑀3 𝑑𝜏(𝑤)) 𝑞(𝑛+2+𝛼) 𝐷(𝑧,𝛾) 𝑢∈𝑄 |1 − ⟨𝑤, 𝑢⟩|𝑞(𝑛+2+𝛼) 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 󵄨 󵄨

𝐼3 = (∫

𝑞/2

𝑞/2

𝐼1 ≤ ( ∫

sup

1/𝑞

,

≤ 𝐶𝛾 ( ∫

(1−|𝑧|2 )

sup

.

𝑞/2

1/𝑞

𝑞/2

(𝐶𝑟|1−⟨𝑧, 𝜁⟩|1/2 +𝑟2 )

𝐷(𝑧,𝛾) 𝑢∈𝑄 |1−⟨𝑧, 𝑢⟩|(𝑞/2)(𝑛+2+𝛼) |1−⟨𝑧, 𝜁⟩|(𝑞/2)(𝑛+2+𝛼)

1/𝑞

𝑞/2

≤ 𝐶𝛾 (

(1 − |𝑧|2 )

1/𝑞

𝑞/2

(𝑟|1 − ⟨𝑧, 𝜁⟩|1/2 + 𝑟2 )

𝑑𝜏 (𝑤) )

)

|1 − ⟨𝑧, 𝜁⟩|𝑞(𝑛+2+𝛼) 1/2

We first estimate 𝐼1 . Note that

≤ 𝐶𝛾

󵄨 󵄨 󵄨 󵄨 𝑀1 ≤ (󵄨󵄨󵄨⟨𝑤, 𝑢 − 𝜁⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨1 − ⟨𝜁, 𝑢⟩󵄨󵄨󵄨) 󵄨𝑛+2+𝛼 󵄨 × 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 |1 − ⟨𝑤, 𝑢⟩|𝑛+2+𝛼

(𝑟|1 − ⟨𝑧, 𝜁⟩|1/2 + 𝑟2 )

|1 − ⟨𝑧, 𝜁⟩|𝑛+(3/2)+𝛼

≤ 𝐶𝛾 (

󵄨1/2 󵄨 󵄨1/2 󵄨 󵄨1/2 󵄨 ≤ (2󵄨󵄨󵄨1 − ⟨𝑢, 𝜁⟩󵄨󵄨󵄨 (󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨1 − ⟨𝑢, 𝜁⟩󵄨󵄨󵄨 )

𝑟1/2 2(𝑛+1+𝛼)+(1/2)

𝑑(𝑧, 𝜁)

+

𝑟 𝑑(𝑧, 𝜁)2(𝑛+1+𝛼)+1

).

(114)

󵄨 󵄨 + 󵄨󵄨󵄨1 − ⟨𝜁, 𝑢⟩󵄨󵄨󵄨 )

Hence,

󵄨𝑛+2+𝛼 󵄨 × 󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 |1 − ⟨𝑤, 𝑢⟩|𝑛+2+𝛼

∫

(2𝑄)𝑐

1 󵄨󵄨 󵄨1/2 󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 ) 2󵄨

+∫

(110)

where the second inequality is the consequence of the following fact which has appeared in the proof of Lemma 14:

󵄨1/2 󵄨 󵄨 󵄨1/2 󵄨 󵄨1/2 ≤ 2󵄨󵄨󵄨1 − ⟨𝑢, 𝜁⟩󵄨󵄨󵄨 (󵄨󵄨󵄨1 − ⟨𝑤, 𝜁⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨1 − ⟨𝑢, 𝜁⟩󵄨󵄨󵄨 ) ;

(111)

the third inequality is obtained by Lemma 7, and the fact 1󵄨 󵄨1/2 󵄨1/2 󵄨󵄨 󵄨󵄨1 − ⟨𝑢, 𝜁⟩󵄨󵄨󵄨 < 𝑟 < 󵄨󵄨󵄨1 − ⟨𝑧, 𝜁⟩󵄨󵄨󵄨 , 2

(112)

for 𝑢 ∈ 𝑄 and 𝑧 ∈ (2𝑄)𝑐 . Since

𝑟 𝑐

𝑑(𝑧, 𝜁)

2(𝑛+1+𝛼)+1

𝑑V𝛼 (𝑧)

(115)

𝑑V𝛼 (𝑧) .

The second term on the right hand side has been estimated in the proof of Lemma 13. The first term can be estimated as follows: 𝑟1/2

∫

𝑑(𝑧,𝜁)>2𝑟

𝑑(𝑧, 𝜁)2(𝑛+1+𝛼)+(1/2)

1

= 𝑟1/2 ∑ ∫

𝑘 𝑘+1 𝑘≥0 2 𝑟 2) with Lipschitz conditions boundary, 𝑎𝑖 and 𝑇𝑖 (𝑖 = 1, 2, . . . , 𝑁) are Carathéodory functions, ℎ𝑖 and 𝑓 satisfy suitable conditions, and 𝜕𝑥𝑖 𝑎𝑖 (𝑥, 𝑢, ∇𝑢) and 𝜕𝑥𝑖 ℎ𝑖 (𝑥) are Einstein Sum; that is, 𝜕𝑥𝑖 𝑎𝑖 (𝑥, 𝑢, ∇𝑢) = 𝑁 ∑𝑁 𝑖=1 (𝜕𝑎𝑖 (𝑥, 𝑢, ∇𝑢)/𝜕𝑥𝑖 ), 𝜕𝑥𝑖 ℎ𝑖 (𝑥) = ∑𝑖=1 (𝜕ℎ𝑖 (𝑥)/𝜕𝑥𝑖 ). We usually use critical theory to obtain the existence of weak solutions. However, since the problem (1) has no variational structure, we cannot define the energy functional for the problem (1). Therefore, based on truncation technique and priori estimates, we prove the existence and uniqueness of 󳨀󳨀󳨀→ 1,𝑝(𝑥)

(Ω). Furtherweak solutions for the problem (1) in 𝑊0 more, we obtain that the weak solution for the problem (1) is locally bounded.

2

Journal of Function Spaces and Applications In particular, for the special case

Now, we recall some results on anisotropic variable expo󳨀󳨀󳨀→

− 𝜕𝑥𝑖 𝑎𝑖 (𝑥, ∇𝑢) = 𝑓 − 𝜕𝑥𝑖 ℎ𝑖 , 𝑢 = 0,

nent Sobolev space 𝑊1,𝑝(𝑥) (Ω) [13]; set

𝑥 ∈ Ω, 𝑥 ∈ 𝜕Ω,

(2)

𝐿∞ + (Ω) := {𝑝 ∈ 𝐿∞ (Ω) : 𝑝 (𝑥) ≥ 1

1,𝛼

we obtain that the weak solution is 𝐶 (Ω). To our knowledge, the above two problems have not been deeply studied in the anisotropic variable exponent Sobolev spaces. The paper is organized as follows. In Section 2, we recall some results on anisotropic variable exponent Sobolev spaces and state our main results. In Section 3, we prove the existence, uniqueness and locally bounded of weak solution for the problem (1). In Section 4, the regularity of weak solutions for the problem (2) is proved.

(8)

for a.e. 𝑥 ∈ Ω} .

󳨀󳨀󳨀→ 𝑁 Denote 𝑝(𝑥) = (𝑝1 (𝑥), 𝑝2 (𝑥), . . . , 𝑝𝑁(𝑥)) ∈ (𝐿∞ + (Ω)) and define ∀𝑥 ∈ Ω, 𝑝− (𝑥) = min {𝑝1 (𝑥) , 𝑝2 (𝑥) , . . . , 𝑝𝑁 (𝑥)} ,

(9)

𝑝+ (𝑥) = max {𝑝1 (𝑥) , 𝑝2 (𝑥) , . . . , 𝑝𝑁 (𝑥)} . The anisotropic variable exponent Sobolev space

2. Preliminary and Main Results This section is dedicated to a general overview on the 󳨀󳨀󳨀→

󳨀󳨀󳨀→

𝑊1,𝑝(𝑥) (Ω)

𝑊1,𝑝(𝑥) (Ω) and 𝐿𝑝(𝑥) (Ω); for a deeper treatment on these spaces, see [5, 7, 8, 13, 14]. Let

= {𝑢 ∈ 𝐿𝑝+ (𝑥) (Ω) : 𝐷𝑖 𝑢 ∈ 𝐿𝑝𝑖 (𝑥) (Ω)

𝐶+ (Ω) := {ℎ : ℎ ∈ 𝐶 (Ω) , ℎ (𝑥) > 1 ∀𝑥 ∈ Ω} .

is a Banach space with respect to the norm

(3)

for 𝑖 = 1, 2, . . . , 𝑁} (10)

𝑁

Set ℎ+ = sup ℎ (𝑥) and ℎ− = inf ℎ (𝑥) , 𝑥∈Ω

𝑥∈Ω

󵄩 󵄩 󳨀󳨀→ = ‖𝑢‖𝑝+ (𝑥) + ∑󵄩󵄩󵄩𝐷𝑖 𝑢󵄩󵄩󵄩𝑝𝑖 (𝑥) . ‖𝑢‖𝑊1,󳨀𝑝(𝑥) (Ω) (4)

󳨀󳨀󳨀→

for any 𝑝(𝑥) ∈ 𝐶+ (Ω), and we define the variable exponent Lebesgue space 𝐿𝑝(⋅) (Ω)

− 1,𝑝(𝑥) (Ω) If 𝑝𝑖 (𝑥) ∈ 𝐿∞ + (Ω), for 𝑖 = 1, 2, . . . , 𝑁, 𝑝𝑖 > 1, then 𝑊 󳨀󳨀󳨀→ 1,𝑝(𝑥)

is reflexive. We define 𝑊0 with respect to the norm ‖𝑢‖

󳨀󳨀→ 1,𝑝(⋅) 𝑊0 (Ω)

= {𝑢 : a measurable real − valued function such that ∫ |𝑢 (𝑥)|𝑝(𝑥) 𝑑𝑥 < ∞} . Ω

(5)

󳨀󳨀󳨀→ 1,𝑝(𝑥)

and 𝑊0 Let

(Ω) as the closure of 𝐶0∞ (Ω)

𝑁󵄩 󵄩󵄩 𝜕𝑢 󵄩󵄩󵄩 󵄩󵄩 = ∑󵄩󵄩󵄩 , 󵄩󵄩 󵄩 𝑖=1 󵄩 𝜕𝑥𝑖 󵄩𝐿𝑝𝑖 (𝑥) (Ω)

𝑁 ∑𝑁 𝑖=1 (1/𝑝𝑖 (𝑥))

,

{ 𝑁𝑝 (𝑥) if 𝑝 (𝑥) < 𝑁, ∗ 𝑝 (𝑥) = { 𝑁 − 𝑝 (𝑥) if 𝑝 (𝑥) ≥ 𝑁, { +∞

(6)

From [6], we have the following:

(12)

(Ω) is a reflexive Banach space (see [9]). 𝑝 (𝑥) =

We define the norm of 𝐿𝑝(𝑥) (Ω): 󵄨󵄨 𝑢 (𝑥) 󵄨󵄨𝑝(𝑥) 󵄨 󵄨󵄨 ‖𝑢‖𝐿𝑝(𝑥) (Ω) = inf {𝜇 > 0 ; ∫ 󵄨󵄨󵄨 󵄨󵄨 𝑑𝑥 ≤ 1} . 󵄨 󵄨󵄨 𝜇 Ω󵄨

(11)

𝑖=1

(13)

∗

𝑝∞ (𝑥) = max {𝑝 (𝑥) , 𝑝+ (𝑥)} .

(1) (𝐿𝑝(𝑥) (Ω), ‖ ⋅ ‖𝑝(𝑥) ) is a Banach space; (2) 𝐿𝑝(𝑥) (Ω) is reflexive, if and only if 1 < 𝑝− ≤ 𝑝+ < ∞;

Hence, we have the following embedding theorem for

(3) Hölder inequality.

𝑊0

𝑝(𝑥)

For all 𝑢, V ∈ 𝐿

󸀠

(Ω).

󳨀󳨀→ Let Ω ⊂ R𝑁 be a bounded domain and 𝑝(⋅) = (𝑝1 (⋅), 𝑁 𝑝2 (⋅), . . . , 𝑝𝑁(⋅)) ∈ (𝐶+0 (Ω)) .

(Ω),

󵄨 󵄨󵄨 󵄨󵄨∫ 𝑢V 𝑑𝑥󵄨󵄨󵄨 ≤ ( 1 + 1 ) ‖𝑢‖ 𝑝(𝑥) ‖V‖ 󸀠 , 󵄨󵄨 󵄨󵄨 𝐿 (Ω) 𝐿𝑝 (𝑥) (Ω) 𝑝− (𝑝󸀠 )− 󵄨 󵄨 Ω

󳨀󳨀󳨀→ 1,𝑝(𝑥)

(7)

where 1/𝑝(𝑥) + 1/𝑝󸀠 (𝑥) = 1, 𝐿𝑝 (𝑥) (Ω) is the conjugate space of 𝐿𝑝(𝑥) (Ω).

Theorem 1 (see [13, Theorem 2.5]). (i) If 𝑞 ∈ 𝐶+0 (Ω) and 𝑞 (𝑥) < 𝑝∞ (𝑥)

∀𝑥 ∈ Ω,

(14)

Journal of Function Spaces and Applications 󳨀󳨀→ 1,𝑝(⋅)

then 𝑊0 compact.

3 (T1) for a.e. 𝑥 ∈ Ω, for all 𝜉 ∈ R𝑁, 𝑠 ∈ R,

(Ω) 󳨅→󳨅→ 𝐿𝑞(⋅) (Ω). The embedding is

(ii) If 𝑝(𝑥) > 𝑁 for all 𝑥 ∈ Ω, then there exists 𝛽 ∈ (0, 1) 󳨀󳨀→ 1,𝑝(⋅) 𝑊0 (Ω)

such that

󳨅→󳨅→ 𝐶0,𝛽 (Ω).

∀𝑥 ∈ Ω.

󸀠

(F2) for a.e. 𝑥 ∈ Ω, 𝑠 ∈ R, for some 𝑘 ∈ {1, 2, . . . , 𝑁}, |𝑓𝑥𝑘 | ≤ 𝑐6 |𝑠|𝑞(𝑥)/2 , where 𝑓𝑥𝑘 = 𝜕𝑓/𝜕𝑥𝑘 ; 󸀠

󳨀󳨀→ 1,𝑝(⋅)

∀𝑢 ∈ 𝑊0

(Ω) ,

(16)

(H2) for a.e. 𝑥 ∈ Ω, 𝑖 = 1, 2, . . . , 𝑁, for some 𝑘 ∈ {1, 2, . . . , 𝑁}, 𝜁 ∈ R𝑁, 𝑠 ∈ R, 𝑁

where 𝐶 is a positive constant independent of 𝑢 ∈

󵄨 󵄨2 󵄨 󵄨2 𝑐7 󵄨󵄨󵄨𝜁󵄨󵄨󵄨 ≤ ∑ℎ𝑥𝑖 𝑘 𝜁𝑖 ≤ 𝑐8 (󵄨󵄨󵄨𝜁󵄨󵄨󵄨 + |𝑠|𝑞(𝑥) ) ,

󳨀󳨀→ 1,𝑝(⋅) 𝑊0 (Ω).

where ℎ𝑥𝑖 𝑘 = 𝜕ℎ𝑖 /𝜕𝑥𝑘 , 𝑐6 , 𝑐7 , 𝑐8 are positive constants.

∗

󳨀󳨀→ 1,𝑝(⋅)

𝑢 ∈ 𝑊0 (Ω), for every 𝑖 = 1, 2, . . . , 𝑁, then ||𝑢||𝐿𝑝𝑖 (𝑥) (Ω) ≤ 𝐶||𝜕𝑥𝑖 𝑢||𝐿𝑝𝑖 (𝑥) (Ω) , where 𝐶 is a positive constant independent 󳨀󳨀→ 1,𝑝(⋅)

of 𝑢 ∈ 𝑊0 (Ω). Assume that 𝑎𝑖 : Ω×R×R𝑁 → R and 𝑇𝑖 : Ω×R𝑁 → R are Carathéodory functions and satisfy the following: (A1) for a.e. 𝑥 ∈ Ω, for all 𝜉 ∈ R𝑁, 𝑠 ∈ R, we have

𝑖=1

󵄨 󵄨𝑝 (𝑥) 𝜆∑󵄨󵄨󵄨𝜉𝑖 󵄨󵄨󵄨 𝑖 ≤ ∑𝑎𝑖 (𝑥, 𝑠, 𝜉) 𝜉𝑖 𝑝∞ (𝑥)/𝑝𝑖󸀠 (𝑥)

and |𝑎𝑖 (𝑥, 𝑠, 𝜉)| ≤ 𝛽[|𝑠| 𝜆, 𝛽 > 0;

], where

𝜉𝑖󸀠 , 𝛾 > 0 and 𝜀 > 0, 𝑎𝑖 (A2) for a.e. 𝑥 ∈ Ω, for all 𝜉𝑖 ≠ satisfies [𝑎𝑖 (𝑥, 𝑠, 𝜉) − 𝑎𝑖 (𝑥, 𝑠, 𝜉 )] (𝜉𝑖 −

𝜉𝑖󸀠 )

󵄨 󵄨 󵄨 󵄨 𝑝𝑖 (𝑥)−2 󵄨󵄨󵄨 󸀠 󵄨󵄨2 ≥ 𝛾(𝜀 + 󵄨󵄨󵄨𝜉𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜉𝑖󸀠 󵄨󵄨󵄨󵄨) 󵄨󵄨𝜉𝑖 − 𝜉𝑖 󵄨󵄨󵄨 ;

(18)

(A3) for a.e. 𝑥 ∈ Ω, for all 𝜉 ∈ R𝑁, 𝜁 ∈ R𝑁, 𝑖, 𝑗 = 1, 2, . . . , 𝑁, 𝑁 𝑁

𝑁

𝑖=1 𝑗=1

𝑖=1

󵄨 󵄨2 󵄨 󵄨𝑝 (𝑥)−2 󵄨󵄨 󵄨󵄨2 𝑐1 󵄨󵄨󵄨𝜁󵄨󵄨󵄨 ≤ ∑ ∑𝑎𝜉𝑖 𝑗 (𝑥, 𝑠, 𝜉) 𝜁𝑖 𝜁𝑗 ≤ 𝑐2 (1 + ∑󵄨󵄨󵄨𝜉𝑖 󵄨󵄨󵄨 𝑖 ) 󵄨󵄨𝜁󵄨󵄨 , (19) where 𝑎𝜉𝑖 𝑗 (𝑥, 𝑠, 𝜉) = 𝜕𝑎𝑖 (𝑥, 𝑠, 𝜉)/𝜕𝜉𝑗 ; (A4) for a.e. 𝑥 ∈ Ω, 𝑖 = 1, 2, . . . , 𝑁, for some 𝑘 ∈ {1, 2, . . . , 𝑁}, for all 𝜁 ∈ R𝑁, 𝑠 ∈ R, 𝑁

󵄨 󵄨2 󵄨 󵄨2 𝑐3 󵄨󵄨󵄨𝜁󵄨󵄨󵄨 ≤ ∑𝑎𝑥𝑖 𝑘 (𝑥, 𝑠, 𝜉) 𝜁𝑖 ≤ 𝑐4 (󵄨󵄨󵄨𝜁󵄨󵄨󵄨 + |𝑠|𝑞(𝑥) ) , 𝑖=1

where 𝑎𝑥𝑖 𝑘 (𝑥,𝑠,𝜉) = 𝜕𝑎𝑖 (𝑥, 𝑠, 𝜉)/𝜕𝑥𝑘 ;

Now, we define the weak solution of the problem (1). A 󳨀󳨀󳨀→ 1,𝑝(𝑥)

function 𝑢 ∈ 𝑊0

+ |𝜉𝑖 |

󸀠

Remark 4. Now, we give a simple example. Let 𝑘 = 1, 𝑥0 = (3, 0), 𝑥 = (𝑥1 , 𝑥2 ) ∈ Ω = 𝐵(𝑥0 , 1) ⊂ R2 , 𝑢(𝑥) = 𝑥12 + 𝑥22 , 𝜁 = 𝑥13/2 +𝑥23/2 , ℎ1 (𝑥) = 𝑥12 +2𝑥1 𝑥2 , ℎ2 (𝑥) = (1/2)𝑥12 +𝑥1 𝑥2 +𝑥22 , 𝑇1 (𝑥, 𝜉) = 𝑥1 + 𝑥2 + ∑2𝑖=1 𝜉𝑖 , 𝑇2 (𝑥, 𝜉) = 𝑥1 𝑥2 + |𝜉|2 , 𝑓(𝑥) = (1/2)𝑥12 + 𝑥1 𝑥2 . By a simple calculation, we obtain that 𝑇𝑖 satisfies (T1), 𝑓 satisfies (F2), and ℎ𝑖 satisfies (H2), where 𝑖 = 1, 2.

(17) 𝑝𝑖 (𝑥)−1

(20)

(22)

𝑖=1

Remark 3. From [13], we know that if 𝑝+ (𝑥) < 𝑝 (𝑥), for all

𝑖=1

(21)

(H1) for 𝑖 = 1, 2, . . . , 𝑁, ℎ𝑖 ∈ 𝐿𝑝𝑖 (𝑥) (Ω);

𝑁󵄩 󵄩󵄩 𝜕𝑢 󵄩󵄩󵄩 󵄩󵄩 ‖𝑢‖𝐿𝑝+ (⋅) (Ω) ≤ 𝐶∑󵄩󵄩󵄩 󵄩󵄩 󵄩 𝑖=1 󵄩 𝜕𝑥𝑖 󵄩𝐿𝑝𝑖 (⋅) (Ω)

𝑁

𝑖=1

(F1) 𝑓 ∈ 𝐿𝑝∞ (𝑥) (Ω);

(15)

Then one has

𝑁

𝑖=1

We enumerate the hypotheses concerning 𝑓 and ℎ𝑖 ,

Theorem 2 (see [13, Theorem 2.6]). Let Ω ⊂ R𝑁 be a bounded 󳨀󳨀→ 𝑁 domain and 𝑝(⋅) = (𝑝1 (⋅), 𝑝2 (⋅), . . . , 𝑝𝑁(⋅)) ∈ (𝐶+0 (Ω)) . Suppose that 𝑝+ (𝑥) < 𝑝 (𝑥)

𝑁

󸀠 (𝑥) − where 𝑚𝑖 ∈ 𝐿𝑏𝑖 (𝑥) (Ω) with 1/𝑏𝑖 (𝑥) = 1/𝑝∞ 󸀠 1/𝑝𝑖 (𝑥), 𝑐1 , . . . , 𝑐5 are positive constants.

The embedding is also compact.

∗

𝑁

󵄨 󵄨 󵄨 󵄨𝑝 (𝑥)−1 , ∑ 󵄨󵄨󵄨𝑇𝑖 (𝑥, 𝜉)󵄨󵄨󵄨 ≤ 𝑐5 |𝑠|𝑞(𝑥)/2 ≤ ∑𝑚𝑖 󵄨󵄨󵄨𝜉𝑖 󵄨󵄨󵄨 𝑖

(1), if for all 𝜑 ∈

(Ω) is a weak solution of the problem

󳨀󳨀󳨀→ 1,𝑝(𝑥) (Ω), 𝑊0

𝑁

𝑁

∑ ∫ [𝑎𝑖 (𝑥, 𝑢, ∇𝑢) 𝜑𝑥𝑖 + 𝑇𝑖 (𝑥, ∇𝑢) 𝜑] = ∫ [𝑓𝜑 + ∑ℎ𝑖 𝜑𝑥𝑖 ] , 𝑖=1 Ω

Ω

𝑖=1

(23) where 𝜑𝑥𝑖 = 𝜕𝜑/𝜕𝑥𝑖 . Theorem 5. Let Ω ⊂ R𝑁 (𝑁 > 2) be a bounded open subset, for every 𝑖 = 1, 2, . . . , 𝑁, 𝑥 ∈ Ω, assume 𝑝(𝑥) < 𝑁, 𝑞 ∈ 𝐶+0 (Ω) ∗ and 𝑞(𝑥) < 𝑝∞ = 𝑝 (𝑥), 𝑝𝑖 (𝑥) satisfies 2 ≤ 𝑝𝑖 (𝑥) ≤

2𝑁𝑟𝑖 (𝑥) 𝑁𝑟𝑖 (𝑥) − 2𝑟𝑖 (𝑥) − 2𝑁

(24)

and ∑𝑁 𝑖=1 (1/𝑝𝑖 (𝑥)) < 2, where 𝑟𝑖 (𝑥) ≥ max{𝑁, 𝑝∞ (𝑥)𝑝𝑖 (𝑥)/ (𝑝∞ (𝑥) − 𝑝𝑖 (𝑥))}. 𝑎𝑖 satisfies the hypotheses (A1), (A2), the hypotheses (F1) and (H1) hold, and 𝑇𝑖 satisfies (T1) and the following Lipschitz condition: 󵄨 󵄨 𝛿 (𝑥) 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝑇𝑖 (𝑥, 𝜉) − 𝑇𝑖 (𝑥, 𝜉󸀠 )󵄨󵄨󵄨 ≤ 𝑘𝑖 (𝑥) (󵄨󵄨󵄨󵄨𝜉𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉𝑖󸀠 󵄨󵄨󵄨) 𝑖 󵄨󵄨󵄨𝜉𝑖 − 𝜉𝑖󸀠 󵄨󵄨󵄨 , 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨

(25)

where 𝑘𝑖 (𝑥) ∈ 𝐿𝑟𝑖 (𝑥) (Ω), 0 ≤ 𝛿𝑖 (𝑥) ≤ 𝑝𝑖 (𝑥)/𝑁 − 𝑝𝑖 (𝑥)/𝑟𝑖 (𝑥) + (𝑝𝑖 (𝑥) − 2)/2. Then there exists a unique weak solution 𝑢 for the problem (1). Furthermore, 𝑢 ∈ 𝐿∞ loc (Ω).

4


Theorem 6. Let Ω ⊂ R𝑁 (𝑁 > 2) be a bounded open subset. Suppose (A3), (A4) and (F2), (H2) hold, for 𝑖 = 1, 2, . . . , 𝑁, ∗ 2 ≤ 𝑝𝑖 (𝑥) ≤ 2𝑁/(𝑁 − 2), 𝑞 ∈ 𝐶+0 (Ω) and 𝑞(𝑥) < 𝑝 (𝑥). If 𝑢 is a weak solution of the problem (2), then 𝑢 ∈ 𝐶1,𝛼 (Ω), for all 0 < 𝛼 < 1.

