The approximation theory, optimization theory, theory of variational inequalities, and fixed point theory ..... Fenchel-Moreau conjugate, 160. Firm function, 139.
Trends in Mathematics
Qamrul Hasan Ansari Editor
Nonlinear Analysis Approximation Theory, Optimization and Applications
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For further volumes: http://www.springer.com/series/4961
Qamrul Hasan Ansari Editor
Nonlinear Analysis Approximation Theory, Optimization and Applications
Editor Qamrul Hasan Ansari Department of Mathematics Aligarh Muslim University Aligarh, Uttar Pradesh India
ISBN 978-81-322-1882-1 ISBN 978-81-322-1883-8 (eBook) DOI 10.1007/978-81-322-1883-8 Springer New Delhi Heidelberg New York Dordrecht London Library of Congress Control Number: 2014939616 Springer India 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)
In memory of Prof. S. P. Singh (d. 2013), Professor of Mathematics (retd.), Memorial University, St.John’s, Canada
Preface
The approximation theory, optimization theory, theory of variational inequalities, and fixed point theory constitute some of the core topics of nonlinear analysis. These topics provide most elegant and powerful tools to solve the problems from diverse branches of science, social science, engineering, management, etc. The theory of best approximation is applicable in a variety of problems arising in nonlinear functional analysis. The well-posedness and stability of minimization problems are topics of continuing interest in the literature on variational analysis and optimization. The variational inequality problem, complementarity problem, and fixed point problem are closely related to each other. However, they have their own applications within mathematics and in diverse areas of science, management, social sciences, engineering, etc. The split feasibility problem is a general form of the inverse problem which arises in phase retrievals and in medical image reconstruction. This book aims to provide the current, up-to-date, and comprehensive literature on different topics from approximation theory, variational inequalities, fixed point theory, optimization, complementarity problem, and split feasibility problem. Each chapter is self-contained and contributed by different authors. All chapters contain several examples and complete references on the topic. Ky Fan’s best approximation theorems, best proximity pair theorems, and best proximity point theorems have been studied in the literature when the fixed point equation Tx ¼ x does not admit a solution. ‘‘Best Proximity Points’’ contains some basic results on best proximity points of cyclic contractions and relatively nonexpansive maps. An application of a best proximity point theorem to a system of differential equations has been discussed here. Although, it is not possible to include all the available interesting results on best proximity points, an attempt has been made to introduce some results involving best proximity points and references of the related work have been indicated. ‘‘Semi-continuity Properties of Metric Projections’’ presents some selected results regarding semi-continuity of metric projections onto closed subspaces of normed linear spaces. Though there are several significant results relevant to this topic, only a limited coverage of the results is undertaken, as an extensive survey is beyond our scope. This exposition is divided into three parts. The first one deals with results from finite dimensional normed linear spaces. The second one deals with results connecting semi-continuity of metric projection maps and duality vii
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maps. The third one deals with subspaces of finite codimension of infinite dimensional normed linear spaces. The purpose of ‘‘Convergence of Slices, Geometric Aspects in Banach Spaces and Proximinality’’ is to discuss some notions of convergence of sequence of slices and relate these notions with certain geometric properties of Banach spaces and also to some known proximinality properties in best approximation theory. The results which are presented here are not new and in fact they are scattered in the literature in different formulations. The geometric and proximinality results discussed in this chapter are presented in terms of convergence of slices, and it is observed that several known results fit naturally in this framework. The presentation of the results in this framework not only unifies several results in the literature, but also it allows us to view the results as geometric results and understand some problems, which remain to be solved in this area. The chapter is in two parts. The first part begins from the classical works of Banach and Šmulian on the characterizations of smooth spaces and uniformly smooth spaces (or uniformly convex spaces) and present similar characterizations for other geometric properties including some recent results. Similarly, the second part begins from the classical results of James and Day on characterizations of reflexivity and strict convexity in terms of some proximinality properties of closed convex subsets and present similar characterizations for other proximinality properties including some recent results. ‘‘Measures of Noncompactness and Well-Posed Minimization Problems’’ is devoted to present some facts concerning the theory of well-posed minimization problems. Some classical results obtained in the framework of that theory are presented but the focus here is mainly on the detailed presentation of the application of the theory of measures of noncompactness to investigations of the well-posedness of minimization problem. ‘‘Well-Posedness, Regularization, and Viscosity Solutions of Minimization Problems’’ is divided into two parts. The first part surveys some classical notions for well-posedness of minimization problems. The main aim here is to synthesize some known results in approximation theory for best approximants, restricted Chebyshev centers and prox points from the perspective of well-posedness of these problems. The second part reviews Tikhonov regularization of ill-posed problems. This leads us to revisit the so-called viscosity methods for minimization problems using the modern approach of variational convergence. Lastly, some of these results are particularized to convex minimization problems, and also to ill-posed inverse problems. In ‘‘Best Approximation in Nonlinear Functional Analysis,’’ some results from fixed point theory, variational inequalities, and optimization theory are presented. At the end, convergence of approximating sequences and the sequence of iterative process are also given. In ‘‘Hierarchical Minimization Problems and Applications,’’ several iterative methods for solving fixed point problems, variational inequalities and zeros of monotone operators are presented. A generalized mixed equilibrium problem is considered. The hierarchical minimization problem over the set of intersection of
