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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 123613, 8 pages
http://dx.doi.org/10.1155/2014/123613
Research Article

Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

1Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 14 September 2013; Accepted 10 December 2013; Published 16 January 2014

Academic Editor: Calogero Vetro

Copyright © 2014 Moosa Gabeleh and Naseer Shahzad. 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.

Abstract

Let A and B be two nonempty subsets of a Banach space X. A mapping T : is said to be cyclic relatively nonexpansive if T(A) and T(B) and for all () . In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.

1. Introduction

Let be a Banach space and . Recall that a mapping is nonexpansive provided that for all . A closed convex subset of a Banach space has normal structure in the sense of Brodskii and Milman [1] if for each bounded, closed, and convex subset of which contains more than one point, there exists a point which is not a diametral point; that is, where is the diameter of . The set is said to have fixed point property (FPP) if every nonexpansive mapping has a fixed point. In 1965, Kirk proved the following famous fixed theorem.

Theorem 1 (see [2]). Let be a nonempty, weakly compact, and convex subset of a Banach space . If has normal structure, then has the FPP.

We mention that every compact and convex subset of a Banach space has normal structure (see [3]) and so has the FPP. Moreover, every bounded, closed, and convex subset of a uniformly convex Banach space has normal structure (see [4]) and then by Theorem 1 has the FPP. It is interesting to note that there exists a weakly compact and convex subset of which does not have the fixed point property (see [5] for more information). In particular, cannot have normal structure.

In the current paper, we introduce a geometric notion of seminormal structure on a nonempty, closed, and convex pair of subsets of a Banach space and present a new fixed point theorem which is an extension of Kirk’s fixed point theorem. We also study the stability of fixed points by using this geometric property. Finally, we establish a best proximity point theorem for a new class of mappings.

2. Preliminaries

In [6], Kirk et al. obtained an interesting extension of Banach contraction principle as follows.

Theorem 2 (see [6]). Let and be nonempty closed subsets of a complete metric space . Suppose that is a cyclic mapping, that is, and . If for some and for all , , has a unique fixed point in .

An interesting feature about the above observation is that continuity of is no longer needed. Indeed, simple examples can be constructed showing that discontinuous mappings can satisfy all the assumptions.

Let and be two nonempty subsets of a normed linear space . A mapping is said to be cyclic relatively nonexpansive if is cyclic and whenever and . It is clear that the class of cyclic relatively nonexpansive mappings contains the class of nonexpansive mappings as a subclass. Indeed, a cyclic relatively nonexpansive mapping need not to be continuous in general. Of course, if , then the cyclic relatively nonexpansive mapping restricted to is nonexpansive. If then the fixed point equation cannot have a solution; instead it is interesting to study the existence of best proximity points; that is, a point such that The relevance of best proximity points is that they provide optimal solutions for the problem of best approximation between two sets. Existence of best proximity points for cyclic relatively nonexpansive mappings was first studied in [7]. Afterwards, Eldred and Veeramani [8] studied the existence, uniqueness, and convergence of a best proximity point for cyclic contraction mappings in uniformly convex Banach spaces. For more information about the existence of best proximity points for various classes of cyclic mappings one can refer to [5, 922]. For other related results, we refer to [23, 24].

To describe our results, we need some definitions and notations. We shall say that a pair of subsets of a Banach space satisfies a property if both and satisfy that property. For example, is convex if and only if both and are convex, , and . We will also adopt the notation

The closed and convex hull of a set will be denoted by . Given a pair of nonempty subsets of a Banach space, then its proximal pair is the pair given by Proximal pairs may be empty but, in particular, if and are nonempty weakly compact and convex then is a nonempty weakly compact convex pair in .

Definition 3. A pair of sets is said to be proximal if and .

In [7], Eldred et al. introduced a geometric concept called proximal normal structure which generalizes the notion of normal structure introduced by Brodskii and Milman [1].

Definition 4. A convex pair in a Banach space is said to have proximal normal structure if for any bounded, closed, and convex proximal pair for which and , there exits such that

It was announced in [7] that every nonempty, bounded, closed, and convex pair of subsets of a uniformly convex Banach space has proximal normal structure (see Proposition 2.1 of [7]). We mention that a weaker notion than proximal normal structure which is called proximal quasinormal structure was introduced in [25] in order to study the existence of best proximity points for cyclic relatively Kannan nonexpansive mappings.

The next best proximity point theorem was established in [7].