𝑁 𝑡

≤ 𝑐2 ∑ ∑

𝑖=1 𝑠=1 + 󵄨 󵄨𝑝𝑖 (𝑥) 󵄨󵄨 󵄨󵄨1/𝑝𝑖− −1/𝑝∞ × [∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑠 󵄨󵄨󵄨󵄨 󵄨󵄨Ω𝑠 󵄨󵄨 Ω 𝑠

𝑝+ /𝑁 󵄨󵄨

𝑁

We consider the following problem: 𝑁

−𝜕𝑥𝑖 𝑎𝑖 (𝑥, V𝑛 , ∇V𝑛 ) + ∑𝑇𝑛𝑖 (𝑥, ∇V𝑛 ) = 𝑓 − 𝜕𝑥𝑖 ℎ𝑖 , 𝑖=1

𝑢 = 0,

−

𝑖=1

𝑥 ∈ Ω,

󵄨 󵄨𝑝𝑖 (𝑥) × [ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨󵄨 Ω

𝑥 ∈ 𝜕Ω,

𝑡

where 𝑇𝑛𝑖 (𝑥, ∇V) is the truncation at levels ±𝑛 of 𝑇𝑖 . Due to [15], we obtain that there exists a weak solution V𝑛 ∈ 󳨀󳨀󳨀→ 1,𝑝(𝑥)

𝑡−1 󵄨 󵄨𝑝𝑖 (𝑥) 𝑝+ /𝑁 + ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑠 󵄨󵄨󵄨󵄨 + 𝐴𝑡+ ] , Ω 𝑠=1

(Ω) for the problem (26). 󳨀󳨀󳨀→ 1,𝑝(𝑥)

(Ω) is a weak solution of the problem (26);

𝑁 󵄨 󵄨𝑝𝑖 (𝑥) ≤ 𝐶. ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑛 󵄨󵄨󵄨󵄨 Ω

Proof. Let 𝐾 ∈ R+ ; 𝐾 will be chosen later. There exist 𝑚 measurable subsets Ω1 , . . . , Ω𝑚 of Ω and 𝑚 functions 𝑗, Ω𝑖 ⋂ Ω𝑗 = Ø. For 𝑡 ∈ {1, 2, . . . , 𝑚− V1 , . . . , V𝑚 such that if 𝑖 ≠ 0} ⊂ Ω𝑡 , 1}, |Ω𝑡 | = 𝐾, if 𝑡 = 𝑚, |Ω𝑡 | ≤ 𝐾. Let {𝑥 ∈ Ω : |∇V𝑡 | ≠ ∇V = ∇V𝑡 a.e. in Ω𝑡 , ∇(V1 + V2 + ⋅ ⋅ ⋅ + V𝑡 )V𝑡 = (∇V)V𝑡 , 0, V1 + V2 + ⋅ ⋅ ⋅ + V𝑡 = V in Ω. For 𝑡 ∈ {1, 2, . . . , 𝑚}, if V𝑡 ≠ sign(V) = sign(V𝑡 ). Choose V𝑡 as test function of the problem (26) and fix 𝑡 ∈ {1, 2, . . . , 𝑚}. Using Young inequality, Hölder inequality, the 󳨀󳨀󳨀→ 1,𝑝(𝑥)

embedding theorem of the 𝑊0 (A1), (F1), and (H1), we have

+ where 𝑝𝑖− = inf 𝑥∈Ω 𝑝𝑖 (𝑥), 𝑝++ = sup𝑥∈Ω 𝑝+ (𝑥), 𝑝∞ = sup𝑥∈Ω 𝑝∞ (𝑥). Combining (28) with (29), we have

(27)

𝑖=1

𝑁

𝑁 󵄨 󵄨𝑝𝑖 (𝑥) ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨󵄨 Ω 𝑖=1

𝑁

󵄨 󵄨𝑝󸀠 (𝑥) 󵄩 󵄩 󸀠 1/𝑁 ≤ 𝑐1 {󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝∞ + ∑ ∫ 󵄨󵄨󵄨ℎ𝑖 󵄨󵄨󵄨 𝑖 (𝑥) 𝐴 𝑡 Ω 𝑖=1

𝑁

−

+

+ 𝑐2 ∑𝐾1/𝑝𝑖 −1/𝑝∞

(30)

𝑖=1

󵄨 󵄨𝑝𝑖 (𝑥) × [ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨󵄨 Ω

(Ω), and the hypotheses

𝑡

𝑡−1 󵄨 󵄨𝑝𝑖 (𝑥) 𝑝+ /𝑁 + ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑠 󵄨󵄨󵄨󵄨 + 𝐴 𝑡 + ]} . Ω

󵄨󵄨 󵄨𝑝 (𝑥) 󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨 𝑖 󵄨 󵄨 Ω

∑∫ 𝑖=1

𝑠

(29)

Lemma 7. If 𝑝(𝑥) < 𝑁, (A1), (A2), (T1), (F1), and (H1) hold. Assume V𝑛 ∈ 𝑊0 then one has

+

≤ 𝑐2 ∑𝐾1/𝑝𝑖 −1/𝑝∞

(26)

𝑊0

+ 󵄨1/𝑝− −1/𝑝∞ ] 󵄨󵄨Ω𝑠 󵄨󵄨󵄨 𝑖

+ 𝐴𝑡+

3. The Proof of Theorem 5

𝑠=1

𝑁

󵄨 󵄨󵄨 𝑖 󵄨󵄨𝑇 (𝑥, ∇V)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨V𝑡 󵄨󵄨󵄨󵄨 󵄨 󵄨 Ω

󵄩 󵄩 󸀠 1/𝑁 ≤ 𝑐1 (󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝∞ + ∑∫ (𝑥) 𝐴 𝑡 𝑖=1

(28)

𝑠

−

+

1/𝑝𝑖 −1/𝑝∞ < 1; then we have Let 𝐾 satisfy 𝑐1 𝑐2 ∑𝑁 𝑖=1 𝐾

𝑁

󵄨 󵄨𝑝󸀠 (𝑥) +∑ ∫ 󵄨󵄨󵄨ℎ𝑖 󵄨󵄨󵄨 𝑖 ) , Ω 𝑖=1

where 𝑐1 > 0 is a constant, 𝐴 𝑡 = ∏𝑁 𝑖=1 ||𝜕𝑥𝑖 V𝑡 ||𝐿𝑝𝑖 (𝑥) (Ω) . By (T1), ̈ inequality, we obtain, for 𝑐2 > 0, Young inequality and Holder 󵄨󵄨 󵄨󵄨 𝑁 󵄨 󵄨󵄨 󵄨󵄨∑ ∫ 𝑇𝑖 (𝑥, ∇V) V𝑡 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑖=1 Ω 𝑁 󵄨 󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨 󵄨󵄨 ≤ ∑𝑚𝑖 ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨 󵄨󵄨V𝑡 󵄨󵄨 Ω 𝑖=1

𝑁 𝑡 󵄨 󵄨𝑝𝑖 (𝑥)−1 󵄨󵄨 󵄨󵄨 ≤ ∑𝑚𝑖 ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑠 󵄨󵄨󵄨󵄨 󵄨󵄨V𝑡 󵄨󵄨 Ω 𝑖=1

𝑠=1

𝑠

𝑁 󵄨 󵄨𝑝𝑖 (𝑥) ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨󵄨 Ω 𝑖=1

𝑁

󵄨 󵄨𝑝󸀠 (𝑥) 󵄩 󵄩 󸀠 1/𝑁 ≤ 𝑐3 {󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝∞ + ∑ ∫ 󵄨󵄨󵄨ℎ𝑖 󵄨󵄨󵄨 𝑖 (𝑥) 𝐴 𝑡 Ω 𝑖=1

𝑡−1 𝑁

+ ∑ ∑𝐾

+ 1/𝑝𝑖− −1/𝑝∞

𝑠=1 𝑖=1 𝑁

󵄨 󵄨𝑝𝑗 (𝑥) ( ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑠 󵄨󵄨󵄨󵄨 ) Ω 𝑗=1

−

+

𝑝+ /𝑁

+ ∑𝐾1/𝑝𝑖 −1/𝑝∞ 𝐴 𝑡 + 𝑖=1

(31)

𝑁

},

𝑠


5

where 𝑐3 > 0 is a constant. Let 𝑡 = 1; we obtain

From (38), we obtain

󵄨 󵄨𝑝𝑖 (𝑥) ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V1 󵄨󵄨󵄨󵄨 Ω 𝑁

𝑎𝑖 (𝑥, 𝑢𝑛 , ∇𝑢𝑛 ) 󳨀→ 𝑎𝑖 (𝑥, 𝑢, ∇𝑢) 𝑇𝑛𝑖 (𝑥, ∇𝑢𝑛 ) 󳨀→ 𝑇𝑛𝑖 (𝑥, ∇𝑢)

󵄨󵄨 󵄨𝑝 (𝑥) 󵄨󵄨𝜕𝑥𝑖 V1 󵄨󵄨󵄨 𝑖 󵄨 󵄨

≤ ∑∫

𝑖=1 Ω

(32)

𝑁

󵄨 󵄨𝑝󸀠 (𝑥) 󵄩 󵄩 󸀠 1/𝑁 ≤ 𝑐3 [󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝∞ + ∑ ∫ 󵄨󵄨󵄨ℎ𝑖 󵄨󵄨󵄨 𝑖 (𝑥) 𝐴 1 Ω 𝑖=1

𝑁

+ ∑𝐾

+ 1/𝑝𝑖− −1/𝑝∞

𝑖=1

𝑁

𝑖=1

≤ 𝑐 [(∫ |𝑢|𝑝∞ (𝑥) )

𝜅𝑖 (𝑥)/𝑝𝑖󸀠 (𝑥)

Ω

1 𝑁 ∑𝑎 . 𝑁 𝑖=1 𝑖

(33)

󸀠

− 󸀠− 󵄨󵄨 󵄨𝑝 (𝑥) 𝜅𝑖 (𝑥)/𝑝𝑖 (𝑥) (𝑝󸀠+ 𝑖 −𝜅𝑖 )/𝑝𝑖 󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨 𝑖 ) ] , |Ω| 󵄨 󵄨 Ω

+ (∫ ∫

Ω

(40)

󵄨󵄨 𝑖 󵄨𝜅 (𝑥) 󵄨󵄨𝑇𝑛 (𝑥, ∇𝑢𝑛 )󵄨󵄨󵄨 𝑖 󵄨 󵄨 󸀠

− 󸀠− 󵄨 󵄨𝑝𝑖 (𝑥) 𝜅𝑖 (𝑥)/𝑝𝑖 (𝑥) (𝑝󸀠+ ≤ 𝑐[∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 ] |Ω| 𝑖 −𝜅𝑖 )/𝑝𝑖 . Ω

𝑁

𝑁

1 𝑁󵄩 󵄩 󵄩 󵄩 𝐴 1 = ∏󵄩󵄩󵄩󵄩𝜕𝑥𝑖 V1 󵄩󵄩󵄩󵄩𝐿𝑝𝑖 (𝑥) (Ω) ≤ ( ∑󵄩󵄩󵄩󵄩𝜕𝑥𝑖 V1 󵄩󵄩󵄩󵄩𝐿𝑝𝑖 (𝑥) (Ω) ) . 𝑁

(34)

𝑖=1

Combining (32) with (34), there exists a constant 𝑐4 such that 𝑁

󵄨 󵄨𝑝𝑖 (𝑥) ≤ 𝑐4 . ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V1 󵄨󵄨󵄨󵄨 Ω

By Vitali Theorem, we have 𝑎𝑖 (𝑥, 𝑢𝑛 , ∇𝑢𝑛 ) 󳨀→ 𝑎𝑖 (𝑥, 𝑢, ∇𝑢) 𝑇𝑛𝑖 (𝑥, ∇𝑢𝑛 ) 󳨀→ 𝑇𝑖 (𝑥, ∇𝑢)

strongly in 𝐿𝜅𝑖 (𝑥) (Ω) , strongly in 𝐿𝜅𝑖 (𝑥) (Ω) , (41)

(35)

𝑖=1

for any 𝜅𝑖 (𝑥) ∈ [1, 𝑝𝑖󸀠 (𝑥)]. Hence, we complete the proof of Lemma 8.

Furthermore, put (35) in (31) and iterate on 𝑡; we have

∗

𝑁 󵄨 󵄨𝑝𝑖 (𝑥) ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨󵄨 Ω

Lemma 9. If 𝑝(𝑥) < 𝑁, 𝑝∞ (𝑥) = 𝑝 (𝑥), 𝑎𝑖 satisfies the hypotheses (A1), (A2), the hypotheses (F1) and (H1) hold, for 𝑖 = 1, 2, . . . , 𝑁, 𝑥 ∈ Ω, 𝑝𝑖 (𝑥) satisfies

𝑖=1

𝑁

󵄨 󵄨𝑝󸀠 (𝑥) 󵄩 󵄩 󸀠 1/𝑁 ≤ 𝑐5 [󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝∞ + ∑ ∫ 󵄨󵄨󵄨ℎ𝑖 󵄨󵄨󵄨 𝑖 (𝑥) 𝐴 𝑡 Ω

(36)

2 ≤ 𝑝𝑖 (𝑥) ≤

𝑖=1

𝑁

−

+

𝑝+ /𝑁

+ 1 + ∑𝐾1/𝑝𝑖 −1/𝑝∞ 𝐴 1+ Therefore, we obtain 𝑁

󵄨 󵄨𝑝𝑖 (𝑥) ≤ 𝑐6 , ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V𝑡 󵄨󵄨󵄨󵄨 Ω

(37)

𝑖=1

for some constant 𝑐6 > 0. Lemma 8. If 𝑝(𝑥) < 𝑁, and 𝑎𝑖 satisfies the hypotheses (A1), (A2), the hypotheses (T1), (F1), and (H1) hold. Then there exists at least a weak solution for the problem (1). Proof. By (27), we obtain that there exists a sequence 𝜕𝑥𝑖 𝑢𝑛 which is bounded in 𝐿𝑝𝑖 (𝑥) (Ω); we have for 𝑖 = 1, 2, . . . , 𝑁, 𝜕𝑥𝑖 𝑢𝑛 ⇀ 𝜕𝑥𝑖 𝑢 weakly in 𝐿𝑝𝑖 (𝑥) (Ω) and 𝑢𝑛 → 𝑢 strongly in 𝐿𝑝𝑖 (𝑥) (Ω). We argue as in [16] to prove a.e. in Ω for 𝑖 = 1, 2, . . . , 𝑁.

2𝑁𝑟𝑖 (𝑥) , 𝑁𝑟𝑖 (𝑥) − 2𝑟𝑖 (𝑥) − 2𝑁

(42)

where 𝑟𝑖 (𝑥) ≥ max{𝑁, 𝑝∞ (𝑥)𝑝𝑖 (𝑥)/(𝑝∞ (𝑥) − 𝑝𝑖 (𝑥))}, 𝑇𝑖 satisfies (T1) and (25). Then there exists a unique weak solution for the problem (1).

].

𝑖=1

𝜕𝑥𝑖 𝑢𝑛 󳨀→ 𝜕𝑥𝑖 𝑢

(39)

󵄨 󵄨𝜅 (𝑥) ∫ 󵄨󵄨󵄨𝑎𝑖 (𝑥, 𝑢𝑛 , ∇𝑢𝑛 )󵄨󵄨󵄨 𝑖 Ω

𝑝+ /𝑁 𝐴 1+ ] .

From (33), we have

𝑖=1

a.e. in Ω.

By the hypotheses (A1) and (T1), for 𝜅𝑖 (𝑥) ∈ [1, 𝑝𝑖󸀠 (𝑥)] and 𝑐 > 0, we have

Now, we recall the following classical inequality, for 𝑎1 , 𝑎2 , . . . , 𝑎𝑁 are positive numbers, ∏𝑎𝑖1/𝑁 ≤

a.e. in Ω,

(38)

Proof. Suppose that there exist two distinct weak solutions 𝑢 and V for the problem (1). Denote 𝜑 = max{𝑢 − V, 0}, and for 𝑡 ∈ [1, sup 𝜑], let Ω𝑡 = {𝑥 ∈ Ω : 𝑡 < 𝜑 < sup 𝜑}. Otherwise, if 𝑡 ∈ [0, 1), we choose 𝑚 = 1/𝑡 and let Ω𝑚 = {𝑥 ∈ Ω : 𝑚 < 𝜑 < sup 𝜑}. We choose 𝜑𝑡 = {

𝜑 − 𝑡, 𝜑 > 𝑡, 0, otherwise,

(43)

as test function. From (A2) and (25), we have 𝑁 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 𝑝𝑖 (𝑥)−2 ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝜑𝑡 󵄨󵄨󵄨󵄨 (𝜀 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) Ω 𝑖=1

𝑡

1𝑁 󵄨 󵄨 󵄨 󵄨 󵄨 𝛿𝑖 (𝑥) 󵄨 ≤ ∑ ∫ 𝑘𝑖 (𝑥) (󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝜑𝑡 󵄨󵄨󵄨󵄨 𝜑𝑡 𝛾 𝑖=1 Ω𝑡

(44)

6


when 𝛿+ (𝑥) ≥ (𝑝𝑖 (𝑥) − 2)/2. By Young inequality, for 𝑐 independent on 𝑡, we obtain 𝑁 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 𝑝𝑖 (𝑥)−2 ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝜑𝑡 󵄨󵄨󵄨󵄨 (𝜀 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) Ω 𝑖=1

𝑡

(45)

𝑁

󵄨 󵄨 󵄨 󵄨 2𝛿+ (𝑥)−𝑝𝑖 (𝑥)+2 2 ≤ 𝑐∑ ∫ 𝑘𝑖 (𝑥)2 (󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) 𝜑𝑡 . 𝑖=1 Ω𝑡

Lemma 10. If (A1), (T1), (F1), and (H1) hold, for 𝑖 = 1, 2, . . ., 0 𝑁, 2 ≤ 𝑝𝑖 (𝑥) and ∑𝑁 𝑖=1 (1/𝑝𝑖 (𝑥)) < 2, 𝑞 ∈ 𝐶+ (Ω) and 𝑞(𝑥) < ∗ 𝑝∞ = 𝑝 (𝑥), and 𝑢 is a weak solution of the problem (1), then 𝑢 ∈ 𝐿∞ loc (Ω). 󳨀󳨀󳨀→ 1,𝑝(𝑥)

Proof. We denote 𝐸 = 𝑊0 (Ω). Let 𝑡 > 0, 𝜑 = max{𝑢 − 𝑡, 0}, and choose 𝜑 as test function to the problem (1). We have 𝑁

∑ ∫ [𝑎𝑖 (𝑥, 𝑢, ∇𝑢) 𝜑𝑥𝑖 + 𝑇𝑖 (𝑥, ∇𝑢) 𝜑]

Then we have

𝑖=1 Ω

2/2∗

1 (∫ 𝜑2 ) 𝐶 Ω𝑡 𝑡 ∗

= ∫ [𝑓𝜑 + ∑ℎ𝑖 𝜑𝑥𝑖 ] . Ω

1/𝑁

𝑁

󵄨 󵄨2 ≤ ∏(∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝜑𝑡 󵄨󵄨󵄨󵄨 ) Ω 𝑖=1

𝑖=1

𝑁

∑∫

𝑡

𝑁

𝑖=1

Let 𝑥0 ∈ Ω, fix 𝑘 > 1, 𝐵(𝑥0 , 𝑅) be a ball, and let 𝛿 ∈ (0, 1), 𝛿𝑅 < 𝑠 < 𝑡 < 𝑅, and 𝐴 𝑘,𝑡 = {𝑥 ∈ 𝐵(𝑥0 , 𝑡), 𝑢(𝑥) > 𝑘}. Using ̈ inequality, we obtain (51), (A1), (F1), (H1), and Holder

𝑡

𝑁 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 𝑝𝑖 (𝑥)−2 1/𝑁 ≤ 𝑐1 ∏(∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝜑𝑡 󵄨󵄨󵄨󵄨 (𝜀 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) ) Ω

𝑖=1 𝐴 𝑘,𝑠

󵄨 󵄨 󵄨 󵄨 2𝛿+ (𝑥)−𝑝𝑖 (𝑥)+2 2 ≤ 𝑐1 ∑ ∫ 𝑘𝑖 (𝑥) (󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) 𝜑𝑡 Ω

󵄨󵄨 󵄨𝑝 (𝑥) 𝑁 󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨 𝑖 ≤ ∑ ∫ 󵄨 󵄨 𝐴 𝑖=1

2

𝑖=1

𝑘,𝑡

2/2∗

≤ 𝑐1 (∫ 𝜑𝑡2 )

𝑁

Ω𝑡

+ ∑∫

𝑖=1 𝐴 𝑘,𝑡

󵄨 󵄨 󵄨 󵄨 (2𝛿+ (𝑥)−𝑝𝑖 (𝑥)+2)(𝑁/2) 2/𝑁 × ∑(∫ 𝑘𝑖 (𝑥) (󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) ) , Ω 𝑁

𝑖=1

𝑘,𝑡

󵄨󵄨 󵄨𝑝 (𝑥) 󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨 𝑖 󵄨 󵄨

󵄩 󵄩 󸀠 ≤ 𝑐 [󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝∞ ‖𝑢 − 𝑘‖𝐿𝑝∞ (𝐴 𝑘,𝑡 ) (𝐴 )

𝑡

∗

𝑁

𝑁

+∑ ∫

𝑡

(46)


𝑁

𝑁 1 󵄨 󵄨 󵄨 󵄨 (2𝛿+ (𝑥)−𝑝𝑖 (𝑥)+2)(𝑁/2) 2/𝑁 ) . ≤ ∑(∫ 𝑘𝑖 (𝑥)𝑁(󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) 𝐶 𝑖=1 Ω𝑡 (47)

∑∫


𝑡 → sup 𝜑 Ω 𝑡

(48) Therefore, (48) leads to a contradiction. On the other hand, if 𝛿+ (𝑥) < (𝑝𝑖 (𝑥) − 2)/2, from (44), and Young inequality, for some constant 𝑐2 (independent on 𝑡), we have 𝑁 󵄨 󵄨2 󵄨 󵄨 󵄨 2𝛿+ (𝑥) 2 󵄨 𝜑𝑡 . (49) ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝜑𝑡 󵄨󵄨󵄨󵄨 ≤ 𝑐2 ∑ ∫ 𝑘𝑖 (𝑥)2 (󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) Ω Ω

∗

󵄨󵄨 𝑢 − 𝑘 󵄨󵄨𝑝(𝑥) 1 𝑁 󵄨 󵄨󵄨 ≤ 𝑐 ∫ 󵄨󵄨󵄨 + ∑ ∫ |𝑢|𝑞(𝑥) , 󵄨󵄨 󵄨 󵄨 𝑡 − 𝑠 2 𝑖=1 𝐴 𝑘,𝑡 𝐴 𝑘,𝑡 󵄨 󵄨 𝑁

1𝑁 1𝑁 󵄨 󵄨2 ℎ𝑖 𝜕𝑥𝑖 𝑢 ≤ ∑ ∫ ℎ𝑖2 + ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 2 𝑖=1 𝐴 𝑘,𝑡 2 𝑖=1 𝐴 𝑘,𝑡 𝐴 𝑘,𝑡

∑∫ 𝑖=1

1𝑁 1𝑁 󵄨 󵄨𝑝𝑖 (𝑥) ≤ ∑ ∫ ℎ𝑖2 + ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 . 2 𝑖=1 𝐴 𝑘,𝑡 2 𝑖=1 𝐴 𝑘,𝑡

𝑡

Argue as in (46), for some constant 𝑐3 (independent on 𝑡), we have (50)

Since 𝑁/𝑟𝑖 (𝑥) + 𝑁𝛿𝑖 (𝑥)/𝑝𝑖 (𝑥) ≤ 1, we obtain a contradiction.

󵄨 󵄨󵄨 󵄨󵄨𝑇𝑖 (𝑥, ∇𝑢)󵄨󵄨󵄨 |𝑢 − 𝑘|

1𝑁 1𝑁 ≤ ∑ ∫ |𝑢 − 𝑘|𝑝𝑖 (𝑥) + ∑ ∫ |𝑢|𝑞(𝑥) 2 𝑖=1 𝐴 𝑘,𝑡 2 𝑖=1 𝐴 𝑘,𝑡

󵄨 󵄨 󵄨 󵄨 (2𝛿+ (𝑥)−𝑝𝑖 (𝑥)+2)(𝑁/2) = 0. lim ∫ 𝑘𝑖 (𝑥)𝑁(󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨)

𝑁 1 󵄨 󵄨 󵄨 󵄨 𝑁𝛿+ (𝑥) 2/𝑁 ) . ≤ 𝑐3 ∑(∫ 𝑘𝑖 (𝑥)𝑁(󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 V󵄨󵄨󵄨󵄨) 𝐶 Ω𝑡 𝑖=1

ℎ𝑖 𝜕𝑥𝑖 𝑢] .

𝑁 1 1 󵄨2 󵄨 ≤ ∑ ( ∫ |𝑢 − 𝑘|2 + ∫ 󵄨󵄨󵄨𝑇𝑖 (𝑥, ∇𝑢)󵄨󵄨󵄨 ) 2 2 𝐴 𝐴 𝑘,𝑡 𝑘,𝑡 𝑖=1

Since 𝑁/𝑟𝑖 (𝑥) + ((2𝛿𝑖 (𝑥) − 𝑝𝑖 (𝑥) + 2)/𝑝𝑖 (𝑥))(𝑁/2) ≤ 1, we have

𝑖=1

(52) 󵄨 󵄨󵄨 󵄨󵄨𝑇𝑖 (𝑥, ∇𝑢)󵄨󵄨󵄨 |𝑢 − 𝑘|

For 2 ≤ 𝑝𝑖 (𝑥) and (T1), by Young inequality, we have

where 𝑐1 is a constant independent on 𝑡. Hence, we have

𝑡

(51)

𝑁

(53) As 2 ≤ 𝑝𝑖 (𝑥) and ∑𝑁 𝑖=1 (1/𝑝𝑖 (𝑥)) < 2, if 𝑝(𝑥) < 𝑁, there exists ∗ 𝜀 = 1/2 − 𝑝(𝑥)/𝑝(𝑥) > 0 such that 1/2

𝑐(∫

𝐴 𝑘,𝑡

ℎ𝑖2 )

∗

󵄨󵄨 󵄨1/2 󵄨𝑝(𝑥)/𝑝(𝑥) 󵄨 󵄨󵄨𝐴 𝑘,𝑡 󵄨󵄨󵄨 ≤ 𝑐󵄨󵄨󵄨𝐴 𝑘,𝑡 󵄨󵄨󵄨

+𝜀

.

(54)


7

If 𝑝(𝑥) ≥ 𝑁, a similar estimate is true; just only choose a ∗ suitable 𝑝(𝑥) . ∗ Due to 𝑞 ∈ 𝐶+0 (Ω) and 𝑞(𝑥) < 𝑝∞ (𝑥) = 𝑝(𝑥) , we have 󵄩󵄩 󵄩󵄩 󸀠 𝑞(𝑥) 󵄩󵄩𝑓󵄩󵄩𝐿𝑝∞ ‖𝑢 − 𝑘‖𝐿𝑝∞ ≤ 𝑐 ∫ |𝑢| . 𝐴

(55)

𝑘,𝑡

𝐴 𝑘,𝑡

|𝑢|𝑞(𝑥) = ∫

𝐴 𝑘,𝑡

|(𝑢 − 𝑘) + 𝑘|𝑞(𝑥)

≤ 𝑐∫

𝐴 𝑘,𝑡

𝑁

󵄨 󵄨 |𝑢 − 𝑘|𝑞(𝑥) + 𝑐𝑘𝑞 󵄨󵄨󵄨𝐴 𝑘,𝑡 󵄨󵄨󵄨

Ω

For the definition of

=

󵄨󵄨 󵄨𝑝 (𝑥) 󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨 𝑖 󵄨 󵄨

∗

󵄨 󵄨 󵄨 󵄨𝑝(𝑥)/ 𝑝(𝑥) + 𝑐𝑘 󵄨󵄨󵄨𝐴 𝑘,𝑡 󵄨󵄨󵄨 + 𝑐󵄨󵄨󵄨𝐴 𝑘,𝑡 󵄨󵄨󵄨

+𝜀

∗

(57)

≐ 𝑒𝑖 + 𝑔𝑖𝑗 𝐷𝑗 (Δ𝑘𝑚 𝑢) , 1

1

𝑖 where 𝑒𝑖 = ∫0 𝑎𝑥𝑖 𝑘 𝑑𝑡, 𝑔𝑖𝑗 = ∫0 ∑𝑁 𝑗=1 𝑎𝜉𝑗 𝑑𝑡, and

.

