Algebra-Valued -Metric Spaces - Hindawi

Hindawi Publishing Corporation Journal of Function Spaces Volume 2016, Article ID 7827040, 16 pages http://dx.doi.org/10.1155/2016/7827040

Research Article Fixed Point Theorems for Cyclic Contractions in 𝐶∗-Algebra-Valued 𝑏-Metric Spaces Chakkrid Klin-eam1,2 and Prondanai Kaskasem1 1

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Research Center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand

2

Correspondence should be addressed to Chakkrid Klin-eam; [email protected] Received 10 November 2015; Revised 28 December 2015; Accepted 6 January 2016 Academic Editor: Adrian Petrusel Copyright © 2016 C. Klin-eam and P. Kaskasem. 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 fundamental properties of 𝐶∗ -algebra-valued 𝑏-metric space which was introduced by Ma and Jiang (2015) and give some fixed point theorems for cyclic mapping with contractive and expansive condition on such space analogous to the results presented in Ma and Jiang, 2015.

1. Introduction Firstly, we begin with the basic concept of 𝐶∗ -algebras. A real or a complex linear space A is algebra if vector multiplication is defined for every pair of elements of A satisfying two conditions such that A is a ring with respect to vector addition and vector multiplication and for every scalar 𝛼 and every pair of elements 𝑥, 𝑦 ∈ A, 𝛼(𝑥𝑦) = (𝛼𝑥)𝑦 = 𝑥(𝛼𝑦). A norm ‖ ⋅ ‖ on A is said to be submultiplicative if ‖𝑎𝑏‖ ≤ ‖𝑎‖‖𝑏‖ for all 𝑎, 𝑏 ∈ A. In this case (A, ‖ ⋅ ‖) is called normed algebra. A complete normed algebra is called Banach algebra. An involution on algebra A is conjugate linear map 𝑎 󳨃→ 𝑎∗ on A such that 𝑎∗∗ = 𝑎 and (𝑎𝑏)∗ = 𝑏∗ 𝑎∗ for all 𝑎, 𝑏 ∈ A. (𝐴, ∗) is called ∗-algebra. A Banach ∗-algebra A is ∗-algebra A with a complete submultiplicative norm such that ‖𝑎∗ ‖ = ‖𝑎‖ for all 𝑎 ∈ A. 𝐶∗ -algebra is Banach ∗-algebra such that ‖𝑎∗ 𝑎‖ = ‖𝑎‖2 . There are many examples of 𝐶∗ -algebra, such as the set of complex numbers, the set of all bounded linear operators on a Hilbert space 𝐻, 𝐿(𝐻), and the set of 𝑛 × 𝑛-matrices, 𝑀𝑛 (C). If a normed algebra A admits a unit 𝐼, 𝑎𝐼 = 𝐼𝑎 = 𝑎 for all 𝑎 ∈ A, and ‖𝐼‖ = 1, we say that A is a unital normed algebra. A complete unital normed algebra A is called unital Banach algebra. For properties in 𝐶∗ -algebras, we refer to [1– 3] and the references therein. Let (𝑋, 𝑑) be a complete metric space. The well-known Banach’s contraction principle, which appeared in the Ph.D. dissertation of S. Banach in 1920, runs as follows: a mapping

𝑇 : 𝑋 → 𝑋 is said to be a contraction if there exists 𝑟 ∈ [0, 1) such that 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟𝑑 (𝑥, 𝑦)

∀𝑥, 𝑦 ∈ 𝑋.

(1)

Then, 𝑇 has a unique fixed point in 𝑋 which was published in 1922 [4]. Banach’s contraction principle has become one of the most important tools used for the existence of solutions of many nonlinear problems in many branches of science and has been extensively studied in many spaces which are more general than metric space by serveral mathematictians; see, for example, quasimetric spaces [5, 6], dislocated metric spaces [7], dislocated quasimetric spaces [8], 𝐺-metric spaces [9–11], 𝑏-metric spaces [12–14], metric-type spaces [15, 16], metric-like spaces [17], 𝑏-metric-like spaces (or dislocated 𝑏-metric spaces) [18, 19], quasi 𝑏-metric spaces [20], and dislocated quasi-𝑏-metric spaces [21]. Note that the Banach contraction principle requires that mapping 𝑇 satisfies the contractive condition that each point of 𝑋 × 𝑋 and ranges of 𝑇 are positive real numbers. Consider the operator equation ∞

𝑋 − ∑ 𝐿∗𝑛 𝑋𝐿 𝑛 = 𝑄,

(2)

𝑛=1

where {𝐿 1 , 𝐿 2 , . . . , 𝐿 𝑛 } is subset of the set of linear bounded operators on Hilbert space 𝐻, 𝑋 ∈ 𝐿(𝐻), and 𝑄 ∈ 𝐿(𝐻)+ is positive linear bounded operators on Hilbert space 𝐻. Then,

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Journal of Function Spaces

we convert the operator equation to the mapping 𝐹 : 𝐿(𝐻) → 𝐿(𝐻) which is defined by ∞

𝐹 (𝑋) = ∑ 𝐿∗𝑛 𝑋𝐿 𝑛 + 𝑄.

(3)

𝑛=1

Observe that the range of mapping 𝐹 is not real numbers but it is linear bounded operators on Hilbert space 𝐻. Therefore, the Banach contraction principle can not be applied with this problem. Afterward, does such mapping have a fixed point which is equivalent to the solution of operator equation? In 2014, Ma et al. [22] introduced new spaces, called 𝐶∗ -algebravalued metric spaces, which are more general than metric space, replacing the set of real numbers by 𝐶∗ -algebras, and establish a fixed point theorem for self-maps with contractive or expansive conditions on such spaces, analogous to the Banach contraction principle. As applications, existence and uniqueness results for a type of integral equation and operator equation are given and were able to solve the above problem 2 if 𝐿 1 , 𝐿 2 , . . . , 𝐿 𝑛 ∈ 𝐿(𝐻) satisfy ∑∞ 𝑛=1 ‖𝐿 𝑛 ‖ < 1. Later, many authors extend and improve the result of Ma et al. For example, in [23], Batul and Kamran generalized the notation of 𝐶∗ -valued contraction mappings by weakening the contractive condition introduced by Ma et al. (the mapping is called 𝐶∗ -valued contractive type mappings) and establish a fixed point theorem for such mapping which is more generalized than the result of Ma et al.; in [24], Shehwar and Kamran extend and improve the result of Ma et al. [22] and Jachymski [25] by proving a fixed point theorem for self-mappings on 𝐶∗ -valued metric spaces satisfying the contractive condition for those pairs of elements from the metric space which form edges of a graph in the metric space. In 2015, Ma and Jiang [26] introduced a concept of 𝐶∗ algebra-valued 𝑏-metric spaces which generalize an ordinary 𝐶∗ -algebra-valued metric space and give some fixed point theorems for self-map with contractive condition on such spaces. As applications, existence and uniqueness results for a type of operator equation and an integral equation are given. Generally, in order to use the Banach contraction principle, a self-mapping 𝑇 must be Lipschitz continuous, with the Lipschitz constant 𝑟 ∈ [0, 1). In particular, 𝑇 must be continuous at all elements of its domain. That is one major drawback. Next, many authors could find contractive conditions which imply the existence of fixed point in complete metric space but not imply continuity. We refer to [27, 28] (Kannan-type mappings) and [29] (Chatterjea-type mapping). Theorem 1 (see [27]). If (𝑋, 𝑑) is a complete metric space and mapping 𝑇 : 𝑋 → 𝑋 satisfies 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] ,

(4)

where 0 ≤ 𝑟 < 1/2 and 𝑥, 𝑦 ∈ 𝑋, then 𝑇 has a unique fixed point. Theorem 2 (see [29]). If (𝑋, 𝑑) is a complete metric space and mapping 𝑇 : 𝑋 → 𝑋 satisfies 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟 [𝑑 (𝑥, 𝑇𝑦) + 𝑑 (𝑦, 𝑇𝑥)] ,

(5)

where 0 ≤ 𝑟 < 1/2 and 𝑥, 𝑦 ∈ 𝑋, then 𝑇 has a unique fixed point. In 2003, Kirk et al. [30] introduced the following notation of a cyclic representation and characterized the Banach contraction principle in context of a cyclic mapping as follows. Theorem 3. Let 𝐴 1 , 𝐴 2 , . . . , 𝐴 𝑚 be nonempty closed subsets of a complete metric space (𝑋, 𝑑). Assume that a mapping 𝑇 : 𝑚 ⋃𝑚 𝑖=1 𝐴 𝑖 → ⋃𝑖=1 𝐴 𝑖 satisfies the following conditions: (i) 𝑇(𝐴 𝑖 ) ⊆ 𝐴 𝑖+1 for all 1 ≤ 𝑖 ≤ 𝑚 and 𝐴 𝑚+1 = 𝐴 1 . (ii) There exists 𝑘 ∈ [0, 1) such that 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝑘𝑑(𝑥, 𝑦) for all 𝑥 ∈ 𝐴 𝑖 and 𝑦 ∈ 𝐴 𝑖+1 for 1 ≤ 𝑖 ≤ 𝑚. Then, 𝑇 has a unique fixed point. In 2011, Karapinar and Erhan [31] introduced Kannantype cyclic contraction and Chatterjea-type cyclic contraction. Moreover, they derive some fixed point theorems for such cyclic contractions in complete metric spaces as follows. Theorem 4 (fixed point theorem for Kannan-type cyclic contraction). Let 𝐴 and 𝐵 be nonempty subsets of metric spaces (𝑋, 𝑑) and a cyclic mapping 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 satisfies 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑘 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] , ∀𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵,

(6)

where 0 ≤ 𝑘 < 1/2. Then, 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵. Theorem 5 (fixed point theorem for Chatterjea-type cyclic contraction). Let 𝐴 and 𝐵 be nonempty subsets of a metric spaces (𝑋, 𝑑) and a cyclic mapping 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 satisfies 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑘 [𝑑 (𝑥, 𝑇𝑦) + 𝑑 (𝑦, 𝑇𝑥)] , ∀𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵,

(7)

where 0 ≤ 𝑘 < 1/2. Then, 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵. The purpose of this paper is to study fundamental properties of 𝐶∗ -algebra-valued 𝑏-metric space which was introduced by Ma and Jiang [26] and give some fixed point theorems for cyclic mapping with contractive and expansive condition on such space analogous to the results presented in [26].

2. Preliminaries In this section, we recollect some basic notations, defintions, and results that will be used in main result. Firstly, we begin with the concept of 𝑏-metric spaces. Definition 6 (see [12, 13]). Let 𝑋 be a nonempty set. A mapping 𝑑 : 𝑋 × 𝑋 → R is called 𝑏-metric if there exists a real number 𝑏 ≥ 1 such that, for every 𝑥, 𝑦, 𝑧 ∈ 𝑋, we have (i) 𝑑(𝑥, 𝑦) ≥ 0,

Journal of Function Spaces

3

(ii) 𝑑(𝑥, 𝑦) = 0 if and only if 𝑥 = 𝑦,

Theorem 11 (see [3]). Let A be a unital Banach algebra and let 𝑎 be an element of A such that ‖𝑎‖ < 1. Then, 𝐼 − 𝑎 ∈ Inv(A) and

(iii) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥), (iv) 𝑑(𝑥, 𝑧) ≤ 𝑏[𝑑(𝑥, 𝑦) + 𝑑(𝑦, 𝑧)].

∞

(𝐼 − 𝑎)−1 = ∑ 𝑎𝑛 .

In this case, the pair (𝑋, 𝑑) is called a 𝑏-metric space. The class of 𝑏-metric spaces is larger than the calass of metric spaces, since a 𝑏-metric space is a metric when 𝑏 = 1 in the fourth condition in the above definition. There exist many examples in some work showing that the class of 𝑏-metric is efficiently larger than those metric spaces (see also [12, 14, 32, 33]). Example 7 (see [12]). The set 𝑙𝑝 (R) with 0 < 𝑝 < 1, where 𝑝 𝑙𝑝 (R) fl {{𝑥𝑛 } ⊆ R : ∑∞ 𝑛=1 |𝑥𝑛 | < ∞}, together with the function 𝑑 : 𝑙𝑝 (R) × 𝑙𝑝 (R) → R, 1/𝑝

∞

󵄨 󵄨𝑝 𝑑 (𝑥, 𝑦) = ( ∑ 󵄨󵄨󵄨𝑥𝑛 − 𝑦𝑛 󵄨󵄨󵄨 )

,

(8)

𝑛=1

where 𝑥 = {𝑥𝑛 }, 𝑦 = {𝑦𝑛 } ∈ 𝑙𝑝 (R), is a 𝑏-metric space with coefficient 𝑏 = 21/𝑝 > 1. Observe that the result holds for the general case 𝑙𝑝 (𝑋) with 0 < 𝑝 < 1, where 𝑋 is a Banach space. Example 8 (see [12]). The space 𝐿 𝑝 (0 < 𝑝 < 1) of all real 1

functions 𝑥(𝑡), 𝑡 ∈ [0, 1], such that ∫0 |𝑥(𝑡)|𝑝 𝑑𝑡 < ∞, together with the function 1

󵄨𝑝 󵄨 𝑑 (𝑥, 𝑦) = (∫ 󵄨󵄨󵄨𝑥 (𝑡) − 𝑦 (𝑡)󵄨󵄨󵄨 𝑑𝑡) 0

1/𝑝

, (9)

is a 𝑏-metric space with 𝑏 = 21/𝑝 . Example 9 (see [33]). Let (𝑋, 𝑑1 ) be a metric space and 𝑑2 (𝑥, 𝑦) = (𝑑1 (𝑥, 𝑦))𝑝 , where 𝑝 > 1 is natural numbers. Then, 𝑑2 is a 𝑏-metric with 𝑏 = 2𝑝−1 . The notation convergence, compactness, closedness, and completeness in 𝑏-metric space are given in the same way as in metric space. Next, we give concept of spectrum of element in 𝐶∗ algebra A. Definition 10 (see [3]). We say that 𝑎 ∈ A is invertible if there is an element 𝑏 ∈ A such that 𝑎𝑏 = 𝑏𝑎 = 𝐼. In this case, 𝑏 is unique and written 𝑎−1 . The set (10)

is a group under multiplication. We define spectrum of an element 𝑎 to be the set 𝜎 (𝑎) = 𝜎𝐴 (𝑎) = {𝜆 ∈ C | 𝜆𝐼 − 𝑎 ∉ Inv (A)} .

