Best Proximity Point Theorems for Cyclic Contractions Mappings in Banach Algebras

Abstract

In this paper, we present some new best proximity point theorems for three operators acting in Banach algebras. An application is given to show the usefulness and the applicability of the obtained results.

1. Introduction

The study of functional integral equations and differential equations is the main object of research in nonlinear functional analysis. These equations occur in physical, biological, and economic problems. Some of these equations can be formulated into nonlinear operator equations:in suitable Banach algebras.

Recently, many authors are interested on the study of equation (NOE) and obtained some interesting results (see for instance [1–5]). In 2010, Ben Amar et al. [6] proved some existence fixed point theorems which allowed them to solve equation (NOE) where the involved operators are weakly sequentially continuous.

Let be a Banach algebra with a norm . Let be a pair of nonempty subsets of . Given two mappings and defined on and an operator . Under suitable conditions, we define the operator such that and . If is nonempty, then the mapping restricted to is a self mapping. Then, a solution of equation (NOE) is a fixed point of . Furthermore, if the fixed point equation does not possess a solution it is natural to explore to find an satisfyingwhere . This point is said to be the best proximity point of . Note that a point is the best proximity point of if is a solution of the minimization problem

The best proximity point notion can be viewed as a generalization of fixed point, since most fixed point theorems can be derived as corollaries of the best proximity point theorems.

The first result of this kind is due to Fan (see [7], Theorem 2) which is stated in normed spaces for continuous mappings. In [8], Eldred and Veeramani introduced the concept of cyclic contraction mappings and gave the best proximity point results for this class of mappings. They also gave an algorithm to reach this best proximity point where the space is uniformly convex. Furthermore, in [9], Taghafi and Shahzad proved the existence of the best proximity point for a cyclic contraction mapping in a reflexive Banach space.

For noncyclic mappings, i.e., and , Gabelah and Künzi in [10] established some best proximity point results in the framework of complete CAT(0) spaces. In addition, they gave an approach to reach this best proximity point by means of an algorithm. Regarding the relationship between the noncyclic and cyclic results, the authors in [11] proved that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces. For more on the best proximity point results, the interested reader can consult [12–18].

The paper is organized as follows. After some preliminaries, in Section 3, we prove the existence of the best proximity point where the involving operators are -Lipschitzs and cyclic contraction (see Theorem 1). Also, an example is given to illustrate the obtained result. In Theorem 1, we consider the case where is a uniformly convex Banach algebra. In Section 4, we show the applicability of our result (Theorem 1) to the theory of nonlinear integral equations:where and two subsets of the Banach algebra of all continuous functions from to .

2. Preliminaries

Definition 1. An algebra is a vector space endowed with an internal composition law noted by i.e.,which is associative and bilinear.
A normed algebra is an algebra endowed with a norm satisfying the following property for all ; . A complete normed algebra is called a Banach algebra.

Definition 2. Let be a Banach space with norm . A mapping is called -Lipschitz if there exists a continuous nondecreasing function satisfyingfor all with . In the special case when for some , is called lipschitzian mapping with a Lipschitz constant .

Definition 3. Let be a Banach space. We say that is uniformly convex if for every ,The function is known as the modulus of uniform convexity of . Note that any uniformly convex Banach space is reflexive.

Theorem 4 (see [17]). Let be a uniformly convex Banach space. Let be a nonempty closed bounded convex subset of such that is compact, and be a nonempty closed convex subset of . Let be a relatively nonexpansive mapping. Then, there exists such that .

The authors in [8] introduced the following notion of cyclic contraction.

Definition 5. Let and be nonempty subsets of a metric space . A mapping is said to be a cyclic contraction if it satisfies:(1) and (2)for some , , for all , Since , for and , for all , i.e., is relatively nonexpansive.
We conclude this section by recalling some best proximity point results for this class of mappings.

Theorem 6 (see [8]). Let and be nonempty closed subsets of a complete metric space . Let be a cyclic contraction mapping and . Define , . Suppose has a convergent subsequence in , then there exists such that .

Theorem 7 (see [8, 19]). Let and be nonempty closed convex subsets of a uniformly convex Banach space. Suppose is a cyclic contraction mapping. Then, has a unique best proximity point in . Further, if and , then the sequence converges to the best proximity point.

3. Main Results

We start this section by introducing the notion of -monotone property for a pair of functions.

Definition 8. Let and be two mappings. We say that the pair has the property -monotone if(i),(ii) is nondecreasing on and

Remark 9. If has the property -monotone and are continuous, then the mapping is invertible.

Example 1. Let be the mappings defined by:So, the mapping has the property -monotone.
Recall that an operator from a Banach algebra is said to be regular on if maps into the set of all invertible elements of .