Δ𝑘𝑚 𝑓 =

𝑖=1 𝐴 𝑘,𝛿𝑅

1 1 𝑑 𝑓 (𝑥 + 𝑡𝑚𝑒𝑘 ) 𝑑𝑡 ∫ 𝑚 0 𝑑𝑡 1

󵄨󵄨 󵄨𝑝 (𝑥) 󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨 𝑖 󵄨 󵄨

∑∫

(61)

𝑁

= ∫ [𝑎𝑥𝑖 𝑘 + ∑𝑎𝜉𝑖 𝑗 𝐷𝑗 (Δ𝑘𝑚 𝑢)] 𝑑𝑡 0 𝑗=1 ] [

Using Lemma 3.1 in [17], and (57), we have 𝑁

and [19], we have

1 1 𝑑 𝑖 [𝑎 (𝑥 + 𝑡𝑚𝑒𝑘 , Du + 𝑡𝑚Δ𝑘𝑚 𝐷𝑢)] 𝑑𝑡 ∫ 𝑚 0 𝑑𝑡 1

𝑞+

Δ𝑘𝑚

Δ𝑘𝑚 𝑎𝑖 (𝑥, 𝐷𝑢)

and (52)–(55), we obtain

󵄨󵄨 𝑢 − 𝑘 󵄨󵄨𝑝(𝑥) 1𝑁 󵄨 󵄨𝑝𝑖 (𝑥) 󵄨 󵄨󵄨 ≤ ∑ ∫ 󵄨󵄨󵄨󵄨𝜕𝑥𝑖 𝑢󵄨󵄨󵄨󵄨 + 𝑐 ∫ 󵄨󵄨󵄨 󵄨󵄨 2 𝑖=1 𝐴 𝑘,𝑡 𝐴 𝑘,𝑡 󵄨󵄨 𝑡 − 𝑠 󵄨󵄨

𝑖=1 Ω

(60)

(56)

󵄨󵄨 𝑢 − 𝑘 󵄨󵄨𝑝(𝑥) + 󵄨󵄨 󵄨󵄨 󵄨 󵄨 + 𝑐𝑘𝑞 󵄨󵄨󵄨𝐴 𝑘,𝑡 󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 𝐴 𝑘,𝑡 󵄨󵄨 𝑡 − 𝑠 󵄨󵄨

𝑖=1 𝐴 𝑘,𝑠

𝑁

𝑖=1 Ω

+

≤ 𝑐∫

𝑁

𝑘

∑ ∫ Δ𝑘𝑚 𝑎𝑖 (𝑥, 𝐷𝑢) 𝜙𝑥𝑖 − ∫ Δ𝑘𝑚 𝑓𝜙 − ∑ ∫ Δ𝑘𝑚 ℎ𝑖 𝜙𝑥𝑖 = 0.

∗

∑∫

𝑔 (𝑥 + 𝑚𝑒𝑘 ) − 𝑔 (𝑥) (59) , 𝑚 where 𝑒𝑘 = (0, 0, . . . , 1, . . . , 0). Hence, if for any 𝜙 ∈ 𝐸, 𝜙 has Δ𝑘𝑚 𝑔 (𝑥) =

compact support in Ω and |𝑚| < (1/2) dist(supp 𝜙, 𝜕Ω); from the definition of weak solution for the problem (1), we have

From ∫

any function with compact support in Ω, for 1 ≤ 𝑘 ≤ 𝑁, for small enough 𝑚, we define

= ∫ 𝑓𝑥𝑘 𝑑𝑡 ≐ 𝐹, 0

∗

󵄨󵄨 𝑢 − 𝑘 󵄨󵄨𝑝(𝑥) + 󵄨󵄨 󵄨󵄨 󵄨 󵄨 + 𝑐𝑘𝑞 󵄨󵄨󵄨𝐴 𝑘,𝑅 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝐴 𝑘,𝑅 󵄨 𝑅 − 𝛿𝑅 󵄨

≤ 𝑐∫

󵄨 󵄨𝑝(𝑥)/𝑝(𝑥) + 𝑐󵄨󵄨󵄨𝐴 𝑘,𝑅 󵄨󵄨󵄨

∗

+𝜀

Δ𝑘𝑚 ℎ𝑖

(58)

1 1 𝑑 𝑖 = ∫ ℎ (𝑥 + 𝑡𝑚𝑒𝑘 ) 𝑑𝑡 𝑚 0 𝑑𝑡

(62)

1

= ∫ ℎ𝑥𝑖 𝑘 𝑑𝑡 ≐ 𝐻𝑖 . 0

.

By Lemma 2.4 in [18], (58), we obtain that 𝑢 is bounded from above on 𝐵(𝑥0 , 𝑅/2). Note that −𝑢 is also a weak solution of the problem (1), where 𝑎̃𝑖 (𝑥, 𝑢, ∇𝑢) = 𝑎𝑖 (𝑥, −𝑢, −∇𝑢) and ̃𝑖 (𝑥, ∇𝑢) = 𝑇𝑖 (𝑥, −∇𝑢). Hence, −𝑢 is also bounded from 𝑇 above on 𝐵(𝑥0 , 𝑅/2), and 𝑢 ∈ 𝐿∞ (𝐵(𝑥0 , 𝑅/2)). This implies that 𝑢 ∈ 𝐿∞ loc (Ω). Proof of Theorem 5. From Lemmas 8, 9, and 10, we obtain that 󳨀󳨀󳨀→ 1,𝑝(𝑥) 𝑊0 (Ω)

there exists a uniqueness weak solution 𝑢 ∈ for the problem (1), and what is more, 𝑢 ∈ 𝐿∞ loc (Ω). Hence we complete the proof of Theorem 5.

Then, combining (60) with (61)-(62), we obtain 𝑁

𝑁

∑ ∫ 𝑒𝑖 𝐷𝑖 𝜙 + ∑ ∫ 𝑔𝑖𝑗 𝐷𝑗 (Δ𝑘𝑚 𝑢) 𝐷𝑖 𝜙 𝑖=1 Ω

𝑖,𝑗=1 Ω

(63)

𝑁

− ∫ 𝐹𝜙 − ∑ ∫ 𝐻𝑖 𝐷𝑖 𝜙 = 0. Ω

𝑖=1 Ω

Due to (A4), (F2), (H2), (63), Young inequality, we have 𝑁

∑ ∫ 𝑔𝑖𝑗 𝐷𝑗 (Δ𝑘𝑚 𝑢) 𝐷𝑖 𝜙

𝑖,𝑗=1 Ω

𝑁

𝑁

𝑖=1 Ω

𝑖=1 Ω

= ∫ 𝐹𝜙 + ∑ ∫ 𝐻𝑖 𝐷𝑖 𝜙 − ∑ ∫ 𝑒𝑖 𝐷𝑖 𝜙

4. The Proof of Theorem 6

Ω

In this section, we prove the regularity of weak solutions for the problem (2). 󳨀󳨀󳨀→ 1,𝑝(𝑥)

(Ω). For underProof of Theorem 6. We denote 𝐸 = 𝑊0 𝑖 standing, we write 𝑎 (𝑢, 𝐷𝑢) = 𝑎𝑖 (𝑢, ∇𝑢), and ℎ𝑖 = ℎ𝑖 . If 𝑔 is

󵄨 󵄨2 ≤ 𝑐 ∫ 󵄨󵄨󵄨𝐷𝜙󵄨󵄨󵄨 + 𝑐 ∫ |𝑢|𝑞(𝑥) + 𝑐 ∫ |𝑢|𝑞(𝑥)/2 𝜙 Ω Ω Ω 󵄨 󵄨2 ≤ 𝑐 ∫ 󵄨󵄨󵄨𝐷𝜙󵄨󵄨󵄨 + 𝑐 ∫ |𝑢|𝑞(𝑥) + 𝑐 ∫ 𝜙2 ; Ω Ω Ω

(64)

8


that is,

By (A3), we obtain 𝑁

∑∫

𝑖,𝑗=1 Ω

󵄨 󵄨2 ∫ 𝜓2 󵄨󵄨󵄨󵄨𝐷 (Δ𝑘𝑚 𝑢)󵄨󵄨󵄨󵄨 Ω

𝑔𝑖𝑗 𝐷𝑗 (Δ𝑘𝑚 𝑢) 𝐷𝑖 𝜙 (65)

󵄨 󵄨2 ≤ 𝑐 ∫ 󵄨󵄨󵄨𝐷𝜙󵄨󵄨󵄨 + 𝑐 ∫ |𝑢|𝑞(𝑥) + 𝑐 ∫ 𝜙2 . Ω Ω Ω

𝑁

󵄨 󵄨𝑝 (𝑥)−2 ≤ 𝑐 ∫ [1 + ∑ (󵄨󵄨󵄨𝐷𝑖 𝑢 (𝑥)󵄨󵄨󵄨 𝑖 Ω

𝑖=1

(68) 󵄨 󵄨𝑝 (𝑥)−2 + 󵄨󵄨󵄨𝐷𝑖 𝑢 (𝑥 + 𝑚𝑒𝑘 )󵄨󵄨󵄨 𝑖 )]

Choose 𝜙 = 𝜓2 Δ𝑘𝑚 𝑢, where 𝜓 ∈ 𝐶01 (Ω), |𝜓| ≤ 1, and (65), we obtain

󵄨 󵄨2 󵄨 󵄨2 × 󵄨󵄨󵄨󵄨Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝐷𝜓󵄨󵄨󵄨 + 𝑐 ∫ |𝑢|𝑞(𝑥) . Ω

𝑁

∑ ∫ 𝑔𝑖𝑗 𝜓2 𝐷𝑖 (Δ𝑘𝑚 𝑢) 𝐷𝑗 (Δ𝑘𝑚 𝑢)

𝑖,𝑗=1 Ω

𝑁 󵄨 󵄨2 ≤ −2 ∑ ∫ 𝜓𝑢𝑔𝑖𝑗 𝐷𝑖 𝜓𝐷𝑗 (Δ𝑘𝑚 𝑢) + 𝑐 ∫ 󵄨󵄨󵄨󵄨𝐷𝜓2 Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 Ω Ω 𝑖,𝑗=1

2

𝑖=1

Ω

𝑁 󵄨 󵄨 ∑ ∫ 󵄨󵄨󵄨󵄨𝑔𝑖𝑗 𝜓2 𝐷𝑖 (Δ𝑘𝑚 𝑢) 𝐷𝑗 (Δ𝑘𝑚 𝑢)󵄨󵄨󵄨󵄨 Ω

𝑁 + 󵄨2 󵄨 󵄨𝑝 (𝑥)−2 󵄨󵄨 󵄨󵄨2 󵄨󵄨󵄨 ≤ 𝑐 ∫ (1 + ∑󵄨󵄨󵄨𝐷𝑖 𝑢󵄨󵄨󵄨 𝑖 ) 󵄨󵄨𝐷𝜓󵄨󵄨 󵄨󵄨(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨

𝑁

󵄨 󵄨 ≤ 2 ∑ ∫ 󵄨󵄨󵄨󵄨𝜓𝑢𝑔𝑖𝑗 𝐷𝑖 𝜓𝐷𝑗 (Δ𝑘𝑚 𝑢)󵄨󵄨󵄨󵄨 Ω

Ω

𝑖,𝑗=1

󵄨 󵄨2 󵄨2 󵄨 󵄨4 󵄨 +𝑐 ∫ 󵄨󵄨󵄨󵄨𝐷𝜓2 Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 +𝑐 ∫ |𝑢|𝑞(𝑥) +𝑐 ∫ 󵄨󵄨󵄨𝜓󵄨󵄨󵄨 󵄨󵄨󵄨󵄨Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 Ω Ω Ω

Ω

1/2 2 ×[∫ 𝑔𝑖𝑗 (Δ𝑘𝑚 𝑢) 𝐷𝑖 𝜓𝐷𝑗 𝜓] +𝑐 ∫ Ω Ω

+𝑐 ∫ |𝑢| Ω

𝑖=1

𝑁−1

+ 󵄨2 󵄨 󵄨𝑝 (𝑥)−2 󵄨󵄨 󵄨󵄨2 󵄨󵄨󵄨 ≤ 𝑐 ∫ (1 + ∑ 󵄨󵄨󵄨𝐷𝑖 𝑢󵄨󵄨󵄨 𝑖 ) 󵄨󵄨𝐷𝜓󵄨󵄨 󵄨󵄨(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨 Ω 𝑖=1

+ 󵄨2 󵄨 󵄨𝑝 −2 󵄨 󵄨2 󵄨 + 𝑐 ∫ 󵄨󵄨󵄨𝐷𝑁𝑢󵄨󵄨󵄨 𝑁 󵄨󵄨󵄨𝐷𝜓󵄨󵄨󵄨 󵄨󵄨󵄨󵄨(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨 Ω

1/2

≤ 2 ∑ [∫ 𝑔𝑖𝑗 𝜓2 𝐷𝑖 (Δ𝑘𝑚 𝑢) 𝐷𝑖 (Δ𝑘𝑚 𝑢)]

𝑞(𝑥)

1,2 Hence, 𝐷𝑁𝑢 ∈ 𝑊loc (Ω). If we take 𝜙 = 𝜓2 (Δ𝑁𝑚 𝑢 − 𝑟)+ , where 𝑟 > 0, |𝜓| < 1, (Δ𝑁𝑚 𝑢 − 𝑟)+ = max{Δ𝑁𝑚 𝑢 − 𝑟, 0}. Since 𝐷𝑁𝑢 ∈ 1,2 (Ω), 𝐷𝑁𝑢 ∈ 𝐿2𝑁/(𝑁−2) (Ω), we have 𝑊loc loc

󵄨 + 󵄨2 ∫ 𝜓2 󵄨󵄨󵄨󵄨𝐷(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨 Ω

𝑖,𝑗=1

𝑖,𝑗=1

(69)

(66)

From (66), (A3), Cauchy inequality and Young inequality, we have

𝑁

𝑁

󵄨 󵄨 󵄨2 󵄨𝑝 (𝑥)−2 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨2 ) 󵄨󵄨𝐷𝜓󵄨󵄨 󵄨󵄨𝐷𝑁𝑢󵄨󵄨󵄨 . ∫ 𝜓2 󵄨󵄨󵄨𝐷 (𝐷𝑁𝑢)󵄨󵄨󵄨 ≤ 𝑐 ∫ (1 + ∑󵄨󵄨󵄨𝐷𝑖 𝑢󵄨󵄨󵄨 𝑖 Ω Ω

+ 𝑐 ∫ |𝑢|𝑞(𝑥) + 𝑐 ∫ (𝜓2 Δ𝑘𝑚 𝑢) . Ω

Assume 𝑝𝑁 = 𝑝++ = sup𝑥∈Ω p+ (𝑥); we have for 𝑖 = 1, 2, . . . , 𝑁, ∗ 𝑝𝑖 (𝑥) ≤ 𝑝𝑁. For 𝑞(𝑥) < 𝑝 (𝑥), let 𝑘 = 𝑁 and take 𝑚 → 0; then we have

󵄨󵄨 2 𝑘 󵄨󵄨2 󵄨󵄨𝐷𝜓 Δ 𝑚 𝑢󵄨󵄨 󵄨 󵄨

󵄨2 󵄨 󵄨4 󵄨 +𝑐 ∫ 󵄨󵄨󵄨𝜓󵄨󵄨󵄨 󵄨󵄨󵄨󵄨Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 Ω

󵄨 󵄨 + 󵄨𝑝𝑁 󵄨 + 󵄨2 󵄨2 ≤ 𝑐 ∫ 󵄨󵄨󵄨󵄨(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝐷𝜓󵄨󵄨󵄨 + 𝑐𝑟𝑝𝑁 −2 ∫ 󵄨󵄨󵄨󵄨(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨 Ω Ω 𝑁 󵄨 + 󵄨2𝑁/(𝑁−2) 󵄨󵄨 󵄨2𝑁/(𝑁−2) (𝑁−2)/𝑁 + 𝑐∑ ∫ (󵄨󵄨󵄨󵄨(𝐷𝑁𝑢 − 𝑟) 󵄨󵄨󵄨󵄨 ) 󵄨󵄨𝐷𝜓󵄨󵄨󵄨 Ω 𝑖=1

𝑁

𝜀 1 ≤ 2 ∑ [ (∫ 𝑔𝑖𝑗 𝜓2 𝐷𝑖 (Δ𝑘𝑚 𝑢) 𝐷𝑖 (Δ𝑘𝑚 𝑢)) + 2𝜀 Ω 𝑖,𝑗=1 2 2 ×(∫ 𝑔𝑖𝑗 (Δ𝑘𝑚 𝑢) 𝐷𝑖 𝜓𝐷𝑗 𝜓)]+𝑐 ∫ Ω Ω

×(∫

{𝐷𝑁 𝑢>𝑘} ⋂ supp 𝜓

𝑁/2 2/𝑁

)

.

(70)

𝑞(𝑥)

|𝑢|

For 2 ≤ 𝑝𝑖 (𝑥) ≤ 2𝑁/(𝑁 − 2), we have

󵄨 󵄨2 + 𝑐 ∫ 𝜓2 󵄨󵄨󵄨󵄨𝐷 (Δ𝑘𝑚 𝑢)󵄨󵄨󵄨󵄨 Ω 󵄨 󵄨 󵄨2 󵄨 󵄨2 󵄨2 + 𝑐 ∫ 󵄨󵄨󵄨󵄨Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝐷𝜓󵄨󵄨󵄨 + 𝑐 ∫ 󵄨󵄨󵄨󵄨Δ𝑘𝑚 𝑢󵄨󵄨󵄨󵄨 . Ω Ω

󵄨 󵄨𝑝 (𝑥)−2 (1 + 󵄨󵄨󵄨𝐷𝑖 𝑢󵄨󵄨󵄨 𝑖 )

(∫

{𝐷𝑁 𝑢>𝑘} ⋂ supp 𝜓

(67)

2/𝑁

󵄨 󵄨𝑝 (𝑥)−2 𝑁/2 (1 + 󵄨󵄨󵄨𝐷𝑖 𝑢󵄨󵄨󵄨 𝑖 ) )

󵄨 󵄨2/𝑁−2/𝑝−− +1 ≤ 𝑐󵄨󵄨󵄨󵄨{𝐷𝑁𝑢 > 𝑘} ⋂ supp 𝜓󵄨󵄨󵄨󵄨 .

(71)


9

If we fix 𝐵(𝑥0 , 𝑅), supp 𝜓 ⊂ 𝐵(𝑥0 , 𝑅), where 𝑅 < 1; let 𝑟 ≥ 1, 0 < 𝛿 < 1, and take 𝜓 ∈ 𝐶01 (𝐵(𝑥0 , 𝑅)), 𝜓 ≡ 1 on 𝐵(𝑥0 , 𝛿𝑅); 0 ≤ 𝜓 ≤ 1, |𝐷𝜓| ≤ 𝑐/(𝑅 − 𝛿𝑅), then we have ∫

𝐴 𝑟,𝛿𝑅

[11]

󵄨󵄨 󵄨2 󵄨󵄨𝐷(𝐷𝑁𝑢 − 𝑟)+ 󵄨󵄨󵄨 󵄨 󵄨

󵄨 2𝑁/(𝑁−2) 󵄨󵄨 󵄨𝐷 𝑢 − 𝑟󵄨󵄨󵄨 ≤ 𝑐 ∫ (󵄨 𝑁 ) 𝑅 − 𝛿𝑅 𝐴 𝑟,𝑅 󵄨 2𝑁/(𝑁−2) (𝑁−2)/𝑁 󵄨󵄨 󵄨󵄨𝐷𝑁𝑢 − 𝑟󵄨󵄨󵄨 + 𝑐(∫ ( ) ) 𝑅 − 𝛿𝑅 𝐴 𝑟,𝑅 󵄨󵄨2/𝑁−2/𝑝−− +1

󵄨 × 󵄨󵄨󵄨𝐴 𝑟,𝑅 󵄨󵄨

[12]

(72)

[13]

[14]

󵄨 󵄨 + 𝑐𝑟2𝑁/(𝑁−2) 󵄨󵄨󵄨𝐴 𝑟,𝑅 󵄨󵄨󵄨 .

Hence, (72) implies that 𝐷𝑁𝑢 is in 𝐿∞ loc (Ω). So we obtain 1,∞ 2,2 𝑢 ∈ 𝑊loc (Ω) ⋂ 𝑊loc (Ω). By De Giorgi-Moser regularity theorem, for any 𝜔 ⊂⊂ Ω and 0 < 𝛼 < 1, we have 𝐷𝑁𝑢 ∈ 𝐶0,𝛼 (𝜔); then 𝑢 ∈ 𝐶1,𝛼 (Ω), for 0 < 𝛼 < 1.

[15]

[16]

Acknowledgments The authors express their sincere thanks to the referees for their valuable suggestions. This paper was supported by Shanghai Natural Science Foundation Project (no. 11ZR1424500) and Shanghai Leading Academic Discipline Project (no. XTKX2012).

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Research Article Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set Jinli Xu,1 Baodong Zheng,1 and Hongmei Yao2 1 2

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China College of Science, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Baodong Zheng; [email protected] Received 12 May 2013; Accepted 16 July 2013 Academic Editor: Gen-Qi Xu Copyright © 2013 Jinli Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let 𝐻𝑛 be the set of 𝑛 × 𝑛 complex Hermitian matrices and P𝑛 (resp., T𝑛 ) be the set of all idempotent (resp., tripotent) matrices in 𝐻𝑛 . In l-partite quantum system 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 = ⊗𝑙1 𝐻𝑚𝑖 , ⊗𝑙1 P𝑚𝑖 (resp., ⊗𝑙1 T𝑚𝑖 ) denotes the set of all decomposable elements ⊗𝑙1 𝐴 𝑖 such that 𝐴 𝑖 ∈ P𝑚𝑖 (resp., 𝐴 𝑖 ∈ T𝑚𝑖 ). In this paper, linear maps 𝜙 from 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 to 𝐻𝑛 with 𝑛 ≤ 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 such that 𝜙(⊗𝑙1 P𝑚𝑖 ) ∈ P𝑛 are characterized. As its application, the structure of linear maps 𝜙 from 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 to 𝐻𝑛 with 𝑛 ≤ 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 such that 𝜙(⊗𝑙1 T𝑚𝑖 ) ∈ T𝑛 is also obtained.

1. Introduction Let 𝑀𝑛 be the vector space of 𝑛 × 𝑛 complex matrices and 𝐻𝑛 ⊆ 𝑀𝑛 be the vector space of 𝑛 × 𝑛 complex Hermitian matrices. In quantum information theory, a 𝑙-partite system can be represented as the tensor product space 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 ≡ ⊗𝑙1 𝐻𝑚𝑖 , where ⊗ is the usual Kronecker product of matrices; a quantum state is represented as a positive semidefinite with trace one in 𝐻𝑛 , see [1]. In quantum information science and quantum computing, it is important to understand, characterize, and construct different classes of maps on quantum states [2]. For example, entanglement is one of the main concepts in quantum information theory, and entangled state involves at least bipartite system or multipartite system [3]; to study entangled states, one should construct entanglement witnesses, which are special types of positive maps, see [4]. On this background, the research on the characterizations of maps leaving invariant, some important subsets or quantum properties attracted more and more researchers’ attention, see ([1, 3, 5]). Especially, in [5], Lim characterized the linear and additive maps on tensor products of spaces of Hermitian matrices that carry the set of tensor product of rank one matrices into itself.

Preserver problem is a hot area in Banach algebra; there are many results about this area, see ([6–11]). Specially, the idempotent preservers and the rank one preservers play an important role; therefore, it is meaningful to study the two preservers. Chan et al. [12] first characterized linear transformations on 𝑀𝑛 preserving idempotent matrices. In [13], the authors obtained the following result: linear map 𝑇 : 𝐻𝑛 → 𝐻𝑚 with 𝑛 ≤ 𝑚 satisfies 𝑇(P𝑛 ) ⊂ P𝑚 if and only if 𝑇 = 0 or 𝑇 (𝑋) = 𝑄𝑋𝑄−1

or 𝑇 (𝑋) = 𝑄𝑋𝑇 𝑄−1 ,

∀𝑋 ∈ 𝐻𝑛 , (1)

where 𝑄 is unitary matrix. Since the study of linear transformations, preserving idempotents is important in many aspects of mathematics and physics, see [14–17], and inspired by the above, the purpose of this paper is to study linear maps from 𝑙-partite system to the space of Hermitian matrices that carry the set of tensor product of idempotent matrices into the set of idempotent matrices in the space of Hermitian matrices, that is, 𝜙 : 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 → 𝐻𝑛 with 𝑛 ≤ 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 satisfying 𝜙 (⊗𝑙1 P𝑚𝑖 ) ⊂ P𝑛 ,

(2)

2


where P𝑛 is the set of all idempotent matrices in 𝐻𝑛 , that is, P𝑛 = {𝐴 ∈ 𝐻𝑛 |𝐴2 = 𝐴} and ⊗𝑙1 P𝑚𝑖 denotes the set of all decomposable elements ⊗𝑙1 𝐴 𝑖 such that 𝐴 𝑖 ∈ P𝑚𝑖 . As application, the forms of linear maps from 𝑙-partite system to the space of Hermitian matrices that carry the set of tensor product of tripotent matrices into the set of tripotent matrices in the space of Hermitian matrices are obtained, that is, 𝜙 : 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 → 𝐻𝑛 with 𝑛 ≤ 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 satisfying 𝜙 (⊗𝑙1 T𝑚𝑖 ) ⊂ T𝑛 ,

(3)

where T𝑛 is the set of all tripotent matrices in 𝐻𝑛 , that is, T𝑛 = {𝐴 ∈ 𝐻𝑛 |𝐴3 = 𝐴} and ⊗𝑙1 T𝑚𝑖 denotes the set of all decomposable elements ⊗𝑙1 𝐴 𝑖 such that 𝐴 𝑖 ∈ T𝑚𝑖 . Throughout this paper, we always assume integers 𝑙 ≥ 1 and 𝑛, 𝑚1 , . . ., 𝑚𝑙 ≥ 2. Let 𝐼𝑘 be the 𝑘 × 𝑘 identity matrix, 0 be the zero matrix which order is omitted in different matrices just for simplicity, and 𝑋𝑇 (resp., 𝑋∗ , rank 𝑋) be the transpose (resp., conjugate transpose, rank) of 𝑋. 𝐸𝑖𝑗(𝑛) (1 ≤ 𝑖, 𝑗 ≤ 𝑛) stands for the 𝑛 × 𝑛 matrix with 1 at the (𝑖, 𝑗)th entry and 0 (𝑛) (𝑛) and 𝐾𝑖𝑗(𝑛) = √−1𝐸𝑖𝑗(𝑛) − √−1𝐸𝑗𝑖 , otherwise. 𝐷𝑖𝑗(𝑛) = 𝐸𝑖𝑗(𝑛) + 𝐸𝑗𝑖 where 1 ≤ 𝑖 ≠ 𝑗 ≤ 𝑛. For real numbers 𝑎 and 𝑏 with 𝑎 ≤ 𝑏, let [𝑎, 𝑏] be the set of all integers between 𝑎 and 𝑏. ⊕ is the usual direct sum of matrices. A linear map 𝜋 = ⊗𝑙1 𝜋𝑖 : 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 → 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 is canonical if 𝜋 satisfies 𝜋(⊗𝑙1 𝐴 𝑖 ) = ⊗𝑙1 𝜋𝑖 (𝐴 𝑖 ), where 𝜋𝑖 : 𝐻𝑚𝑖 → 𝐻𝑚𝑖 satisfies 𝜋𝑖 (𝐴 𝑖 ) = 𝐴 𝑖 or 𝜋𝑖 (𝐴 𝑖 ) = 𝐴𝑇𝑖 for all 𝐴 𝑖 ∈ 𝐻𝑚𝑖 .