Theorem 5 (see [7]). Let be a nonempty, weakly compact convex pair in a Banach space , and suppose has proximal normal structure. Assume that is a cyclic relatively nonexpansive mapping. Then has a best proximity point in both and , that is, there exists such that .

As a result of Theorem 5, the following corollary was obtained in [7].

Corollary 6. Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space and let be a cyclic relatively nonexpansive mapping. Then has a best proximity point.

3. Seminormal Structure and a Fixed Point Theorem

In this section motivated by Theorem 2, we prove a fixed point theorem for cyclic relatively nonexpansive mappings which is an extension of Theorem 1 due to Kirk. We begin our main result with the following geometric notion.

Definition 7. A convex pair in a Banach space is said to have seminormal structure if for any bounded, closed and convex pair with , there exits such that

We note that the pair has seminormal structure if and only if has normal structure in the sense of Brodskii and Milman.

Here, we state the main result of this section.

Theorem 8. Let be a nonempty, weakly compact, and convex pair in a Banach space , and suppose has seminormal structure. Assume that is a cyclic relatively nonexpansive mapping. Then has a fixed point.

Proof. Let denote the collection of all nonempty, closed, and convex pairs such that is cyclic on . Since is a weakly compact and convex pair in , the pair is also nonempty, closed, and convex pair in . Moreover, is cyclic on . Indeed, if then there exists an element such that . By the fact that is cyclic relatively nonexpansive, we have that is, and so, . Similarly, we can see that . Thus, and then is nonempty. It follows from Zorn’s lemma that has a minimal element, say . Since , we have and so, Similarly, ; that is, is cyclic on . Minimality of concludes that Observe that if , then we must have for some and so, is a fixed point of and we are finished. So, we may assume that . By seminormal structure there exist and such that Put Note that . We show that is a closed and convex pair in . Let be a sequence in such that . Then for each there exists such that for all . Since is closed, . Let . For all we have which implies that . Hence, . Thereby, is closed. Similarly, we can see that is closed. Now, let and . For all we have Therefore, which deduces that . Hence, is convex. Similarly, is also convex. We assert that is cyclic on . Suppose that . Let and . Since is dense in , there exists such that , and for all and . We have This implies that and thus, . That is, . Similar argument concludes that and so, is cyclic on . Again, by the minimality of we obtain and . Hence, for each we have . Thus, which is a contradiction.

In what follows, we give a sufficient condition which ensures that every nonempty, bounded, closed, and convex pair of subsets of a uniformly convex Banach space has seminormal structure.

Definition 9. A nonempty, bounded, closed, and convex pair of a normed linear space is said to have property provided that for each nonempty, closed, and convex pair we have

Example 10. Let be a nonempty, bounded, closed, and convex pair in a normed linear space such that Then has the property.

Proof. Suppose that is a nonempty, closed, and convex pair. Then we have that is, has the property .

Let be a uniformly convex Banach space with modulus of convexity . Then for . Moreover, if , , and we have Motivated by the fact that every nonempty, bounded, closed, and convex subset of a uniformly convex Banach space has normal structure, we establish the following result concerning seminormal structure in uniformly convex Banach spaces.

Proposition 11. Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space such that has the property . Then has seminormal structure.

Proof. Let be a nonempty, closed, and convex pair. Put, and . Since has the property , we have . There exists such that . Now, for each we have It follows from the uniformly convexity of the Banach space that If we set and , then for each we have which concludes that . Similar argument implies that there exists an element such that , which completes the proof.

The following corollary obtains from Theorem 8 and Proposition 11, immediately.

Corollary 12. Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space such that has the property . Assume that is a cyclic relatively nonexpansive mapping. Then has a fixed point.

It is interesting to note that by admitting property to the assumptions of Corollary 6, we conclude the existence of fixed points for cyclic relatively nonexpansive mappings instead of the existence of best proximity points. Here, we raise the following question.

Question 1. It is interesting to ask whether one considers a better condition than the property in Corollary 12 such that the cyclic relatively nonexpansive mapping has a fixed point.

Example 13. Let us consider with the usual metric. Let and . Define by We claim that is cyclic relatively nonexpansive. In this order, we consider the following two cases.

Case 1. If and , then

Case 2. If and , then Thus, is a cyclic relatively nonexpansive mapping. Corollary 12 guarantees the existence of fixed point for the mapping which is a point . It is interesting to note that the existence of a fixed point for the mapping cannot be obtained from Theorem 1 due to Kirk because is not continuous.

In 1974, Lim proved the following common fixed point theorem.