Theorem 12 (see [3]). Let A be a unital 𝐶∗ -algebra with a unit 𝐼, then (1) 𝐼∗ = 𝐼, (2) For any 𝑎 ∈ Inv(A), (𝑎∗ )−1 = (𝑎−1 )∗ . (3) For any 𝑎 ∈ A, 𝜎(𝑎∗ ) = 𝜎(𝑎)∗ = {𝜆 ∈ C : 𝜆 ∈ 𝜎(𝑎)}. All over this paper, A means a unital 𝐶∗ -algebra with a unit 𝐼. R is set of real numbers and R+ is the set of nonnegetive real numbers. 𝑀𝑛 (R) is 𝑛 × 𝑛 matrix with entries R. Definition 13 (see [3]). The set of hermitain elements of A is denoted by Aℎ ; that is, Aℎ = {𝑥 ∈ A : 𝑥 = 𝑥∗ }. An element 𝑥 in A is positive element which is denoted by 𝜃 ⪯ 𝑥, where 𝜃 means the zero element in A if and only if 𝑥 ∈ Aℎ and 𝜎(𝑥) is a subset of nonnegative real numbers. We define a partial ordering Aℎ by using definition of positive element as 𝑥 ⪯ 𝑦 if and only if 𝑦 − 𝑥 ⪰ 𝜃. The set of positive elements in A is denoted by A+ = {𝑥 ∈ A : 𝑥 ⪰ 𝜃}. The following are definitions and some properties of positive element of a 𝐶∗ -algebra A. Lemma 14 (see [3]). The sum of two positive elements in a 𝐶∗ algebra is a positive element. Theorem 15 (see [3]). If a is an arbitrary element of a 𝐶∗ algebra A, then 𝑎∗ 𝑎 is positive.

∀𝑥, 𝑦 ∈ 𝐿 𝑝 [0, 1] ,

Inv (A) = {𝑎 ∈ A | 𝑎 is invertible}

(12)

𝑛=0

(11)

We summarise some elementary facts about A+ in the following results. Theorem 16 (see [3]). Let A be a 𝐶∗ -algebra: (1) The set A+ is closed cone in A [a cone 𝐶 in a real or complex vector space is a subset closed under addition and under scalar multiplication by R+ ].

(2) The set A+ is equal to {𝑎∗ 𝑎 : 𝑎 ∈ A}. (3) If 𝜃 ⪯ 𝑎 ⪯ 𝑏, then ‖𝑎‖ ≤ ‖𝑏‖.

(4) If A is unital and 𝑎 and 𝑏 are positive invertible elements, then 𝑎 ⪯ 𝑏 ⇒ 𝜃 ⪯ 𝑏−1 ⪯ 𝑎−1 . Theorem 17 (see [3]). Let A be a 𝐶∗ -algebra. If 𝑎, 𝑏 ∈ A+ and 𝑎 ⪯ 𝑏, then for any 𝑥 ∈ A both 𝑥∗ 𝑎𝑥 and 𝑥∗ 𝑏𝑥 are positive elements and 𝑥∗ 𝑎𝑥 ⪯ 𝑥∗ 𝑏𝑥. Lemma 18 (see [3]). Suppose that A is a unital 𝐶∗ -algebra with a unit 𝐼: (1) If 𝑎 ∈ A+ with ‖𝑎‖ < 1/2, then 𝐼 − 𝑎 is invertible and ‖𝑎(𝐼 − 𝑎)−1 ‖ < 1.

4

Journal of Function Spaces (2) Suppose that 𝑎, 𝑏 ∈ A with 𝑎, 𝑏 ⪰ 𝜃 and 𝑎𝑏 = 𝑏𝑎; then, 𝑎𝑏 ⪰ 𝜃. (3) Define A󸀠 = {𝑎 ∈ A : 𝑎𝑏 = 𝑏𝑎, ∀𝑏 ∈ A}. Let 𝑎 ∈ A󸀠 ; if 𝑏, 𝑐 ∈ A with 𝑏 ⪰ 𝑐 ⪰ 𝜃 and 𝐼 − 𝑎 ∈ A󸀠+ is invertible operator, then (𝐼 − 𝑎)−1 𝑏 ⪰ (𝐼 − 𝑎)−1 𝑐.

(13)

Definition 19 (see [34]). A matrix 𝐴 ∈ 𝑀𝑛 (C) is Hermitian if 𝐴 = 𝐴∗ , where 𝐴∗ is a conjugate transpose matrix of 𝐴. A Hermitian matrix 𝐴 ∈ 𝑀𝑛 (C) is positive definite if 𝑥∗ 𝐴𝑥 > 0 for all nonzero 𝑥 ∈ C𝑛 , and it is positive semidefinite if 𝑥∗ 𝐴𝑥 ≥ 0 for all nonzero 𝑥 ∈ C𝑛 . In 2014, Ma et al. [22] introduced the concept of 𝐶∗ algebra-valued metric space by using the concept of positive elements in A. The following is definition of 𝐶∗ -algebravalued metric. Definition 20 (see [22]). Let 𝑋 be a nonempty set. A mapping 𝑑 : 𝑋 × 𝑋 → A is called 𝐶∗ -algebra-valued metric on 𝑋 if it satisfies the following conditions:

In the same way, the concept of expansive mapping is defined in the following way. Definition 23 (see [22]). Let 𝑋 be a nonempty set. A mapping 𝑇 is a 𝐶∗ -algebra-valued expansive mapping on 𝑋, if 𝑇 : 𝑋 → 𝑋 satisfies (1) 𝑇(𝑋) = 𝑋, (2) 𝑑(𝑇𝑥, 𝑇𝑦) ⪰ 𝜆∗ 𝑑(𝑥, 𝑦)𝜆, for all 𝑥, 𝑦 ∈ 𝑋, where 𝜆 ∈ A is an invertible element and ‖𝜆−1 ‖ < 1. The following is the related fixed point theorem for 𝐶∗ algebra-valued expansive mapping. Theorem 24 (see [22]). Let (𝑋, A, 𝑑) be a complete 𝐶∗ algebra-valued metric space. If a 𝑇 : 𝑋 → 𝑋 satisfies Defintion 23, then 𝑇 has a unique fixed point in 𝑋.

3. Fundamental Properties of 𝐶∗ -Algebra-Valued 𝑏-Metric Spaces

(1) 𝑑(𝑥, 𝑦) ⪰ 𝜃 for all 𝑥, 𝑦 ∈ 𝑋. (2) 𝑑(𝑥, 𝑦) = 𝜃 if and only if 𝑥 = 𝑦.

In this section, we begin with the concept of 𝐶∗ -algebravalued 𝑏-metric space which was introduced by Ma and Jiang [26] as follows.

(3) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋. (4) 𝑑(𝑥, 𝑦) ⪯ 𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then, 𝑑 is called a 𝐶∗ -algebra-valued metric on 𝑋 and (𝑋, A, 𝑑) is called a 𝐶∗ -algebra-valued metric space. We know that range of mapping 𝑑 in metric space is the set of real numbers which is 𝐶∗ -algebra; then, 𝐶∗ -algebravalued metric space generalizes the concept of metric spaces, replacing the set of real numbers by A+ . In such paper, Ma et al. state the notation of convergence, Cauchy sequence, and completeness in 𝐶∗ -algebra-valued metric space. For detail, a sequence {𝑥𝑛 } in a 𝐶∗ -algebra-valued metric space (𝑋, A, 𝑑) is said to converge to 𝑥 ∈ 𝑋 with respect to A if for any 𝜀 > 0 there is 𝑁 ∈ N such that ‖𝑑(𝑥𝑛 , 𝑥)‖ < 𝜀 for all 𝑛 ≥ 𝑁. We write it as lim𝑛 → ∞ 𝑥𝑛 = 𝑥. A sequence {𝑥𝑛 } is called a Cauchy sequence with respect to A if for any 𝜀 > 0 there is 𝑁 ∈ N such that ‖𝑑(𝑥𝑚 , 𝑥𝑛 )‖ < 𝜀 for all 𝑛, 𝑚 ≥ 𝑁. The (𝑋, A, 𝑑) is said to be a complete 𝐶∗ -algebravalued metric space if every Cauchy sequence with respect to A is convergent. Moreover, they introduce definition of contractive and expansive mapping and give some related fixed point theorems for self-maps with 𝐶∗ -algebra-valued contractive and expansive mapping, analogous to Banach contraction principle. The following is the definition of contractive mapping and the related fixed point theorem. Definition 21 (see [22]). Suppose that (𝑋, A, 𝑑) is a 𝐶∗ algebra-valued metric space. A mapping 𝑇 : 𝑋 → 𝑋 is called 𝐶∗ -algebra-valued contractive mapping on 𝑋, if there is an 𝜆 ∈ A with ‖𝜆‖ < 1 such that 𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆∗ 𝑑 (𝑥, 𝑦) 𝜆,

Theorem 22 (see [22]). If (𝑋, A, 𝑑) is a complete 𝐶∗ -algebravalued metric space and 𝑇 : 𝑋 → 𝑋 satisfies Defintion 21, then 𝑇 has a unique fixed point in 𝑋.

∀𝑥, 𝑦 ∈ 𝑋.

(14)

Definition 25 (see [26]). Let 𝑋 be a nonempty set. A mapping 𝑑 : 𝑋 × 𝑋 → A is called 𝐶∗ -algebra-valued 𝑏-metric on 𝑋 if there exists 𝑏 ∈ A󸀠 such that 𝑏 ⪰ 𝐼 satisfies following conditions: (1) 𝑑(𝑥, 𝑦) ⪰ 𝜃 for all 𝑥, 𝑦 ∈ 𝑋. (2) 𝑑(𝑥, 𝑦) = 𝜃 if and only if 𝑥 = 𝑦. (3) 𝑑(𝑥, 𝑦) = 𝑑(𝑦, 𝑥) for all 𝑥, 𝑦 ∈ 𝑋. (4) 𝑑(𝑥, 𝑦) ⪯ 𝑏[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)] for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then, (𝑋, 𝑑, A) is called a 𝐶∗ -algebra-valued 𝑏-metric space. Remark 26. If 𝑏 = 𝐼, then a 𝐶∗ -algebra-valued 𝑏-metric spaces are 𝐶∗ -algebra-valued metric spaces. In particular, if A is set of real numbers and 𝑏 = 1, then the 𝐶∗ -algebra-valued 𝑏-metric spaces is the metric spaces. Definition 27 (see [26]). Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space. A sequence {𝑥𝑛 } in (𝑋, A, 𝑑) is said to converge to 𝑥 if and only if for any 𝜀 > 0 there exists 𝑁 ∈ N such that, for all 𝑛 ≥ 𝑁, ‖𝑑(𝑥𝑛 , 𝑥)‖ ≤ 𝜀. Then, {𝑥𝑛 } is said to be convergent with respect to A and 𝑥 is called limit point of {𝑥𝑛 }. We denote it by lim𝑛 → ∞ 𝑥𝑛 = 𝑥. A sequence {𝑥𝑛 } is called a Cauchy seqeunce with respect to A if and only if for any 𝜀 > 0 there exists 𝑁 ∈ N such that, for all 𝑛, 𝑚 ≥ 𝑁, ‖𝑑(𝑥𝑛 , 𝑥𝑚 )‖ ≤ 𝜀. We say (𝑋, A, 𝑑) is a complete 𝐶∗ -algebra-valued 𝑏metric space if every Cauchy sequence with respect to A is convergent sequence with respect to A.

Journal of Function Spaces

5

The following is an example of complete 𝐶∗ -algebravalued 𝑏-metric space. Example 28 (see [26]). Let 𝑋 = R and let A = 𝑀𝑛 (R). Define 𝑑 (𝑥, 𝑦)

It is easy to verify 𝑑 is a 𝐶∗ -algebra-valued 𝑏-metric space and (𝑋, 𝑀2 (R), 𝑑) is a complete 𝐶∗ -algebra-valued 𝑏-metric space be completeness of R. Proof. An element 𝐴 ∈ A = 𝑀𝑛 (R) is positive element; denote it by

𝑝 󵄨𝑝 󵄨𝑝 󵄨𝑝 󵄨 󵄨 󵄨 = diag ((𝑥 − 𝑦) , 𝛼1 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 , 𝛼2 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 , . . . , 𝛼𝑛−1 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 )

[ [ [ [ [ =[ [ [ [ [ [ [

󵄨𝑝 󵄨󵄨 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 0

0 󵄨𝑝 󵄨󵄨 𝛼1 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨

0

0

.. .

.. .

0

0

0

⋅⋅⋅

0

0

⋅⋅⋅

0

󵄨𝑝 󵄨 𝛼2 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ⋅ ⋅ ⋅

.. .

.. .