𝑋11 𝑋12 𝑋13 [𝑋21 𝑋22 𝑋23 ] ∈ P𝑛 , [𝑋31 𝑋32 𝑋33 ] 0𝑟 0 0 𝑋11 𝑋12 𝑋13 [ 0 𝐼𝑡 0] − [𝑋21 𝑋22 𝑋23 ] ∈ P𝑛 . [ 0 0 0] [𝑋31 𝑋32 𝑋33 ]

(6)

A straightforward computation shows that 𝑋 = 0𝑠 ⊕ [𝑇(𝑋22 ⊕ 0𝑟−𝑡 )𝑇∗ ] ⊕ 0𝑛−𝑟−𝑠 ; therefore, the result holds. Lemma 3. Let 𝐴 = 𝑃𝑟 ⊕ 0𝑠 ⊕ 0𝑛−𝑟−𝑠 and 𝐵 = 0𝑟 ⊕ 𝑃𝑠 ⊕ 0𝑛−𝑟−𝑠 with 𝑃𝑟 ∈ P𝑟 , 𝑃𝑠 ∈ P𝑠 . If 𝛼𝐴 + 𝛽𝐵 + 𝛾𝑋 ∈ P𝑛

󵄨 󵄨 1 for any real 𝛾 with 0 ≠󵄨󵄨󵄨𝛾󵄨󵄨󵄨 ≤ , (7) 2

where 𝑋 ∈ 𝑀𝑛 , 𝛼 and 𝛽 are all roots of 𝑥2 − 𝑥 + 𝛾2 = 0, then, 𝑋2 = 𝐴𝐵, and 𝑋 is of the form 0 𝑋12 ] ⊕ 0𝑛−𝑟−𝑠 , [ 𝑟 𝑋21 0𝑠

(8)

where 𝑋12 ∈ 𝑀𝑟×𝑠 , 𝑋21 ∈ 𝑀𝑠×𝑟 . Proof. It is clear that 𝐴, 𝐵 ∈ P𝑛 and 𝐴𝐵 = 𝐵𝐴 = 0. By a direct computation, one can obtain that 𝑋 = 𝛼 (𝐴𝑋 + 𝑋𝐴) + 𝛽 (𝐵𝑋 + 𝑋𝐵) + 𝛾 (𝑋2 − 𝐴 − 𝐵) . (9) Replacing 𝛾 by −𝛾, we have 𝑋2 = 𝐴 + 𝐵 and

2. Main Results Lemma 1 (see [18]). Suppose 𝑃1 , . . . , 𝑃𝑘 ∈ P𝑛 such that 𝑃𝑖 + 𝑃𝑗 ∈ P𝑛 , ∀1 ≤ 𝑖 < 𝑗 ≤ 𝑘. Let 𝑟𝑖 = rank 𝑃𝑖 . Then, there exists a unitary matrix 𝑈 ∈ 𝑀𝑛 such that 𝑃𝑖 = 𝑈 diag (0, . . . , 0, 1, . . . , 1, 0, . . . , 0) 𝑈∗ ,

where 𝑋11 ∈ 𝐻𝑠 , 𝑋22 ∈ 𝐻𝑡 . Then,

𝑋 = 𝛼 (𝐴𝑋 + 𝑋𝐴) + 𝛽 (𝐵𝑋 + 𝑋𝐵) .

Choosing 𝛾 = 2/5, then 𝛼 = 1/5, 𝛽 = 4/5 or 𝛼 = 4/5, 𝛽 = 1/5, this yields that

(4)

𝑋=

4 1 (𝐴𝑋 + 𝑋𝐴) + (𝐵𝑋 + 𝑋𝐵) , 5 5

where diag(0, . . . , 0, 1, . . . , 1, 0, . . . , 0) is the diagonal matrix in which all diagonal entries are zero except those in the (𝑟1 + ⋅ ⋅ ⋅ + 𝑟𝑖−1 + 1)st to the (𝑟1 + ⋅ ⋅ ⋅ + 𝑟𝑖 )th rows.

𝑋=

1 4 (𝐴𝑋 + 𝑋𝐴) + (𝐵𝑋 + 𝑋𝐵) . 5 5

Lemma 2. Let 𝑃 ∈ P𝑟 . Suppose 𝑋, 0𝑠 ⊕ 𝑃 ⊕ 0𝑛−𝑟−𝑠 − X ∈ P𝑛 . Then, there exists 𝑋𝑟 ∈ P𝑟 such that 𝑋 = 0𝑠 ⊕ 𝑋𝑟 ⊕ 0𝑛−𝑟−𝑠 , where 0 ≤ 𝑟, 𝑠 ≤ 𝑛. Proof. Let 𝑇 ∈ 𝑀𝑟 be a unitary matrix satisfying 𝑃 = 𝑇(𝐼𝑡 ⊕ 0)𝑇∗ . Set 𝐼𝑟 0 𝑋11 𝑋12 𝑋13 0 0 𝐼𝑟 0 [ 0 𝑇∗ 0 ] 𝑋 [ 0 𝑇 0 ] = [𝑋21 𝑋22 𝑋23 ] , [ 0 0 𝐼𝑛−𝑟−𝑠 ] [ 0 0 𝐼𝑛−𝑟−𝑠 ] [𝑋31 𝑋32 𝑋33 ] (5)

(10)

(11)

Thus, 𝑋 = 𝐴𝑋+𝑋𝐴 and 𝑋 = 𝐵𝑋+𝑋𝐵. This, together with the form of 𝐴 and 𝐵, implies that 𝑋 has the form [ 𝑋021𝑟 𝑋012𝑠 ]⊕0𝑛−𝑟−𝑠 , where 𝑋12 ∈ 𝑀𝑟×𝑠 , 𝑋21 ∈ 𝑀𝑠×𝑟 . Lemma 4. Let 𝐽 = 𝑃𝑝 ⊕ −𝑃𝑞 ⊕ 0𝑛−𝑝−𝑞 with 𝑃𝑝 ∈ P𝑝 and 𝑃𝑞 ∈ P𝑞 . Suppose 𝑋 ∈ T𝑛 satisfying 𝐽 − 𝑋 ∈ T𝑛 and 𝐽 − 2𝑋 ∈ T𝑛 . Then, 𝑋 = 𝑋𝑝 ⊕ −𝑋𝑞 ⊕ 0𝑛−𝑝−𝑞 with 𝑋𝑝 ∈ P𝑝 and 𝑋𝑞 ∈ P𝑞 . Proof. By a direct computation, we have 3𝑋 = 𝐽2 𝑋 + 𝐽𝑋𝐽 + 𝑋𝐽2 ,

(12)

3𝑋 = 𝐽𝑋2 + 𝑋𝐽𝑋 + 𝑋2 𝐽.

(13)


3

Let 𝑈 ∈ 𝑀𝑝 and 𝑉 ∈ 𝑀𝑞 be unitary matrices satisfying 𝑃𝑝 = 𝑈(𝐼𝑟 ⊕ 0𝑝−𝑟 )𝑈∗ and 𝑃𝑞 = 𝑉(𝐼𝑠 ⊕ 0𝑞−𝑠 )𝑉∗ , respectively. Then, 𝑈 0 0 0 𝑈∗ 0 [ 0 𝑉∗ 0 ] 0 ] 𝐽 [0 𝑉 0 0 𝐼 0 0 𝐼 𝑛−𝑝−𝑞 ] [ 𝑛−𝑝−𝑞 ] [ =[ [

𝐼𝑟 ⊕ 0𝑝−𝑟 0 0 0 −𝐼𝑠 ⊕ 0𝑞−𝑠 0 ]. 0 0 0𝑛−𝑝−𝑞 ]

(14)

𝑋𝑟 ⊕ 0𝑝−𝑟 0 0 0 −𝑋𝑠 ⊕ 0𝑞−𝑠 0 ], =[ 0 0 0𝑛−𝑝−𝑞 ] [

Theorem 6. Suppose 𝜙 is a linear map from 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 to 𝐻𝑛 with 𝑛 ≤ 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 . Then, 𝜙(⊗𝑙1 P𝑚𝑖 ) ⊂ P𝑛 if and only if either 𝜙 = 0 or 𝑛 = 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 , there exists a unitary matrix 𝑈 ∈ 𝑀𝑛 and a canonical map 𝜋 on 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 such that 𝜙 (𝑋) = 𝑈𝜋 (𝑋) 𝑈∗ ,

By (12), one can assume that 𝑈∗ 0 0 𝑈 0 0 [ 0 𝑉∗ 0 ] 𝑋 [0 𝑉 0 ] 0 0 𝐼 0 0 𝐼 𝑛−𝑝−𝑞 ] 𝑛−𝑝−𝑞 ] [ [

we have 𝜑1 (𝑌)𝑊 = 𝑊𝜑2 (𝑌), thus, by induction hypothesis, 0. Thus, 𝑈 = 𝑊 = 𝜆𝐼𝑚1 ⋅⋅⋅𝑚𝑙−1 . Since 𝑈 is invertible, we have 𝜆 ≠ 𝜆𝐼𝑚1 ⋅⋅⋅𝑚𝑙 and 𝜓1 (𝑋) = (1/𝜆)𝜓1 (𝑋)𝑈 = (1/𝜆)𝑈𝜓2 (𝑋) = 𝜓2 (𝑋) for all 𝑋 ∈ 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 .

(15)

∀𝑋 ∈ 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 .

(20)

Proof. The sufficiency part is obvious. We prove the necessity part by induction on 𝑙. When 𝑙 = 1, it is the result in [13], which has been given in the above introduction. We assume now that the result holds true for 𝑙 − 1 and give the proof of the case 𝑙 by the following five steps. Step 1. Suppose 𝑃 ∈ P𝑚1 and 𝜙(𝑃 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) = 0𝑠 ⊕ 𝐼𝑟 ⊕ 0𝑛−𝑟−𝑠 . Then,

where 𝑋𝑟 ∈ 𝐻𝑟 and 𝑋𝑠 ∈ 𝐻𝑠 . This, together with (13), implies that 𝑋𝑟 ∈ P𝑟 and 𝑋𝑠 ∈ P𝑠 . Thus, 𝑋 = [𝑈 (𝑋𝑟 ⊕ 0𝑝−𝑟 ) 𝑈∗ ] ⊕ − [𝑉 (𝑋𝑠 ⊕ 0𝑞−𝑠 ) 𝑉∗ ] ⊕ 0𝑛−𝑝−𝑞 , (16)

𝜙 (𝑃 ⊗ 𝑋) = 0𝑠 ⊕ 𝜓 (𝑋) ⊕ 0𝑛−𝑟−𝑠 ,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 ,

(21)

where 𝜓 is a linear map from 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 to 𝐻𝑟 satisfying 𝜓(⊗𝑙2 P𝑚𝑖 ) ⊂ P𝑟 . Proof of Step 1. Set

as desired. Lemma 5. Let 𝜓1 , 𝜓2 be canonical maps on 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 , and 𝑈, 𝑉 are invertible matrices of order 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 . If 𝜓1 (𝑋)𝑈 = 𝑉𝜓2 (𝑋) for all 𝑋 ∈ 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 , then 𝜓1 = 𝜓2 and 𝑈 = 𝑉 = 𝜆𝐼𝑚1 ⋅⋅⋅𝑚𝑙 for some 𝜆 ≠ 0. Proof. Let us first point out a simple observation which will be used in our proof. If a matrix 𝑈 ∈ 𝑀𝑚𝑛 commutes with 𝐼𝑚 ⊗ 𝑆 for all real symmetric 𝑆 ∈ 𝑀𝑛 , then 𝑈 has the form 𝑊 ⊗ 𝐼𝑛 with 𝑊 ∈ 𝑀𝑚 . Since 𝜓1 , 𝜓2 are canonical maps on 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 , we assume that 𝜓1 = 𝜏1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝜏𝑙 ,

𝜓2 = 𝜂1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝜂𝑙 ,

(17)

where 𝜏𝑖 and 𝜂𝑖 are the identity maps 𝑋 󳨃 → 𝑋 or the transposition maps 𝑋 󳨃 → 𝑋𝑇 for 𝑖 ∈ [1, 𝑙]. Clearly, 𝜓𝑖 (𝐼𝑚1 ⋅⋅⋅𝑚𝑙 ) = 𝐼𝑚1 ⋅⋅⋅𝑚𝑙 , 𝑖 = 1, 2, hence 𝑈 = 𝑉. We prove 𝑈 = 𝜆𝐼𝑚1 ⋅⋅⋅𝑚𝑙 by induction on 𝑙. The case of 𝑙 = 1 is the well-known fact. We assume that the statement holds true for 𝑙 − 1. We will prove it for 𝑙. For any real symmetric 𝑆 ∈ 𝑀𝑚𝑙 , since 𝜓𝑖 (𝐼𝑚1 ⋅⋅⋅𝑚𝑙−1 ⊗ 𝑆) = 𝐼𝑚1 ⋅⋅⋅𝑚𝑙−1 ⊗ 𝑆, 𝑖 = 1, 2, it follows that 𝑈 = 𝑊 ⊗ 𝐼𝑚𝑙 for some matrix 𝑊 ∈ 𝑀𝑚1 ⋅⋅⋅𝑚𝑙−1 . We define linear maps 𝜑1 = 𝜏1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝜏𝑙−1 ,

𝜑2 = 𝜂1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝜂𝑙−1 .

(18)

It is easy to see that 𝜑1 and 𝜑2 are canonical maps on 𝐻𝑚1 ⋅⋅⋅𝑚𝑙−1 . For any 𝑌 ∈ 𝐻𝑚1 ⋅⋅⋅𝑚𝑙−1 , since

= (𝑊𝜑2 (𝑌)) ⊗ 𝐼𝑚𝑙 ,

Γ1 = {𝑃2 ⊗ 𝐼𝑚3 ⋅⋅⋅𝑚𝑙 : 𝑃2 ∈ P𝑚2 } , Γ2 = {𝑃2 ⊗ 𝑃3 ⊗ 𝐼𝑚4 ⋅⋅⋅𝑚𝑙 : 𝑃𝑖 ∈ P𝑚𝑖 , 𝑖 ∈ [2, 3]} ,

(19)

(22)

⋅⋅⋅ Γ𝑙−1 = {𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑚𝑙 : 𝑃𝑖 ∈ P𝑚𝑖 , 𝑖 ∈ [2, 𝑙]} . We prove by induction on 𝑘 that 𝜙(𝑃 ⊗ 𝑋) = 0𝑠 ⊕ 𝜓(𝑋) ⊕ 0𝑛−𝑟−𝑠 , ∀𝑋 ∈ Γ𝑘 , 𝑘 ∈ [0, 𝑙 − 1]. Assume for a moment that we have already proved this. Then, when 𝜙 is linear, we can complete the proof of Step 1. Now, we prove the assertion. The case of 𝑘 = 0 is just the assumption. Then, we assume that our statement holds true for 𝑘 − 1, and we consider the image of 𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 for 𝑃𝑖 ∈ P𝑚𝑖 , 𝑖 ∈ [2, 𝑘 + 1]. Because 𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ 𝐼𝑚𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 , 𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ 𝑃𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 ,

(23)

𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ (𝐼𝑚𝑘+1 − 𝑃𝑘+1 ) ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 , we obtain using the property of 𝜙 that 𝜙 (𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ 𝐼𝑚𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ) ∈ P𝑛 , 𝜙 (𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ 𝑃𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ) ∈ P𝑛 ,

(𝜑1 (𝑌) 𝑊) ⊗ 𝐼𝑚𝑙 = 𝜓1 (𝑌 ⊗ 𝐼𝑚𝑙 ) 𝑈 = 𝑈𝜓2 (𝑌 ⊗ 𝐼𝑚𝑙 )

Γ0 = {𝐼𝑚2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝐼𝑚𝑙 } ,

𝜙 (𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ 𝐼𝑚𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ) , − 𝜙 (𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ) ∈ P𝑛 .

(24)

4


We obtain by induction hypothesis and Lemma 2 that 𝜙(𝑃 ⊗ 𝑃2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘+1 ⊗ 𝐼𝑚𝑘+2 ⋅⋅⋅𝑚𝑙 ) = 0𝑠 ⊕ 𝑋𝑟 ⊕ 0𝑛−𝑟−𝑠 , where 𝑋𝑟 ∈ P𝑟 , as desired. Step 2. If 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 > 𝑛. Then, 𝜙 = 0. (𝑚1 )

Proof of Step 2. Since 𝐸𝑖𝑖 (𝑚 ) (𝐸𝑖𝑖 1

(𝑚 ) 𝐸𝑗𝑗 1 )

⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 , ∀𝑖 ∈ [1, 𝑚1 ] and

+ ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 , ∀𝑖 ≠ 𝑗 ∈ [1, 𝑚1 ], we obtain using the property of 𝜙 that (𝑚 ) 𝜙 (𝐸𝑖𝑖 1 (𝑚1 )

𝜙 (𝐸𝑖𝑖

⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) ∈ P𝑛 ,

∀𝑖 ∈ [1, 𝑚1 ] ,

(𝑚 )

⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) + 𝜙 (𝐸𝑗𝑗 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) ∈ P𝑛 ,

(25)

∀𝑖 ≠ 𝑗 ∈ [1, 𝑚1 ] . (𝑚 )

Denote by 𝑟𝑖 = rank 𝜙(𝐸𝑖𝑖 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ), ∀𝑖 ∈ [1, 𝑚1 ]. Using Lemma 1 and composing 𝜙 by a similarity transformation, we may obtain that (𝑚1 )

𝜙 (𝐸𝑖𝑖

⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) = diag (0, . . . , 0, 1, . . . , 1, 0, . . . , 0) , (26)

where diag(0, . . . , 0, 1, . . . , 1, 0, . . . , 0) is the diagonal matrix in which all diagonal entries are zero except those in the (𝑟1 + ⋅ ⋅ ⋅ + 𝑟𝑖−1 + 1)st to the (𝑟1 + ⋅ ⋅ ⋅ + 𝑟𝑖 )th rows. Since 𝑛 < 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 , there exists some 𝑟𝑖 < 𝑚2 ⋅ ⋅ ⋅ 𝑚𝑙 . Without loss of generality, we may assume 𝑟1 < 𝑚2 ⋅ ⋅ ⋅ 𝑚𝑙 . (𝑚 ) Thus, 𝜙(𝐸11 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) = 𝐼𝑟1 ⊕ 0𝑛−𝑟1 . By Step 1, we see that there exists a linear map 𝜓 from 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 to 𝐻𝑟1 satisfying (𝑚 )

𝜓(⊗𝑙2 P𝑚𝑖 ) ⊂ P𝑟1 such that 𝜙(𝐸11 1 ⊗ 𝑋) = 𝜓(𝑋) ⊕ 0𝑛−𝑟1 for all 𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 . We obtain by induction hypothesis 𝜓 = 0. Thus, (𝑚 )

𝜙 (𝐸11 1 ⊗ 𝑋) = 0,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 .

(𝑚 )

(𝑚 )

𝜙(𝐷1𝑗 1 ⊗ 𝑃) = 0. By using the similar argument, we see that (𝑚1 )

(𝑚1 )

⊗ 𝑃) = 0, 𝜙(𝐾𝑖𝑗

⊗ 𝑃) = 0, ∀𝑖 ≠ 𝑗 ∈ [1, 𝑚1 ]. Thus,

𝜙 (𝐴 ⊗ 𝑃) = 0,

∀𝐴 ∈ 𝐻𝑚1 .

(28)

By the arbitrariness of 𝑃 ∈ ⊗𝑙2 P𝑚𝑖 , we have 𝜙 (𝐴 ⊗ 𝑋) = 0,

(𝑚 )

𝜙 (𝐹𝑘 ⊗ 𝑋) = 𝑄 [𝐸𝑘𝑘 1 ⊗ 𝜓𝑘 (𝑋)] 𝑄∗ ,

∀𝐴 ∈ 𝐻𝑚1 , 𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 .

𝜓𝑘 (𝑋) = 𝑈𝑘 𝜋𝑘 (𝑋) 𝑈𝑘∗ ,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑘 ∈ [1, 𝑚1 ] . (32)

Set 𝑈 = 𝑄 diag(𝑈1 , . . . , 𝑈𝑚1 ), we complete the proof of Step 3. By Step 3, we may assume that (𝑚 )

(𝑚 )

𝜙 (𝐸𝑘𝑘 1 ⊗ 𝑋) = 𝐸𝑘𝑘 1 ⊗ 𝜋𝑘 (𝑋) , ∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑘 ∈ [1, 𝑚1 ] ,

Step 4. For ∀𝑖 ≠ 𝑗 ∈ [1, 𝑚1 ], 𝜋𝑖 = 𝜋𝑗 and there exists 𝜆 𝑖𝑗 , 𝜇𝑖𝑗 with |𝜆 𝑖𝑗 | = |𝜇𝑖𝑗 | = 1 such that (𝑚1 )

𝜙 (𝐷𝑖𝑗

(𝑚1 )

⊗ 𝑋) = (𝜆 𝑖𝑗 𝐸𝑖𝑗

(𝑚 )

+ 𝜆𝑖𝑗 𝐸𝑗𝑖 1 ) ⊗ 𝜋1 (𝑋) , ∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 ,

(𝑚1 )

𝜙 (𝐾𝑖𝑗

(𝑚1 )

⊗ 𝑋) = (𝜇𝑖𝑗 𝐸𝑖𝑗

(𝑚 )

+ 𝜇𝑖𝑗 𝐸𝑗𝑖 1 ) ⊗ 𝜋1 (𝑋) , ∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 .

Proof of Step 4. Without loss of generality, we only prove the case of 𝑖 = 1, 𝑗 = 2. (𝑚 ) (𝑚 ) (𝑚 ) Set 𝐹1 = (1/2)𝐸11 1 + (1/2)𝐸22 1 + (1/2)𝐷12 1 , 𝐹2 = (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) (1/2)𝐸11 1 +(1/2)𝐸22 1 −(1/2)𝐷12 1 , 𝐹𝑘 = 𝐸𝑘𝑘 1 for 𝑘 ∈ [3, 𝑚1 ] (if 𝑚1 ≥ 3). By Step 3, there exists a unitary matrix 𝑉 and canonical maps 𝜂𝑘 on 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 such that

(29)

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑘 ∈ [1, 𝑚1 ] .

(35)

This, together with (33), implies that

(𝑚 )

where 𝜋𝑘 is a canonical map on 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , ∀𝑘 ∈ [1, 𝑚1 ].

(34)

(𝑚 )

Step 3. Suppose 𝐹1 , . . . , 𝐹𝑚1 ∈ P𝑚1 are of rank one with 𝐹𝑖 + 𝑗 ∈ [1, 𝑚1 ]. Then, there exists a unitary matrix 𝐹𝑗 ∈ P𝑚1 , ∀𝑖 ≠ 𝑈 of order 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 such that

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑘 ∈ [1, 𝑚1 ] ,

(33)

where 𝜋𝑘 is canonical map on 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , ∀𝑘 ∈ [1, 𝑚1 ].

𝜙 (𝐹𝑘 ⊗ 𝑋) = 𝑉 [𝐸𝑘𝑘 1 ⊗ 𝜂𝑘 (𝑋)] 𝑉∗ ,

Therefore, 𝜙 = 0. 0. We next always assume 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 = 𝑛 and 𝜙 ≠

𝜙 (𝐹𝑘 ⊗ 𝑋) = 𝑈 [𝐸𝑘𝑘 1 ⊗ 𝜋𝑘 (𝑋)] 𝑈∗ ,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , (31)

where linear map 𝜓𝑘 : 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 → 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 satisfying 𝜓𝑘 (⊗𝑙2 P𝑚𝑖 ) ⊂ P𝑚2 ⋅⋅⋅𝑚𝑙 , for any 𝑘 ∈ [1, 𝑚1 ]. We obtain by induction hypothesis that there exists unitary matrices 𝑈𝑘 ∈ M𝑚2 ⋅⋅⋅𝑚𝑙 and canonical maps 𝜋𝑘 on 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 such that

(𝑚 )

we have (𝛼𝐸11 1 + 𝛽𝐸𝑗𝑗 1 + 𝛾𝐷1𝑗 1 ) ⊗ 𝑃 ∈ ⊗𝑙1 P𝑚𝑖 , where 𝛼 and 𝛽 are all roots of 𝑥2 − 𝑥 + 𝛾2 = 0. We obtain using the (𝑚 ) property of 𝜙, (27), and Lemma 3 that 𝜙(𝐸𝑗𝑗 1 ⊗ 𝑃) = 0 and 𝜙(𝐷𝑖𝑗

(𝑚 )

𝑄[𝐸𝑘𝑘 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ]𝑄∗ . Thus, applying Step 1, we obtain that

(27)

|𝛾| ≤ 1/2, For any 𝑗 ∈ [2, 𝑚1 ], 𝑃 ∈ ⊗𝑙2 P𝑚𝑖 and real 𝛾 with 0 ≠ (𝑚 )

Proof of Step 3. Using the similar approach as in the proof of Step 2, we may show that rank 𝜙(𝐹𝑘 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) = 𝑚2 ⋅ ⋅ ⋅ 𝑚𝑙 , ∀𝑘 ∈ [1, 𝑚1 ]. In fact, if rank 𝜙(𝐹𝑘 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) > 𝑚2 ⋅ ⋅ ⋅ 𝑚𝑙 for some 𝑘, then, by Lemma 1, there exists rank 𝜙(𝐹𝑖 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) < 𝑚2 ⋅ ⋅ ⋅ 𝑚𝑙 for some 𝑖. Then, using the similar approach as in the proof of Step 2, we have 𝜙 = 0, which is a contradiction, and there exist a unitary matrix 𝑄 such that 𝜙(𝐹𝑘 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) =

(30)

(𝑚 )

(𝑚 )

𝐸𝑘𝑘 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 = 𝑉 [𝐸𝑘𝑘 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ] 𝑉∗ ,

∀𝑘 ∈ [3, 𝑚1 ] . (36)

Thus, we may set 𝑉 𝑉 𝑉 = [ 11 12 ] ⊕ diag (𝑉33 , . . . , 𝑉𝑚1 𝑚1 ) , 𝑉21 𝑉22 where 𝑉𝑘𝑘 ∈ 𝑀𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑘 ∈ [1, 𝑚1 ].

(37)


5 Step 5. There exists 𝜀 ∈ {−1, 1} such that 𝜇𝑖𝑗 = 𝜀√−1𝜆 𝑖𝑗 , ∀𝑖 ≠ 𝑗 ∈ [1, 𝑚1 ]. If 𝑚1 ≥ 3, then 𝜆 𝑖𝑗 𝜆 𝑗𝑘 = 𝜆 𝑖𝑘 , for distinct 𝑖, 𝑗, 𝑘 ∈ [1, 𝑚1 ].

On one hand, noting that (𝑚 )

(𝑚 )

(𝑚 )

𝐷12 1 = 2𝐹1 − 𝐸11 1 − 𝐸22 1 , (𝑚 )

(𝑚 )

(38)

(𝑚 )

𝐷12 1 = 𝐸11 1 + 𝐸22 1 − 2𝐹2 ,

1 (𝑚 ) 1 (𝑚 ) √2 (𝑚1 ) √2 (𝑚1 ) ( 𝐸𝑖𝑖 1 + 𝐸𝑗𝑗 1 + + 𝐷 𝐾 ) 2 2 4 𝑖𝑗 4 𝑖𝑗

we obtain using (33) and (35) that for all 𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , (𝑚 )

𝜙 (𝐷12 1 ⊗ 𝑋) =

∗ 2𝑉 𝜂 (𝑋) 𝑉11 [ 11 1

− 𝜋1 (𝑋) ∗ 2𝑉21 𝜂1 (𝑋) 𝑉11

∗ 2𝑉11 𝜂1 (𝑋) 𝑉21 ∗ 2𝑉21 𝜂1 (𝑋) 𝑉21 − 𝜋2

⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ∈ ] ⊕ 0,

(𝑋)

(𝑚 )

∗ ∗ 𝜋 (𝑋) − 2𝑉12 𝜂2 (𝑋) 𝑉12 −2𝑉12 𝜂2 (𝑋) 𝑉22 =[ 1 ∗ ∗ ] ⊕ 0. −2𝑉22 𝜂2 (𝑋) 𝑉12 𝜋2 (𝑋) − 2V22 𝜂2 (𝑋) 𝑉22 (40)

On the other hand, for any 𝑃 ∈ ⊗𝑙2 P𝑚𝑖 and real 𝛾 with (𝑚 ) (𝑚 ) (𝑚 ) 0 ≠ |𝛾| ≤ 1/2, we have (𝛼𝐸11 1 +𝛽𝐸22 1 +𝛾𝐷12 1 )⊗𝑃 ∈ ⊗𝑙1 P𝑚𝑖 , 2 2

where 𝛼 and 𝛽 are all roots of 𝑥 − 𝑥 + 𝛾 = 0. We obtain using the property of 𝜙, (33), and Lemma 3 that 2

[𝜙 (𝐷12 1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 )] =

(𝑚 ) 𝜙 (𝐸11 1

⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) +

(𝑚 ) 𝜙 (𝐸22 1

𝑌 0 (𝑚 ) 𝜙 (𝐷12 1 ⊗ 𝑋) = [ 𝑚∗1 12 ] ⊕ 0, 𝑌12 0𝑚1

(41) ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ) ,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 ,

(42)

where 𝑌12 ∈ 𝑀𝑚1 . Choosing 𝑋 = 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 , we get using (39), ∗ ∗ ∗ ∗ = 2𝑉22 𝑉22 = 2𝑉21 𝑉21 = 2𝑉12 𝑉12 = (40), and (42) that 2𝑉11 𝑉11 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 . Hence, by (39), (40), and (42) again, one can obtain that −1

∗ ∗ 𝜋1 (𝑋) (𝑉11 ) 𝑉21

−1

∗ ∗ By Lemma 5, we have 𝜋1 = 𝜋2 and (𝑉11 ) 𝑉21 = 𝜆 12 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 with 𝜆 12 ≠ 0. Hence, (𝑚 )

(𝑚 )

𝜙 (𝐷12 1 ⊗ 𝑋) = (𝜆 12 𝐸12 1 + 𝜆12 𝐸21 1 ) ⊗ 𝜋1 (𝑋) , ∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝜆 12 ≠ 0.