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fixed points of a mapping and the set of solutions of a generalized mixed equilibrium problem are considered. A new unified hybrid steepest-descent-like iterative algorithm for finding a common solution of a generalized mixed equilibrium problem and a common fixed point problem of uncountable family of nonexpansive mappings is presented and analyzed. ‘‘Triple Hierarchical Variational Inequalities’’ is devoted to the theory of variational inequalities. A brief introduction of variational inequalities is given. The hierarchical variational inequalities are considered, and several iterative methods are presented. The triple hierarchical variational inequalities are discussed in detail along with several examples. Several solution methods are presented. ‘‘Split Feasibility and Fixed Point Problems’’ is devoted to the theory of split feasibility problems and fixed point problems. The split feasibility problems and multisets split feasibility problems are described. Several solution methods, namely, CQ methods, are presented for these two problems. Mann-type iterative methods are given for finding the common solution of a split feasibility problem and a fixed point problem. Some methods and results are illustrated by examples. The last chapter is devoted to the study of nonlinear complementarity problems in a Hilbert space. A notion of *-isotone is discussed in relation with solvability of nonlinear complementarity problems. The problem of finding nonzero solution of these problems is also presented. We would like to thank our colleagues and friends who, through their encouragement and help, influenced the development of this book. In particular, we are grateful to Prof. Huzoor H. Khan and Prof. Satya Deo Tripathi who encouraged us (me and Prof. S. P. Singh) to hold the special session on Approximation Theory and Optimization in the Indian Mathematical Society Conference which was held at Banara Hindu University, Varanasi, India during January 12–15, 2012. Prof. S. P. Singh could not participate in this conference due to the illness. Most of the authors who contributed to this monograph presented their talks in this special session and agreed to be a part of this project. We would like to convey our special thanks to Mr. Shamim Ahmad, Editor, Mathematics, Springer India for taking keen interest in publishing this book. Last, but not the least, we would like to thank the members of our family for their infinite patience, encouragement, and forbearance. Aligarh, India, February 2014
Qamrul Hasan Ansari
Contents
Best Proximity Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Veeramani and S. Rajesh
1
Semi-continuity Properties of Metric Projections . . . . . . . . . . . . . . . . V. Indumathi
33
Convergence of Slices, Geometric Aspects in Banach Spaces and Proximinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Shunmugaraj
61
Measures of Noncompactness and Well-Posed Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Józef Banas´
109
Well-Posedness, Regularization, and Viscosity Solutions of Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. V. Pai
135
Best Approximation in Nonlinear Functional Analysis . . . . . . . . . . . . S. P. Singh and M. R. Singh
165
Hierarchical Minimization Problems and Applications . . . . . . . . . . . . D. R. Sahu and Qamrul Hasan Ansari
199
Triple Hierarchical Variational Inequalities . . . . . . . . . . . . . . . . . . . . Qamrul Hasan Ansari, Lu-Chuan Ceng and Himanshu Gupta
231
Split Feasibility and Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . Qamrul Hasan Ansari and Aisha Rehan
281
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Isotone Projection Cones and Nonlinear Complementarity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Abbas and S. Z. Németh
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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contributors
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Contributors
M. Abbas Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa Qamrul Hasan Ansari Department of Mathematics, Aligarh Muslim University, Aligarh, India; Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Józef Banas´ Department of Mathematics, Rzeszów University of Technology, Rzeszów, Poland Lu-Chuan Ceng Department of Mathematics, Shanghai Normal University, Shanghai, China; Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China Himanshu Gupta Department of Mathematics, Aligarh Muslim University, Aligarh, India V. Indumathi Department of Mathematics, Pondicherry University, Pondicherry, India S. Z. Németh School of Mathematics, The University of Birmingham, Birmingham, UK D. V. Pai Indian Institute of Technology Gandhinagar, VGEC Campus, Chandkheda, Ahmedabad, India S. Rajesh Department of Mathematics, Indian Institute of Technology Madras, Chennai, India Aisha Rehan Department of Mathematics, Aligarh Muslim University, Aligarh, India D. R. Sahu Department of Mathematics, Banaras Hindu University, Varanasi, India P. Shunmugaraj Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
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Contributors
S. P. Singh Department of Mathematics and Statistics, Memorial University, St. John’s, NL, Canada M. R. Singh Department of Physics and Astronomy, The University of Western Ontario, London, ON, Canada P. Veeramani Department of Mathematics, Indian Institute of Technology Madras, Chennai, India
About the Editor
Qamrul Hasan Ansari is a professor at the Department of Mathematics, Aligarh Muslim University, Aligarh, India. He has been a visiting professor at several universities, namely, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia; National Sun Yat-sen University, Kaohsiung, Taiwan. He has visited several universities across the globe to deliver his talks. His main areas of research are variational analysis, optimization, convex analysis, and fixed-point theory. He has published more than 150 research papers in international journals. He is the author of two books, and editor of four contributed volumes. He is an associate editor of Journal of Optimization Theory and Applications and Fixed Point Theory and Applications. He has also served as an associate editor or a lead guest editor of several journals, namely, Journal of Inequalities and Applications, Journal of Global Optimization, Positivity, Applicable Analysis.
