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

Authors: Jack Markin; Naseer Shahzad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 1. Introduction

LetA, B be nonempty subsets of a Banach space (M,∥·∥). In [1], Eldred et al. considered the best proximity point problem for mappings T:A∪B→A∪B with T(A)⊂B and T(B)⊂A or T(A)⊂A and T(B)⊂B, respectively; that is, they sought conditions on the subsets A, B, the space M, and the mapping T that assure existence of points x0∈A, y0∈B such that
(1)∥x0-T(x0)∥=∥y0-T(y0)∥=dist(A,B),
or
(2)x0=T(x0),y0=T(y0),∥x0-y0∥=dist(A,B),
respectively. In solving this problem they considered a new class of mappings.Definition 1 (see [1]).
LetA, B be nonempty subsets of a metric space (M,d). Then a mapping T:A∪B→A∪B is said to be relatively nonexpansive if
(3)d(T(x),T(y))≤d(x,y)forx∈A,y∈B.
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 calledproximal normal structure, they obtained the following result.Theorem 2 (see [1]).
Let (A,B) be a nonempty weakly compact convex pair in a Banach space (M,∥·∥). Let T:A∪B→A∪B be a relatively nonexpansive mapping such that T(A)⊂B and T(B)⊂A, and suppose that (A,B) has proximal normal structure. Then there exists (x0,y0)∈A×B such that
(4)∥x0-T(x0)∥=∥y0-T(y0)∥=dist(A,B).With the goal of generalizing relatively nonexpansive mappings, Eldred et al. [2] introduced the notion of a relatively u-continuous mapping in Banach spaces, which we state here for a metric space.Definition 3 (see [2]).
LetA, B be nonempty subsets of a metric space (M,d). A mapping T:A∪B→A∪B is said to be relativelyu-continuous if for each ϵ>0, there exists δ>0 such that d(T(x),T(y))<ε+dist(A,B) whenever
(5)d(x,y)<δ+dist(A,B),∀x∈A,y∈B.
Every relatively nonexpansive mapping is relatively u-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 u-continuous mappings.Theorem 4 (see [2]).
LetA, B be nonempty compact convex subsets of a strictly convex Banach space X, and let T:A∪B→A∪B be a relatively u-continuous mapping such that T(A)⊂B and T(B)⊂A. Then there exists
(6)(x0,y0)∈A×Bsuchthat∥x0-T(x0)∥=∥y0-T(y0)∥=dist(A,B).In this paper we show that Theorem4 holds for any Banach space without the assumption of strict convexity as follows.Theorem 5.
Let(M,∥·∥) be a Banach space, and let A, B be nonempty compact convex subsets of M. If  T:A∪B→A∪B is relatively u-continuous such that T(A)⊂B and T(B)⊂A, then there exist points x∈A and y∈B such that ∥x-T(x)∥=∥y-T(y)∥=dist(A,B).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 relativelyu-continuous mappings in Banach and hyperconvex metric spaces. Our results generalize and extend some recent results to relatively u-continuous mappings and to general spaces.