(44)

This, together with (41), implies that |𝜆 12 | = 1. Similarly, (𝑚 )

we obtain using the property of 𝜙, (33), and Step 4 that √2 (𝜆 𝑖𝑗 + 𝜇𝑖𝑗 )] 4 ] ∈ P2 . ] 1 (𝜆𝑖𝑗 + 𝜇𝑖𝑗 ) 2 [ 4 ] [ [ [ √2

1 2

(47)

Hence, (𝜆 𝑖𝑗 +𝜇𝑖𝑗 )(𝜆𝑖𝑗 +𝜇𝑖𝑗 ) = 2. It follows from |𝜆 𝑖𝑗 | = |𝜇𝑖𝑗 | = 1 that 𝜆2𝑖𝑗 = −𝜇𝑖𝑗2 . Thus, there exists 𝜀𝑖𝑗 ∈ {−1, 1} such that 𝜇𝑖𝑗 = 𝜀𝑖𝑗 √−1𝜆 𝑖𝑗 . If 𝑚1 ≥ 3, since 1 (𝑚1 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) + 𝐸𝑗𝑗 1 + 𝐸𝑘𝑘 1 + 𝐷𝑖𝑗 1 + 𝐷𝑖𝑘 1 + 𝐷𝑗𝑘 1 ) (𝐸 3 𝑖𝑖 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 , 1 (𝑚1 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) + 𝐸𝑗𝑗 1 + 𝐸𝑘𝑘 1 + 𝐾𝑖𝑗 1 + 𝐾𝑖𝑘 1 + 𝐷𝑗𝑘 1 ) (𝐸 3 𝑖𝑖 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ∈

(48)

⊗𝑙1 P𝑚𝑖 ,

1 (𝑚1 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) + 𝐸𝑗𝑗 1 + 𝐸𝑘𝑘 1 + 𝐷𝑖𝑗 1 + 𝐾𝑖𝑘 1 + 𝐾𝑗𝑘 1 ) (𝐸 3 𝑖𝑖 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 P𝑚𝑖 , we obtain using the property of 𝜙, (33), and Step 4 that

∗ ∗ −1 = 2𝑉11 𝜂1 (𝑋) 𝑉21 = −2𝑉12 𝜂2 (𝑋) 𝑉22 = −𝑉12 𝑉22 𝜋2 (𝑋) . (43)

(𝑚 )

(46)

⊗𝑙1 P𝑚𝑖 ,

(39)

𝜙 (𝐷12 1 ⊗ 𝑋)

(𝑚 )

Proof of Step 5. Since

(𝑚 )

(𝑚 )

𝜙 (𝐾12 1 ⊗ 𝑋) = (𝜇12 𝐸12 1 + 𝜇12 𝐸21 1 ) ⊗ 𝜋1 (𝑋) , 󵄨 󵄨 ∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 󵄨󵄨󵄨𝜇12 󵄨󵄨󵄨 = 1. This completes the proof of Step 4.

(45)

1 𝜆 𝑖𝑗 𝜆 𝑖𝑘 1[ ] 𝜆 [ 𝑖𝑗 1 𝜆 𝑗𝑘 ] ∈ P3 , 3 [𝜆𝑖𝑘 𝜆𝑗𝑘 1 ]

(49)

1 𝜀𝑖𝑗 𝜆 𝑖𝑗 √−1 𝜀𝑖𝑘 𝜆 𝑖𝑘 √−1 1[ ] √ 1 𝜆 𝑗𝑘 ] ∈ P3 , [ −𝜀𝑖𝑗 𝜆𝑖𝑗 −1 3 𝜆𝑗𝑘 1 [−𝜀𝑖𝑘 𝜆𝑖𝑘 √−1 ]

(50)

1 𝜆 𝑖𝑗 𝜀𝑖𝑘 𝜆 𝑖𝑘 √−1 1[ ] 𝜆𝑖𝑗 1 𝜀𝑗𝑘 𝜆 𝑗𝑘 √−1] ∈ P3 . [ 3 1 [−𝜀𝑖𝑘 𝜆𝑖𝑘 √−1 −𝜀𝑗𝑘 𝜆𝑗𝑘 √−1 ]

(51)

It follows from (49) that 𝜆 𝑖𝑗 𝜆 𝑗𝑘 = 𝜆 𝑖𝑘 . This, together with (50), implies that 𝜀𝑖𝑗 = 𝜀𝑖𝑘 . Similarly, (51) implies that 𝜀𝑖𝑘 = 𝜀𝑗𝑘 . 𝑗 ∈ Thus, all numbers 𝜀𝑖𝑗 have to be the same for any 𝑖 ≠ [1, 𝑚1 ]. This completes the proof of this step.

6


By Steps 3–5, we set 𝑄 = diag(1, 𝜆 12 , . . . , 𝜆 1𝑚1 ), then 𝑄 is unitary such that (𝑚 )

(𝑚 )

𝜙 (𝐸𝑘𝑘 1 ⊗ 𝑋) = [𝑄𝐸𝑘𝑘 1 𝑄∗ ] ⊗ 𝜋1 (𝑋) ,

Γ0 = {𝐼𝑚1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝐼𝑚𝑙 } ,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑘 ∈ [1, 𝑚1 ] , (𝑚 ) 𝜙 (𝐷𝑖𝑗 1

⊗ 𝑋) =

(𝑚 ) [𝑄𝐷𝑖𝑗 1 𝑄∗ ]

⊗ 𝜋1 (𝑋) ,

∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑖 ≠ 𝑗 ∈ [1, 𝑚1 ] , (𝑚 ) 𝜙 (𝐾𝑖𝑗 1

⊗ 𝑋) =

(𝑚 ) 𝜀 [𝑄𝐾𝑖𝑗 1 𝑄∗ ]

Γ1 = {𝑃1 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 : 𝑃1 ∈ P𝑚1 } , (52)

Thus, set 𝑈 = 𝑄 ⊗ 𝐼𝑚2 ⋅⋅⋅𝑚𝑙 , we have (53)

∀𝐴 ∈ 𝐻𝑚1 , 𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 .

(54)

This completes the proof of Theorem 6. Remark 7. When 𝑙 ≥ 2, the linear transformation that maps the set of tensor product of idempotent matrices into idempotent matrix set does not necessarily preserve idempotent matrices. For example, let 𝜙 (𝐴 1 ⊗ 𝐴 2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝐴 𝑙 ) = 𝐴𝑇1 ⊗ 𝐴 2 ⊗ ⋅ ⋅ ⋅ ⊗ 𝐴 𝑙 , 1 (𝑚1 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) ⊗ 𝐸11 2 + 𝐸22 1 ⊗ 𝐸22 2 + 𝐸12 1 (𝐸 2 11 (𝑚 )

(𝑚 )

(55)

(𝑚 )

⊗𝐸12 2 + 𝐸21 1 ⊗ 𝐸21 2 ) ⊗ 𝐼𝑚3 ⋅⋅⋅𝑚𝑙 . Then 𝜙(⊗𝑙1 P𝑚𝑖 ) ⊂ P𝑚1 ⋅⋅⋅𝑚𝑙 and 𝐶 ∈ P𝑚1 ⋅⋅⋅𝑚𝑙 , but 𝜙 (𝐶) =

(𝑚 )

(59)

where 𝜓𝑝 (𝑋) ∈ P𝑝 and 𝜓𝑞 (𝑋) ∈ P𝑞 . We assume that our statement holds true for 𝑘 − 1 and prove it for 𝑘. For any 𝑋 = 𝑃1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘 ⊗ 𝐼𝑚𝑘+1 ⋅⋅⋅𝑚𝑙 ∈ Γ𝑘 , choosing 𝑌 = 𝑃1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑘−1 ⊗ 𝐼𝑚𝑘 ⊗ 𝐼𝑚𝑘+1 ⋅⋅⋅𝑚𝑙 ∈ Γ𝑘−1 , we have by induction hypothesis that 𝜙 (𝑌) = 𝜓𝑝 (𝑌) ⊕ −𝜓𝑞 (𝑌) ⊕ 0𝑛−𝑝−𝑞 ,

(60)

with 𝜓𝑝 (𝑌) ∈ P𝑝 and 𝜓𝑞 (𝑌) ∈ P𝑞 . Since 𝑋 ∈ ⊗𝑙1 T𝑚𝑖 , 𝑌−𝑋 ∈ ⊗𝑙1 T𝑚𝑖 and 𝑌−2𝑋 ∈ ⊗𝑙1 T𝑚𝑖 . Using the property of 𝜙, we have 𝜙(𝑋) ∈ T𝑛 , 𝜙(𝑌) − 𝜙(𝑋) ∈ T𝑛 and 𝜙(𝑌) − 2𝜙(𝑋) ∈ T𝑛 . By Lemma 4, we have (61)

with 𝜓𝑝 (𝑋) ∈ P𝑝 and 𝜓𝑞 (𝑋) ∈ P𝑞 . This implies that (59) holds. As 𝜙 is linear, we can expend 𝜓𝑝 to be a linear map from 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 to 𝐻𝑝 and 𝜓𝑞 to be a linear map from 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 to 𝐻𝑞 , then 𝜙 (𝑋) = 𝜓𝑝 (𝑋) ⊕ −𝜓𝑞 (𝑋) ⊕ 0𝑛−𝑝−𝑞 ,

1 (𝑚1 ) (𝑚 ) (𝑚 ) (𝑚 ) (𝑚 ) ⊗ 𝐸11 2 + 𝐸22 1 ⊗ 𝐸22 2 + 𝐸21 1 (𝐸 2 11 (𝑚 )

∀𝑋 ∈ Γ𝑘 ,

𝜙 (𝑋) = 𝜓𝑝 (𝑋) ⊕ −𝜓𝑞 (𝑋) ⊕ 0𝑛−𝑝−𝑞 ,

∀𝐴 𝑖 ∈ H𝑚𝑖 , 𝑖 ∈ [1, 𝑙] , 𝐶=

It is obvious that Γ𝑙 = ⊗𝑙1 P𝑚𝑖 and Γ𝑘 ⊂ ⊗𝑙1 T𝑚𝑖 , ∀𝑘 ∈ [0, 𝑙]. Since 𝐼𝑚1 ⋅⋅⋅𝑚𝑙 ∈ ⊗𝑙1 T𝑚𝑖 , we have 𝜙(𝐼𝑚1 ⋅⋅⋅𝑚𝑙 ) ∈ T𝑛 . Without loss of generality, we may assume that 𝜙(𝐼𝑚1 ⋅⋅⋅𝑚𝑙 ) = 𝐼𝑝 ⊕ −𝐼𝑞 ⊕ 0𝑛−𝑝−𝑞 . We next prove by induction on 𝑘 that 𝜙 (𝑋) = 𝜓𝑝 (𝑋) ⊕ −𝜓𝑞 (𝑋) ⊕ 0𝑛−𝑝−𝑞 ,

or 𝜙 (𝐴 ⊗ 𝑋) = 𝑈 [𝐴𝑇 ⊗ 𝜋1 (𝑋)] 𝑈∗ ,

(58)

Γ𝑙 = {𝑃1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑃𝑚𝑙 : 𝑃𝑖 ∈ P𝑚𝑖 , 𝑖 ∈ [1, 𝑙]} .

𝑗 ∈ [1, 𝑚1 ] . ∀𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙 , 𝑖 ≠

∀𝐴 ∈ 𝐻𝑚1 , 𝑋 ∈ 𝐻𝑚2 ⋅⋅⋅𝑚𝑙

Γ2 = {𝑃1 ⊗ 𝑃2 ⊗ 𝐼𝑚3 ⋅⋅⋅𝑚𝑙 : 𝑃𝑖 ∈ P𝑚𝑖 , 𝑖 ∈ [1, 2]} , ⋅⋅⋅

⊗ 𝜋1 (𝑋) ,

𝜙 (𝐴 ⊗ 𝑋) = 𝑈 [𝐴 ⊗ 𝜋1 (𝑋)] 𝑈∗ ,

Proof. The sufficiency part is clear. We give the proof of the necessity part. Set

∀𝑋 ∈ 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 , (62)

with 𝜓𝑝 (⊗𝑙1 P𝑚𝑖 ) ⊂ P𝑝 and 𝜓𝑞 (⊗𝑙1 P𝑚𝑖 ) ⊂ P𝑞 , we obtain using Theorem 6 that

(𝑚 )

⊗𝐸12 2 + 𝐸12 1 ⊗ 𝐸21 2 ) ⊗ 𝐼𝑚3 ⋅⋅⋅𝑚𝑙 ∉ P𝑚1 ⋅⋅⋅𝑚𝑙 . (56)

(a) if 𝑛 < 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 , then 𝑝, 𝑞 < 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 , 𝜓𝑝 = 0, 𝜓𝑞 = 0, thus, 𝜙 = 0, and

As the application of Theorem 6, we give the following theorem.

(b) if 𝜙 ≠ 0, then 𝑛 = 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 and 𝑝 = 𝑛 or 𝑞 = 𝑛, thus 𝜙 = 𝜓𝑝 or 𝜙 = −𝜓𝑞 .

Theorem 8. Suppose 𝜙 is a linear map from 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 to 𝐻𝑛 with 𝑛 ≤ 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 . Then, 𝜙(⊗𝑙1 T𝑚𝑖 ) ⊂ T𝑛 if and only if either 𝜙 = 0 or 𝑛 = 𝑚1 ⋅ ⋅ ⋅ 𝑚𝑙 , there exist a unitary matrix 𝑈 ∈ 𝑀𝑛 , 𝜀 ∈ {−1, 1} and a canonical map 𝜋 on 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 such that ∗

𝜙 (𝑋) = 𝜀𝑈𝜋 (𝑋) 𝑈 ,

∀𝑋 ∈ 𝐻𝑚1 ⋅⋅⋅𝑚𝑙 .

(57)

This completes the proof of Theorem 8.

Acknowledgments The authors show great thanks to the referee for his/her careful reading of the paper and valuable comments which greatly improved the readability of the paper. Jinli Xu is supported

Journal of Function Spaces and Applications in part by NSFC (11171294), Natural Science Foundation of Heilongjiang Province of China (Grant no. A201013). Baodong Zheng is supported by the National Natural Science Foundation Grants of China (Grant no. 10871056). Hongmei Yao is supported by the Fundamental Research Funds for the Central Universities.

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Research Article Exponential Stability of a Linear Distributed Parameter Bioprocess with Input Delay in Boundary Control Fu Zheng,1 LingLing Fu,1 and MingYan Teng2 1 2

Department of Mathematics, Bohai University, Jinzhou 121013, China Beijing Institute of Information and Control, Beijing 100037, China

Correspondence should be addressed to Fu Zheng; [email protected] Received 9 April 2013; Accepted 20 July 2013 Academic Editor: Gen-Qi Xu Copyright © 2013 Fu Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a linear distributed parameter bioprocess with boundary control input possessing a time delay. Using a simple boundary feedback law, we show that the closed-loop system generates a uniformly bounded 𝐶0 -semigroup of linear operators under a certain condition with respect to the feedback gain. After analyzing the spectrum configuration of closed-loop system and verifying the spectrum determined growth assumption, we show that the closed-loop system is exponentially stable. Thus, we demonstrate that the linear distributed parameter bioprocess preserves the exponential stability for arbitrary time delays.

1. Introduction In a practical control system, there is often a time delay between the controller to be implemented and the information via the observation of the system. These hereditary effects are sometime unavoidable because they might turn a wellbehaved system into a wild one. A simple example can be found in Gumowski and Mira [1], where they demonstrated that the occurrence of delays could destroy the stability and cause periodic oscillations in a system governed by differential equation. Datko [2, 3] illustrated that an arbitrary small time delay in the control could destabilize a boundary feedback hyperbolic control system as well. On the other side, the inclusion of an appropriate time delay effect can sometime improve the performance of the system (e.g., see [3–7]). When the time delay appears, redesigning a stabilizing controller becomes thereby sometimes necessary because the stabilization by the PI output feedback becomes defective or the stabilization is not robust to time delay. The stabilization with time delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. However, this does not mean that there is no stabilizing controller in the presence of time delay. You can refer to [8–12] for some successful examples.

Motivated by these works, we will introduce time delays to a linear distributed parameter bioprocess and investigate the effect of time delays on exponential stability of the system. The linear distributed parameter bioprocess treated here was firstly discussed by Bourrel and Dochain in [13]. They showed that the system with zero boundary input is exponentially stable. Following [13], Sano considered the linear distributed parameter bioprocess from the feedback control point of view in [14]. Namely, the control input and the measured output were imposed on the boundaries, and a simply proportional feedback controller was designed. By using Huang’s result in [15], He showed that the closed-loop system is exponentially stable under a certain condition with respect to the feedback gain and further that the exponential decay rate of the system with zero input was derived by letting the feedback gain tend to zero. However, if time delays in the boundary input arise in this linear distributed parameter bioprocess, we want to pose a question. Is the stabilization robust to time delays for the proportional feedback controller? The present paper is devoted to answering this question. The content of this paper is organized as follows. In Section 2 we will introduce the linear distributed parameter bioprocess mentioned previously and formulate our problem in a suitable Hilbert space. We show that the closed-loop

2


system generates a uniformly bounded 𝐶0 -semigroup of linear operators and obtain the wellposedness of the system as well. In Section 3, we carry out a spectral analysis and obtain the spectrum configuration of the closed-loop system. From verifying the spectrum determined growth assumption, we show that the closed-loop system is exponentially stable. In the last section, a concise conclusion is given.

𝜕𝑧2 (𝑡, 𝑥) = 𝑎3 𝑧1 (𝑡, 𝑥) − 𝑎4 𝑧2 (𝑡, 𝑥) , 𝜕𝑡 (𝑡, 𝑥) ∈ (0, ∞) × (0, 1) , 𝜕𝑧 𝜕𝑧3 (𝑡, 𝑥) = −𝜏−1 3 (𝑡, 𝑥) , 𝜕𝑡 𝜕𝑥 𝑧3 (𝑡, 0) = 𝑧1 (𝑡, 1) ,

𝑧1 (𝑡, 0) = −𝑘𝑧3 (𝑡, 1) ,

𝑧1 (0, 𝑥) = 𝑧10 (𝑥) ,

2. System Description and Wellposedness of the System

𝑡 ∈ (0, ∞) ,

𝑧2 (0, 𝑥) = 𝑧20 (𝑥) ,

𝑧3 (0, 𝑥) = 𝑧30 (−𝜏𝑥) ,

We will consider the following type of linear distributed parameter bioprocess model in which time delays occur in boundary control input:

(𝑡, 𝑥) ∈ (0, ∞) × (0, 1) ,

𝑥 ∈ (0, 1) . (3)

We take the state Hilbert space H, 3

H = 𝐿2 [0, 1] × 𝐿2 [0, 1] × 𝐿2 [0, 1] = (𝐿2 [0, 1]) , 𝜕𝑧1 𝜕𝑧 (𝑡, 𝑥) = −] 1 (𝑡, 𝑥) − 𝑎1 𝑧1 (𝑡, 𝑥) − 𝑎2 𝑧2 (𝑡, 𝑥) , 𝜕𝑡 𝜕𝑥 (𝑡, 𝑥) ∈ (0, ∞) × (0, 1) , 𝜕𝑧2 (𝑡, 𝑥) = 𝑎3 𝑧1 (𝑡, 𝑥) − 𝑎4 𝑧2 (𝑡, 𝑥) , 𝜕𝑡 (𝑡, 𝑥) ∈ (0, ∞) × (0, 1) ,

equipped with inner product

𝜕𝑧 𝜕𝑧1 (𝑡, 𝑥) = −] 1 (𝑡, 𝑥) − 𝑎1 𝑧1 (𝑡, 𝑥) − 𝑎2 𝑧2 (𝑡, 𝑥) , 𝜕𝑡 𝜕𝑥 (𝑡, 𝑥) ∈ (0, ∞) × (0, 1) ,

0

0

(5) 𝑓, 𝑔 ∈ H.

+ ∫ 𝑓3 (𝑥) 𝑔3 (𝑥)𝑑𝑥, 0

Define the operator 𝐴 : 𝐷(𝐴) ⊂ H → H as −]

𝜕𝑧1 𝜕𝑧 (𝑡, 𝑥) = −] 1 (𝑡, 𝑥) − 𝑎1 𝑧1 (𝑡, 𝑥) − 𝑎2 𝑧2 (𝑡, 𝑥) , 𝜕𝑡 𝜕𝑥 (𝑡, 𝑥) ∈ (0, ∞) × (0, 1) ,

Setting 𝑧3 (𝑡, 𝑥) = 𝑧1 (𝑡 − 𝑥𝜏, 1), (2) is equivalent to

1

1

(1)

where 𝑧1 (𝑡, 𝑥), 𝑧2 (𝑡, 𝑥) ∈ R are the deviations of substrate and biomass concentrations from steady-state values at the time 𝑡 and at the point 𝑥 ∈ (0, 1), respectively. And 𝑢(𝑡) ∈ R is the control input, 𝑦(𝑡) ∈ R is the measured output, ] > 0 is the fluid superficial velocity, 𝑎1 , 𝑎2 , 𝑎3 , and 𝑎4 are positive constants, and 𝜏 ≥ 0 is the length of time delay. As usual, we adopt the simple feedback control law 𝑢(𝑡) = −𝑘𝑦(𝑡) with 𝑘 > 0 which results in the following closed-loop system:

𝑧1 (𝑡, 0) = −𝑘𝑧1 (𝑡 − 𝜏, 1) , 𝑡 ∈ (0, ∞) , 𝑧1 (0, 𝑥) = 𝑧10 (𝑥) , 𝑧2 (0, 𝑥) = 𝑧20 (𝑥) , 𝑥 ∈ (0, 1) .

1

⟨𝑓, 𝑔⟩H = ∫ 𝑓1 (𝑥) 𝑔1 (𝑥)𝑑𝑥 + ∫ 𝑓2 (𝑥) 𝑔2 (𝑥)𝑑𝑥

𝑧1 (𝑡, 0) = 𝑢 (𝑡 − 𝜏) , 𝑡 ∈ (0, ∞) , 𝑧1 (0, 𝑥) = 𝑧10 (𝑥) , 𝑧2 (0, 𝑥) = 𝑧20 (𝑥) , 𝑥 ∈ (0, 1) , 𝑦 (𝑡) = 𝑧1 (𝑡, 1) , 𝑡 ∈ (0, ∞) ,

𝜕𝑧2 (𝑡, 𝑥) = 𝑎3 𝑧1 (𝑡, 𝑥) − 𝑎4 𝑧2 (𝑡, 𝑥) , 𝜕𝑡 (𝑡, 𝑥) ∈ (0, ∞) × (0, 1) ,

(4)

( 𝐴𝑓 = (

𝑑 − 𝑎1 −𝑎2 𝑑𝑥 𝑎3 −𝑎4 0

0

(

0 𝑓1 ) 𝑓 ) ( 2) , 𝑓3 𝑑 −𝜏−1 𝑑𝑥 ) 0

⊥

𝑓 = (𝑓1 , 𝑓2 , 𝑓3 ) ∈ 𝐷 (𝐴) , 𝐷 (𝐴) = {𝑓 ∈ 𝐻1 (0, 1) × 𝐿2 [0, 1] ×𝐻1 (0, 1) | 𝑓1 (0) = −𝑘𝑓3 (1) , 𝑓1 (1) = 𝑓3 (0)} . (6) Then the system (3) can be written as 𝑧1 (𝑡) 𝑧1 (0) 𝑧10 𝑧1 (𝑡) 𝑑 (𝑧2 (𝑡)) = 𝐴 (𝑧2 (𝑡)) , (𝑧2 (0)) = (𝑧20 ) , 𝑑𝑡 𝑧 (𝑡) 𝑧 (𝑡) 𝑧 (0) 𝑧

(2)

3

3

3

(7)

30

2

where 𝑧10 , 𝑧20 , 𝑧30 ∈ 𝐿 [0, 1]. Therefore, if the operator 𝐴 generates a 𝐶0 -semigroup 𝑇(𝑡) on H, then a unique solution of (7) is expressed as 𝑧10 𝑧1 (𝑡) (𝑧2 (𝑡)) = 𝑇 (𝑡) (𝑧20 ) , 𝑧3 (𝑡) 𝑧30

∀𝑡 ≥ 0,

(8)

which means that the unique solution to (2) or (3) exists. Let us define the operator 𝑀 ∈ 𝐿(H) as √𝑎2 0 0 𝑀 = ( 0 √𝑎3 0) 0 0 1

(9)


3

and consider the properties of a semigroup generated by the operator 𝑀−1 𝐴𝑀. The operator 𝑀−1 𝐴𝑀 is expressed as

Next, for all 𝑓 ∈ 𝐷(𝑀−1 𝐴𝑀) and 𝑔 ∈ H, we have ⟨𝑀−1 𝐴𝑀𝑓, 𝑔⟩H

−1

𝑀 𝐴𝑀𝑓 𝑑 − 𝑎1 −√𝑎2 𝑎3 𝑑𝑥 −𝑎4 ( √𝑎2 𝑎3 =(

=

−]

0

0

(

0 𝑓1 0 ) ) (𝑓2 ) , 𝑓3 𝑑 −𝜏−1 𝑑𝑥 )

⊥

=

−1

𝑓 = (𝑓1 , 𝑓2 , 𝑓3 ) ∈ 𝐷 (𝑀 𝐴𝑀) , 𝐷 (𝑀−1 𝐴𝑀) = {𝑓 ∈ 𝐻1 (0, 1) × 𝐿2 [0, 1] × 𝐻1 (0, 1) | 𝑓1 (0) = −𝑘𝑓3 (1) , 𝑓1 (1) = 𝑓3 (0) } . (10)

1

1

1 ∫ (−]𝑓1󸀠 (𝑥) − 𝑎1 𝑓1 (𝑥)) 𝑔1 (𝑥)𝑑𝑥 ] 0 −

𝑎4 1 √𝑎2 𝑎3 1 ∫ 𝑓2 (𝑥) 𝑔2 (𝑥)𝑑𝑥 − ∫ 𝑓2 (𝑥) 𝑔1 (𝑥)𝑑𝑥 ] 0 ] 0

+

1 √𝑎2 𝑎3 1 ∫ 𝑓1 (𝑥) 𝑔2 (𝑥)𝑑𝑥 − ∫ 𝑓3󸀠 (𝑥) 𝑔3 (𝑥)𝑑𝑥 ] 0 0

1 1 ∫ 𝑓 (𝑥) (]𝑔1󸀠 (𝑥) − 𝑎1 𝑔1 (𝑥) + √𝑎2 𝑎3 𝑔2 (𝑥)) 𝑑𝑥 ] 0 1 −

1 1 ∫ 𝑓 (𝑥) (𝑎4 𝑔2 (𝑥) + √𝑎2 𝑎3 𝑔1 (𝑥)) 𝑑𝑥 ] 0 2

+

1 1 ∫ 𝑓 (𝑥) 𝑔3󸀠 (𝑥)𝑑𝑥 𝜏 0 3

1 1 1 − [𝑓1 (𝑥) 𝑔1󸀠 (𝑥)] − [𝑓3 (𝑥) 𝑔3 (𝑥)]0 . 0 𝜏

Firstly, we have the following result. Theorem 1. Suppose that the feedback gain 𝑘 is chosen such that 0 < 𝑘 < 1. Then, the operator 𝐴 defined by (6) generates a uniformly bounded 𝐶0 -semigroup 𝑇(𝑡) on H. Proof. In order to prove that 𝐴 generates a uniformly bounded 𝐶0 -semigroup, we introduce a new equivalent inner product in H:

From the definition of the adjoint operator and the conditions in the domain of the 𝑀−1 𝐴𝑀, we know that 𝑑 − 𝑎1 √𝑎2 𝑎3 𝑑𝑥 −1 (𝑀 𝐴𝑀) 𝑓 = ( −√𝑎2 𝑎3 −𝑎4 ]

∗

0 ⟨𝑓, 𝑔⟩H1 =

1

1

1 1 ∫ 𝑓 (𝑥) 𝑔1 (𝑥)𝑑𝑥 + ∫ 𝑓2 (𝑥) 𝑔2 (𝑥)𝑑𝑥 ] 0 1 ] 0 1

+ 𝜏 ∫ 𝑓3 (𝑥) 𝑔3 (𝑥)𝑑𝑥, 0

⊥

because it is easily verified that 𝑀−1 𝐴𝑀 is a closed and densely defined linear operator. Thus, by similar arguments as previously, we obtain that, for all 𝑓 ∈ 𝐷((𝑀−1 𝐴𝑀)∗ ), ∗

H1

𝑎1 󵄩󵄩 󵄩󵄩 𝑎 󵄩 󵄩 − 4 󵄩󵄩𝑓 󵄩󵄩 2 󵄩𝑓 󵄩 2 ] 󵄩 1 󵄩𝐿 (0,1) ] 󵄩 2 󵄩𝐿 (0,1) 󵄨2 󵄨 󵄨2 󵄨2 󵄨 󵄨2 󵄨 󵄨 + (󵄨󵄨󵄨𝑓1 (1)󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑓1 (0)󵄨󵄨󵄨 ) + (󵄨󵄨󵄨𝑓3 (1)󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑓3 (0)󵄨󵄨󵄨 ) 𝑎 󵄩 󵄩 𝑎 󵄩 󵄩 󵄨2 󵄨 = − 1 󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩𝐿2 (0,1) − 4 󵄩󵄩󵄩𝑓2 󵄩󵄩󵄩𝐿2 (0,1) − (1 − 𝑘2 ) 󵄨󵄨󵄨𝑓1 (0)󵄨󵄨󵄨 ≤ 0. ] ]

=−

hold for all 𝑓 ∈ 𝐷(𝑀−1 𝐴𝑀). If the feedback gain 𝑘 satisfies 0 < 𝑘 < 1, then it is easy to see

1

| 𝑓3 (1) = −𝑘𝑓1 (0) , 𝑓3 (0) = 𝑓1 (1) } (15)

Re ⟨(𝑀−1 𝐴𝑀) 𝑓, 𝑓⟩

𝑎1 󵄩󵄩 󵄩󵄩 𝑎 󵄩 󵄩 󵄨2 󵄨 − 4 󵄩󵄩𝑓 󵄩󵄩 2 − (1 − 𝑘2 ) 󵄨󵄨󵄨𝑓3 (1)󵄨󵄨󵄨 󵄩𝑓 󵄩 2 ] 󵄩 1 󵄩𝐿 (0,1) ] 󵄩 2 󵄩𝐿 (0,1) (12)

∀𝑓 ∈ 𝐷 (𝑀−1 𝐴𝑀) .