Theorem 10. Let be a nonempty closed pair of a Banach algebra . Let and be three operators which satisfy the following conditions:(1) is regular on and (2) and are -Lipschitzs with the -functions and , respectively, is bounded with bound , and for all , and is nondecreasing(3) is cyclic contraction mapping on (4)suppose there exists a sequence of such that , and the sequence has a convergent subsequence in ,(5)Then, there exists such that

Proof. Let be fixed in and let us define the mapping on byLet . The use of assumption leads toNow an application of Boyd and Wong’s fixed point theorem [20], Theorem 1 leads to the existence of a unique point such that . Hence, the operator is well defined.
Moreover, assumption implies that and . Indeed, let and such that , so . Similarly, for all , . Hence, is cyclic on .
is cyclic contraction on . Indeed, let , the use of assumption and and the fact that for all leads toSince has the property -monotone, we haveBy , there exists a sequence of such that , and the sequence has a convergent subsequence in .
Thus, by Theorem 6, there exists such that .
Let . By , is cyclic contraction on , and , soSimilarly, , where .

Example 2. Let endowed with the usual norm and let , .(i)Let the function defined on byLet , we have .Thus, is cyclic contraction on and .(ii)Let the function defined on by , for all . The function is -Lipschitz with the -function , and (iii)Let the function defined on byFor each ,The function is -Lipschitz with the -function defined by , for all . We have is nondecreasing, and for all ,(iv)Let and . Suppose and . We have(v)For all , , so for each , . Thus, for any sequence of such that , , the sequence has a convergent subsequenceHence, by Theorem 1, there exists such thatwhere .

Theorem 11. Let be a nonempty closed pair of a Banach algebra . Let and be two operators which satisfy the following conditions:(1) is -Lipschitz with the -function , , for all (2) is cyclic contraction mapping on (3) is relatively nonexpansive mapping on (4)suppose there exists a sequence of such that , and the sequence has a convergent subsequence in (5)Then, there exists such that

Proof. By , we show that is well defined. Moreover, the use of assumptions , , and shows that is cyclic contraction on . Indeed, by , it is cyclic. Let , the use of assumptions and leads to ,By , there exists a sequence of such that , and the sequence has a convergent subsequence in .
Thus, by Theorem 6, there exists such thatLet . We have and . By , is cyclic contraction on , soSimilarly, we get , where .

Remark 12. Under the same hypotheses of the previous theorem where , we obtain the classical result. That is, there exists such that

Theorem 13. Let be a nonempty closed convex pair of a uniformly convex Banach algebra . Let and be three operators which satisfy the following conditions:(1) is regular on and (2) and are -Lipschitzs with the -functions and , respectively, is bounded with bound , and for all and is nondecreasing(3) is cyclic contraction on (4)Then, there exists a unique such thatFurther, if and , , then the sequence converges to the best proximity point.

Proof. By , , , and , we show that is well defined and cyclic contraction on .
Thus, by Theorem 7 and , there exists a unique such thatand if with , i.e., for all , then the sequence converges to the best proximity point.

4. Application

Let the Banach algebra of all continuous functions from to , endowed with the sup-norm , defined byfor each . Let and suppose .

We consider the closed and nonempty sets

For any and for all , we haveso We consider the following two nonlinear functional integral equationswhere and .

The integral equations (FIS1)–(FIS2) may be written, respectively, as:where and . To simplify the notations, we put

The goal of this section is to apply our main result to investigate the existence of an optimum solution of the (FIS1)–(FIS2) problem in the sense that the pair satisfies:

Note that, if is a solution of (FIS1) and is a solution of (FIS2), then the pair need not form an optimum solution see [21], pp 27-31 for more details.

We consider the following assumptions:(i)(a)The function is continuous, for all (b)There is a continuous function with bound such that , for all and (ii)(a)The functions are measurables and(b)Let . For all ,(c)For all , (iii)(a) is -Lipschitz with the -function , such that is differentiable on and , , where (b) is regular on , and (iv)The family is equicontinuous and closed

Theorem 14. Assume the hypotheses hold. for all and for all . Moreover,Then, there exists an optimum solution for (FIS1)–(FIS2) problem.

Proof. (1)(i)By , we have for all , so for all , where , for all (ii)-Let , We have, for all Similarly, for all and we get . Hence, .(iii)-Furthermore, by hypothesis we have , for all . So(iv)We show that is nondecreasing. Let such that .Since, is nondecreasing and differentiable on , with , sothus,That is, is nondecreasing.
(2)(i)Let such that ,Thus, is Lipschitzian, so . Let and . By , , We haveHence, . Similarly, we get .(ii)-Let and By , , we have , for all , soThen,Thus, which shows that is cyclic contraction.
(3) Let and such that . We show that . We have, for all , , and , soLet and such that . We show that . We have, for all , , and , so(4) As , so the family is uniformly bounded; by , this family is equicontinuous. Therefore, by Arzela-Ascoli’s theorem, lies in a compact subset of . Let be a sequence of such that , i.e., . We have has a convergent subsequence . Let . As is closed, there exists such that . We obtain, for each Thus,Hence,Which prove that the sequence is convergent.
Thus, by Theorem 1, there exists such that

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his thorough reviews and valuable remarks, especially for the suggestions to complete references by adding papers [10, 11, 13, 18].