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Index
Symbols (L , K )-isotone mapping, 329 V -rotund, 141 Z -matrix, 332 δ-convergence, 151 ε-complementarity problem, 326 *-decreasing mapping, 338 *-increasing mapping, 338 *-isotone projection cone, 329 *-order weekly L-Lipschitz mapping, 341 *-order weekly nonexpansive mapping, 342 *-pseudomonotone decreasing mapping, 338
A Almost Chebyshev set, 140 Approximatively w-compact set, 140 Approximatively compact set, 102, 140, 169 Asymptotically regular map, 194 Auxiliary mixed equilibrium problem, 210 Averaged mapping, 287
B Banach contraction principle, 9, 173 Banach space, 3 Best approximant, 1, 145 Best approximation, 34, 62, 95, 167, 202, 289 Best approximation operator, 168 Best proximity pair, 2 Best proximity point, 2, 18 Best simultaneous approximant, 142, 146 Best simultaneous approximation, 145 Bilevel programming problem, 200 Bishop-Phelps-Bollobas theorem, 92 Boundedly compact set, 3
Boundedly relatively w-compact set, 140 Brouwer’s fixed point theorem, 7, 172 Browder-Kirk-Göhde fixed point theorem, 18
C Cent-compact set, 142 Chebyshev center, 16 Chebyshev radius, 16, 142, 145 Chebyshev set, 34, 97, 140, 168 Closed unit ball, 3 Closest point, 167 Coisotone cone, 329 Complementarity problem, 166 Cone, 327 dual, 325 pointed, 328 polar, 325, 328 positive, 329 regular, 329 superdual, 328 Continuous relation, 329 Continuous set-valued map, 78 Contraction mapping, 8, 173, 287 Contractive mapping, 174 Convex feasibility problem, 282, 316 Convex minimization problem, 200 Convex set, 327 Convexly constrained linear inverse problem, 282 CQ algorithm, 296 CQ-method for split feasibility problems, 294 Cyclic contraction map, 10
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350 D Demiclosed mapping, 175, 236 Demiclosed principle, 202 Densifying map, 192 Diameter of a set, 16 Duality map, 41, 92
E Effective domain, 58 Elliptic regularization method, 150 Epi-convergence, 151 Equilibrium problem, 166 Euler equation, 162 Extragradient method, 307
F Fan’s best approximation theorem, 179, 180 Fenchel-Moreau conjugate, 160 Firm function, 139 Firmly nonexpansive mapping, 287 Fixed point, 172 Forcing function, 139
G Generalized complementarity problem, 325 Generalized mixed equilibrium problem, 208, 209 Generalized multivalued complementarity problem, 325 Generalized well-posed problem, 139
H Haar condition, 146 Haar subspace, 146 Hartman–Stampacchia theorem, 181 Hausdorff convergence, 72 Hausdorff distance, 119, 137 Hausdorff excess, 137 Hausdorff index of noncompactness, 76 Hausdorff lower semi-continuous set-valued map, 35 Hausdorff measure of noncompactness, 121 Hausdorff metric, 73 Hausdorff semi-continuous set-valued map, 35 Hausdorff strongly best simultaneous approximant, 146 Hausdorff strongly uniquely multifunction, 145 Hausdorff upper semi-continuous set-valued map, 35, 78 Hierarchial minimization, 157
Index Hierarchical fixed point problem, 200 Hierarchical minimization problem, 200, 242 Hierarchical variational inequality problem, 238 Hyperplane, 63
I Improved relaxed CQ method, 303 Inf-bounded function, 138 Inf-compact function, 138 Intensity-modulated radiation therapy, 283 Inverse strongly monotone mapping, 201, 288 Isotone projection cone, 329 Iterated contraction map, 176
J James theorem, 66
K Kadec-Klee property, 82 Kakutani’s theorem, 8 Kannan mapping, 178 Kernal of measure of noncompactness, 120 Kransnosel’skíi theorem, 22 Kuratowski measure of noncompactness, 115