∗

∗

1

Re ⟨𝑀−1 𝐴𝑀𝑓, 𝑓⟩H ≤ 0,

𝑓1 ) (𝑓2 ) , 𝑑 𝑓3 𝜏−1 𝑑𝑥 0

𝐷 ((𝑀−1 𝐴𝑀) ) = {𝑓 ∈ 𝐻1 (0, 1) × 𝐿2 [0, 1] × 𝐻1 (0, 1)

𝑓, 𝑔 ∈ H.

𝑎 󵄩 󵄩 𝑎 󵄩 󵄩 = − 1 󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩𝐿2 (0,1) − 4 󵄩󵄩󵄩𝑓2 󵄩󵄩󵄩𝐿2 (0,1) ] ] 󵄨󵄨2 󵄨󵄨 󵄨2 󵄨2 󵄨 󵄨2 󵄨󵄨 󵄨 − (󵄨󵄨𝑓1 (1)󵄨󵄨 − 󵄨󵄨𝑓1 (0)󵄨󵄨󵄨 ) − (󵄨󵄨󵄨𝑓3 (1)󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑓3 (0)󵄨󵄨󵄨 ) =−

0

0

𝑓 = (𝑓1 , 𝑓2 , 𝑓3 ) ∈ 𝐷 ((𝑀−1 𝐴𝑀) ) ,

(11)

From the domain of the operator 𝑀−1 𝐴𝑀 it follows that the identities Re ⟨𝑀−1 𝐴𝑀𝑓, 𝑓⟩H

(14)

(13)

(16) It follows from Re ⟨𝑀−1 𝐴𝑀𝑓, 𝑓⟩H1 ≤ 0 and Re ⟨(𝑀−1 𝐴𝑀)∗ 𝑓, 𝑓⟩H1 ≤ 0 that the operators 𝑀−1 𝐴𝑀

4


and (𝑀−1 𝐴𝑀)∗ are dissipative. According to Proposition 3.1.11 of [16], the closed operator 𝑀−1 𝐴𝑀 is m-dissipative. Therefore, Lumer-Philips theorem implies that 𝑀−1 𝐴𝑀 generates contraction semigroups 𝑆(𝑡) on state space H. For all 𝑡 ≥ 0, if we define 𝑇(𝑡) by 𝑇(𝑡) = 𝑀𝑆(𝑡)𝑀−1 , then the semigroups 𝑇(𝑡) and 𝑆(𝑡) are similar. This means that the 𝐶0 -semigroups 𝑇(𝑡) are uniformly bounded (i.e., a 𝐶0 semigroup with the operator norm bound ‖ 𝑇(𝑡)‖L(H1 ) ≤ 𝑀, for some 𝑀 > 0 and ∀𝑡 ≥ 0) and their generator is 𝐴. Thus, the proof of the theorem is complete since the new inner product is equivalent to the original one.

3. Exponential Stability of the System (7) In order to show the exponential stability of the system (7), we will verify that the operator 𝐴 satisfies the conditions of Theorem 1.1 of [14], which is a summarized edition of Huang’s result on the spectrum determined growth assumption in [15]. To this end, we should analyze the spectrum configuration of the operator 𝐴 and show that the norm of the resolvent is uniformly bounded in any given right half-plane. All these results are collected in the following two lemmas.

where 𝜇(𝑘, 𝜏) is defined by 𝜇 (𝑘, 𝜏) 𝑎 − ] log 𝑘 { , 𝛽1 (𝑘, 𝜏) , 𝛽2 (𝑘, 𝜏)} max {−𝑎4 , − 1 { { ]𝜏 + 1 { { { { 2 { { (𝑎1 + 𝑎4 − ] log 𝑘) { { > 4 [𝑎2 𝑎3 + 𝑎4 (𝑎1 − ] log 𝑘)] , if { { { ]𝜏 + 1 := { 2 { { 𝑎1 − ] log 𝑘 (𝑎1 + 𝑎4 − ] log 𝑘) { { max {−𝑎4 , − ,− } { { ]𝜏 + 1 2 (]𝜏 + 1) { { { { 2 { { { (𝑎1 + 𝑎4 − ] log 𝑘) if = 4 [𝑎2 𝑎3 + 𝑎4 (𝑎1 − ] log 𝑘)] , { ]𝜏 + 1 𝜇 (𝑘, 𝜏) := max {−𝑎4 ,

2

if

(17)

(𝑎1 + 𝑎4 − ] log 𝑘) ]𝜏 + 1

< 4 [𝑎2 𝑎3 + 𝑎4 (𝑎1 − ] log 𝑘)] ,

Lemma 2. Suppose that the assumption of Theorem 1 is satisfied. Then, the following inequality holds: sup {Re (𝜆) : 𝜆 ∈ 𝜎 (𝐴)} ≤ 𝜇 (𝑘, 𝜏) ,

−𝑎1 + ] log 𝑘 } ]𝜏 + 1

(18) with 𝛽1 (𝑘, 𝜏) and 𝛽2 (𝑘, 𝜏) being 2

𝛽1 (𝑘, 𝜏) =

− (𝑎1 + 𝑎4 − ] log 𝑘) + √(𝑎1 + 𝑎4 − ] log 𝑘) − 4 (]𝜏 + 1) [𝑎2 𝑎3 + 𝑎4 (𝑎1 − ] log 𝑘)] 2 (]𝜏 + 1)

, (19)

2

𝛽2 (𝑘, 𝜏) =

− (𝑎1 + 𝑎4 − ] log 𝑘) − √(𝑎1 + 𝑎4 − ] log 𝑘) − 4 (]𝜏 + 1) [𝑎2 𝑎3 + 𝑎4 (𝑎1 − ] log 𝑘)] 2 (]𝜏 + 1)

Proof. First, let us calculate the eigenvalues of the operator 𝐴. It is easy to see that, for 𝜆 ∈ C and 𝑓 = (𝑓1 (𝑥), 𝑓2 (𝑥), 𝑓3 (𝑥)) ∈ 𝐷(𝐴), 𝐴𝑓 = 𝜆𝑓 is equivalent to −]𝑓1󸀠 (𝑥) − 𝑎1 𝑓1 (𝑥) − 𝑎2 𝑓2 (𝑥) = 𝜆𝑓1 (𝑥) ,

(20)

𝑎3 𝑓1 (𝑥) (𝑥) − 𝑎4 𝑓2 (𝑥) = 𝜆𝑓2 (𝑥) ,

(21)

−𝜏−1 𝑓3󸀠

(𝑥) = 𝜆𝑓3 (𝑥) ,

𝑓1 (0) = −𝑘𝑓3 (1) ,

𝑓3 (0) = 𝑓1 (1) .

.

Set 𝑎𝑎 1 𝛼 (𝜆) = − (𝜆 + 𝑎1 + 2 3 ) , ] 𝜆 + 𝑎4

𝛽 (𝜆) = 𝜏𝜆 − 𝛼 (𝜆) . (26)

Substituting (24) to (20), we have

(22) (23)

By using similar argument of the appendix of [14], it is easy to know that 𝜆 = −𝑎4 belongs to the continuous spectrum 𝜎𝐶(𝐴) of 𝐴. When 𝜆 ≠− 𝑎4 , solving (21) and (22), we have 𝑎3 𝑓 (𝑥) , 𝑓2 (𝑥) = 𝜆 + 𝑎4 1

(24)

𝑓3 (𝑥) = 𝑓3 (0) 𝑒−𝜏𝜆𝑥 .

(25)

𝑓1 (𝑥) = 𝑓1 (0) 𝑒𝛼(𝜆)𝑥 .

(27)

It follows from (23), (25), and (27) that 𝑒𝛽(𝜆) = −𝑘.

(28)

In order to solve (28) with respect to 𝜆, let us set 𝜆 = 𝑥+𝑖𝑦, 𝑥, 𝑦 ∈ R, and 𝑢 (𝑥, 𝑦) = (]𝜏 + 1) 𝑥 + 𝑎1 +

V (𝑥, 𝑦) = 𝑦 [1 + ]𝜏 −

𝑎2 𝑎3 (𝑥 + 𝑎4 ) 2

(𝑥 + 𝑎4 ) + 𝑦2 𝑎2 𝑎3 2

(𝑥 + 𝑎4 ) + 𝑦2

].

, (29)


5

Then, (28) becomes 𝑒𝛽(𝜆) = 𝑒(1/])[𝑢(𝑥,𝑦)+𝑖V(𝑥,𝑦)] = −𝑘,

(30)

which is equivalent to 1 𝑒(1/])𝑢(𝑥,𝑦) cos V (𝑥, 𝑦) = −𝑘, ]

(𝑥 + 𝑎4 ) (𝑥 −

V (𝑥, 𝑦) = (2𝑛 + 1) ]𝜋,

× (𝑥2 +

𝑛 ∈ Z, (32) +

which are equivalent to

𝑦 [1 + ]𝜏 −

𝑎2 𝑎3 (𝑥 + 𝑎4 )

= ] log 𝑘,

(33)

] = (2𝑛 + 1) ]𝜋.

(34)

2

(𝑥 + 𝑎4 ) + 𝑦2 𝑎2 𝑎3 2

(𝑥 + 𝑎4 ) +

𝑦2

Combining (33) with (34), we get (2𝑛 + 1) ]𝜋 (𝑥 + 𝑎4 ) 𝑦= , 2 (]𝜏 + 1) 𝑥 + 𝑎1 + 𝑎4 (]𝜏 + 1) − ] log 𝑘

𝑛 ∈ Z.

𝑎2 𝑎3 (𝑥 + 𝑎4 ) 2 − (𝑥 + 𝑎4 ) . 𝑦 = ±√ ] log 𝑘 − (]𝜏 + 1) 𝑥 − 𝑎1

(35)

(36)

As a result, introducing two sets (2𝑛 + 1) ]𝜋 (𝑥 + 𝑎4 ) , 2 (]𝜏 + 1) 𝑥 + 𝑎1 + 𝑎4 (]𝜏 + 1) − ] log 𝑘

𝑥, 𝑦 ∈ R, 𝑥 ≠𝑎4 , 𝑛 ∈ Z} ,

𝑎1 − ] log 𝑘 + 𝑎4 𝑥 ]𝜏 + 1

𝑎2 𝑎3 + 𝑎4 (𝑎1 − ] log 𝑘) ) ≤ 0. ]𝜏 + 1

From the inequality and the definition of the 𝜇(𝑘, 𝜏), it is obvious that sup {Re (𝜆) : 𝜆 ∈ 𝜎 (𝐴)} ≤ 𝜇 (𝑘, 𝜏) .

𝑎2 𝑎3 (𝑥 + 𝑎4 ) 2 − (𝑥 + 𝑎4 ) , ] log 𝑘 − (]𝜏 + 1) 𝑥 − 𝑎1

in which 𝜇(𝑘, 𝜏) is the number defined in Lemma 2. Proof. In Theorem 1, it is shown that the operator 𝐴 generates a uniformly bounded 𝐶0 -semigroup 𝑇(𝑡) on H when the feedback gain 𝑘 is chosen such that 𝑘2 < 1/(]𝜏) < 1. Then it follows from Theorem 5.3 and Remark 5.4 of [17] that, for any 𝜀 > 0, there exists some constant 𝑀 such that 𝑀 𝑀 󵄩 󵄩󵄩 󵄩󵄩(𝜆𝐼 − 𝐴)−1 󵄩󵄩󵄩 󵄩L(H) ≤ Re (𝜆) ≤ 𝜀 󵄩

(43)

(44)

In order to apply Theorem 1.1 of [14], it must be shown that 󵄩 󵄩 sup {󵄩󵄩󵄩󵄩(𝜆𝐼 − 𝐴)−1 󵄩󵄩󵄩󵄩L(H) : 𝜆 ∈ 𝐸1 } < ∞.

(37)

we see that the point spectrum 𝜎𝑝 (𝐴) of 𝐴 is given by 𝜎𝑝 (𝐴) = 𝑆1 ∩ 𝑆2 . But we remark that the resolvent set 𝜌(𝐴) of 𝐴 is (𝑆1𝑐 ∪ 𝑆2𝑐 ) \ {−𝑎4 } (see the Appendix). This means that 𝜎 (𝐴) = (𝑆1 ∩ 𝑆2 ) ∪ {−𝑎4 } .

(41)

󵄩 󵄩 sup {󵄩󵄩󵄩󵄩(𝜆𝐼 − 𝐴)−1 󵄩󵄩󵄩󵄩L(H) : 𝜆 ∈ C and Re 𝜆 ≥ 𝜇 (𝑘, 𝜏) + 𝜀} < ∞, (42)

𝐸1 = {𝜆 ∈ C : 𝜇 (𝑘, 𝜏) + 𝜀 ≤ Re (𝜆) ≤ 𝜀} .

} 𝑥, 𝑦 ∈ R, 𝑥 ≠𝑎4 } , }

(40)

holds for all 𝜆 ∈ C with Re(𝜆) ≥ 𝜀. Now, let the subset 𝐸1 of the complex domain C be given by

{ 𝑆2 = {𝑥 + 𝑖𝑦 : 𝑦 { = ±√

] log 𝑘 − 𝑎1 ) ]𝜏 + 1

Lemma 3. Suppose that the assumption of Theorem 1 is satisfied. Then, for any 𝜀 > 0, the following holds:

On the other hand, solving (33) with respect to 𝑦, we have

𝑆1 = {𝑥 + 𝑖𝑦 : 𝑦 =

(39)

which is equivalent to

Thus, it follows from the previous equations that

(]𝜏 + 1) 𝑥 + 𝑎1 +

𝑎2 𝑎3 (𝑥 + 𝑎4 ) 2 − (𝑥 + 𝑎4 ) ≥ 0, ] log 𝑘 − (]𝜏 + 1) 𝑥 − 𝑎1

(31)

1 sin V (𝑥, 𝑦) = 0. ]

𝑢 (𝑥, 𝑦) = ] log 𝑘,

When 𝜆 = 𝑥 + 𝑖𝑦 ∈ 𝜎𝑝 (𝐴), from the definition of the set 𝑆2 , 𝑥 must satisfy the inequality

(38)

(45)

First, for each 𝜆 ∈ 𝐸1 and each 𝑔 = (𝑔1 , 𝑔2 , 𝑔3 )𝑇 ∈ H, we consider the resolvent equation (𝜆𝐼 − 𝐴)𝑓 = 𝑔, which is equivalent to ]𝑓1󸀠 (𝑥) + (𝜆 + 𝑎1 ) 𝑓1 (𝑥) + 𝑎2 𝑓2 (𝑥) = 𝑔1 (𝑥) ,

(46)

−𝑎3 𝑓1 (𝑥) (𝑥) + (𝜆 + 𝑎4 ) 𝑓2 (𝑥) = 𝑔2 (𝑥) ,

(47)

𝑓3󸀠 (𝑥) + 𝜏𝜆𝑓3 (𝑥) = 𝜏𝑔3 (𝑥) .

(48)

6


Solving (47) and (48), we have 𝑎3 1 𝑓 (𝑥) + 𝑔 (𝑥) , 𝜆 + 𝑎4 1 𝜆 + 𝑎4 2

𝑓2 (𝑥) =

(49)

Also, 𝑓2 (𝑥) can be obtained from the previous equations and (49). Next, we will estimate a bound of H norm of 𝑓 = (𝑓1 , 𝑓2 , 𝑓3 )𝑇 . It follows from

𝑥

𝑓3 (𝑥) = 𝑓3 (0) 𝑒−𝜏𝜆𝑥 + ∫ 𝜏𝑒−𝜏𝜆(𝑥−𝑠) 𝑔3 (𝑠) 𝑑𝑠. 0

𝜀 ≤ 𝜇 (𝑘, 𝜏) + 𝜀 + 𝑎4

(50)

󵄨 󵄨 󵄨 󵄨 = 󵄨󵄨󵄨𝜇 (𝑘, 𝜏) + 𝜀 + 𝑎4 󵄨󵄨󵄨 ≤ Re 𝜆 + 𝑎4 = 󵄨󵄨󵄨𝜆 + 𝑎4 󵄨󵄨󵄨

Substituting (49) with (46) and solving it, we have that

𝑥

𝑓1 (𝑥) = 𝑓1 (0) 𝑒𝛼(𝜆)𝑥 + ∫ 𝜏𝑒𝛼(𝜆)(𝑥−𝑠) 0

𝑎2 1 × [ 𝑔1 (𝑠) − 𝑔 (𝑠)] 𝑑𝑠. ] ] (𝜆 + 𝑎4 ) 2

(51)

󵄨󵄨 −(1/])(𝑎2 𝑎3 /(𝜆+𝑎4 ))𝑥 󵄨󵄨 󵄨󵄨𝑒 󵄨󵄨 󵄨 󵄨 2

2

Since 𝑓 = (𝑓1 , 𝑓2 , 𝑓3 ) belongs to the domain 𝐷(𝐴) of 𝐴, 𝑓1 (𝑥) and 𝑓3 (𝑥) should satisfy the relations 𝑓1 (1) = 𝑓3 (0) .

(52)

1

𝛼(𝜆)(1−𝑠)

× [∫ 𝑒 0

1

𝜏𝜆𝑠

+ ∫ 𝜏𝑒 0

Noting that ] log 𝑘 + (1 + ]𝜏) 𝜀 ≤ 𝑎1 + (1 + ]𝜏) [𝜇 (𝑘, 𝜏) + 𝜀]

𝑎2 1 ( 𝑔1 (𝑠) − 𝑔 (𝑠)) 𝑑𝑠 ] ] (𝜆 + 𝑎4 ) 2

(53)

𝑔3 (𝑠) 𝑑𝑠] ,

󵄨 󵄨 = 𝑒−(1/])[Re 𝜆+𝑎1 ] 󵄨󵄨󵄨󵄨𝑒−(1/])(𝑎2 𝑎3 /(𝜆+𝑎4 ))𝑥 󵄨󵄨󵄨󵄨

−𝑘𝑒−𝜏𝜆 𝑒𝛼(𝜆)𝑥 1 + 𝑘𝑒−𝛽(𝜆) 0

𝑎2 1 ( 𝑔1 (𝑠) − 𝑔 (𝑠)) 𝑑𝑠 ] ] (𝜆 + 𝑎4 ) 2

0

𝑥

𝑎2 1 + ∫ 𝜏𝑒𝛼(𝜆)(𝑥−𝑠) [ 𝑔1 (𝑠) − 𝑔 (𝑠)] 𝑑𝑠, ] ] (𝜆 + 𝑎4 ) 2 0 𝑒𝜏𝜆𝑥 1 + 𝑘𝑒−𝛽(𝜆) 1 𝑎2 1 × [∫ 𝑒𝛼(𝜆)(1−𝑠) ( 𝑔1 (𝑠) − 𝑔 (𝑠)) 𝑑𝑠 ] ] (𝜆 + 𝑎4 ) 2 0 1

+ ∫ 𝜏𝑒−𝜏𝜆(1−𝑠) 𝑔3 (𝑠) 𝑑𝑠] 1

− ∫ 𝜏𝑒−𝜏𝜆(𝑥−𝑠) 𝑔3 (𝑠) 𝑑𝑠. 𝑥

for all 𝑥 ∈ [0, 1]. Putting 𝑥 = 1 in (58), we have

(54)

(60)

Moreover, the continuous function ℎ(𝑥, 𝑦) := 𝑒−𝜏𝑥𝑦 defined on the compact set [0, 1] × [𝜇(𝑘, 𝜏) + 𝜀, 𝜀] has absolute maximum and absolute minimum, which are denoted by 𝐿 and 𝑙, respectively. Thus, we have 󵄨󵄨 −𝜏𝜆 󵄨󵄨 󵄨󵄨 𝛼(𝜆)𝑥 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑘 󵄨󵄨󵄨𝑒 󵄨󵄨󵄨 󵄨󵄨󵄨𝑒 󵄨󵄨 󵄨 󵄨󵄨𝑓1 (𝑥)󵄨󵄨 ≤ 󵄨󵄨 󵄨 󵄨󵄨1 + 𝑘𝑒−𝛽(𝜆) 󵄨󵄨󵄨 1 𝑎 󵄨 󵄨 1󵄨 󵄨 󵄨 󵄨 × [∫ 󵄨󵄨󵄨󵄨𝑒𝛼(𝜆)(1−𝑠) 󵄨󵄨󵄨󵄨 ( 󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 󵄨󵄨 2 󵄨󵄨 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨) 𝑑𝑠 ] ] 󵄨󵄨𝜆 + 𝑎4 󵄨󵄨 0 1 󵄨 󵄨󵄨 󵄨 + ∫ 𝜏 󵄨󵄨󵄨󵄨𝑒𝜏𝜆𝑠 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑠] 0

0

(59)

≤ 𝑒−(1/])[] log 𝑘+𝜀]𝑥 = 𝑘−1 𝑒−(1/])𝜀𝑥 < 𝑘−1 , 󵄨 󵄨 𝑘 󵄨󵄨󵄨󵄨𝑒−𝛽(𝜆) 󵄨󵄨󵄨󵄨 ≤ 𝑒−(1/])(1+]𝜏)𝜀 < 1.

1

+ ∫ 𝜏𝑒𝜏𝜆𝑠 𝑔3 (𝑠) 𝑑𝑠]

𝑓3 (𝑥) =

(58)

󵄨󵄨 𝛼(𝜆)𝑥 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑒 󵄨 󵄨

If substitute them into (50) and (51), then we have

× [∫ 𝑒

󵄨 󵄨 = 𝑒−(1/])[(1+]𝜏) Re 𝜆+𝑎1 ] 󵄨󵄨󵄨󵄨𝑒−(1/])(𝑎2 𝑎3 /(𝜆+𝑎4 ))𝑥 󵄨󵄨󵄨󵄨

= 𝑘−1 𝑒−(1/])(1+]𝜏)𝜀𝑥 < 𝑘−1 ,

0

𝛼(𝜆)(1−𝑠)

we have 󵄨󵄨 −𝛽(𝜆)𝑥 󵄨󵄨 󵄨󵄨𝑒 󵄨󵄨 󵄨 󵄨

≤ 𝑒−(1/])[] log 𝑘+(1+]𝜏)𝜀]𝑥

1

1

(57)

] log 𝑘 + 𝜀 ≤ ] log 𝑘 + 𝜀 + ]𝜏 (𝜀 − Re 𝜆) ≤ 𝑎1 + Re 𝜆,

𝑓3 (0) = 𝑒𝜏𝜆 𝑓3 (1) − ∫ 𝜏𝑒𝜏𝜆𝑠 𝑔3 (𝑠) 𝑑𝑠.

𝑓1 (𝑥) =

2

≤ 𝑎1 + (1 + ]𝜏) Re 𝜆,

𝑒−𝜏𝜆 1 + 𝑘𝑒−𝛽(𝜆)

(56)

≤ 𝑒−(1/])𝑎2 𝑎3 𝜀𝑥/((Re 𝜆+𝑎4 ) +(Im 𝜆) ) ≤ 1.

