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.