.. .

d

] ] ] ] ] ], ] ] ] ] ]

𝐴 ⪰ 𝜃, iff 𝐴 is positive semidefinite. We define a partial ordering ⪯ on A as follows: (15)

󵄨𝑝 󵄨 ⋅ ⋅ ⋅ 𝛼𝑛−1 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ]

0

where 𝑥, 𝑦 ∈ R and 𝛼𝑖 > 0 for all 𝑖 = 1, 2, . . . , 𝑛 − 1 are constants and 𝑝 is a natural number such that 𝑝 ≥ 2. A norm ‖ ⋅ ‖ on A is defined by 󵄨 󵄨1/𝑝 ‖𝐴‖ = max 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 , 𝑖,𝑗

∗

𝑎11 𝑎21 ⋅ ⋅ ⋅ 𝑎𝑛1 [𝑎 𝑎 ⋅ ⋅ ⋅ 𝑎 ] [𝑎 𝑎 ⋅ ⋅ ⋅ 𝑎 ] [ 21 22 [ 12 22 2𝑛 ] 𝑛2 ] ] ] [ [ 𝐴∗ = [ . ] =[ . . . . .. ] ] ] [ . [ . . . . . d . ] . d . ] [ . [ . [𝑎𝑛1 𝑎𝑛2 ⋅ ⋅ ⋅ 𝑎𝑛𝑛 ]

𝐴 ⪯ 𝐵 iff 𝜃 ⪯ 𝐵 − 𝐴,

[ [ [ 𝑏=[ [ [ [

⋅⋅⋅ ⋅⋅⋅

2

0

] 0 ] ] ] .. ] . ] ]

.. .

.. .

d

[ 0

0

⋅ ⋅ ⋅ 2𝑝−1 ]𝑛×𝑛

.

1 = (|𝑎|𝑝 + |𝑐|𝑝 ) 2

(20)

(21)

and hence |𝑎 + 𝑐|𝑝 ≤ 2𝑝−1 (|𝑎|𝑝 + |𝑐|𝑝 ) for all 𝑎, 𝑐 ∈ R. We substitute 𝑎 = 𝑥 − 𝑦 and 𝑐 = 𝑦 − 𝑧; then,

[𝑎1𝑛 𝑎2𝑛 ⋅ ⋅ ⋅ 𝑎𝑛𝑛 ]

[𝑎 𝑎 ⋅ ⋅ ⋅ 𝑎 ] [ 12 22 𝑛2 ] ] [ . =[ . .. .. ] ] [ . . d . ] [ .

󵄨𝑝 󵄨 |𝑥 − 𝑧|𝑝 = 󵄨󵄨󵄨𝑥 − 𝑦 + 𝑦 − 𝑧󵄨󵄨󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨 ≤ 2𝑝−1 (󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑦 − 𝑧󵄨󵄨󵄨 ) .

(22)

Hence, setting 𝑀0 = (|𝑥 − 𝑦|𝑝 + |𝑦 − 𝑧|𝑝 ) and 𝑀1 = |𝑥 − 𝑧|𝑝 , we obtain that

[𝑎1𝑛 𝑎2𝑛 ⋅ ⋅ ⋅ 𝑎𝑛𝑛 ]

[

0

0 𝑝−1

󵄨𝑝 󵄨󵄨 𝑎 + 𝑐 󵄨󵄨𝑝 󵄨󵄨 1 󵄨󵄨 = 󵄨󵄨 𝑎 + (1 − 1 ) 𝑐󵄨󵄨󵄨 ≤ 1 |𝑎|𝑝 + (1 − 1 ) |𝑐|𝑝 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 2 󵄨󵄨 2 2 󵄨2 󵄨 2 󵄨

𝑎11 𝑎21 ⋅ ⋅ ⋅ 𝑎𝑛1

2𝑝−1 𝑀0 − 𝑀1 0

2𝑝−1

Since function 𝑓(𝑥) = |𝑥|𝑝 is convex function for all 𝑝 ≥ 2 and 𝑥 ∈ R, this implies that

(17)

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [

(19)

where 𝜃 mean the zero matrix in 𝑀𝑛 (R). Firstly, it clears that ⪯ is partially order relation. Next, we show that 𝑑 is a 𝐶∗ algebra-valued 𝑏-metric space. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋. It is easy to see that 𝑑 satifies conditions (1), (2), and (3) of Definition 25. We will only show condition (4) where 𝑑(𝑥, 𝑦) ⪯ 𝑏[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)] with

(16)

where 𝐴 = (𝑎𝑖𝑗 )𝑛×𝑛 ∈ A. The involution is given by 𝐴∗ = (𝐴)𝑇 , conjugate transpose of matrix 𝐴: 𝑎11 𝑎12 ⋅ ⋅ ⋅ 𝑎1𝑛

(18)

𝑝−1

𝛼1 (2

0

0

⋅⋅⋅

𝑀0 − 𝑀1 )

0

⋅⋅⋅

𝛼2 (2𝑝−1 𝑀0 − 𝑀1 ) ⋅ ⋅ ⋅

0

0

.. .

.. .

.. .

d

0

0

0

⋅⋅⋅

0

] ] ] 0 ] ] ] .. ] ] . ] ] ] .. ] ] . ] ] 𝑝−1 𝛼𝑛−1 (2 𝑀0 − 𝑀1 )]

6

Journal of Function Spaces

[ [ [ [ [ =[ [ [ [ [ [

2𝑝−1 𝑀0

0

0

𝑝−1

[

[ [ [ =[ [ [ [

𝛼1 2

𝑀0

0

⋅⋅⋅

0

⋅⋅⋅

𝛼2 2𝑝−1 𝑀0 ⋅ ⋅ ⋅

0

0

.. .

.. .

.. .

d

0

0

0

⋅⋅⋅

2𝑝−1

0 𝑝−1

0

2

.. .

.. .

[ 0

0

⋅⋅⋅

0

[ ][ [ ] ⋅⋅⋅ 0 ][ ][ . ][ [ d .. ] ][ [ [ ⋅ ⋅ ⋅ 2𝑝−1 ] [

0

] [ ] [ 0 ] [ ] [ ] [ .. ]−[ . ] [ ] [ ] [ .. ] [ . ] 𝑝−1 𝛼𝑛−1 2 𝑀0 ] [

𝑀1

0

0

⋅⋅⋅

0

0

𝛼1 𝑀1

0

⋅⋅⋅

0

0

0

𝛼2 𝑀1 ⋅ ⋅ ⋅

.. .

.. .

.. .

.. .

d

0

0

0

⋅ ⋅ ⋅ 𝛼𝑛−1 𝑀1 ]

𝑀1

0

0

⋅⋅⋅

0

0

𝛼1 𝑀1

0

⋅⋅⋅

0

0

0

𝛼2 𝑀1 ⋅ ⋅ ⋅

.. .

.. .

.. .

.. .

d

[ 0

0

0

⋅ ⋅ ⋅ 𝛼𝑛−1 𝑀1 ]

𝑀0

0

0

⋅⋅⋅

0

0

𝛼1 𝑀0

0

⋅⋅⋅

0

0

0

𝛼2 𝑀0 ⋅ ⋅ ⋅

.. .

.. .

.. .

.. .

d

0

0

0

⋅ ⋅ ⋅ 𝛼𝑛−1 𝑀0 ]

.. .

] [ ] [ ] [ ] [ ] [ ]−[ ] [ ] [ ] [ ] [

.. .

] ] ] ] ] ] ] ] ] ]

.. .

] ] ] ] ] ] ] ] ] ]

= 𝑏 (𝑑 (𝑥, 𝑦) + 𝑑 (𝑦, 𝑧)) − 𝑑 (𝑥, 𝑧) (23)

implies that each eigenvalue of 𝑏[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)] − 𝑑(𝑥, 𝑦) is nonnegative. Since each eigenvalue of a positive semidefinite matrix is a nonnegative real number, we have that 𝑏[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)] − 𝑑(𝑥, 𝑦) is positive semidefinite; that is, 𝑏[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)] − 𝑑(𝑥, 𝑦) ⪰ 𝜃, that is, 𝑑(𝑥, 𝑦) ⪯ 𝑏[𝑑(𝑥, 𝑧) + 𝑑(𝑧, 𝑦)], where 𝑏 = 2𝑝−1 𝐼 ∈ A󸀠 and 𝑏 ⪰ 𝐼 by 2𝑝−1 > 1. But |𝑥 − 𝑦|𝑝 ≤ |𝑥 − 𝑧|𝑝 + |𝑧 − 𝑦|𝑝 is impossible for all 𝑥, 𝑦, 𝑧 ∈ R. Hence, (𝑋, 𝑀𝑛 (R), 𝑑) is 𝐶∗ -algebra-valued 𝑏-metric spaces but not 𝐶∗ -algebra-valued metric spaces. Finally, we show that (𝑋, A, 𝑑) is a complete 𝐶∗ -algebravalued 𝑏-metric space. Suppose that {𝑥𝑛 } is a Cauchy sequence with respect to A. Then, for any 𝜀 > 0, there exists 𝑁 ∈ N such that ‖𝑑(𝑥𝑚 , 𝑥𝑛 )‖ ≤ 𝜀 for all 𝑚, 𝑛 ≥ 𝑁; that is, 󵄨 󵄨𝑝 1/𝑝 󵄨 󵄨𝑝 1/𝑝 max {(󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 ) , (𝛼1 󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 ) , 󵄨 󵄨𝑝 1/𝑝 󵄨 󵄨𝑝 1/𝑝 (𝛼2 󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 ) , . . . , (𝛼𝑛−1 󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 ) } ≤ 𝜀

(24)

󵄨 󵄨󵄨 󵄨 󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 = (󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨 )

󵄨󵄨𝑝 1/𝑝

󵄨 ≤ max {(󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨 )

,

󵄨 󵄨𝑝 1/𝑝 󵄨 󵄨𝑝 1/𝑝 (𝛼1 󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 ) , (𝛼2 󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨󵄨 ) , . . . , 󵄨󵄨𝑝 1/𝑝

󵄨 (𝛼𝑛−1 󵄨󵄨󵄨𝑥𝑚 − 𝑥𝑛 󵄨󵄨 )

(25)

}≤𝜀

for all 𝑚, 𝑛 ≥ 𝑁. Hence, {𝑥𝑛 } is a Cauchy sequnce in R. By completeness of R, there exists 𝑥 ∈ R such that lim𝑛 → ∞ 𝑥𝑛 = 𝑥; that is, lim𝑛 → ∞ |𝑥𝑛 − 𝑥| = 0. Then, we have that 󵄨 󵄩󵄩 󵄨 󵄩 󵄨𝑝 1/𝑝 󵄨𝑝 1/𝑝 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 = max {(󵄨󵄨󵄨𝑥𝑛 − 𝑥󵄨󵄨󵄨 ) , (𝛼1 󵄨󵄨󵄨𝑥𝑛 − 𝑥󵄨󵄨󵄨 ) , 󵄨 󵄨 󵄨𝑝 1/𝑝 󵄨𝑝 1/𝑝 (𝛼2 󵄨󵄨󵄨𝑥𝑛 − 𝑥󵄨󵄨󵄨 ) , . . . , (𝛼𝑛−1 󵄨󵄨󵄨𝑥𝑛 − 𝑥󵄨󵄨󵄨 ) }

Next, we disscus some fundamental properties of 𝐶∗ algebra-valued 𝑏-metric spaces. Theorem 29. Let (𝑋, A, 𝑑) be 𝐶∗ -algebra-valued 𝑏-metric space. If {𝑥𝑛 } is a convergent sequence with respect to A, then {𝑥𝑛 } is Cauchy sequence with respect to A. Proof. Assume that {𝑥𝑛 } is a convergent sequence with respect to A; then, there exists a 𝑥 ∈ 𝑋 such that lim𝑛 → ∞ 𝑥𝑛 = 𝑥. Let 𝜀 > 0, there is 𝑁 ∈ N such that, for all 𝑛 ≥ 𝑁, 𝜀 󵄩 󵄩󵄩 . 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ (27) 2 ‖𝑏‖ For 𝑚, 𝑛 ∈ 𝑁, we get that 𝑑 (𝑥𝑚 , 𝑥𝑛 ) ⪯ 𝑏 [𝑑 (𝑥𝑚 , 𝑥) + 𝑑 (𝑥, 𝑥𝑛 )] .

for all 𝑚, 𝑛 ≥ 𝑁. Therefore, 󵄨󵄨𝑝 1/𝑝

converges to 0 as 𝑛 → ∞. Therefore, {𝑥𝑛 } is convergent with respect to A and {𝑥𝑛 } converging to 𝑥, so (𝑋, A, 𝑑) is a complete 𝐶∗ -algebra-valued 𝑏-metric space.

(26)

By Theorem 16, for 𝑚, 𝑛 ≥ 𝑁, we have 󵄩󵄩󵄩𝑑 (𝑥𝑚 , 𝑥𝑛 )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑏 [𝑑 (𝑥𝑚 , 𝑥) + 𝑑 (𝑥, 𝑥𝑛 )]󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ ‖𝑏‖ 󵄩󵄩𝑑 (𝑥𝑚 , 𝑥) + 𝑑 (𝑥, 𝑥𝑛 )󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑚 , 𝑥)󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥, 𝑥𝑛 )󵄩󵄩󵄩 𝜀 𝜀 ≤ ‖𝑏‖ + ‖𝑏‖ = 𝜀. 2 ‖𝑏‖ 2 ‖𝑏‖

(28)

(29)

This implies that {𝑥𝑛 } is Cauchy sequence with respect to A. Definition 30. A subset 𝑆 of a 𝐶∗ -algebra-valued 𝑏-metric space (𝑋, A, 𝑑) is bounded with respect to A if there exists 𝑥 ∈ 𝑋 and a nonnegetive real number 𝑀 such that ‖𝑑 (𝑥, 𝑥)‖ ≤ 𝑀,

∀𝑥 ∈ 𝑋.