Putting 𝑥 = 1 in (50) and (51), respectively, we obtain 𝑓3 (1) =

2

= 𝑒−(1/])𝑎2 𝑎3 (Re 𝜆+𝑎4 )𝑥/((Re 𝜆+𝑎4 ) +(Im 𝜆) )

𝑇

𝑓1 (0) = −𝑘𝑓3 (1) ,

(55)

𝑥 󵄨 󵄨 + ∫ 𝜏 󵄨󵄨󵄨󵄨𝑒𝛼(𝜆)(𝑥−𝑠) 󵄨󵄨󵄨󵄨 0


7

𝑎 1󵄨 󵄨 󵄨 󵄨 × [ 󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 󵄨󵄨 2 󵄨󵄨 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨] 𝑑𝑠 ] ] 󵄨󵄨𝜆 + 𝑎4 󵄨󵄨

Similarly, we have 󵄨󵄨 𝜏𝜆𝑥 󵄨󵄨 󵄨󵄨𝑒 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨𝑓3 (𝑥)󵄨󵄨 ≤ 󵄨󵄨 󵄨 −𝛽(𝜆) 󵄨󵄨 󵄨󵄨1 + 𝑘𝑒 󵄨󵄨

𝑒−𝜏 Re 𝜆 ≤ 󵄨 󵄨 1 − 󵄨󵄨󵄨𝑘𝑒−𝛽(𝜆) 󵄨󵄨󵄨 × [∫

1

0

1 󵄨 󵄨 × [∫ 󵄨󵄨󵄨󵄨𝑒𝛼(𝜆)(1−𝑠) 󵄨󵄨󵄨󵄨 0

1 󵄨󵄨 󵄨 𝑎 󵄨 󵄨 󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 2 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨 𝑑𝑠 𝑘] 𝑘]𝜀 1

󵄨 󵄨 + ∫ 𝜏𝑒𝜏𝑠 Re 𝜆 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑠]

𝑎 1󵄨 󵄨 󵄨 󵄨 × ( 󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 󵄨󵄨 2 󵄨󵄨 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨) 𝑑𝑠 ] ] 󵄨󵄨𝜆 + 𝑎4 󵄨󵄨

𝜏 󵄨󵄨 󵄨 𝜏𝑎 󵄨 󵄨 󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 2 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨 𝑑𝑠 𝑘] 𝑘]𝜀

1 󵄨󵄨 󵄨 󵄨 + ∫ 𝜏 󵄨󵄨󵄨󵄨𝑒−𝜏𝜆(1−𝑠) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑠] 0

0

+∫

1

0

≤

𝐿

1

1−

× [∫

1

0

1 󵄨󵄨 󵄨 𝑎 󵄨 󵄨 󵄨𝑔 (𝑠)󵄨󵄨 + 2 󵄨󵄨𝑔 (𝑠)󵄨󵄨 𝑑𝑠 𝑘] 󵄨 1 󵄨 𝑘]𝜀 󵄨 2 󵄨 1

+∫

0

+∫

1

0

≤(

+

󵄨󵄨 󵄨 󵄨 + ∫ 𝜏 󵄨󵄨󵄨󵄨𝑒−𝜏𝜆(𝑥−𝑠) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑠 𝑥

𝑒−(1/])(1+]𝜏)𝜀 ≤

𝜏 󵄨󵄨 󵄨 󵄨𝑔 (𝑠)󵄨󵄨 𝑑𝑠] 𝑙󵄨 3 󵄨

1 1󵄨 󵄨 𝑎 󵄨 󵄨 × [∫ ( 󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 2 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨) 𝑑𝑠 ] ]𝜀 0

𝜏 󵄨󵄨 󵄨 𝜏𝑎 󵄨 󵄨 󵄨𝑔 (𝑠)󵄨󵄨 + 2 󵄨󵄨𝑔 (𝑠)󵄨󵄨 𝑑𝑠 𝑘] 󵄨 1 󵄨 𝑘]𝜀 󵄨 2 󵄨 𝐿

𝑘] [1 −

𝑒−(1/])(1+]𝜏)𝜀 ]

+

1

󵄨 󵄨 + ∫ 𝜏𝑒−𝜏 Re 𝜆(1−𝑠) 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑠] 0

𝜏 √ 1 󵄨󵄨 󵄨2 ) ∫ 󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 𝑑𝑥 𝑘] 0

1

󵄨 󵄨 + ∫ 𝜏𝑒−𝜏 Re 𝜆(𝑥−𝑠) 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑠 𝑥

1 𝑎2 𝐿 √ ∫ 󵄨󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨󵄨2 𝑑𝑥 −(1/])(1+]𝜏)𝜀 𝑘]𝜀 [1 − 𝑒 ] 0

+(

𝑒𝜏𝑥 Re 𝜆 1 − 𝑒−(1/])(1+]𝜏)𝜀

≤

1 𝑙 [1 − 𝑒−(1/])(1+]𝜏)𝜀 ] 1 1󵄨 󵄨 𝑎 󵄨 󵄨 × [∫ ( 󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 + 2 󵄨󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨) 𝑑𝑠 ] ]𝜀 0

1

𝜏𝑎 𝜏𝐿 󵄨2 󵄨 + 2 ) √ ∫ 󵄨󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑥. 𝑙 [1 − 𝑒−(1/])(1+]𝜏)𝜀 ] 𝑘]𝜀 0 (61)

+∫

1

𝜏𝐿 󵄨󵄨 󵄨 󵄨𝑔 (𝑠)󵄨󵄨 𝑑𝑠] 𝑙 󵄨 3 󵄨

0

+∫

The Cauchy-Schwarz inequality is applied in the last step. This means that 󵄨 󵄨󵄨 󵄩 󵄩 󵄨󵄨𝑓1 (𝑥)󵄨󵄨󵄨 ≤ 𝑀1 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H

󵄩 󵄩 󵄩 󵄩 or 󵄩󵄩󵄩𝑓1 (𝑥)󵄩󵄩󵄩𝐿2 [0,1] ≤ 𝑀1 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H , (62)

0

≤

𝑀1 = max {

𝐿 𝑘] [1 − 𝑒−(1/])(1+]𝜏)𝜀 ]

+

𝜏 , 𝑘]

𝑎2 𝐿 , 𝑘]𝜀 [1 − 𝑒−(1/])(1+]𝜏)𝜀 ] 𝜏𝐿 𝑙 [1 −

𝑒−(1/])(1+]𝜏)𝜀 ]

+

𝜏𝐿 󵄨󵄨 󵄨 󵄨𝑔 (𝑠)󵄨󵄨 𝑑𝑠 𝑙 󵄨 3 󵄨 1

1 𝑙] [1 − +

in which

1

𝑒−(1/])(1+]𝜏)𝜀 ]

𝑙]𝜀 [1 −

+(

√ ∫ 󵄨󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨󵄨2 𝑑𝑥 0

1 𝑎2 √∫ −(1/])(1+]𝜏)𝜀 𝑒 ] 0

𝜏𝐿 𝑙2 [1 − 𝑒−(1/])(1+]𝜏)𝜀 ]

+

󵄨2 󵄨󵄨 󵄨󵄨𝑔2 (𝑠)󵄨󵄨󵄨 𝑑𝑥

𝜏𝐿 √ 1 󵄨󵄨 󵄨2 ) ∫ 󵄨󵄨𝑔3 (𝑠)󵄨󵄨󵄨 𝑑𝑥. 𝑙 0 (64)

(63) This means that 𝜏𝑎2 }. 𝑘]𝜀

󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨𝑓3 (𝑥)󵄨󵄨󵄨 ≤ 𝑀3 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H

󵄩 󵄩 󵄩 󵄩 or 󵄩󵄩󵄩𝑓3 (𝑥)󵄩󵄩󵄩𝐿2 [0,1] ≤ 𝑀3 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H , (65)

8


4. Conclusion

in which 𝑀3 = max {

𝑎2 1 , , 𝑙] [1 − 𝑒−(1/])(1+]𝜏)𝜀 ] 𝑙]𝜀 [1 − 𝑒−(1/])(1+]𝜏)𝜀 ] 𝜏𝐿 𝑙2

[1 −

𝑒−(1/])(1+]𝜏)𝜀 ]

+

𝜏𝐿 }. 𝑙 (66)

It follows from (49) and (62) that 1󵄨 󵄨 𝑎 󵄨 󵄨 1󵄨 󵄨 𝑎𝑀 󵄩 󵄩 󵄨 󵄨󵄨 󵄨󵄨𝑓2 (𝑥)󵄨󵄨󵄨 ≤ 3 󵄨󵄨󵄨𝑓1 (𝑥)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔2 (𝑥)󵄨󵄨󵄨 ≤ 3 1 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H + 󵄨󵄨󵄨𝑔2 (𝑥)󵄨󵄨󵄨 . 𝜀 𝜀 𝜀 𝜀 (67) This implies that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑓2 (𝑥)󵄩󵄩󵄩𝐿2 [0,1] ≤ 𝑀2 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H

(68)

Appendix

(69)

Let 𝑆1 and 𝑆2 be the sets defined in the proof of Lemma 2. To show that the resolvent set 𝜌(𝐴) of 𝐴 is (𝑆1𝑐 ∪ 𝑆2𝑐 ) \ {−𝑎4 }, we have to prove that the operator 𝜆𝐼 − 𝐴 is bijective for each 𝜆 ∈ (𝑆1𝑐 ∪ 𝑆2𝑐 ) \ {−𝑎4 }. Thus, for each 𝜆 ∈ (𝑆1𝑐 ∪ 𝑆2𝑐 ) \ {−𝑎4 } and each 𝑔 = (𝑔1 , 𝑔2 , 𝑔3 )𝑇 ∈ H, we consider the resolvent equation (𝜆𝐼 − 𝐴)𝑓 = 𝑔, which is equivalent to

in which 𝑀2 =

𝑎3 𝑀1 1 + . 𝜀 𝜀

Therefore, by using (62), (65), and (68), we can estimate a bound of H norm of 𝑓 = (𝑓1 , 𝑓2 , 𝑓3 )𝑇 as follows: 󵄩2 󵄩2 󵄩2 󵄩󵄩 󵄩󵄩2 󵄩 󵄩 󵄩 󵄩󵄩𝑓󵄩󵄩H = 󵄩󵄩󵄩𝑓1 (𝑥)󵄩󵄩󵄩𝐿2 [0,1] + 󵄩󵄩󵄩𝑓2 (𝑥)󵄩󵄩󵄩𝐿2 [0,1] + 󵄩󵄩󵄩𝑓3 (𝑥)󵄩󵄩󵄩𝐿2 [0,1] 󵄩 󵄩 ≤ (𝑀12 + 𝑀22 + 𝑀32 ) 󵄩󵄩󵄩𝑔󵄩󵄩󵄩H ,

(70)

𝜆∈𝐸1

(A.1)

𝑓3󸀠 (𝑥) + 𝜏𝜆𝑓3 (𝑥) = 𝜏𝑔3 (𝑥) . (71)

Since the inequality holds for all 𝑔 ∈ H and for all 𝜆 ∈ 𝐸1 , we have 󵄩 󵄩 sup 󵄩󵄩󵄩󵄩(𝜆𝐼 − 𝐴)−1 󵄩󵄩󵄩󵄩𝐿(H) ≤ √𝑀12 + 𝑀22 + 𝑀32 < ∞.

]𝑓1󸀠 (𝑥) + (𝜆 + 𝑎1 ) 𝑓1 (𝑥) + 𝑎2 𝑓2 (𝑥) = 𝑔1 (𝑥) , −𝑎3 𝑓1 (𝑥) (𝑥) + (𝜆 + 𝑎4 ) 𝑓2 (𝑥) = 𝑔2 (𝑥) ,

which is equivalent to 󵄩󵄩 󵄩 2 2 󵄩󵄩(𝜆𝐼 − 𝐴)−1 󵄩󵄩󵄩 √ 2 󵄩 󵄩𝐿(H) ≤ 𝑀1 + 𝑀2 + 𝑀3 .

In the present paper, we have considered a linear distributed parameter bioprocess with boundary control input possessing a time delay. Using a simple boundary feedback law, we have shown that the closed-loop system generates a uniformly bounded 𝐶0 -semigroup of linear operators if the feedback gain 𝑘 satisfies 0 < 𝑘 < 1. After analyzing the spectrum configuration of the closed-loop system and verifying the spectrum determined growth assumption, we have demonstrated that the closed-loop system becomes exponentially stable. Our main result implies that the linear distributed parameter bioprocess preserves the exponential stability for arbitrary time delay. This means that the answer to the question posed in Section 1 is positive.

(72)

This shows that (45) holds. In this way, we finally obtain 󵄩 󵄩 sup {󵄩󵄩󵄩󵄩(𝜆𝐼 − 𝐴)−1 󵄩󵄩󵄩󵄩L(H) : 𝜆 ∈ C and Re 𝜆 ≥ 𝜇 (𝑘, 𝜏) + 𝜀} < ∞.

It follows from the proof of Lemma 3 that 𝑓1 (𝑥) = 𝑓1 (0) 𝑒𝛼(𝜆)𝑥 𝑥 𝑎2 1 + ∫ 𝜏𝑒𝛼(𝜆)(𝑥−𝑠) [ 𝑔1 (𝑠) − 𝑔 (𝑠)] 𝑑𝑠, ] ] (𝜆 + 𝑎4 ) 2 0

𝑓2 (𝑥) =

𝑎3 1 𝑓1 (𝑥) + 𝑔 (𝑥) , 𝜆 + 𝑎4 𝜆 + 𝑎4 2 𝑥

𝑓3 (𝑥) = 𝑓3 (0) 𝑒−𝜏𝜆𝑥 + ∫ 𝜏𝑒−𝜏𝜆(𝑥−𝑠) 𝑔3 (𝑠) 𝑑𝑠, 0

(73)

Theorem 4. Suppose that the assumption of Theorem 1 is satisfied. Then, for any 𝜀 > 0, there exists a constant 𝑀𝜀,𝜏 such that ‖𝑇 (𝑡)‖L(H) ≤ 𝑀𝜀,𝜏 𝑒[𝜇(𝑘,𝜏)+𝜀]𝑡

(74)

in which 𝜇(𝑘, 𝜏) is defined in Lemma 2. Proof. According to Theorem 1.1 of [14], Theorem 4 is direct consequence of Lemmas 2 and 3.

(1 + 𝑘𝑒−𝛽(𝜆) ) 𝑓3 (1) 1 𝑎2 1 = 𝑒−𝜏𝜆 [∫ 𝑒𝛼(𝜆)(1−𝑠) ( 𝑔1 (𝑠) − 𝑔 (𝑠)) 𝑑𝑠 ] ] (𝜆 + 𝑎4 ) 2 0 1

+ ∫ 𝜏𝑒𝜏𝜆𝑠 𝑔3 (𝑠) 𝑑𝑠] . 0

(A.2)

It is easy to see that 𝜆𝐼 − 𝐴 is bijective if and only if 1 + 𝑘𝑒−𝛽(𝜆) ≠0. From the proof of Lemma 3, we know that 𝑒𝛽(𝜆) = −𝑘 for 𝜆 ∈ 𝑆1 ∩ 𝑆2 . This implies that 𝜌(𝐴) = (𝑆1𝑐 ∪ 𝑆2𝑐 ) \ {−𝑎4 }.


Acknowledgment This work was supported by the National Natural Science Foundation of China (11201037).

References [1] I. Gumowski and C. Mira, Optimization in Control Theory and Practice, Cambridge University Press, Cambridge, UK, 1968. [2] R. Datko, J. Lagnese, and M. P. Polis, “An example on the effect of time delays in boundary feedback stabilization of wave equations,” SIAM Journal on Control and Optimization, vol. 24, no. 1, pp. 152–156, 1986. [3] R. Datko, “Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,” SIAM Journal on Control and Optimization, vol. 26, no. 3, pp. 697–713, 1988. [4] I. H. Suh and Z. Bien, “Use of time delay action in the controller design,” IEEE Transactions on Automatic Control, vol. 25, no. 3, pp. 600–603, 1980. [5] W. H. Kwon, G. W. Lee, and S. W. Kim, “Performance improvement using time delays in multivariable controller design,” International Journal of Control, vol. 52, no. 6, pp. 1455–1473, 1990. [6] N. Jalili and N. Olgac, “Optimum delayed feedback vibration absorber for MDOF mechanical structures,” in Proceedings of the 37th IEEE Conference on Decision and Control (CDC ’98), pp. 4734–4739, Tampa, Fla, USA, December 1998. [7] W. Aernouts, D. Roose, and R. Sepulchre, “Delayed control of a Moore-Greitzer axial compressor model,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 10, no. 5, pp. 1157–1164, 2000. [8] H. Logemann, R. Rebarber, and G. Weiss, “Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop,” SIAM Journal on Control and Optimization, vol. 34, no. 2, pp. 572–600, 1996. [9] B.-Z. Guo and K.-Y. Yang, “Dynamic stabilization of an EulerBernoulli beam equation with time delay in boundary observation,” Automatica, vol. 45, no. 6, pp. 1468–1475, 2009. [10] B.-Z. Guo, C.-Z. Xu, and H. Hammouri, “Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation,” European Series in Applied and Industrial Mathematics, vol. 18, no. 1, pp. 22–35, 2012. [11] S. Nicaise and C. Pignotti, “Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,” SIAM Journal on Control and Optimization, vol. 45, no. 5, pp. 1561–1585, 2006. [12] G. Q. Xu, S. P. Yung, and L. K. Li, “Stabilization of wave systems with input delay in the boundary control,” European Series in Applied and Industrial Mathematics , vol. 12, no. 4, pp. 770–785, 2006. [13] S. Bourrel and D. Dochain, “Stability analysis of two linear distributed parameter bioprocess models,” Mathematical and Computer Modelling of Dynamical Systems, vol. 6, no. 3, pp. 267– 281, 2000. [14] H. Sano, “Boundary control of a linear distributed parameter bioprocess,” Journal of the Franklin Institute, vol. 340, no. 5, pp. 293–306, 2003.

9 [15] F. L. Huang, “Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,” Annals of Differential Equations, vol. 1, no. 1, pp. 43–56, 1985. [16] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, Switzerland, 2009. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.


Research Article On the Gauss Map of Surfaces of Revolution with Lightlike Axis in Minkowski 3-Space Minghao Jin,1,2 Donghe Pei,1 and Shu Xu3 1

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China 3 Rear Services Office, Chongqing Police College, Chongqing 401331, China 2

Correspondence should be addressed to Donghe Pei; [email protected] Received 29 March 2013; Accepted 2 July 2013 Academic Editor: Ji Gao Copyright © 2013 Minghao Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By studying the Gauss map G and Laplace operator Δℎ of the second fundamental form h, we will classify surfaces of revolution with a lightlike axis in 3-dimensional Minkowski space and also obtain the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere that satisfy condition Δℎ 𝐺 = Λ𝐺, Λ being a 3 × 3 real matrix.

1. Introduction The Gauss map is a useful tool for studying surfaces in Euclidean space and pseudo-Euclidean space. Suppose that 𝑀 is a connected surface in R3 and 𝐺 is the Gauss map on 𝑀. According to a theorem proved by Ruh and Vilms [1], 𝑀 has constant mean curvature if and only if Δ𝐺 = ‖𝑑𝐺‖2 𝐺,

(1)

where Δ is the Laplace operator on 𝑀 that corresponds to the metric induced on 𝑀 from R3 . A special case of (1) is given by Δ𝐺 = 𝜆𝐺,

(2)

where the Gauss map 𝐺 is an eigenfunction of the Laplacian Δ on 𝑀. As a more general form of (1), Dillen et al. [2] proved that a surface of revolution 𝑀 in R3 satisfies the condition Δ𝐺 = Λ𝐺,

Λ ∈ Mat (3, R) ,

(3)

if and only if 𝑀 is a plane, sphere, or cylinder. Baikoussis and Blair [3] proved that a ruled surface 𝑀 in R3 satisfies condition (3) if and only if 𝑀 is a plane, helicoidal surface, or spiral surface in R3 . Additionally, Choi and Al´ıas et al.

[4–6] completely classified the surfaces of revolution and ruled surfaces in 3-dimensional Minkowski space that satisfy condition (3). Kim and Yoon [7] studied ruled surfaces in R𝑚 1 such that Δ𝐺 = Λ𝐺,

Λ ∈ Mat (𝑁, R) ,

𝑚 𝑁 = ( ). 2

(4)

Recently, an interesting question was raised: what surfaces of revolution without parabolic points in Euclidean or pseudo-Euclidean space satisfy the following condition? Δℎ 𝐺 = Λ𝐺,

Λ ∈ Mat (3, R) ,

(5)

where Δℎ is the Laplace operator with respect to the second fundamental form ℎ of the surface. This operator is formally defined by Δℎ = −

2

𝜕 𝜕 (√|H|ℎ𝑖𝑗 𝑗 ) 𝜕𝑥 √|H| 𝑖,𝑗=1 𝜕𝑥𝑖 1

∑

(6)

for the components ℎ𝑖𝑗 (𝑖, 𝑗 = 1, 2) of the second fundamental form ℎ on 𝑀, and we denote by (ℎ𝑖𝑗 ) (resp., H) the inverse matrix (resp., the determinant) of the matrix (ℎ𝑖𝑗 ). In [8], the authors studied surfaces of revolution without parabolic points in Euclidean 3-space R3 and presented

2


some classification theorems. In this paper, we will consider surfaces of revolution with lightlike axis in R31 and present some classification results.

2. Preliminaries Let R31 be a 3-dimensional Minkowski space with the scalar product and Lorentz cross-product defined as ⟨x, y⟩ = −𝑥02 + 𝑥12 + 𝑥22 , x × y = (𝑥2 𝑦1 − 𝑥1 𝑦2 , 𝑥2 𝑦0 − 𝑥0 𝑦2 , 𝑥0 𝑦1 − 𝑥1 𝑦0 )

(7)

for every vector x = (𝑥0 , 𝑥1 , 𝑥2 ) and y = (𝑦0 , 𝑦1 , 𝑦2 ) in R31 . A vector x of R31 is said to be spacelike if ⟨x, x⟩ > 0 or x = 0, timelike if ⟨x, x⟩ < 0 and lightlike or null if ⟨x, x⟩ = 0 and x ≠ 0. A timelike or lightlike vector in R31 is said to be causal. Let 𝛾 : 𝐼 → R31 be a smooth curve in R31 , where 𝐼 is an interval in R. We call 𝛾 spacelike, timelike, or lightlike curve if the tangent vector 𝛾󸀠 at any point is spacelike, timelike, or lightlike, respectively. Let 𝐼 be an open interval and 𝛾 : 𝐼 → Π a plane curve lying in a plane Π of R31 and 𝑙 a straight line in Π which does not intersect with the curve 𝛾. A surface of revolution 𝑀 with axis 𝑙 in R31 is defined to be invariant under the group of motions in R31 , which fixes each point of the line 𝑙 [9]. Because the present paper discusses the case of lightlike axis, without loss of generality, we may assume that the axis is the line spanned by vector (1, 1, 0) in the plane 𝑂𝑥0 𝑥1 . So, we choose the line spanned by the vector (1, 1, 0) as axis and express the suppose curve 𝛾 as follows: 𝛾 (𝑢) = (𝑓 (𝑢) , 𝑔 (𝑢) , 0) ,

(8)

where 𝑓(𝑢) is a smooth positive function and 𝑔(𝑢) is a smooth function such that ℎ(𝑢) = 𝑓(𝑢) − 𝑔(𝑢) ≠ 0. Then, the surface of revolution 𝑀 with such axis may be given by 𝑥 (𝑢, V) = (𝑓 (𝑢) +

V2 V2 ℎ (𝑢) , 𝑔 (𝑢) + ℎ (𝑢) , ℎ (𝑢) V) . 2 2 (9)

Now, let us consider the Gauss map 𝐺 on a surface 𝑀 in R31 . The map 𝐺 : 𝑀 → 𝑄2 (𝜀) ⊂ R31 , which sends each point of 𝑀 to the unit normal vector to 𝑀 at that point, is called the Gauss map of surface 𝑀. Here, 𝜀(= ±1) denotes the sign of the vector field 𝐺 and 𝑄2 (𝜀) is a 2-dimensional space form as follows: 𝑆12

(1) 𝑄2 (𝜀) = { 2 𝐻 (−1)

in in

R31 R31

if 𝜀 = 1, if 𝜀 = −1.

(10)

A surface 𝑀 ⊂ R31 is called minimal if and only if its mean curvature 𝐻 is zero. As de Woestijne ([10]) proved, we have the following theorems. Theorem 1 (see [10]). Every minimal, spacelike surface of revolution 𝑀 ⊂ R31 is congruent to a part of one of the following surfaces:

(1) a spacelike plane; (2) the catenoid of the 1st kind; (3) the catenoid of the 2nd kind; (4) the surface of Enneper of the 2nd kind. Theorem 2 (see [10]). Every minimal, timelike surface of revolution 𝑀 ⊂ R31 is congruent to a part of one of the following surfaces: (1) a Lorentzian plane; (2) the catenoid of the 3rd kind; (3) the catenoid of the 4th kind; (4) the catenoid of the 5th kind; (5) the surface of Enneper of the 3rd kind. Now, we consider some examples of surfaces of revolution which are mentioned in our theorems. Example 1 (The surface of Enneper of the 2nd kind is shown in Figure 1). The surface of Enneper of the 2nd kind is parameterized by 𝑥 (𝑢, V) = (𝑢3 − 𝑢 − V2 𝑢, 𝑢3 + 𝑢 − V2 𝑢, −2𝑢V)

(11)

for 𝑢 < 0. Then, the components of the first and the second fundamental forms are given by 𝑔11 = 12𝑢2 ,

𝑔12 = 𝑔21 = 0,

𝑔22 = 4𝑢2 ,

−24𝑢2 ℎ11 = 󵄨󵄨 󵄨, 󵄨󵄨𝑥𝑢 × 𝑥V 󵄨󵄨󵄨

ℎ12 = ℎ21 = 0,

8𝑢2 ℎ22 = 󵄨󵄨 󵄨. 󵄨󵄨𝑥𝑢 × 𝑥V 󵄨󵄨󵄨 (12)

So, the mean curvature 𝐻 on the surface is (−24𝑢2 ) (4𝑢2 ) + (8𝑢2 ) (12𝑢2 ) (13) 𝐻= = 0. 󵄨 󵄨 2 (12𝑢2 ) (4𝑢2 ) 󵄨󵄨󵄨𝑥𝑢 × 𝑥V 󵄨󵄨󵄨 Therefore, the surface of Enneper of the 2nd kind is minimal. Example 2 (The surface of Enneper of the 3rd kind is shown in Figure 2). The surface of Enneper of the 3rd kind is parameterized by 𝑥 (𝑢, V) = (−𝑢3 − 𝑢 − V2 𝑢, −𝑢3 + 𝑢 − V2 𝑢, −2𝑢V)

(14)

for 𝑢 < 0. Then, the components of the first and the second fundamental forms are given by 𝑔11 = −12𝑢2 , 24𝑢2 ℎ11 = 󵄨󵄨 󵄨, 󵄨󵄨𝑥𝑢 × 𝑥V 󵄨󵄨󵄨

𝑔12 = 𝑔21 = 0, ℎ12 = ℎ21 = 0,

𝑔22 = 4𝑢2 , 8𝑢2 ℎ22 = 󵄨󵄨 󵄨. 󵄨󵄨𝑥𝑢 × 𝑥V 󵄨󵄨󵄨 (15)

So, the mean curvature 𝐻 on the surface is (24𝑢2 ) (4𝑢2 ) + (8𝑢2 ) (−12𝑢2 ) (16) = 0. 󵄨 󵄨 2 (−12𝑢2 ) (4𝑢2 ) 󵄨󵄨󵄨𝑥𝑢 × 𝑥V 󵄨󵄨󵄨 Therefore, the surface of Enneper of the 3rd kind is minimal. 𝐻=


3

Figure 1: The surface of Enneper of the 2nd kind. Figure 3: The de Sitter pseudosphere.

Figure 2: The surface of Enneper of the 3rd kind.

Example 3 (The de Sitter pseudosphere is shown in Figure 3). The de Sitter pseudosphere with radius 1 can be expressed as 𝑥 (𝑢, V) = (sinh 𝑢, cosh 𝑢 cos V, cosh 𝑢 sin V) .

(17) Then, its Gauss map 𝐺 and Laplacian are given by

Then, its Gauss map 𝐺 and Laplacian are given by

𝐺 = (− cosh 𝑢, − sinh 𝑢 cos V, − sinh 𝑢 sin V) ,

𝐺 = (− sinh 𝑢, − cosh 𝑢 cos V, − cosh 𝑢 sin V) , Δℎ =

𝜕2 𝜕2 1 sinh 𝑢 𝜕 − + . 2 2 2 𝜕𝑢 cosh 𝑢 𝜕𝑢 cosh 𝑢 𝜕V

(18)

Δℎ = −

𝜕2 𝜕2 1 cosh 𝑢 𝜕 − − . 2 2 2 𝜕𝑢 𝜕V sinh 𝑢 𝜕𝑢 sinh 𝑢

(22)

By a straight computation, we get

By a straight computation, we get Δℎ 𝐺 = (−2 sinh 𝑢, −2 cosh 𝑢 cos V, −2 cosh 𝑢 sin V) ,

Figure 4: The future hyperbolic pseudosphere.

(19)

Δℎ 𝐺 = (2 cosh 𝑢, 2 sinh 𝑢 cos V, 2 sinh 𝑢 sin V) .

(23)

So, we have

which means 2 0 0 Δℎ 𝐺 = (0 2 0) 𝐺; 0 0 2

(20)

(24)

that is, the hyperbolic pseudosphere satisfies condition (1).

that is, the de Sitter pseudosphere satisfies condition (1). Example 4 (The hyperbolic pseudosphere is shown in Figure 4). The hyperbolic pseudosphere with radius 1 is parameterized by 𝑥 (𝑢, V) = (cosh 𝑢, sinh 𝑢 cos V, sinh 𝑢 sin V) .

−2 0 0 Δℎ 𝐺 = ( 0 −2 0 ) 𝐺; 0 0 −2

(21)

3. The Surface of Revolution with Lightlike Axis In this section, we will classify the surfaces of revolution with lightlike axis in R31 that satisfy condition (5).

4


Theorem 3. The only surfaces of revolution with lightlike axis in R31 , whose Gauss map 𝐺 satisfies

we obtain the components of the first and the second fundamental forms of the surface as follows: 𝑔11 = ⟨𝑥𝑢 , 𝑥𝑢 ⟩ = 1,

Δℎ 𝐺 = Λ𝐺,

Λ ∈ Mat (3, R) ,

𝑔12 = 𝑔21 = ⟨𝑥𝑢 , 𝑥V ⟩ = 0,

(25)

𝑔22 = ⟨𝑥V , 𝑥V ⟩ = ℎ2 ,

are locally the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere.