Theorem 11 (see [26]). Let be a nonempty, weakly compact, and convex subset of a Banach space . If has normal structure, then any family of commuting nonexpansive mappings on into itself admits a common fixed point.

The following result is a generalized version of Lim’s theorem.

Theorem 12. Let be a nonempty, weakly compact and convex pair of subsets of a Banach space such that has seminormal structure. Suppose that is a family of commuting cyclic relatively nonexpansive mappings on with some index set . Then has a common fixed point, that is, there exists an element such that

At the end of this section, we consider cyclic mappings which do not increase large distances. Our purpose is not to seek fixed points but rather to determine what can be said about minimal displacement for such cyclic mappings.

Definition 13. Let be a nonempty pair of subsets of a normed linear space. A mapping is said to be cyclic relatively -nonexpansive for if

Our result is the following. The surprising aspect of the conclusion of the following theorem is the fact that cyclic relatively -nonexpansive mappings have minimal displacement strictly less than in the presence of seminormality.

Theorem 14. Let be a nonempty, weakly compact, and convex pair in a Banach space such that has seminormal structure. Let be a cyclic relatively h-nonexpansive mapping for . Then there exists an element such that

Proof. Proceeding in a similar way as in Theorem 8, we obtain, by minimality, that and . If , then there is nothing to prove. So, assume that . Since has seminormal structure, there exist and such that By the fact that is a weakly compact and convex pair in , the pair is nonempty, closed, and convex. Let . Then and . If , thus and again we are finished. So, we assume that . Let . If , then On the other hand, if , then Therefore, in either case . Hence, which concludes that . That is, . Similarly, we can see that . Thereby, is cyclic on . It follows from the minimality of that and . Now, for each we have and so, Also, we can see that . Hence, which is a contradiction.

4. A Best Proximity Point Theorem

Recently, Kosuru and Veeramani introduced a concept of pointwise cyclic contractions as follows.

Definition 15 (see [27]). Let be a pair of subsets of a metric space . A mapping is said to be a pointwise cyclic contraction if is cyclic and for each there exist , such that

The following best proximity point theorem was proved in [27] by using a geometric notion of proximal normal structure.

Theorem 16 (Theorem 4.1 of [27]). Let be a nonempty, weakly compact, and convex pair of a Banach space . If is a pointwise cyclic contraction mapping, then has a best proximity point.

To establish our results, we introduce the following class of cyclic mappings.

Definition 17. Let be a pair of subsets of a metric space . A mapping is said to be a generalized pointwise cyclic contraction if is cyclic and for each there exist , , , and such that , and where for all .

Obviously, every generalized pointwise cyclic contraction is a cyclic relatively nonexpansive mapping. Also, the class of generalized pointwise cyclic contractions contains the class of pointwise cyclic contractions as a subclass. The next example shows that the reverse implication does not hold.

Example 18. Let with the usual metric. For , define with Then is generalized pointwise cyclic contraction. Indeed, if either , or , then the result follows, easily. Also, if and , then for every if , we have which implies that is generalized pointwise cyclic contraction. It is interesting to note that is not pointwise cyclic contraction. In fact, if and and if is pointwise cyclic contraction then we must have which deduces that which is a contradiction. Besides, we mention that is not nonexpansive because is not continuous.

Here, we state the main result of this section.

Theorem 19. Let be a nonempty, weakly compact, and convex pair of a Banach space . If is a generalized pointwise cyclic contraction mapping, then has a best proximity point.

Proof. Since is cyclic relatively nonexpansive, we can apply Theorem 8 to deduce the existence of a minimal weakly compact convex pair such that and . Let be fixed. For each we have Hence, for all , we have , and then Therefore, which concludes that Similar argument implies that Now, let be an arbitrary element in . We may suppose that . Put . Let We note that and by the fact that we have . Moreover, it is easy to see that that is, is a nonempty, closed, and convex pair in . Besides, if then we have which implies that and so, . Similarly, . Therefore, is cyclic on . Now, by the minimality of we must have and . Thereby, which implies that Hence, for all we have If , then is a best proximity point of and we are finished. So, let . We have which is a contradiction. Therefore, for all we must have

Remark 20. Note that Theorem 19 was proved directly and without using the notion of proximal normal structure.

Remark 21. Theorem 19 holds once the minimal sets and have been fixed and the cyclic mapping satisfies the condition that for each there exist ,  ,  , and   such that ,   and where for all .

Conflict of Interests

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

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Naseer Shahzad acknowledges with thanks DSR for financial support.

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