(30)

Journal of Function Spaces

7

Theorem 31. Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space and let {𝑥𝑛 } be a sequence in 𝑋 and 𝑥 ∈ 𝑋. Then, (1) 𝑥𝑛 → 𝑥 if and only if 𝑑(𝑥𝑛 , 𝑥) → 𝜃, (2) a convergent sequence in 𝑋 is bounded with respect to A and its limit is unique, (3) a Cauchy sequence in 𝑋 is bounded with respect to A. Proof. (1) Assume that 𝑥𝑛 → 𝑥. Let 𝜀 > 0 is given. Then, there exists 𝑁0 ∈ N such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥) − 𝜃󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ 𝜀.

(31)

This implies that 𝑑(𝑥𝑛 , 𝑥) → 𝜃. Conversely, assume that 𝑑(𝑥𝑛 , 𝑥) → 𝜃. Then, for any 𝜀 > 0, there exists 𝑁1 ∈ N such that 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥) − 𝜃󵄩󵄩󵄩 ≤ 𝜀 󳨐⇒ (32) 󵄩󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ 𝜀; that is, 𝑥𝑛 → 𝑥. (2) Let {𝑥𝑛 } be a convergent sequence with respect to A. Suppose that 𝑥𝑛 → 𝑥. Then, taking 𝜀 = 1, we can find 𝑁 ∈ N such that 𝑑 (𝑥𝑛 , 𝑥) ≤ 1,

∀𝑛 ≥ 𝑁.

(33)

Let 𝐾 = max{‖𝑑(𝑥1 , 𝑥)‖, ‖𝑑(𝑥2 , 𝑥)‖, . . . , ‖𝑑(𝑥𝑁, 𝑥)‖}. Setting 𝑀 = max{1, 𝐾}. This implies that 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ 𝑀,

∀𝑛 ∈ N.

(34)

Next, suppose that 𝑥𝑛 → 𝑥 and 𝑥𝑛 → 𝑦. Consider, 𝑑(𝑥, 𝑦) ⪯ 𝑏[𝑑(𝑥, 𝑥𝑛 ) + 𝑑(𝑥𝑛 , 𝑦)]; by Theorem 16, we have 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ ‖𝑏‖ [󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑦)󵄩󵄩󵄩] .

(35)

From (1), letting 𝑛 → ∞, we obtian that ‖𝑑(𝑥, 𝑦)‖ = 0; that is 𝑥 = 𝑦. (3) Assume that {𝑥𝑛 } is a Cauchy sequence with respect to A. In particular, 𝜀 = 1; there exists 𝑁1 ∈ N such that 󵄩󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑚 , 𝑥𝑛 )󵄩󵄩󵄩 ≤ 1 ∀𝑚, 𝑛 ≥ 𝑁1 .

(36)

Let 𝐾 = max{‖𝑑(𝑥1 , 𝑥𝑁1 )‖, ‖𝑑(𝑥2 , 𝑥𝑁1 )‖, . . . , ‖𝑑(𝑥𝑁1 −1 , 𝑥𝑁1 )‖}. Then, 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥𝑁1 )󵄩󵄩󵄩 ≤ 𝐾 󵄩 󵄩

∀𝑛 < 𝑁1 .

(37)

Set 𝑀 = max{1, 𝐾}. Then, we get that 󵄩󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥𝑁1 )󵄩󵄩󵄩 ≤ 𝑀 ∀𝑛 ∈ N. 󵄩 󵄩

(38)

Proof. Let 𝜀 > 0 be given. Then, there exists 𝑁 ∈ N such that 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ 𝜀, ∀𝑛 ≥ 𝑁.

(39)

Since 𝑛1 < 𝑛2 < ⋅ ⋅ ⋅ < 𝑛𝑘 < ⋅ ⋅ ⋅ is an increasing sequence of natural numbers, it is easily proved (by induction) that 𝑛𝑘 ≥ 𝑘. Hence, if 𝑘 ≥ 𝑁, we also have 𝑛𝑘 ≥ 𝑘 ≥ 𝑁 so that 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥𝑛𝑘 , 𝑥)󵄩󵄩󵄩 ≤ 𝜀, ∀𝑛𝑘 ≥ 𝑁. 󵄩 󵄩

(40)

Therefore, subsequence {𝑥𝑛𝑘 } also converges to 𝑥. Theorem 33. Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space. Then, every subsequence of a Cauchy sequence is Cauchy sequence. Proof. Let {𝑥𝑛𝑘 } be a subsequence of Cauchy sequence {𝑥𝑛 } in a 𝐶∗ -algebra-valued 𝑏-metric space. Then, for every 𝜀 > 0, there is 𝑁 ∈ N such that, for all 𝑟, 𝑠 ≥ 𝑁, we have ‖𝑑(𝑥𝑟 , 𝑥𝑠 )‖ ≤ 𝜀. Similar to the facts in proof of previous theorem, we have 𝑛𝑟 ≥ 𝑟 ≥ 𝑁 and 𝑛𝑠 ≥ 𝑠 ≥ 𝑁. Hence, we obtain that ‖𝑑(𝑥𝑛𝑟 , 𝑥𝑛𝑠 )‖ ≤ 𝜀. Therefore, {𝑥𝑛𝑘 } is Cauchy sequence. Theorem 34. Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space and let {𝑥𝑛 } be a Cauchy sequence with respect to A. If {𝑥𝑛 } contains its convergent subsequence, then {𝑥𝑛 } is convergent sequence. Proof. Let 𝜀 > 0. Since {𝑥𝑛 } is a Cauchy sequence with respect to A, there exists an 𝑁0 ∈ N such that 1 󵄩󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑚 , 𝑥𝑝 )󵄩󵄩󵄩 ≤ 󵄩 󵄩 2 ‖𝑏‖ 𝜀, ∀𝑚, 𝑝 ≥ 𝑁0 .

(41)

Let {𝑥𝑛𝑘 } be a convergent subsequence of {𝑥𝑛 } and 𝑥𝑛𝑘 → 𝑥 (𝑘 → ∞). Then, there exists 𝑁1 ∈ N such that 1 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥𝑛𝑘 , 𝑥)󵄩󵄩󵄩 ≤ 󵄩 2 ‖𝑏‖ 𝜀, 󵄩

∀𝑛𝑘 ≥ 𝑁1 .

(42)

Let 𝑁 = max{𝑁0 , 𝑁1 }. For 𝑛, 𝑘 ≥ 𝑁, we have 𝑑 (𝑥𝑛 , 𝑥) ⪯ 𝑏 [𝑑 (𝑥𝑛 , 𝑥𝑛𝑘 ) + 𝑑 (𝑥𝑛𝑘 , 𝑥)] .

(43)

By Theorem 16, we also have 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥𝑛𝑘 ) + 𝑑 (𝑥𝑛𝑘 , 𝑥)󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥𝑛𝑘 )󵄩󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑑 (𝑥𝑛𝑘 , 𝑥)󵄩󵄩󵄩󵄩 ≤ ‖𝑏‖ [

(44)

𝜀 𝜀 + ] ≤ 𝜀. 2 ‖𝑏‖ 2 ‖𝑏‖

Therefore, 𝑥𝑛 → 𝑥 as 𝑛 → ∞. Theorem 32. Let {𝑥𝑛 } be a convergent sequence in a 𝐶∗ algebra-valued 𝑏-metric space (𝑋, A, 𝑑) and lim𝑛 → ∞ 𝑥𝑛 = 𝑥. Then, every subsequence {𝑥𝑛𝑘 } of {𝑥𝑛 } is convergent and has the same limit 𝑥.

Theorem 35. Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space. Suppose that {𝑥𝑛 } and {𝑦𝑛 } are convergent with respect to A and converge to 𝑥 and 𝑦, respectively. Then, 𝑑(𝑥𝑛 , 𝑦𝑛 ) converges to 𝑏2 𝑑(𝑥, 𝑦).

8

Journal of Function Spaces

Proof. Let 𝜀 > 0. Since 𝑥𝑛 → 𝑥 and 𝑦𝑛 → 𝑦, there exist 𝑁0 , 𝑁1 ∈ N such that 𝜀 󵄩 󵄩󵄩 , ∀𝑛 ≥ 𝑁0 , 󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 ≤ 2 ‖𝑏‖ 𝜀 󵄩󵄩 󵄩 , ∀𝑛 ≥ 𝑁1 . 󵄩󵄩𝑑 (𝑦𝑛 , 𝑦)󵄩󵄩󵄩 ≤ 2 ‖𝑏‖2

(45)

Since 𝑑(𝑥𝑛 , 𝑦𝑛 ) ⪯ 𝑏𝑑(𝑥𝑛 , 𝑥) + 𝑏2 𝑑(𝑥, 𝑦) + 𝑏2 𝑑(𝑦, 𝑦𝑛 ), by Theorem 16, we have 󵄩󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑦𝑛 ) − 𝑏2 𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 󵄩 󵄩

(46)

Theorem 36. Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space. Suppose that {𝑥𝑛 } and {𝑦𝑛 } are convergent with respect to A and converge to 𝑥 and y, respectively. Then,

Taking the limit as 𝑛 → ∞ in this inequality, we obtain that lim𝑛 → ∞ ‖𝑑(𝑥𝑛 , 𝑦𝑛 )‖ = 0. Since 𝑑 (𝑥, 𝑧) ⪯ 𝑏 [𝑑 (𝑥, 𝑥𝑛 ) + 𝑑 (𝑥𝑛 , 𝑧)] ,

󵄩 󵄩 ≤ lim sup 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑦𝑛 )󵄩󵄩󵄩 𝑛→∞

(47)

󵄩 󵄩 ≤ ‖𝑏‖2 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 . In particular, if 𝑥 = 𝑦, then we have lim𝑛 → ∞ ‖𝑑(𝑥𝑛 , 𝑦𝑛 )‖ = 0. Moreover, for any 𝑧 ∈ 𝑋, we have 1 󵄩 󵄩 inf 󵄩󵄩𝑑 (𝑥𝑛 , 𝑧)󵄩󵄩󵄩 ‖𝑑 (𝑥, 𝑧)‖ ≤ lim 𝑛→∞ 󵄩 ‖𝑏‖ 󵄩 󵄩 ≤ lim sup 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑧)󵄩󵄩󵄩 ≤ ‖𝑏‖ ‖𝑑 (𝑥, 𝑧)‖ . 𝑛→∞

(48)

Proof. By defintion of 𝐶∗ -algebra-valued 𝑏-metric space, it easy to see that

(49) 2

𝑑 (𝑥𝑛 , 𝑦𝑛 ) ⪯ 𝑏𝑑 (𝑥𝑛 , 𝑥) + 𝑏 𝑑 (𝑥, 𝑦) + 𝑏 𝑑 (𝑦, 𝑦𝑛 ) . Using Theorem 16, we have 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 2󵄩 󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥, 𝑥𝑛 )󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑦𝑛 )󵄩󵄩󵄩

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 2󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑦𝑛 )󵄩󵄩󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩

󵄩 󵄩 󵄩 󵄩 ‖𝑑 (𝑥, 𝑧)‖ ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥, 𝑥𝑛 )󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑧)󵄩󵄩󵄩 , 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑧)󵄩󵄩󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 + ‖𝑏‖ ‖𝑑 (𝑥, 𝑧)‖ .

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Definition 37. Let (𝑋, A, 𝑑) be a 𝐶∗ -algebra-valued 𝑏-metric space. A subset 𝐹 of (𝑋, A, 𝑑) is called a closed set if a sequence {𝑥𝑛 } in 𝑋 and 𝑥𝑛 → 𝑥 with respect to A imply 𝑥 ∈ 𝐹.

4. Fixed Point Theorems for Cyclic Contractions Theorem 38. Let 𝐴 and 𝐵 be nonempty closed subset of a complete 𝐶∗ -algebra-valued 𝑏-metric space (𝑋, A, 𝑑). Assume that 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 is cyclic mapping that satisfies 𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆∗ 𝑑 (𝑥, 𝑦) 𝜆,

∀𝑥 ∈ 𝐴, ∀𝑦 ∈ 𝐵,

(54)

where 𝜆 ∈ A with ‖𝜆‖ < 1/‖𝑏‖. Then, 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵.

𝑑 (𝑥, 𝑦) ⪯ 𝑏𝑑 (𝑥, 𝑥𝑛 ) + 𝑏2 𝑑 (𝑥𝑛 , 𝑦𝑛 ) + 𝑏2 𝑑 (𝑦𝑛 , 𝑦) ,

󵄩 󵄩 + ‖𝑏‖2 󵄩󵄩󵄩𝑑 (𝑦𝑛 , 𝑦)󵄩󵄩󵄩 ,

𝑑 (𝑥𝑛 , 𝑧) ⪯ 𝑏 [𝑑 (𝑥𝑛 , 𝑥) + 𝑑 (𝑥, 𝑧)] ,

Again taking the lower limit as 𝑛 → ∞ in the first inequality and the upper limit as 𝑛 → ∞ in the second inequality, we obtain the second desired result.

1 󵄩󵄩 󵄩 󵄩 󵄩 inf 󵄩󵄩𝑑 (𝑥𝑛 , 𝑦𝑛 )󵄩󵄩󵄩 󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ lim 𝑛→∞ 󵄩 ‖𝑏‖2

󵄩 󵄩 + ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑦, 𝑦𝑛 )󵄩󵄩󵄩 .