L Linear complementarity problem, 325 Linearly conditioned problem, 145 Lipschitz continuous mapping, 287 Lower semi-continuous set-valued map, 5, 35, 78 approximate, 36
M Mann’s iterative algorithm, 292 Mann’s iterative method, 292 Mann-type extragradient-like algorithm, 310, 312 Mann-type iterative method, 311 Mann-type relaxed CQ algorithm, 315 Mann-type viscosity algorithm, 313 Maximal monotone operator, 207 Measure of noncompactness, 76, 118, 119 Metric projection, 4, 140, 168, 289 Metric projection multifunction, 145 Metrically well-set, 139 Michael selection theorem, 35 Minimizing sequence, 101, 111, 169
Index Modified CQ algorithm, 299 Modified relaxed CQ algorithm, 299 Modulus of convexity, 3 Modulus of near convexity, 117 Monotone mapping, 207, 288 Mosco convergence, 137 Mosco-Beer topology, 137 Multifunction, 4 Multivalued map, 4 Mutually polar cones, 328
N Nadezhkina and Takahashi extragradient algorithm, 308 Nearest point, 167 Nearly best approximation, 62, 95 Nearly strictly convex space, 113 Nearly uniformly convex, 117 Nonexpansive mapping, 17, 174, 287 generalized, 194 iterative, 177 quasi, 189 Nonsymmetric Hausdorff distance, 119 Normal structure, 16
O Opial condition, 202 Optimal value function, 138 Optimization problem, 336 Order complementarity problem, 326 Order relation, 328 Ordered vector space, 329
P Polyhedral, 38 Positive stable matrix, 332 Pre-duality map, 41, 92 Projection, 170 Projection gradient algorithm, 318 Projection gradient method, 206, 237, 290, 291 Projection order weekly L-Lipschitz mapping, 342 Projection order weekly nonexpansive mapping, 342 Proper function, 138 Prox pair, 144 Proximal normal structure, 18 Proximal pair, 18 Proximinal set, 34, 97, 168 Proximity map, 170
351 Q QP-space, 47 Quasi-polyhedral point, 47 R Radius of a set, 16 Radon-Riesz property, 82 Relatively nonexpansive map, 18, 24 Relaxed CQ algorithm, 298 Relaxed extragradient method, 309 Resolvent operator, 207 Restricted (Chebyshev) center, 142 Restricted center, 146 Retract set, 201 Retraction mapping, 201 S Scale invariant relation, 329 Schauder fixed point theorem, 172 Sequentially Kadec set, 141 Set-valued map, 4 Slice, 62, 65 Smooth space, 68 Split feasibility problem, 281 constrained multiple-sets, 318 multiple-sets, 282, 284, 316 Stable problem, 139 Strictly convex norm, 3 Strictly convex space, 3, 68, 168 Strongly best simultaneous approximant, 146 Strongly convex space, 82 Strongly monotone mapping, 201, 288 Strongly proximinal set, 34, 102 Strongly subdifferentiable function, 47 Strongly subdifferentiable norm, 87 Subdifferential, 70, 160 Sublevel set, 142 Sunny mapping, 202 Sunny nonexpansive retract, 202 Support functional, 65 Supporting hyperplane, 65 T T-regular set, 17 Tikhonov ill-posed problem, 149 Tikhonov regularization, 136 Tikhonov well-posed problem, 138 Translation invariant relation, 329 U Uniformly convex space, 3, 71, 168
352 Uniformly quasi-convex function, 148 Unit sphere, 3 Upper semi-continuous set-valued map, 4, 35, 78 V Variational inequality, 171 Variational inequality problem, 200, 205, 237, 290, 335 Vector lattice, 329 Vietoris convergence, 72 Vietoris topology, 72
Index Viscosity selection criterion, 154, 158 Viscosity solution, 150, 155
W W-cent-compact set, 142 Weak topology, 4 Well posedness, 111 Well-posed in the sense of Furi and Vignoli, 116 Well-posed minimization problem, 113 Wijsman topology, 138
