Abstract and Applied Analysis

Volume 2013 (2013), Article ID 680186, 5 pages

http://dx.doi.org/10.1155/2013/680186

Research Article

Best Proximity Points for Relatively -Continuous Mappings in Banach and Hyperconvex Spaces

Jack Markin¹ and Naseer Shahzad²

¹528 Rover Boulevard, Los Alamos, NM 87544, USA

²Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 21 May 2013; Accepted 9 August 2013

Academic Editor: Adrian Petrusel

Copyright © 2013 Jack Markin 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.

We prove some best proximity point results for relatively -continuous mappings in Banach and hyperconvex metric spaces. Our results generalize and extend some recent results to relatively -continuous mappings and to general spaces.

1. Introduction

Let , be nonempty subsets of a Banach space (). In [1], Eldred et al. considered the best proximity point problem for mappings with and or and , respectively; that is, they sought conditions on the subsets , , the space , and the mapping that assure existence of points , such that

or

respectively. In solving this problem they considered a new class of mappings.

Definition 1 (see [1]). Let , be nonempty subsets of a metric space (). Then a mapping is said to be relatively nonexpansive if

The assumption that a mapping is relatively nonexpansive is weaker than the assumption that it is nonexpansive and does not even imply continuity [1].

Introducing a geometric condition for Banach spaces called proximal normal structure, they obtained the following result.

Theorem 2 (see [1]). Let () be a nonempty weakly compact convex pair in a Banach space . Let be a relatively nonexpansive mapping such that and , and suppose that has proximal normal structure. Then there exists such that

With the goal of generalizing relatively nonexpansive mappings, Eldred et al. [2] introduced the notion of a relatively -continuous mapping in Banach spaces, which we state here for a metric space.

Definition 3 (see [2]). Let , be nonempty subsets of a metric space (). A mapping is said to be relatively -continuous if for each , there exists such that whenever

Every relatively nonexpansive mapping is relatively -continuous. For an example showing that the converse is not true see [2, Example 2.1].

Eldred et al. [2] were able to extend some of the results of [1] to include the class of relatively -continuous mappings.

Theorem 4 (see [2]). Let , be nonempty compact convex subsets of a strictly convex Banach space , and let be a relatively -continuous mapping such that and . Then there exists

In this paper we show that Theorem 4 holds for any Banach space without the assumption of strict convexity as follows.

Theorem 5. Let be a Banach space, and let , be nonempty compact convex subsets of . If   is relatively -continuous such that and , then there exist points and such that .

Some interesting best proximity point theorems for various kinds of mappings have been accomplished in [3–8]. Other related results on cyclical mappings can be found in [9, 10].

The aim of this paper is to prove some best proximity point results for relatively -continuous mappings in Banach and hyperconvex metric spaces. Our results generalize and extend some recent results to relatively -continuous mappings and to general spaces.

2. Preliminaries

Let and be nonempty subsets of a metric space (). Define

Definition 6. A metric space () is hyperconvex if given any family of points in and any family of nonnegative real numbers satisfying for all , then , where

Definition 7. The admissible subsets of are sets of the form , that is, the family of ball intersections in . For a subset of , denotes the closed -hull of ; that is, , where .

If is an admissible set, then is also an admissible set [11]. For recent progress in hyperconvex metric spaces, we refer the reader to [12].

Definition 8. Let be a metric space and a multivalued mapping with nonempty values. Then is said to be almost lower semicontinuous at a point if for each there is an open neighborhood of and a point such that, for ,

In establishing existence of best proximity points for relatively -continuous mappings in Banach and hyperconvex spaces, we apply the following continuous selection and fixed point theorems.

Theorem 9 (see [13]). Let be a paracompact space and a normed linear space. Let be a multivalued mapping with nonempty closed convex values. Then is an almost lower semicontinuous mapping if and only if for each , has a continuous -approximate selection; that is, a function such that for every , .

Theorem 10 (see [14]). Let be a paracompact topological space, a hyperconvex metric space, and an almost lower semicontinuous mapping with admissible values. Then has a continuous selection; that is, there is a continuous mapping such that for each .

Theorem 11 (see [15, 16]). Let be a compact hyperconvex metric space and a continuous mapping. Then has a fixed point.

3. Best Proximity Points in Banach Spaces

The following theorem extends the best proximity point result of Eldred et al. [2, Theorem  3.1] for strictly convex Banach spaces to any Banach space.