(51)

by Theorem 16, we have

Therefore, 𝑑(𝑥𝑛 , 𝑦𝑛 ) → 𝑏2 𝑑(𝑥, 𝑦).

2

󵄩 󵄩 󵄩 󵄩 󵄩󵄩 2󵄩 󵄩󵄩𝑑 (𝑥𝑛 , 𝑦𝑛 )󵄩󵄩󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑦, 𝑦𝑛 )󵄩󵄩󵄩 .

(52)

󵄩 󵄩 󵄩 󵄩 ≤ ‖𝑏‖ 󵄩󵄩󵄩𝑑 (𝑥𝑛 , 𝑥)󵄩󵄩󵄩 + ‖𝑏‖2 󵄩󵄩󵄩𝑑 (𝑦, 𝑦𝑛 )󵄩󵄩󵄩 ≤ 𝜀.

2

Taking the lower limit as 𝑛 → ∞ in the first inequality and the upper limit as 𝑛 → ∞ in the second inequality, this completes the first result. In particular, if 𝑥 = 𝑦, we have

Proof. Let 𝑥0 be any point in 𝐴. Since 𝑇 is cyclic mapping, we have 𝑇𝑥0 ∈ 𝐵 and 𝑇2 𝑥0 ∈ 𝐴. Using the contractive condition of mapping 𝑇, we get 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) = 𝑑 (𝑇𝑥0 , 𝑇 (𝑇𝑥0 )) ⪯ 𝜆∗ 𝑑 (𝑥0 , 𝑇𝑥0 ) 𝜆. (55) For all 𝑛 ∈ N, we have

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𝑛

𝑑 (𝑇𝑛 𝑥0 , 𝑇𝑛+1 𝑥0 ) ⪯ (𝜆∗ ) 𝑑 (𝑥0 , 𝑇𝑥0 ) 𝜆𝑛 (56) ∗ 𝑛

𝑛

= (𝜆 ) 𝛽𝜆 ,

Journal of Function Spaces

9

where 𝛽 = 𝑑(𝑥0 , 𝑇𝑥0 ). Consider, for any 𝑚, 𝑛 ∈ N such that 𝑚 ≤ 𝑛; then, 𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑛 𝑥0 )

Next, we will complete the proof by showing that 𝑥 is a unique fixed point of 𝑇. Since 𝜃 ⪯ 𝑑 (𝑇𝑥, 𝑥) ⪯ 𝑏 [𝑑 (𝑇𝑥, 𝑇2𝑛 𝑥0 ) + 𝑑 (𝑇2𝑛 𝑥0 , 𝑥)]

⪯ 𝑏 [𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑚+1 𝑥0 ) + 𝑑 (𝑇𝑚+1 𝑥0 , 𝑇𝑛 𝑥0 )]

⪯ 𝑏 [𝜆∗ 𝑑 (𝑥, 𝑇2𝑛−1 𝑥0 ) 𝜆 + 𝑑 (𝑇2𝑛 𝑥0 , 𝑥)]

⪯ 𝑏𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑚+1 𝑥0 )

by Theorem 16, we obtain that

+ 𝑏2 [𝑑 (𝑇𝑚+1 𝑥0 , 𝑇𝑚+2 𝑥0 ) + 𝑑 (𝑇𝑚+2 𝑥0 , 𝑇𝑛 𝑥0 )] ⪯ ⋅⋅⋅ ⪯ 𝑏𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑚+1 𝑥0 ) + 𝑏2 𝑑 (𝑇𝑚+1 𝑥0 , 𝑇𝑚+2 𝑥0 )

0 ≤ ‖𝑑 (𝑇𝑥, 𝑥)‖ (57)

𝑚+1

⪯ 𝑏 (𝜆∗ ) 𝛽𝜆𝑚 + 𝑏2 (𝜆∗ ) 𝑛−1

+ 𝑏𝑛−𝑚 (𝜆∗ )

(𝑛 󳨀→ ∞) .

𝜃 ⪯ 𝑑 (𝑥, 𝑦) = 𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆∗ 𝑑 (𝑥, 𝑦) 𝜆, 𝑘

𝛽𝜆𝑛−1 = ∑ 𝑏𝑘−𝑚+1 (𝜆∗ ) 𝛽𝜆𝑘 .

we have 󵄩 󵄩 󵄩 ∗ 󵄩 󵄩 ∗󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝜆 𝑑 (𝑥, 𝑦) 𝜆󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝜆 󵄩󵄩󵄩 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ‖𝜆‖ 󵄩 󵄩 󵄩 󵄩 = ‖𝜆‖2 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 < 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 .

𝑘=𝑚

From Theorem 16, we have 󵄩󵄩 𝑛−1 󵄩󵄩 󵄩󵄩 𝑘 󵄩󵄩 󵄩 󵄩󵄩 𝑚 𝑛 𝑘−𝑚+1 (𝜆∗ ) 𝛽𝜆𝑘 󵄩󵄩󵄩 󵄩󵄩𝑑 (𝑇 𝑥0 , 𝑇 𝑥0 )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ 𝑏 󵄩󵄩 󵄩󵄩󵄩𝑘=𝑚 󵄩

(61)

(62)

This is a contradiction. Therefore, 𝑥 = 𝑦 which implies that the fixed point is unique.

𝑛−1 󵄩 𝑘 󵄩 󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩𝑏𝑘−𝑚+1 (𝜆∗ ) 𝛽𝜆𝑘 󵄩󵄩󵄩 󵄩 󵄩 𝑘=𝑚

Example 39. Let 𝑋 be a set of real numbers and A = 𝑀2×2 (R) with ‖𝐴‖ = max𝑖,𝑗 |𝑎𝑖𝑗 |, where 𝑎𝑖𝑗 are entries of the matrix 𝐴 ∈ 𝑀2×2 (R). Then, (𝑋, A, 𝑑) is a 𝐶∗ -algebra-valued 𝑏metric space with 𝑏 = [ 20 02 ], where the involution is given by 𝐴∗ = (𝐴)𝑇 ,

𝑛−1 󵄩 󵄩 󵄩󵄩 𝑘󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩󵄩𝑏𝑘−𝑚+1 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩(𝜆∗ ) 󵄩󵄩󵄩 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝜆𝑘 󵄩󵄩󵄩󵄩 󵄩 󵄩 𝑘=𝑚

𝑛−1 󵄩 󵄩󵄩 󵄩2 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ 󵄩󵄩󵄩󵄩𝑏𝑘−𝑚+1 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩(𝜆)𝑘 󵄩󵄩󵄩󵄩 𝑘=𝑚

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We have 𝑇𝑥 = 𝑥; that is, 𝑥 is a fixed point of 𝑇. Suppose that 𝑦 is fixed point of 𝑇 and 𝑦 ≠ 𝑥. Since

𝛽𝜆𝑚+1 + ⋅ ⋅ ⋅

𝑛−1

󵄩 󵄩 󵄩 󵄩 ≤ ‖𝑏‖ ‖𝜆‖2 󵄩󵄩󵄩󵄩𝑑 (𝑥, 𝑇2𝑛−1 𝑥0 )󵄩󵄩󵄩󵄩 + ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑑 (𝑇2𝑛 𝑥0 , 𝑥)󵄩󵄩󵄩󵄩 󳨀→ 0

+ ⋅ ⋅ ⋅ + 𝑏𝑛−𝑚 𝑑 (𝑇𝑛−1 𝑥0 , 𝑇𝑛 𝑥0 ) 𝑚

(59)

(58)

𝑛−1

󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ ‖𝑏‖𝑘−𝑚+1 ‖𝜆‖2𝑘

󵄨2 󵄨󵄨 0 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 [ ] 𝑑 (𝑥, 𝑦) = 󵄨󵄨2 󵄨󵄨 𝑥 − 𝑦 0 󵄨 󵄨 󵄨] 󵄨 [ and partial ordering on A is given as

𝑘=𝑚 𝑛−1

󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ ‖𝑏‖2𝑘 ‖𝜆‖2𝑘 𝑘=𝑚 ∞

󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ (‖𝑏‖ ‖𝜆‖)2𝑘 𝑘=𝑚

2𝑚 󵄩 󵄩 (‖𝑏‖ ‖𝜆‖) = 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 . 1 − (‖𝑏‖ ‖𝜆‖)

Since 0 ≤ ‖𝜆‖ < 1/‖𝑏‖, we have ‖𝛽‖(‖𝑏‖‖𝜆‖)2𝑚 /(1 − (‖𝑏‖‖𝜆‖)) → 0 as 𝑚 → ∞. Therefore, {𝑇𝑛 𝑥0 } is Cauchy sequence with respect to A. By the completeness of (𝑋, A, 𝑑), there exists an element 𝑥 ∈ 𝑋 such that 𝑥 = lim𝑛 → ∞ 𝑇𝑛 𝑥0 . Since {𝑇2𝑛 𝑥0 } is a sequence in 𝐴 and {𝑇2𝑛−1 𝑥0 } is a sequence in 𝐵, we obtain that both sequences converge to the same limit 𝑥. Since 𝐴 and 𝐵 are closed set, this implies that 𝑥 ∈ 𝐴 ∩ 𝐵.

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𝑎11 𝑎12

[

𝑎21

𝑏11 𝑏12 ]⪯[ ] ⇐⇒ 𝑎22 𝑏21 𝑏22 𝑎𝑖𝑗 ≤ 𝑏𝑖𝑗

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∀𝑖, 𝑗 = 1, 2, 3, 4.

Define a mapping 𝑇 : 𝑋 → 𝑋 by 1 󵄨󵄨 1 1 𝑥 + 1/3 󵄨󵄨󵄨 󵄨󵄨sin ( )󵄨󵄨󵄨 − ; 𝑥 ∈ (∞, − ] { − { 󵄨 󵄨 { 3 𝑥 3 3 󵄨 󵄨 { { { { { { 1 1 𝑇𝑥 = {− ; 𝑥 ∈ (− , 0] { 3 3 { { { { { { { 1 𝑥 ∈ (0, +∞) . − ; { 2

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It is clear that 𝑇 is not continuous at all elements of 𝑋. Therefore, Theorem 22 cannot imply the existence of fixed point of mapping 𝑇.

10

Journal of Function Spaces

Suppose that 𝐴 = [−1/2, −1/3] and 𝐵 = [−1/3, 0]. Firstly, we will show that 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 is cyclic mapping. Let 𝑥 ∈ 𝐵; that is, −1/3 ≤ 𝑥 ≤ 0. Then, 𝑇𝑥 = −1/3 ∈ 𝐴. Again, let 𝑦 ∈ 𝐴; that is, −1/2 ≤ 𝑥 ≤ −1/3. Indeed, we consider 1 1 − ≤ 𝑥 ≤ − 󳨐⇒ 2 3

∀𝑥, 𝑦 ∈ 𝑋,

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Proof. Putting 𝐴 = 𝐵 = 𝑋, by Theorem 38, this implies that 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵 = 𝑋.

1 𝑥 + 1/3 − ≤ ≤ 0 󳨐⇒ 18 3 𝑥 + 1/3 1 0 ≤ −( )≤ 󳨐⇒ 3 18

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1 󵄨󵄨 1 󵄨󵄨󵄨 1 󵄨󵄨 1 𝑥 + 1/3 󵄨󵄨󵄨 󵄨󵄨sin ( )󵄨󵄨󵄨 ≤ ) 󵄨󵄨󵄨sin ( )󵄨󵄨󵄨󵄨 ≤ 󳨐⇒ 3 𝑥 󵄨 18 󵄨󵄨 𝑥 󵄨󵄨 18 󵄨

(2) 𝑑(𝑇𝑥, 𝑇𝑦) ⪰ 𝜆∗ 𝑑(𝑥, 𝑦)𝜆 for all 𝑥, 𝑦 ∈ 𝑋,

this implies that 𝑇𝑥 ∈ [−1/3, 0] = 𝐵. For any 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵, since −1/2 ≤ 𝑥 ≤ −1/3 and −1/3 ≤ 𝑦, we have 1/9 ≤ −𝑥/3 ≤ 1/6 and −1/9 ≤ 𝑦/3. Hence, we obtain that 𝑥 1 𝑥 𝑦 − ≤− + . 3 9 3 3

Theorem 41. Suppose that (𝑋, A, 𝑑) is a complete 𝐶∗ -algebravalued 𝑏-metric space. Assume that a mapping 𝑇 : 𝑋 → 𝑋 satisfies (1) 𝑇(𝑋) = 𝑋;

1 󵄨󵄨 1 1 𝑥 + 1/3 󵄨󵄨󵄨 1 1 − ≤ −( ) 󵄨󵄨󵄨sin ( )󵄨󵄨󵄨󵄨 − ≤ − ≤ 0; 3 3 𝑥 󵄨 3 18 3 󵄨

0≤−

𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆∗ 𝑑 (𝑥, 𝑦) 𝜆,

where 𝜆 ∈ A with ‖𝜆‖ < 1/‖𝑏‖. Then, 𝑇 has a unique fixed point in 𝑋.