ℎ11 = ⟨𝑥𝑢𝑢 , 𝐺⟩ = 𝑓󸀠󸀠 𝑔󸀠 − 𝑓󸀠 𝑔󸀠󸀠 = 𝑡󸀠 ,

Proof. Let 𝑀 be a surface of revolution with lightlike axis as (9); then we may assume that the profile curve 𝛾 is of unit speed; thus

ℎ22 = ⟨𝑥VV , 𝐺⟩ = −ℎℎ󸀠 ,

⟨𝛾󸀠 , 𝛾󸀠 ⟩ = −𝑓󸀠2 (𝑢) + 𝑔󸀠2 (𝑢) = 𝜀 (±1) .

(26)

ℎ12 = ℎ21 = ⟨𝑥𝑢V , 𝐺⟩ = 0,

where Gauss map 𝐺 is defined by (𝑥𝑢 × 𝑥V )/|𝑥𝑢 × 𝑥V | = (−𝑔󸀠 + (V2 /2)ℎ󸀠 , −𝑓󸀠 + (V2 /2)ℎ󸀠 , Vℎ󸀠 ). So, the matrix (ℎ𝑖𝑗 ) is composed by second fundamental form ℎ as follows:

Without lost of generality, we assume that ℎ = 𝑓(𝑢)−𝑔(𝑢) > 0 and give a detailed proof just for the case 𝜀 = 1. Then, we may put 𝑓󸀠 (𝑢) = sinh 𝑡,

𝑔󸀠 (𝑢) = cosh 𝑡

(27)

for the smooth function 𝑡 = 𝑡(𝑢). Using the natural frame {𝑥𝑢 , 𝑥V } of 𝑀 defined by

𝑥𝑢 = (𝑓󸀠 +

V2 󸀠 󸀠 V2 󸀠 󸀠 ℎ , 𝑔 + ℎ , ℎ V) , 2 2

𝑥𝑢𝑢 = (𝑓󸀠󸀠 +

2

𝑥V = (Vℎ, Vℎ, ℎ) ,

𝑡󸀠 0 ℎ ℎ ). ( 11 12 ) = ( ℎ21 ℎ22 0 −ℎℎ󸀠

ℎ2 𝑡󸀠 − ℎℎ󸀠 1 󸀠 ℎ󸀠 (31) = (𝑡 − ) . 2ℎ2 2 ℎ By a straightforward computation, the Laplacian Δℎ of the second fundamental form ℎ on 𝑀 with the help of (2), (27), and (29) turns out to be 𝐻=

Δℎ = −

V 󸀠󸀠 󸀠󸀠 V 󸀠󸀠 󸀠󸀠 ℎ , 𝑔 + ℎ , ℎ V) , 2 2

1 𝜕2 1 𝜕2 + 󸀠 2 󸀠 2 𝑡 𝜕𝑢 ℎℎ 𝜕V

𝑡󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 𝜕 + ( 󸀠2 − − ) . 2𝑡 2ℎ𝑡󸀠 2ℎ󸀠 𝑡󸀠 𝜕𝑢

𝑥VV = (ℎ, ℎ, 0) , (28)

(30)

2 Since H = ℎ11 ℎ22 − ℎ12 = 0 makes Laplacian Δℎ degenerate, so we can assume that H ≠ 0 for every 𝑡. Then, the mean curvature 𝐻 on 𝑀 is given by

2

𝑥𝑢V = (Vℎ󸀠 , Vℎ󸀠 , 0) ,

(29)

(32)

Accordingly, we get

𝑔󸀠󸀠󸀠 𝑡󸀠󸀠 𝑔󸀠󸀠 ℎ󸀠 𝑔󸀠󸀠 ℎ󸀠󸀠 𝑔󸀠󸀠 1 ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 + 󸀠2 − 󸀠 − 󸀠 󸀠 ) V2 + 󸀠 − 󸀠2 + 󸀠 + 󸀠 󸀠 + 󸀠 2𝑡 4𝑡 4𝑡 ℎ 4𝑡 ℎ 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ ℎ 󸀠󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠 󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠󸀠󸀠 󸀠󸀠 󸀠󸀠 󸀠 󸀠󸀠 󸀠󸀠2 𝑓 𝑡 ℎ 𝑓 𝑓 𝑓 ℎ ℎ ℎ ℎ 𝑡 ℎ 1) ℎ ( Δℎ 𝐺 = ((− 󸀠 + 󸀠2 − 󸀠 − 󸀠 󸀠 ) V2 + 󸀠 − 󸀠2 + 󸀠 + 󸀠 󸀠 + ) . 2𝑡 4𝑡 4𝑡 ℎ 4𝑡 ℎ 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ ℎ ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 (− 󸀠 + 󸀠2 − 󸀠 − 󸀠 󸀠 ) V 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ ( ) (−

By the assumption (25) and the above equation, we get the following system of differential equations: 𝑎 + 𝑎12 󸀠 2 ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 (− 󸀠 + 󸀠2 − 󸀠 − 󸀠 󸀠 − 11 ℎ )V 2𝑡 4𝑡 4𝑡 ℎ 4𝑡 ℎ 2 𝑔󸀠󸀠󸀠 𝑡󸀠󸀠 𝑔󸀠󸀠 ℎ󸀠 𝑔󸀠󸀠 ℎ󸀠󸀠 𝑔󸀠󸀠 − 𝑎13 ℎ V + 󸀠 − 󸀠2 + 󸀠 + 󸀠 󸀠 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ 󸀠

+ (−

1 + 𝑎11 𝑔󸀠 + 𝑎12 𝑓󸀠 = 0, ℎ

𝑎 + 𝑎22 󸀠 2 ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 + − − 21 − ℎ )V 2𝑡󸀠 4𝑡󸀠2 4𝑡󸀠 ℎ 4𝑡󸀠 ℎ󸀠 2 − 𝑎23 ℎ󸀠 V +

𝑓󸀠󸀠󸀠 𝑡󸀠󸀠 𝑓󸀠󸀠 ℎ󸀠 𝑓󸀠󸀠 ℎ󸀠󸀠 𝑓󸀠󸀠 − 󸀠2 + 󸀠 + 󸀠 󸀠 𝑡󸀠 2𝑡 2𝑡 ℎ 2𝑡 ℎ

(33)


5

1 + 𝑎21 𝑔󸀠 + 𝑎22 𝑓󸀠 = 0, ℎ 𝑎 + 𝑎32 󸀠 2 − 31 ℎV 2

By the computation (37) × cosh 𝑡− (38) × sinh 𝑡 and using 𝑓󸀠 = sinh 𝑡, 𝑓󸀠󸀠 = 𝑡󸀠 cosh 𝑡, 𝑓󸀠󸀠󸀠 = 𝑡󸀠2 sinh 𝑡 + 𝑡󸀠󸀠 cosh 𝑡, 𝑔󸀠 = cosh 𝑡, 𝑔󸀠󸀠 = 𝑡󸀠 sinh 𝑡, and 𝑔󸀠󸀠󸀠 = 𝑡󸀠2 cosh 𝑡 + 𝑡󸀠󸀠 sinh 𝑡, we easily get

+

+ (−

ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 + 󸀠2 − 󸀠 − 󸀠 󸀠 − 𝑎33 ℎ󸀠 ) V 󸀠 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ

𝑡󸀠 −

+ 𝑎31 𝑔󸀠 + 𝑎32 𝑓󸀠 = 0,

ℎ󸀠 = −𝜆 cosh2 𝑡 + 𝜇 sinh2 𝑡 + (𝜆 − 𝜇) sinh 𝑡 cosh 𝑡. ℎ

(40)

(34)

On the other hand, substituting ℎ󸀠󸀠 = −ℎ󸀠 𝑡󸀠 and ℎ󸀠󸀠󸀠 = ℎ󸀠 (𝑡󸀠2 − 𝑡󸀠󸀠 ) into (39) equivalently, we get the following equation:

where 𝑎𝑖𝑗 (𝑖, 𝑗 = 1, 2, 3) denote the components of the matrix Λ given by (25). In order to prove the theorem, we have to solve the above system of ordinary differential equations. So, we get three systems of ODE, equivalently:

ℎ󸀠 󸀠 (41) 𝑡 = (𝜆 + 𝜇) 𝑡󸀠 . ℎ Now, we discuss five cases according to the constants 𝜆 and 𝜇.

−

𝑎 + 𝑎12 󸀠 ℎ󸀠󸀠2 ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 + 󸀠2 − 󸀠 − 󸀠 󸀠 − 11 ℎ = 0, 󸀠 2𝑡 4𝑡 4𝑡 ℎ 4𝑡 ℎ 2 −𝑎13 ℎ󸀠 = 0,

𝑔󸀠󸀠󸀠 𝑡󸀠󸀠 𝑔󸀠󸀠 ℎ󸀠 𝑔󸀠󸀠 ℎ󸀠󸀠 𝑔󸀠󸀠 1 − 󸀠2 + 󸀠 + 󸀠 󸀠 + + 𝑎11 𝑔󸀠 + 𝑎12 𝑓󸀠 = 0, 𝑡󸀠 2𝑡 2𝑡 ℎ 2𝑡 ℎ ℎ −

𝑎 + 𝑎22 󸀠 ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 + 󸀠2 − 󸀠 − 󸀠 󸀠 − 21 ℎ =0 󸀠 2𝑡 4𝑡 4𝑡 ℎ 4𝑡 ℎ 2

𝑡󸀠󸀠 − 3𝑡󸀠2 +

Case 1 (𝜆 = 𝜇 = 0). In this case, we easily get 𝑡󸀠 − (ℎ󸀠 /ℎ) = 0, which implies that the mean curvature 𝐻 vanishes identically because of (31). Therefore, the surface is minimal; from Theorem 1 it is the surface of Enneper of the 2nd kind. Furthermore, a surface of Enneper of the 2nd kind satisfies the condition (25). Case2 (𝜆 = 𝜇 ≠ 0). By (40), we get ℎ󸀠 − 𝜆. ℎ Differentiating (42) with respect to 𝑢, we have 𝑡󸀠 =

󸀠

−𝑎23 ℎ = 0, 𝑓󸀠󸀠󸀠 𝑡󸀠󸀠 𝑓󸀠󸀠 ℎ󸀠 𝑓󸀠󸀠 ℎ󸀠󸀠 𝑓󸀠󸀠 1 − 󸀠2 + 󸀠 + 󸀠 󸀠 + + 𝑎21 𝑔󸀠 + 𝑎22 𝑓󸀠 = 0, 𝑡󸀠 2𝑡 2𝑡 ℎ 2𝑡 ℎ ℎ 𝑎 + 𝑎32 󸀠 − 31 ℎ = 0, 2

𝑡󸀠󸀠 = −

(43)

2

4(

ℎ󸀠 ℎ󸀠 ) + 4𝜆 + 𝜆2 = 0 ℎ ℎ

(44)

from which (35)

From (35), we easily deduce that 𝑎13 = 𝑎23 = 𝑎31 = 𝑎32 = 0 and 𝑎33 = (𝑎11 + 𝑎22 )/2 = 𝑎11 + 𝑎12 = 𝑎21 + 𝑎22 . We put 𝑎11 = 𝜆 and 𝑎22 = 𝜇. Therefore, the matrix Λ satisfies 1 (𝜇 − 𝜆) 0 2 1 Λ = ( (𝜆 − 𝜇) ). (36) 𝜇 0 2 1 (𝜆 + 𝜇) 0 0 2 Then, three systems (35) now reduce to the following equations: 𝜆

𝜇−𝜆 󸀠 𝑔󸀠󸀠󸀠 𝑡󸀠󸀠 𝑔󸀠󸀠 ℎ󸀠 𝑔󸀠󸀠 ℎ󸀠󸀠 𝑔󸀠󸀠 1 − 󸀠2 + 󸀠 󸀠 + 󸀠 󸀠 + = −𝜆𝑔󸀠 − 𝑓 , (37) 󸀠 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ ℎ 2 𝜆−𝜇 󸀠 𝑓󸀠󸀠󸀠 𝑡󸀠󸀠 𝑓󸀠󸀠 ℎ󸀠 𝑓󸀠󸀠 ℎ󸀠󸀠 𝑓󸀠󸀠 1 − 󸀠2 + 󸀠 󸀠 + 󸀠 + = −𝜇𝑓󸀠 − 𝑔 , (38) 󸀠 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ ℎ 2 𝜇+𝜆 󸀠 ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 − 󸀠 + 󸀠2 − 󸀠 − 󸀠 󸀠 = ℎ. 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ 2

2

ℎ󸀠 󸀠 ℎ󸀠 𝑡 −( ) . ℎ ℎ

Substituting (42) and (43) into (41), we get

ℎ󸀠󸀠󸀠 𝑡󸀠󸀠 ℎ󸀠󸀠 ℎ󸀠 ℎ󸀠󸀠 ℎ󸀠󸀠2 − 󸀠 + 󸀠2 − 󸀠 − 󸀠 󸀠 − 𝑎33 ℎ󸀠 = 0, 𝑡 2𝑡 2𝑡 ℎ 2𝑡 ℎ 𝑎31 𝑔󸀠 + 𝑎32 𝑓󸀠 = 0.

(42)

(39)

ℎ󸀠 𝜆 = . ℎ 2

(45)

Furthermore, (45) together with (42) becomes 𝑡󸀠 = −(𝜆/2); that is, 𝜆 𝑡 (𝑢) = − 𝑢 + 𝑘, 𝑘 ∈ R. 2 On the other hand, by (27), (45), and (46), we have 2 𝜆 𝑓 (𝑢) = − cosh (− 𝑢 + 𝑘) + 𝑐, 𝜆 2 2 𝜆 𝑔 (𝑢) = − sinh (− 𝑢 + 𝑘) + 𝑐, 𝜆 2

(46)

(47)

𝑐 ∈ R.

Then, the surface 𝑀 has the following expression: 2 𝜆 V2 𝑥 (𝑢, V) = (− cosh (− 𝑢 + 𝑘) + ℎ + 𝑐, 𝜆 2 2 2 𝜆 V2 − sinh (− + 𝑘) + ℎ + 𝑐, ℎV) , 𝜆 2 2

(48)

6


where ℎ = 𝑓 − 𝑔 = −(2/𝜆)𝑒(𝜆/2)−𝑘 , 𝑐, 𝑘 ∈ R. From this, we easily get 2 2 ⟨𝑥 (𝑢, V) − C, 𝑥 (𝑢, V) − C⟩ = −( ) , 𝜆

Ψ2 = 𝜆 (−4 sinh8 𝑡 + 44 sinh7 𝑡 cosh 𝑡 − 220 sinh6 𝑡 cosh2 𝑡 + 612 sinh5 𝑡 cosh3 𝑡 − 1020 sinh4 𝑡 cosh4 𝑡 + 1044 sinh3 𝑡 cosh5 𝑡

C = (𝑐, 𝑐, 0) . (49)

This equation means that the surface 𝑀 is contained in the hyperbolic pseudosphere 𝐻2 (−(2/|𝜆|)) centered at C with radius 2/|𝜆| . Also, the hyperbolic pseudosphere satisfies condition (25).

− 644 sinh2 𝑡 cosh6 𝑡 + 220 sinh 𝑡 cosh7 𝑡 −32 cosh8 𝑡) , Ψ3 = 8 sinh7 𝑡 − 64 sinh6 𝑡 cosh 𝑡

Case 3 (𝜆 ≠ 0, 𝜇 = 0). In this case, (40) becomes 𝑡󸀠 − (ℎ󸀠 /ℎ) = −𝜆 cosh2 𝑡 + 𝜆 sinh 𝑡 cosh 𝑡; that is, 𝑡󸀠 =

ℎ󸀠 − 𝜆 cosh2 𝑡 + 𝜆 sinh 𝑡 cosh 𝑡 ℎ

+ 208 sinh5 𝑡cosh2 𝑡 − 360 sinh4 𝑡 cosh3 𝑡 + 360 sinh3 𝑡 cosh4 𝑡 − 208 sinh2 𝑡 cosh5 𝑡 + 64 sinh 𝑡 cosh6 𝑡 − 8 cosh7 𝑡.

(50)

(55)

and thus Combining (52) and (54), we show that 𝑡󸀠󸀠 =

󸀠󸀠

󸀠

2

ℎ ℎ − ( ) − 2𝜆𝑡󸀠 sinh 𝑡 cosh 𝑡 ℎ ℎ + 𝜆𝑡󸀠 sinh2 𝑡 + 𝜆𝑡󸀠 cosh2 𝑡.

(56)

where 𝜒1 = Φ2 Ψ1 − Φ1 Ψ2 , 𝜒2 = Φ3 Ψ1 − Φ1 Ψ3 . Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in sinh 𝑡 and cosh 𝑡 satisfying

Substituting (50) and (51) into (41), we get Φ1 ℎ2 + Φ2 ℎ + Φ3 = 0,

𝜒1 𝑓 + 𝜒2 = 0,

(51)

(52) 31

𝜇2 (∑𝑐𝑖 sinh31−𝑖 𝑡 cosh5+𝑖 𝑡) = 0,

where we put

(57)

𝑖=1

Φ1 = 𝜆2 (−3 cosh4 𝑡 + 8 sinh 𝑡 cosh3 𝑡 2

2

3

−7 sinh 𝑡 cosh 𝑡 + 2 sinh 𝑡 cosh 𝑡) , Φ2 = 𝜆 (−6 cosh3 𝑡 + 14 sinh 𝑡 cosh2 𝑡

(53)

−10 sinh2 𝑡 cosh 𝑡 + 2 sinh3 𝑡) ,

where 𝑐1 = 1024, 𝑐2 = −24064, . . ., and 𝑐31 = 170496 are nonzero coefficients of the function sinh31−𝑖 𝑡cosh5+𝑖 𝑡. Since this polynomial is equal to zero for every 𝑡, all its coefficients must be zero. Thus, we have 𝜇 = 0, which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis in this case. Case 4 (𝜆 = 0, 𝜇 ≠ 0). In this case, (40) becomes 𝑡󸀠 − (ℎ󸀠 /ℎ) = 2 𝜇 sinh 𝑡 − 𝜇 sinh 𝑡 cosh 𝑡; that is,

Φ3 = −4 sinh2 𝑡 + 8 sinh 𝑡 cosh 𝑡 − 4 cosh2 𝑡. Differentiating (52) and using (50), we find 2

Ψ1 𝑓 + Ψ2 𝑓 + Ψ3 = 0,

𝑡󸀠 = (54)

ℎ󸀠 + 𝜇 sinh2 𝑡 − 𝜇 sinh 𝑡 cosh 𝑡 ℎ

(58)

and thus

where Ψ1 = 𝜆2 (−4 sinh8 𝑡 cosh 𝑡 + 40 sinh7 𝑡 cosh2 𝑡 − 182 sinh6 𝑡 cosh3 𝑡 + 474 sinh5 𝑡 cosh4 𝑡 4

5

𝑡󸀠󸀠 =

2

ℎ󸀠󸀠 ℎ󸀠 − ( ) + 2𝜇𝑡󸀠 sinh 𝑡 cosh 𝑡 ℎ ℎ 󸀠

2

󸀠

(59)

2

− 𝜇𝑡 sinh 𝑡 − 𝜇𝑡 cosh 𝑡.

− 760 sinh 𝑡cosh 𝑡 + 764 sinh3 𝑡 cosh6 𝑡 − 470 sinh2 𝑡 cosh7 𝑡 +162 sinh 𝑡 cosh8 𝑡 − 24 cosh9 𝑡) ,

Substituting (58) and (59) into (41), we get 𝜄1 ℎ2 + 𝜄2 ℎ + 𝜄3 = 0,

(60)


7 coefficients must be zero. Thus, we have 𝜇 = 0, which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis.

where we put 𝜄1 = 𝜇2 (−3 sinh4 𝑡 + 8 sinh3 𝑡 cosh 𝑡 −7 sinh2 𝑡 cosh2 𝑡 + 2 sinh 𝑡 cosh3 𝑡) , 𝜄2 = 𝜇 (−6 sinh3 𝑡 + 14 sinh2 𝑡 cosh 𝑡

Case 5 (𝜆 ≠ 0, 𝜇 ≠ 0, 𝜆 ≠ 𝜇). In this case, (40) is unchanged; that is, (61) 𝑡󸀠 =

− 10 sinh 𝑡 cosh2 𝑡 + 2 cosh3 𝑡) , 𝜄3 = −4 sinh2 𝑡 + 8 sinh 𝑡 cosh 𝑡 − 4 cosh2 𝑡.

ℎ󸀠 − 𝜆 cosh2 𝑡 + 𝜇 sinh2 𝑡 + (𝜆 − 𝜇) sinh 𝑡 cosh 𝑡 ℎ

(66)

and thus

Differentiating (60) and using (58), we find 𝜅1 𝑓2 + 𝜅2 𝑓 + 𝜅3 = 0,

𝑡󸀠󸀠 =

(62)

where 2

9

2

ℎ󸀠󸀠 ℎ󸀠 − ( ) − 2 (𝜆 − 𝜇) 𝑡󸀠 sinh 𝑡 cosh 𝑡 ℎ ℎ

(67)

+ (𝜆 − 𝜇) 𝑡󸀠 sinh2 𝑡 + (𝜆 − 𝜇) 𝑡󸀠 cosh2 𝑡.

8

𝜅1 = 𝜇 (108 sinh 𝑡 − 246 sinh 𝑡 cosh 𝑡

Substituting (66) and (67) into (41), we get

+ 106 sinh7 𝑡cosh2 𝑡 − 64 sinh6 𝑡 cosh3 𝑡 + 424 sinh5 𝑡cosh4 𝑡 − 530 sinh4 𝑡 cosh5 𝑡

𝑃1 ℎ2 + 𝑃2 ℎ + 𝑃3 = 0,

+ 238 sinh3 𝑡 cosh6 𝑡 − 40 sinh2 𝑡 cosh7 𝑡

(68)

where we put

8

+4 sinh 𝑡 cosh 𝑡) , 𝑃1 = (2𝜆𝜇 − 5𝜇2 ) sinh4 𝑡

𝜅2 = 𝜇 (116 sinh8 𝑡 − 220sinh7 𝑡 cosh 𝑡 − 28 sinh6 𝑡 cosh2 𝑡 + 76 sinh5 𝑡 cosh3 𝑡 + 432 sinh4 𝑡 cosh4 𝑡 − 612 sinh3 𝑡 cosh5 𝑡

+ (2𝜆2 − 12𝜆𝜇 + 10𝜇2 ) sinh3 𝑡 cosh 𝑡 (63)

+ (−7𝜆2 + 18𝜆𝜇 − 5𝜇2 ) sinh2 𝑡 cosh2 𝑡 + (8𝜆2 − 8𝜆𝜇) sinh 𝑡 cosh3 𝑡 − 3𝜆2 cosh4 𝑡,

+ 276 sinh2 𝑡 cosh6 𝑡 7

𝑃2 = (−4𝜆 − 2𝜇) sinh3 𝑡

8

−44 sinh 𝑡 cosh 𝑡 + 4 cosh 𝑡) ,

+ (−4𝜆 + 10𝜇) sinh2 𝑡 cosh 𝑡

𝜅3 = 8 sinh7 𝑡 − 64 sinh6 𝑡 cosh 𝑡

+ (14𝜆 − 8𝜇) sinh 𝑡 cosh2 𝑡 − 6𝜆cosh3 𝑡

+ 208 sinh5 𝑡 cosh2 𝑡 − 360 sinh4 𝑡 cosh3 𝑡

𝑃3 = −4 sinh2 𝑡 + 8 sinh 𝑡 cosh 𝑡 − 4 cosh2 𝑡.

+ 360 sinh3 𝑡 cosh4 𝑡 − 208 sinh2 𝑡 cosh5 𝑡 + 64 sinh 𝑡 cosh6 𝑡 − 8 cosh7 𝑡.

Differentiating (68) and using (66), we find

Combining (60) and (62), we show that 𝜔1 𝑓 + 𝜔2 = 0,

𝑄1 𝑓2 + 𝑄2 𝑓 + 𝑄3 = 0,

(64)

where 𝜔1 = 𝜄2 𝜅1 − 𝜄1 𝜅2 , 𝜔2 = 𝜄3 𝜅1 − 𝜄1 𝜅3 . Differentiating once again this equation and using the same method above, we find the following trigonometric polynomial in sinh 𝑡 and cosh 𝑡 satisfying 31

𝜇2 (∑𝑐𝑖 sinh37−𝑖 𝑡 cosh𝑖−1 𝑡) = 0,

(69)

(65)

𝑖=1

where 𝑐1 = 86420736, 𝑐2 = −4471635456, . . ., and 𝑐31 = −8192 are nonzero coefficients of the function sinh37−𝑖 𝑡cosh𝑖−1 𝑡. Since this polynomial is equal to zero for every 𝑡, all its

where 𝑄1 = (8𝜆3 𝜇 − 44𝜆2 𝜇2 + 16𝜆𝜇3 + 20𝜇4 ) sinh9 𝑡 + (8𝜆4 − 100𝜆3 𝜇 + 144𝜆2 𝜇2 +118𝜆𝜇3 − 170𝜇4 ) sinh8 𝑡 cosh 𝑡 + ⋅ ⋅ ⋅ + (−48𝜆4 + 48𝜆3 𝜇 + 42𝜆2 𝜇2 −24𝜆𝜇3 − 18𝜇4 ) cosh9 𝑡,

(70)

8

Journal of Function Spaces and Applications 𝑄2 = (8𝜆3 − 52𝜆2 𝜇 + 4𝜆𝜇2 − 20𝜇3 ) sinh8 𝑡

References

+ (−64𝜆3 + 180𝜆2 𝜇 − 28𝜆𝜇2 − 40𝜇3 ) × sinh7 𝑡 cosh 𝑡 + ⋅ ⋅ ⋅ + (−80𝜆3 + 80𝜆2 𝜇 + 48𝜆𝜇2 − 24𝜇3 ) × cosh8 𝑡, 𝑄3 = (8𝜆2 − 32𝜆𝜇) sinh7 𝑡 + (−64𝜆2 + 176𝜆𝜇 − 40𝜇2 ) sinh6 𝑡 cosh 𝑡 + ⋅ ⋅ ⋅ + (−32𝜆2 + 32𝜆𝜇 + 24𝜇2 ) cosh7 𝑡. (71) Combining (68) and (70), we show that 𝑅1 𝑓 + 𝑅2 = 0,

(72)

where 𝑅1 = 𝑃2 𝑄1 − 𝑃1 𝑄2 , 𝑅2 = 𝑃3 𝑄1 − 𝑃1 𝑄3 . Differentiating once again this equation and using the same algebraic techniques above, we find the following trigonometric polynomial in sinh 𝑡 and cosh 𝑡 satisfying 37

∑𝑐𝑖 (𝜆, 𝜇) sinh37−𝑖 𝑡 cosh𝑖−1 𝑡 = 0,

(73)

𝑖=1

where 𝑐1 (𝜆, 𝜇) = − 331776𝜆11 𝜇3 + 6819840𝜆10 𝜇4 + ⋅ ⋅ ⋅ − 4352000𝜇14 , .. .

(74)

𝑐37 (𝜆, 𝜇) = − 18874368𝜆14 + 54263808𝜆13 𝜇1 + ⋅ ⋅ ⋅ + 2985984𝜇14 , where 𝑐𝑖 (𝜆, 𝜇) (𝑖 = 1, . . . , 37) are the known polynomials in 𝜆 and 𝜇. Since this polynomial is equal to zero for every 𝑡, all its coefficients must be zero. Therefore, 𝜆 = 𝜇 = 0, which is a contradiction. Consequently, there are no surfaces of revolution with lightlike axis in this case. When 𝜀 = −1, we can assume that 𝑓󸀠 (𝑢) = cosh 𝑡 and 󸀠 𝑔 (𝑢) = sinh 𝑡. Using the same algebraic techniques as for 𝜀 = 1, we easily prove from theorem (9) that the surfaces of Enneper of the 3rd kind and the de Sitter pseudosphere satisfy condition (25). This completes the proof.

Acknowledgments The authors were supported by NSF of China (no. 11271063) and NSF of Heilongjiang Institute of Technology (no. 2012QJ1 9).

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