Proof of Theorem 5. Since , are compact convex subsets, , are nonempty compact convex subsets. By [2, Proposition 3.1] and .

By -continuity of , for any , such that and any positive integer there is a and a neighborhood of in defined as

such that implies that

For each positive integer , define a multivalued mapping by

for . Since , is nonempty. As the intersection of closed convex sets, each is also closed convex.

By (11), for each , which implies that the mapping is almost lower semicontinuous. By the approximate selection result of Deutsch et al. [13] (see Theorem 9), for any , has a continuous -approximate selection; that is, there is a continuous such that . Choosing , by the definition of the selection satisfies

Since the mapping is continuous and is a compact convex subset of a Banach space, the Schauder fixed point theorem implies that has a fixed point ; that is, there is a point such that .

By (13), , and by compactness of and , we can assume that and . Therefore, , and by -continuity of , . It follows that

which implies that .

The following proposition follows by a slight change in the proof in [2, Proposition 3.1].

Proposition 12. Let , be nonempty subsets of a normed linear space , and let be a relatively -continuous mapping such that and . Then and .

Proposition 13 (see [17]). Let be a strictly convex Banach space, a nonempty compact convex subset of , and a nonempty closed convex subset of . Let be a sequence in and . If

In [1] a best proximity result was given for relatively nonexpansive mappings in a uniformly convex space. The following result is a version of that result for relatively -continuous mappings in a strictly convex space.

Theorem 14. Let be a strictly convex Banach space, and let , be compact convex subsets of . If is relatively -continuous such that and , then there exist points and such that , and .

Proof. Since , are compact convex sets, and are nonempty compact convex sets, and by Proposition 12, and .

By -continuity of , for any positive integer there is a such that

implies that , for and . For define , and let . Then implies that

and therefore, by -continuity of ,

For each positive integer , define a map by

for . As the intersection of closed convex sets, is also closed convex. By (18), for , which implies that is nonempty and also that is an almost lower semicontinuous mapping.

Since is a normed linear space, by Theorem 9 for any , has a continuous -approximate selection; that is, there is a continuous such that , for . Choosing , by the definition of the selection satisfies

for .

Consider the metric projection operator . Since and , the map sends into . Since is continuous and is compact and convex, by the Schauder fixed point theorem there is a fixed point . Let , and assume by compactness that , converge to , , respectively. By continuity of , .

By definition of the map , , and since we have

Therefore, by Proposition 13,

By -continuity of , for any there is a such that

Since , choose sufficiently large that . Then

which implies that

Since is arbitrary,

Therefore, by Proposition 13,

By the relations (22) and (27), converges to both and . Therefore, . Since , , and by strict convexity of , .

Since , we have by -continuity of that . Therefore, , and since , this implies that .

4. Best Proximity Points in Hyperconvex Spaces

The following is a best proximity point result for relatively -continuous mappings in hyperconvex metric spaces. Best proximity point/pair results were obtained in the setting of hyperconvex spaces by some authors in [18–21].

Theorem 15. Let , be admissible subsets of a hyperconvex metric space , let be a compact subset of and let be a relatively -continuous mapping such that , and . Then there is an such that .

Proof. By a result of Kirk et al. [18], the sets and are nonempty and hyperconvex. For , choose such that . Then, by -continuity of , for any there is a such that for , ,

It follows that . This implies that for .

Define an open neighborhood of in by .

Then implies that

and therefore, by -continuity of ,

Define a multivalued by

for . Since for , is a nonempty subset of , and since is admissible, is also admissible.

We show that is almost lower semicontinuous by establishing that for . By (30) and the hyperconvexity of , for ,

Since , we have

Any point in the intersection (33) is in since . Therefore,

By (32), (33), and the fact that , the sets , , and have pairwise nonempty intersection. Since all of these sets are ball intersections, the hyperconvexity of the space implies that

Further, by (34), the intersection in (35) is contained in . It follows from (35) that for . This implies that the mapping is almost lower semicontinuous.

By the selection theorem in Markin [14] (see Theorem 10), an almost lower semicontinuous mapping on a hyperconvex space with nonempty admissible values has a continuous selection; that is, there is a continuous such that for . By Theorem 11, a continuous self-mapping on a compact hyperconvex space has a fixed point. Therefore, there is a such that . By the definition of ,

Acknowledgment

The authors thank the referees for providing useful comments and suggestions that improved the paper.

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