1 1 − ≤ 𝑥 + ≤ 0 󳨐⇒ 6 3

0 ≤ −(

Corollary 40. Suppose that (𝑋, A, 𝑑) is a 𝐶∗ -algebra-valued 𝑏-metric space. Assume that 𝑇 : 𝑋 → 𝑋 is called a 𝐶∗ algebra-valued 𝑏-contractive mapping on 𝑋; that is, 𝑇 satisfies

(67)

where 𝜆 ∈ A is an invertible element and ‖𝜆−1 ‖ < 1/‖𝑏‖ such that 𝑇 is a 𝐶∗ -algebra-valued 𝑏-expansive mapping on 𝑋. Then, 𝑇 has a unique fixed point in 𝑋. Proof. We will begin to prove this theorem by showing that 𝑇 is injective. Let 𝑥, 𝑦 be an element in 𝑋 such that 𝑥 ≠ 𝑦; that is, 𝑑(𝑥, 𝑦) ≠ 0. Assume that 𝑇𝑥 = 𝑇𝑦. We have 𝜃 = 𝑑 (𝑇𝑥, 𝑇𝑦) ⪰ 𝜆∗ 𝑑 (𝑥, 𝑦) 𝜆

Next, we consider 󵄨 1 󵄨󵄨 1 1 󵄨󵄨󵄨2 󵄨2 󵄨󵄨 𝑥 + 1/3 󵄨󵄨󵄨󵄨 󵄨󵄨 ) 󵄨󵄨sin ( )󵄨󵄨󵄨󵄨 − − (− )󵄨󵄨󵄨 󵄨󵄨𝑇𝑥 − 𝑇𝑦󵄨󵄨󵄨 = 󵄨󵄨󵄨− ( 󵄨󵄨 3 𝑥 󵄨 3 3 󵄨󵄨 󵄨 󵄨󵄨 𝑥 + 1/3 󵄨󵄨 1 󵄨󵄨󵄨󵄨󵄨2 󵄨 = 󵄨󵄨󵄨− ( ) 󵄨󵄨󵄨󵄨sin ( )󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨 3 𝑥 󵄨󵄨󵄨 󵄨 (68) 󵄨󵄨 𝑥 + 1/3 󵄨󵄨2 󵄨󵄨 𝑥 1 󵄨󵄨2 󵄨 󵄨 ≤ 󵄨󵄨󵄨− ( )󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨− − 󵄨󵄨󵄨󵄨 3 󵄨 3 9󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑥 𝑦 󵄨󵄨2 1 󵄨 󵄨2 ≤ 󵄨󵄨󵄨󵄨− + 󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 . 9 󵄨 3 3󵄨 Then, we have 󵄨2 󵄨󵄨 0 󵄨󵄨𝑇𝑥 − 𝑇𝑦󵄨󵄨󵄨 [ ] 𝑑 (𝑇𝑥, 𝑇𝑦) = 󵄨󵄨2 󵄨󵄨 𝑇𝑥 − 𝑇𝑦 0 󵄨󵄨 ] 󵄨󵄨 [ 1 󵄨󵄨 󵄨2 𝑥 − 𝑦󵄨󵄨󵄨 0 [ 9 󵄨󵄨 ] ⪯[ ] 1 󵄨󵄨 󵄨󵄨2 0 󵄨󵄨𝑥 − 𝑦󵄨󵄨 [ ] 9

1/2

= 𝜆∗ 𝑑 (𝑥, 𝑦)

1/2

= (𝑑 (𝑥, 𝑦)

1/2

𝑑 (𝑥, 𝑦)

𝜆

∗

𝜆) (𝑑 (𝑥, 𝑦)

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1/2

𝜆) ⪰ 𝜃.

This implies that 𝜆∗ 𝑑(𝑥, 𝑦)𝜆 = 𝜃. Since 𝜆 is invertible, we have 𝑑(𝑥, 𝑦) = 𝜃 which leads to contradiction. Thus, 𝑇 is injective. By the first condition of mapping 𝑇, we obtain that 𝑇 is bijective which implies that 𝑇 is invertibe and 𝑇−1 is bijective. Next, we will show that 𝑇 has a unique fixed point in 𝑋. In fact, since 𝑇 is 𝐶∗ -algebra-valued 𝑏-expansive and invertible mapping, we substitute 𝑥, 𝑦 with 𝑇−1 𝑥, 𝑇−1 𝑦 in the second condition of 𝑇, respectively, which implies that 𝑑 (𝑇 (𝑇−1 𝑥) , 𝑇 (𝑇−1 𝑦)) ⪰ 𝜆∗ 𝑑 (𝑇−1 𝑥, 𝑇−1 𝑦) 𝜆, ∀𝑥, 𝑦 ∈ 𝑋.

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That is (69)

1 1 󵄨󵄨2 󵄨󵄨 0 0 0 [ 3 ] [󵄨󵄨𝑥 − 𝑦󵄨󵄨 [ ] 3 ] = [ 1] 󵄨󵄨2 [ 1 ] 󵄨󵄨 0 󵄨󵄨𝑥 − 𝑦󵄨󵄨 ] 0 0 [ 3] [ [ 3] = 𝜆∗ 𝑑 (𝑥, 𝑦) 𝜆, 0 where 𝜆 = [ 1/3 0 1/3 ]. Then, ‖𝜆‖ = 1/3 < 1/2 = 1/‖𝑏‖. Thus, 𝑇 satisfies contraction of Theorem 38 implying that 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵; that is, {−1/3} = 𝐹(𝑇).

𝑑 (𝑥, 𝑦) ⪰ 𝜆∗ 𝑑 (𝑇−1 𝑥, 𝑇−1 𝑦) 𝜆,

∀𝑥, 𝑦 ∈ 𝑋.

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Since 𝑑(𝑥, 𝑦) and 𝜆∗ 𝑑(𝑇−1 𝑥, 𝑇−1 𝑦)𝜆 are positive elements in A, 𝜆∗ 𝑑(𝑇−1 𝑥, 𝑇−1 𝑦) ⪯ 𝜆𝑑(𝑥, 𝑦) and 𝜆−1 ∈ A. By condition (2) of Theorem 12 and Theorem 17, we have ∗

𝑑 (𝑇−1 𝑥, 𝑇−1 𝑦) = (𝜆𝜆−1 ) 𝑑 (𝑇−1 𝑥, 𝑇−1 𝑦) 𝜆 (𝜆−1 ) ∗

= (𝜆−1 ) 𝜆∗ 𝑑 (𝑇−1 𝑥, 𝑇−1 𝑦) 𝜆 (𝜆−1 ) ∗

⪯ (𝜆−1 ) 𝑑 (𝑥, 𝑦) 𝜆−1 .

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Journal of Function Spaces

11

Therefore, 𝑇−1 is 𝑏-contractive mapping. Using Corollary 40, there exists a unique 𝑥 such that 𝑇−1 𝑥 = 𝑥, which means it has a unique fixed point 𝑥 ∈ 𝑋 such that 𝑇𝑥 = 𝑇(𝑇−1 𝑥) = (𝑇𝑇−1 )𝑥 = 𝐼𝑥 = 𝑥. Theorem 42 (cyclic Kannan-type). Let 𝐴 and 𝐵 be nonempty closed subset of a complete 𝐶∗ -algebra-valued 𝑏-metric space (𝑋, A, 𝑑). Assume that 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 is cyclic mapping that satisfies 𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] , ∀𝑥 ∈ 𝐴, ∀𝑦 ∈ 𝐵,

Continuing this process, we have 𝑛

𝑑 (𝑇𝑛 𝑥0 , 𝑇𝑛+1 𝑥0 ) ⪯ [(𝐼 − 𝜆)−1 𝜆] 𝑑 (𝑥0 , 𝑇𝑥0 ) = 𝛼𝑛 𝛽, (83) where 𝛼 = (𝐼 − 𝜆)−1 𝜆 and 𝛽 = 𝑑(𝑥0 , 𝑇𝑥0 ). Next, we will show that {𝑇𝑛 𝑥0 } is Cauchy sequence with respect to A. Consider for any 𝑚, 𝑛 ∈ N and 𝑚 ≤ 𝑛 that we have 𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑛 𝑥0 ) ⪯ 𝑏𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑚+1 𝑥0 ) + 𝑏2 𝑑 (𝑇𝑚+1 𝑥0 , 𝑇𝑚+2 𝑥0 ) + ⋅ ⋅ ⋅

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+ 𝑏𝑛−𝑚 𝑑 (𝑇𝑛−1 𝑥0 , 𝑇𝑛 𝑥0 )

where 𝜆 ∈ A󸀠+ with ‖𝜆‖ < 1/2‖𝑏‖. Then, 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵. Proof. Without loss of generality, we can assume that 𝜆 ≠ 𝜃. Since 𝜆 ∈ A󸀠+ and 𝜃 ⪯ 𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦), by the second condition of Lemma 18, we have 𝜃 ⪯ 𝜆{𝑑(𝑥, 𝑇𝑥) + 𝑑(𝑦, 𝑇𝑦)}. Let 𝑥0 be any element in 𝐴. Since 𝑇 is cyclic mapping, we have 𝑇𝑥0 ∈ 𝐵 and 𝑇2 𝑥0 ∈ 𝐴. Consider

𝑚

⪯ 𝑏𝛼 𝛽 + 𝑏 𝛼

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𝛽 + ⋅⋅⋅ 𝑛−1

+ 𝑏𝑛−𝑚 𝛼𝑛−1 𝛽 = ∑ 𝑏𝑘−𝑚+1 𝛼𝑘 𝛽. 𝑘=𝑚

From Theorem 16, we get that 󵄩󵄩 𝑛−1 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 𝑚 𝑛 𝑘−𝑚+1 𝑘 󵄩 𝛼 𝛽󵄩󵄩󵄩 󵄩󵄩𝑑 (𝑇 𝑥0 , 𝑇 𝑥0 )󵄩󵄩 ≤ 󵄩󵄩 ∑ 𝑏 󵄩󵄩𝑘=𝑚 󵄩󵄩 󵄩 󵄩

𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) = 𝑑 (𝑇𝑥0 , 𝑇 (𝑇𝑥0 )) ⪯ 𝜆 [𝑑 (𝑥0 , 𝑇𝑥0 ) + 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 )]

2 𝑚+1

𝑛−1 󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩󵄩𝑏𝑘−𝑚+1 𝛼𝑘 𝛽󵄩󵄩󵄩󵄩

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𝑘=𝑚

= 𝜆𝑑 (𝑥0 , 𝑇𝑥0 ) + 𝜆𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) ;

𝑛−1

󵄩 󵄩 ≤ ∑ ‖𝑏‖𝑘−𝑚+1 ‖𝛼‖𝑘 󵄩󵄩󵄩𝛽󵄩󵄩󵄩

that is, (𝐼 − 𝜆) 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) ⪯ 𝜆𝑑 (𝑥0 , 𝑇𝑥0 ) .

𝑘=𝑚

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𝑛−1

󵄩 󵄩 ≤ ∑ ‖𝑏‖𝑘 ‖𝛼‖𝑘 󵄩󵄩󵄩𝛽󵄩󵄩󵄩

A󸀠+ and ‖𝜆‖

Since 𝜆 ∈ < 1/2‖𝑏‖ < 1/2, by the first condition of Lemma 18, we have that 𝐼−𝜆 is invertible and ‖(𝐼−𝜆)−1 𝜆‖ < 1. From the third condition of Lemma 18, we have 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) ⪯ (𝐼 − 𝜆)−1 𝜆𝑑 (𝑥0 , 𝑇𝑥0 ) .

𝑘=𝑚

𝑛−1

󵄩 󵄩 = 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ (‖𝑏‖ ‖𝛼‖)𝑘

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𝑘=𝑚 ∞

󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ (‖𝑏‖ ‖𝛼‖)𝑘

Similarly, we get that 𝑑 (𝑇2 𝑥0 , 𝑇3 𝑥0 ) ⪯ (𝐼 − 𝜆)−1 𝜆𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) . −1

(85)

𝑘=𝑚

(79)

𝑚 󵄩 󵄩 (‖𝑏‖ ‖𝛼‖) . = 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 1 − (‖𝑏‖ ‖𝛼‖)

−1

A󸀠+

Since (𝐼 − 𝜆) 𝜆 ∈ and 𝜃 ⪯ (𝐼 − 𝜆) 𝜆𝑑(𝑥0 , 𝑇𝑥0 ) − 𝑑(𝑇𝑥0 , 𝑇2 𝑥0 ), the second condition of Lemma 18, we have 𝜃 ⪯ (𝐼 − 𝜆)−1 ⋅ 𝜆 {(𝐼 − 𝜆)−1 𝜆𝑑 (𝑥0 , 𝑇𝑥0 ) − 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 )} ;

(80)

that is, (𝐼 − 𝜆)−1 𝜆𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) 2

⪯ [(𝐼 − 𝜆)−1 𝜆] 𝑑 (𝑥0 , 𝑇𝑥0 ) .

(81)

Hence, 𝑑 (𝑇2 𝑥0 , 𝑇3 𝑥0 ) ⪯ (𝐼 − 𝜆)−1 𝜆𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) −1

2

⪯ [(𝐼 − 𝜆) 𝜆] 𝑑 (𝑥0 , 𝑇𝑥0 ) .

(82)

Consider 󵄩 󵄩 󵄩 󵄩 ‖𝑏‖ ‖𝛼‖ = ‖𝑏‖ 󵄩󵄩󵄩󵄩𝜆 (𝐼 − 𝜆)−1 󵄩󵄩󵄩󵄩 ≤ ‖𝑏‖ ‖𝜆‖ 󵄩󵄩󵄩󵄩(𝐼 − 𝜆)−1 󵄩󵄩󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 = ‖𝑏‖ ‖𝜆‖ 󵄩󵄩󵄩∑ (𝜆)𝑖 󵄩󵄩󵄩 ≤ ‖𝑏‖ ‖𝜆‖ ∑ ‖(𝜆)‖𝑖 󵄩󵄩𝑖=0 󵄩󵄩 𝑖=0 󵄩 󵄩

< ‖𝑏‖ (

(86)

1 1 1 1 ) < = 1. 2 ‖𝑏‖ 1 − ‖𝜆‖ 2 1 − 1/2

Therefore, ‖𝛽‖(‖𝑏‖‖𝛼‖)2𝑚 /(1 − (‖𝑏‖‖𝛼‖)) → 0 as 𝑚 → ∞. Therefore, {𝑇𝑛 𝑥0 } is Cauchy sequence with respect to A. By the completeness of (𝑋, A, 𝑑), there exists an element 𝑥 ∈ 𝑋 such that 𝑥 = lim𝑛 → ∞ 𝑇𝑛 𝑥0 . Since {𝑇2𝑛 𝑥0 } is a sequence in 𝐴 and {𝑇2𝑛−1 𝑥0 } is a sequence in 𝐵, we obtain that both sequences converge to

12

Journal of Function Spaces

the same limit 𝑥. Since 𝐴 and 𝐵 are closed set, this implies 𝑥 ∈ 𝐴 ∩ 𝐵. Next, we will show that 𝑥 is a unique fixed point of 𝑇. Consider 𝑑 (𝑇𝑥, 𝑥) ⪯ 𝑏 [𝑑 (𝑇𝑥, 𝑇2𝑛 𝑥0 ) + 𝑑 (𝑇2𝑛 𝑥0 , 𝑥)] = 𝑏𝑑 (𝑇𝑥, 𝑇 (𝑇2𝑛−1 𝑥0 )) + 𝑏𝑑 (𝑇2𝑛 𝑥0 , 𝑥) ⪯ 𝑏𝜆 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑇2𝑛−1 𝑥0 , 𝑇2𝑛 𝑥0 )] (87) + 𝑏𝑑 (𝑇2𝑛 𝑥0 , 𝑥) ⪯ 𝑏𝜆𝑑 (𝑥, 𝑇𝑥) + 𝑏2 𝜆𝑑 (𝑇2𝑛−1 𝑥0 , 𝑥)

which leads to contradiction. Therefore, 𝑥 = 𝑦; we complete the proof. Example 43. Let 𝑋 = [−1, 1] and A = 𝑀2×2 (R) with ‖𝐴‖ = max𝑖,𝑗 |𝑎𝑖𝑗 | where 𝑎𝑖𝑗 are entries of the matrix 𝐴 ∈ 𝑀2×2 (R). Then, (𝑋, A, 𝑑) is a 𝐶∗ -algebra-valued 𝑏-metric space with 𝑏 = [ 20 02 ], where the involution is given by 𝐴∗ = (𝐴)𝑇 : 󵄨2 󵄨󵄨 0 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ] 𝑑 (𝑥, 𝑦) = [ 󵄨󵄨2 , 󵄨󵄨 𝑥 − 𝑦 0 󵄨 󵄨 󵄨 󵄨 ] [ and partial ordering on A is given as 𝑎11 𝑎12

[

+ 𝑏2 𝜆𝑑 (𝑥, 𝑇2𝑛 𝑥0 ) + 𝑏𝑑 (𝑇2𝑛 𝑥0 , 𝑥) ;

𝑎21

‖𝑑 (𝑇𝑥, 𝑥)‖ ≤ ‖𝑏‖ ‖𝜆‖ ‖𝑑 (𝑥, 𝑇𝑥)‖ 󵄩 󵄩 + ‖𝑏‖2 ‖𝜆‖ 󵄩󵄩󵄩󵄩𝑑 (𝑇2𝑛−1 𝑥0 , 𝑥)󵄩󵄩󵄩󵄩 󵄩 󵄩 + ‖𝑏‖ ‖𝜆‖ 󵄩󵄩󵄩󵄩𝑑 (𝑥, 𝑇2𝑛 𝑥0 )󵄩󵄩󵄩󵄩

(88)

󵄩 󵄩 + ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑑 (𝑇2𝑛 𝑥0 , 𝑥)󵄩󵄩󵄩󵄩 .

∀𝑖, 𝑗 = 1, 2, 3, 4.

Suppose that 𝐴 = [−1, 0] and 𝐵 = [0, 1]. Define a mapping 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 by 𝑇𝑥 = −𝑥/4. Firstly, we will show that 𝑇 is cyclic mapping. Let 𝑥 be an element in 𝐴; that is, −1 ≤ 𝑥 ≤ 0. Then, 0 ≤ −𝑥/4 ≤ 1 implies 𝑇𝑥 ∈ 𝐵. Similarly, let 𝑦 ∈ 𝐵, so 0 ≤ 𝑦 ≤ 1. Then, −1/4 ≤ −𝑦/4 ≤ 0. Hence, 𝑇𝑦 ∈ 𝐴. For any 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵, we consider

(89)

≤

1 1 󵄨 󵄨2 󵄨 󵄨2 (|𝑥| + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨) ≤ (2 |𝑥|2 + 2 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ) 16 16

1 1 ‖𝑑 (𝑥, 𝑇𝑥)‖ < ‖𝑑 (𝑥, 𝑇𝑥)‖ . (90) 2 ‖𝑏‖ 2

=

2 25 2 25 󵄨󵄨 󵄨󵄨2 ( |𝑥| + 󵄨𝑦󵄨 ) 25 16 16 󵄨 󵄨

=

𝑦 󵄨󵄨2 𝑥 󵄨󵄨2 󵄨󵄨 2 󵄨󵄨󵄨 (󵄨󵄨󵄨𝑥 + 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑦 + 󵄨󵄨󵄨󵄨 ) 25 󵄨 4󵄨 󵄨 4󵄨

=

2 󵄨2 󵄨 (|𝑥 − 𝑇𝑥|2 + 󵄨󵄨󵄨𝑦 − 𝑇𝑦󵄨󵄨󵄨 ) . 25

‖𝑑 (𝑇𝑥, 𝑥)‖ ≤ ‖𝑏‖ ‖𝜆‖ ‖𝑑 (𝑥, 𝑇𝑥)‖ , and so

This implies that ‖𝑑(𝑇𝑥, 𝑥)‖ = 0; that is, 𝑑(𝑇𝑥, 𝑥) = 𝜃 and so 𝑇𝑥 = 𝑥. That is, 𝑥 is fixed point of 𝑇. Now if 𝑦 is another fixed point of 𝑇 and 𝑦 ≠ 𝑥, then 𝜃 ⪯ 𝑑 (𝑥, 𝑦) = 𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆 (𝑑 (𝑥, 𝑇𝑥) +, 𝑑 (𝑦, 𝑇𝑦)) (91) = 𝜆 (𝑑 (𝑥, 𝑥) + 𝑑 (𝑦, 𝑦)) = 𝜃,

(93)

1 󵄨󵄨 󵄨2 󵄨2 󵄨󵄨 −𝑥 −𝑦 󵄨󵄨󵄨󵄨2 󵄨󵄨 − 󵄨𝑥 − 𝑦󵄨󵄨󵄨 󵄨󵄨𝑇𝑥 − 𝑇𝑦󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨 󵄨 = 4 󵄨󵄨 16 󵄨 󵄨 4

Letting 𝑛 → ∞, we get that

‖𝑑 (𝑇𝑥, 𝑥)‖ ≤ ‖𝑏‖

𝑏11 𝑏12 ]⪯[ ] ⇐⇒ 𝑎22 𝑏21 𝑏22 𝑎𝑖𝑗 ≤ 𝑏𝑖𝑗

by Theorem 16 and submultiplicative, we obtian that

2

(92)

(94)

Then, we have

2 󵄨2 󵄨 󵄨2 󵄨󵄨 0 (|𝑥 − 𝑇𝑥|2 + 󵄨󵄨󵄨𝑦 − 𝑇𝑦󵄨󵄨󵄨 ) 0 󵄨󵄨𝑇𝑥 − 𝑇𝑦󵄨󵄨󵄨 ] [ [ ] [ 25 ] 𝑑 (𝑇𝑥, 𝑇𝑦) = [ ]⪯[ ] 2 󵄨󵄨2 󵄨󵄨 2 󵄨 󵄨 2 󵄨 󵄨 𝑇𝑥 − 𝑇𝑦 0 󵄨 󵄨 𝑦 − 𝑇𝑦 + ) 0 (|𝑥 − 𝑇𝑥| 󵄨 󵄨 󵄨 󵄨 [ ] [ 󵄨 ] 󵄨 25 2 󵄨󵄨2 󵄨󵄨 2 0 0 ] [(|𝑥 − 𝑇𝑥| + 󵄨󵄨𝑦 − 𝑇𝑦󵄨󵄨 ) [ 25 ] ][ =[ ] = 𝜆 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] , [ 󵄨󵄨2 󵄨󵄨 2] 2 𝑦 − 𝑇𝑦 0 (|𝑥 − 𝑇𝑥| + ) 󵄨󵄨 ] 󵄨󵄨 0 [ 25 ] [

(95)

Journal of Function Spaces

13

0 where 𝜆 = [ 2/25 0 2/25 ]. Then, ‖𝜆‖ = 2/25 < 1/4 = 1/2‖𝑏‖. Thus, 𝑇 satisfies contraction of Theorem 42 implying that 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵; that is, {0} = 𝐹(𝑇).

Theorem 44 (cyclic Chatterjea-type). Let 𝐴 and 𝐵 be nonempty closed subset of a complete 𝐶∗ -algebra-valued 𝑏metric space (𝑋, A, 𝑑). Assume that 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 is cyclic mapping that satisfies

Hence, 𝑑 (𝑇2 𝑥0 , 𝑇3 𝑥0 ) ⪯ (𝑏𝜆) (𝐼 − 𝑏𝜆)−1 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) (103)

−1 2

⪯ [(𝑏𝜆) (𝐼 − 𝑏𝜆) ] 𝑑 (𝑥0 , 𝑇𝑥0 ) . Continuing this process, we have 𝑛

𝑑 (𝑇𝑥, 𝑇𝑦) ⪯ 𝜆 [𝑑 (𝑦, 𝑇𝑥) + 𝑑 (𝑥, 𝑇𝑦)] , ∀𝑥 ∈ 𝐴, ∀𝑦 ∈ 𝐵,

(96)

Proof. Without loss of generality, we can assume that 𝜆 ≠ 𝜃. Since 𝜆 ∈ A󸀠+ and 𝜃 ⪯ 𝑑(𝑦, 𝑇𝑥) + 𝑑(𝑥, 𝑇𝑦), by the second condition of Lemma 18, we have 𝜃 ⪯ 𝜆{𝑑(𝑦, 𝑇𝑥) + 𝑑(𝑥, 𝑇𝑦)}. Let 𝑥0 be any element in 𝐴, Since 𝑇 is cyclic mapping, we have 𝑇𝑥0 ∈ 𝐵 and 𝑇2 𝑥0 ∈ 𝐴. Consider 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) = 𝑑 (𝑇𝑥0 , 𝑇 (𝑇𝑥0 )) ⪯ 𝜆 [𝑑 (𝑇𝑥0 , 𝑇𝑥0 ) + 𝑑 (𝑥0 , 𝑇 𝑥0 )]

(104) = 𝜔𝑛 𝛽,

where 𝜆 ∈ A󸀠+ with ‖𝜆‖ < 1/2‖𝑏‖2 . Then, 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵.

2

𝑑 (𝑇𝑛 𝑥0 , 𝑇𝑛+1 𝑥0 ) ⪯ [(𝑏𝜆) (𝐼 − 𝑏𝜆)−1 ] 𝑑 (𝑥0 , 𝑇𝑥0 )

where 𝜔 = (𝑏𝜆)(𝐼 − 𝑏𝜆)−1 and 𝛽 = 𝑑(𝑥0 , 𝑇𝑥0 ). Next, we will show that {𝑇𝑛 𝑥0 } is Cauchy sequence with respect to A. Consider for any 𝑚, 𝑛 ∈ N and 𝑚 ≤ 𝑛; we have 𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑛 𝑥0 ) ⪯ 𝑏𝑑 (𝑇𝑚 𝑥0 , 𝑇𝑚+1 𝑥0 ) + 𝑏2 𝑑 (𝑇𝑚+1 𝑥0 , 𝑇𝑚+2 𝑥0 ) + ⋅ ⋅ ⋅ + 𝑏𝑛−𝑚 𝑑 (𝑇𝑛−1 𝑥0 , 𝑇𝑛 𝑥0 )

(97)

(105)

⪯ 𝑏𝜔𝑚 𝛽 + 𝑏2 𝜔𝑚+1 𝛽 + ⋅ ⋅ ⋅

⪯ 𝑏𝜆 [𝑑 (𝑥0 , 𝑇𝑥0 ) + 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 )] ;

𝑛−1

that is,

+ 𝑏𝑛−𝑚 𝜔𝑛−1 𝛽 = ∑ 𝑏𝑘−𝑚+1 𝜔𝑘 𝛽. (𝐼 − 𝑏𝜆) 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) ⪯ 𝑏𝜆𝑑 (𝑥0 , 𝑇𝑥0 ) .

Since 𝜆 ∈ A󸀠+ and 𝑏 ∈ A󸀠+ , from the second condition of Lemma 18, we get that 𝑏𝜆 ∈ A󸀠+ . Since ‖𝑏𝜆‖ < ‖𝑏‖(1/2‖𝑏‖2 ) ≤ 1/2 and 𝑏𝜆 ∈ A󸀠+ , by the first condition of Lemma 18, we have (𝐼 − 𝑏𝜆)−1 ∈ A󸀠+ and (𝑏𝜆)(𝐼 − 𝑏𝜆)−1 ∈ A󸀠+ with ‖(𝑏𝜆)(𝐼 − 𝑏𝜆)−1 ‖ < 1. From the third condition of Lemma 18, we have 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) ⪯ (𝑏𝜆) (𝐼 − 𝑏𝜆)−1 𝑑 (𝑥0 , 𝑇𝑥0 ) .

𝑘=𝑚

(98) From Theorem 16, we get that

󵄩󵄩 𝑛−1 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩󵄩 𝑚 𝑛 𝑘−𝑚+1 𝑘 󵄩 𝜔 𝛽󵄩󵄩󵄩 󵄩󵄩𝑑 (𝑇 𝑥0 , 𝑇 𝑥0 )󵄩󵄩 ≤ 󵄩󵄩 ∑ 𝑏 󵄩󵄩𝑘=𝑚 󵄩󵄩 󵄩 󵄩 𝑛−1 󵄩 󵄩 ≤ ∑ 󵄩󵄩󵄩󵄩𝑏𝑘−𝑚+1 𝜔𝑘 𝛽󵄩󵄩󵄩󵄩 𝑘=𝑚

(99) 𝑛−1

󵄩 󵄩 ≤ ∑ ‖𝑏‖𝑘−𝑚+1 ‖𝜔‖𝑘 󵄩󵄩󵄩𝛽󵄩󵄩󵄩

Similarly, we get that 𝑑 (𝑇2 𝑥0 , 𝑇3 𝑥0 ) ⪯ (𝑏𝜆) (𝐼 − 𝑏𝜆)−1 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) . −1

𝑘=𝑚

(100) 𝑛−1

−1

A󸀠+

Since (𝑏𝜆)(𝐼 − 𝑏𝜆) ∈ and 𝜃 ⪯ (𝑏𝜆)(𝐼 − 𝑏𝜆) 𝑑(𝑥0 , 𝑇𝑥0 ) − 2 𝑑(𝑇𝑥0 , 𝑇 𝑥0 ), the second condition of Lemma 18, we have

󵄩 󵄩 ≤ ∑ ‖𝑏‖𝑘 ‖𝜔‖𝑘 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 𝑘=𝑚

𝑛−1

𝜃 ⪯ (𝑏𝜆) (𝐼 − 𝑏𝜆)−1 ⋅ {(𝑏𝜆) (𝐼 − 𝑏𝜆)−1 𝑑 (𝑥0 , 𝑇𝑥0 ) − 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 )} ;

(101)

󵄩 󵄩 = 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ (‖𝑏‖ ‖𝜔‖)𝑘 𝑘=𝑚 ∞

󵄩 󵄩 = 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 ∑ (‖𝑏‖ ‖𝜔‖)𝑘

that is,

𝑘=𝑚

(𝑏𝜆) (𝐼 − 𝑏𝜆)−1 𝑑 (𝑇𝑥0 , 𝑇2 𝑥0 ) −1 2

⪯ [(𝑏𝜆) (𝐼 − 𝑏𝜆) ] 𝑑 (𝑥0 , 𝑇𝑥0 ) .

(102)

𝑚 󵄩 󵄩 (‖𝑏‖ ‖𝜔‖) = 󵄩󵄩󵄩𝛽󵄩󵄩󵄩 . 1 − (‖𝑏‖ ‖𝜔‖)

(106)

14

Journal of Function Spaces This implies that ‖𝑑(𝑇𝑥, 𝑥)‖ = 0; that is, 𝑑(𝑇𝑥, 𝑥) = 𝜃 and so 𝑇𝑥 = 𝑥. That is, 𝑥 is fixed point of 𝑇. Now if 𝑦 is another fixed point of 𝑇 and 𝑦 ≠ 𝑥, then

Consider 󵄩 󵄩 ‖𝑏‖ ‖𝜔‖ = ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑏𝜆 (𝐼 − 𝑏𝜆)−1 󵄩󵄩󵄩󵄩 󵄩 󵄩 ≤ ‖𝑏‖ ‖𝑏𝜆‖ 󵄩󵄩󵄩󵄩(𝐼 − 𝑏𝜆)−1 󵄩󵄩󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩 󵄩󵄩󵄩 𝑖󵄩 = ‖𝑏‖ ‖𝑏𝜆‖ 󵄩󵄩∑ (𝑏𝜆) 󵄩󵄩󵄩 ≤ ‖𝑏‖ ‖𝑏𝜆‖ ∑ ‖(𝑏𝜆)‖𝑖 󵄩󵄩𝑖=0 󵄩󵄩 𝑖=0 󵄩 󵄩

𝜃 ⪯ 𝑑 (𝑥, 𝑦) = 𝑑 (𝑇𝑥, 𝑇𝑦)

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩2𝜆𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ 2 ‖𝜆‖ 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩

Therefore, ‖𝛽‖(‖𝑏‖‖𝜔‖)2𝑚 /(1 − (‖𝑏‖‖𝜔‖)) → 0 as 𝑚 → ∞. Therefore, {𝑇𝑛 𝑥0 } is Cauchy sequence with respect to A. By the completeness of (𝑋, A, 𝑑), there exists an element 𝑥 ∈ 𝑋 such that 𝑥 = lim𝑛 → ∞ 𝑇𝑛 𝑥0 . Since {𝑇2𝑛 𝑥0 } is a sequence in 𝐴 and {𝑇2𝑛−1 𝑥0 } is a sequence in 𝐵, we obtain that both sequences converge to the same limit 𝑥. Since 𝐴 and 𝐵 are closed set, this implies 𝑥 ∈ 𝐴 ∩ 𝐵. Next, we will complete the proof by showing that 𝑥 is a unique fixed point of 𝑇. Since 𝑑 (𝑇𝑥, 𝑥) ⪯ 𝑏 [𝑑 (𝑇𝑥, 𝑇2𝑛 𝑥0 ) + 𝑑 (𝑇2𝑛 𝑥0 , 𝑥)] = 𝑏𝑑 (𝑇𝑥, 𝑇 (𝑇

1 󵄩 󵄩 󵄩 󵄩 < 2 ( 󵄩󵄩 2 󵄩󵄩 ) 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑑 (𝑥, 𝑦)󵄩󵄩󵄩 , 2 󵄩󵄩𝑏 󵄩󵄩

Example 45. Let 𝑋 = [0, 1] and A = 𝑀2×2 (R) with ‖𝐴‖ = max𝑖,𝑗 |𝑎𝑖𝑗 |, where 𝑎𝑖𝑗 are entries of the matrix 𝐴 ∈ 𝑀2×2 (R). Then, (𝑋, A, 𝑑) is a 𝐶∗ -algebra-valued 𝑏-metric space with 𝑏 = [ 20 02 ], where the involution is given by 𝐴∗ = (𝐴)𝑇 : 𝑑 (𝑥, 𝑦) = [ [

𝑥0 )) + 𝑏𝑑 (𝑇 𝑥0 , 𝑥)

⪯ 𝑏𝜆 [𝑑 (𝑥, 𝑇2𝑛 𝑥0 ) + 𝑑 (𝑇2𝑛−1 𝑥0 , 𝑇𝑥)]

= 𝑏𝜆𝑑 (𝑥, 𝑇2𝑛 𝑥0 ) + 𝑏𝜆𝑑 (𝑇2𝑛−1 𝑥0 , 𝑇𝑥)

󵄨2 󵄨󵄨 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨

[

𝑎21

𝑏11 𝑏12 ]⪯[ ] ⇐⇒ 𝑎22 𝑏21 𝑏22 𝑎𝑖𝑗 ≤ 𝑏𝑖𝑗

⪯ 𝑏𝜆𝑑 (𝑥, 𝑇2𝑛 𝑥0 ) + 𝑏2 𝜆𝑑 (𝑇2𝑛−1 𝑥0 , 𝑥) + 𝑏 𝜆𝑑 (𝑥, 𝑇𝑥) + 𝑏𝑑 (𝑇 𝑥0 , 𝑥) , by Theorem 16, we have 󵄩 󵄩 ‖𝑑 (𝑇𝑥, 𝑥)‖ ≤ ‖𝑏‖ ‖𝜆‖ 󵄩󵄩󵄩󵄩𝑑 (𝑥, 𝑇2𝑛 𝑥0 )󵄩󵄩󵄩󵄩

+ ‖𝑏‖ ‖𝜆‖ ‖𝑑 (𝑥, 𝑇𝑥)‖

(114)

(115)

∀𝑖, 𝑗 = 1, 2, 3, 4.

Suppose that 𝐴 = [0, 1] and 𝐵 = [0, 1/2]. Define a mapping 𝑇 : 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 by 𝑇𝑥 = 𝑥/5. Firstly, we will show that 𝑇 is cyclic mapping. Let 𝑥 ∈ 𝐴; that is, 0 ≤ 𝑥 ≤ 1. Then, 0 ≤ 𝑥/5 ≤ 1/5 implies 𝑇𝑥 ∈ 𝐵. Similarly, let 𝑦 ∈ 𝐵, so 0 ≤ 𝑦 ≤ 1/2. Then, 0 ≤ 𝑦/5 ≤ 1/10. Hence, 𝑇𝑦 ∈ 𝐴. Now, we will show that 𝑇 satisfies the contraction of Theorem 44. Consider

2𝑛

2

] 󵄨2 , 󵄨󵄨 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ]

𝑎11 𝑎12

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+ 𝑏𝑑 (𝑇 𝑥0 , 𝑥)

󵄩 󵄩 + ‖𝑏‖2 ‖𝜆‖ 󵄩󵄩󵄩󵄩𝑑 (𝑇2𝑛−1 𝑥0 , 𝑥)󵄩󵄩󵄩󵄩

0

0

and partial ordering on A is given as

2𝑛

2

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which leads to a contradiction. Therefore, 𝑥 = 𝑦 which implies that the fixed point is unique.

2𝑛

+ 𝑏𝑑 (𝑇2𝑛 𝑥0 , 𝑥)

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From Theorem 16, we get that

1 1 1 ‖𝑏‖ < ‖𝑏‖ ( ) < = 1. 2 1 − ‖𝑏𝜆‖ 2 1 − 1/2 2 ‖𝑏‖

2𝑛−1

⪯ 𝜆 (𝑑 (𝑦, 𝑇𝑥) + 𝑑 (𝑥, 𝑇𝑦)) = 2𝜆𝑑 (𝑥, 𝑦) .

(107)

(109)

󵄩 󵄩 + ‖𝑏‖ 󵄩󵄩󵄩󵄩𝑑 (𝑇2𝑛 𝑥0 , 𝑥)󵄩󵄩󵄩󵄩 .

(𝑥 − 𝑦) 1 (6 (𝑥 − 𝑦)) 1 𝑦 𝑥 = = (𝑥 − + − 𝑦) 5 6 5 6 5 5

(116)

and so 2

Letting 𝑛 → ∞, we get that

(

‖𝑑 (𝑇𝑥, 𝑥)‖ ≤ ‖𝑏‖2 ‖𝜆‖ ‖𝑑 (𝑥, 𝑇𝑥)‖ ,

(110)

2 (𝑥 − 𝑦) 𝑦 𝑥 1 ) = ( (𝑥 − + − 𝑦)) 5 6 5 5

=

and so 1 ‖𝑑 (𝑇𝑥, 𝑥)‖ ≤ ‖𝑏‖ ‖𝑑 (𝑥, 𝑇𝑥)‖ 2 ‖𝑏‖2 2

1 < ‖𝑑 (𝑥, 𝑇𝑥)‖ . 2

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2 𝑦 1 𝑥 ((𝑥 − ) + ( − 𝑦)) 36 5 5

2 𝑦 2 1 𝑥 ≤ (2 (𝑥 − ) + 2 ( − 𝑦) ) 36 5 5

=

1 󵄨󵄨 󵄨2 󵄨2 󵄨 (󵄨󵄨𝑥 − 𝑇𝑦󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑇𝑥 − 𝑦󵄨󵄨󵄨 ) . 18

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Journal of Function Spaces

15

Then, we have

𝑑 (𝑇𝑥, 𝑇𝑦) =

󵄨2 󵄨󵄨 󵄨𝑇𝑥 − 𝑇𝑦󵄨󵄨󵄨 [󵄨 [

0

1 󵄨󵄨 󵄨2 󵄨2 󵄨 0 (󵄨󵄨𝑥 − 𝑇𝑦󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑇𝑥 − 𝑦󵄨󵄨󵄨 ) ] [ ] ⪯ [ 18 ] 2 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 󵄨 󵄨 󵄨 󵄨󵄨𝑇𝑥 − 𝑇𝑦󵄨󵄨 ] 0 (󵄨󵄨𝑥 − 𝑇𝑦󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑇𝑥 − 𝑦󵄨󵄨󵄨 ) ] [ 18 0

1 󵄨󵄨2 󵄨󵄨2 󵄨󵄨 󵄨󵄨 0 0 [ 18 ] [(󵄨󵄨𝑥 − 𝑇𝑦󵄨󵄨 + 󵄨󵄨𝑇𝑥 − 𝑦󵄨󵄨 ) ] =[ 󵄨󵄨2 󵄨󵄨 󵄨󵄨2 = 𝜆 [𝑑 (𝑥, 𝑇𝑦) + 𝑑 (𝑦, 𝑇𝑥)] , 󵄨󵄨 1] 𝑥 − 𝑇𝑦 𝑇𝑥 − 𝑦 0 ( + ) 󵄨 󵄨 󵄨 󵄨 0 󵄨 󵄨 󵄨 ] 󵄨 [ 18 ] [ 0 2 where 𝜆 = [ 1/18 0 1/18 ]. Then, ‖𝜆‖ = 1/18 < 1/8 = 1/2‖𝑏‖ . Thus, 𝑇 satisfies contraction of Theorem 44 implying that 𝑇 has a unique fixed point in 𝐴 ∩ 𝐵.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The authors would like to thank Science Achievement Scholarship of Thailand and Faculty of Science, Naresuan University.

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