## 2. Preliminaries

LetA and B be nonempty subsets of a metric space (M,d). Define
(7)dist(A,B)=inf{d(x,y):x∈A,y∈B},A0={x∈A:d(x,y)=dist(A,B)for somey∈B},B0={y∈B:d(x,y)=dist(A,B)forsomex∈A}.Definition 6.
A metric space (M,d) is hyperconvex if given any family {xα:α∈I} of points in M and any family {rα} of nonnegative real numbers satisfying d(xα,xβ)≤rα+rβ for all α,β∈I, then ∩B(xα;rα)≠∅, where
(8)B(x;r)={y∈M:d(x,y)≤r}.Definition 7.
Theadmissible subsets of M are sets of the form ∩B(xα;rα), that is, the family of ball intersections in M. For a subset X of M, Nε(X) denotes the closed ε-hull of X; that is, Nε(X)={x∈M:dist(x,X)≤ε}, where dist(x,X)=inf{d(x,y):x∈X}.IfX is an admissible set, then Nε(X) is also an admissible set [11]. For recent progress in hyperconvex metric spaces, we refer the reader to [12].Definition 8.
Let(M,d) be a metric space and F:M→2M a multivalued mapping with nonempty values. Then F is said to be almost lower semicontinuous at a point x∈M if for each ε>0 there is an open neighborhood U(x) of x and a point z∈M such that, for y∈U(x),
(9)B(z;ε)∩F(y)≠∅.In establishing existence of best proximity points for relativelyu-continuous mappings in Banach and hyperconvex spaces, we apply the following continuous selection and fixed point theorems.Theorem 9 (see [13]).
LetX be a paracompact space and Y a normed linear space. Let F:X→2Y be a multivalued mapping with nonempty closed convex values. Then F is an almost lower semicontinuous mapping if and only if for each ϵ>0, F has a continuous ϵ-approximate selection; that is, a function f:X→Y such that for every x∈X, dist(f(x),F(x))<ϵ.Theorem 10 (see [14]).
LetX be a paracompact topological space, (M,d) a hyperconvex metric space, and F:X→2M an almost lower semicontinuous mapping with admissible values. Then F has a continuous selection; that is, there is a continuous mapping f:X→M such that f(x)∈F(x) for each x∈X.Theorem 11 (see [15, 16]).
Let(M,d) be a compact hyperconvex metric space and f:M→M a continuous mapping. Then f 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 Theorem5.
SinceA, B are compact convex subsets, A0, B0 are nonempty compact convex subsets. By [2, Proposition 3.1] T(A0)⊂B0 and T(B0)⊂A0.
Byu-continuity of T, for any x∈A, y∈B such that ∥x-y∥=dist(A,B) and any positive integer n there is a δn>0 and a neighborhood of x in A0 defined as
(10)U(x,δn)={u∈A0:∥u-x∥<δn},
such that u∈U(x,δn) implies that
(11)∥T(u)-T(y)∥≤(1n)+dist(A,B).
For each positive integern, define a multivalued mapping Fn:A0→2A0 by
(12)Fn(v)=B(T(v);(1n)+dist(A,B))∩A0,
for v∈A0. Since T(v)∈B0, Fn(v) is nonempty. As the intersection of closed convex sets, each Fn(v) is also closed convex.
By (11), T(y)∈Fn(u) for each u∈U(x,δn), which implies that the mapping Fn is almost lower semicontinuous. By the approximate selection result of Deutsch et al. [13] (see Theorem 9), for any α>0, Fn has a continuous α-approximate selection; that is, there is a continuous fn:A0→A0 such that dist(fn(v),Fn(v))≤α. Choosing α=1/n, by the definition of Fn the selection fn satisfies
(13)∥T(v)-fn(v)∥≤(2n)+dist(A,B),forv∈A0.
Since the mappingfn is continuous and A0 is a compact convex subset of a Banach space, the Schauder fixed point theorem implies that fn has a fixed point xn; that is, there is a point xn∈A0 such that xn=fn(xn).
By (13), ∥T(xn)-xn∥→dist(A,B), and by compactness of A0 and B0, we can assume that xn→x∈A0 and T(xn)→p∈B0. Therefore, ∥x-p∥=dist(A,B), and by u-continuity of T, ∥T(xn)-T(p)∥→dist(A,B). It follows that
(14)dist(A,B)≤∥p-T(p)∥≤∥p-T(xn)∥+∥T(xn)-T(p)∥⟶dist(A,B),
which implies that ∥p-T(p)∥=dist(A,B).The following proposition follows by a slight change in the proof in [2, Proposition 3.1].Proposition 12.
LetA, B be nonempty subsets of a normed linear space M, and let T:A∪B→A∪B be a relatively u-continuous mapping such that T(A)⊂A and T(B)⊂B. Then T(A0)⊂A0 and T(B0)⊂B0.Proposition 13 (see [17]).
Let(M,∥·∥) be a strictly convex Banach space, A a nonempty compact convex subset of M, and B a nonempty closed convex subset of M. Let {xn} be a sequence in A and y∈B. If
(15)∥xn-y∥⟶dist(A,B),thenxn⟶PA(y).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 u-continuous mappings in a strictly convex space.Theorem 14.
Let(M,∥·∥) be a strictly convex Banach space, and let A, B be compact convex subsets of M. If T:A∪B→A∪B is relatively u-continuous such that T(A)⊂A and T(B)⊂B, then there exist points x0∈A and y0∈B such that x0=T(x0), y0=T(y0) and ∥x0-y0∥=dist(A,B).Proof.
SinceA, B are compact convex sets, A0 and B0 are nonempty compact convex sets, and by Proposition 12, T(A0)⊂A0 and T(B0)⊂B0.
Byu-continuity of T, for any positive integer n there is a δn>0 such that
(16)∥x-y∥≤δn+dist(A,B)
implies that ∥T(x)-T(y)∥<(1/n)+dist(A,B), for x∈A and y∈B. For x∈A0 define U(x,δn)={u∈A0:∥u-x∥<δn}, and let y=PB(x). Then u∈U(x,δn) implies that
(17)∥u-y∥≤∥u-x∥+∥x-y∥<δn+dist(A,B),
and therefore, by u-continuity of T,
(18)∥T(u)-T(y)∥≤(1n)+dist(A,B).
For each positive integern, define a map Fn:A0→2B0 by
(19)Fn(v)=B(T(v);(1n)+dist(A,B))∩B0,
for v∈A0. As the intersection of closed convex sets, Fn(v) is also closed convex. By (18), T(y)∈Fn(u) for u∈U(x,δn), which implies that Fn(u) is nonempty and also that Fn is an almost lower semicontinuous mapping.
SinceM is a normed linear space, by Theorem 9 for any α>0, Fn has a  continuous α-approximate selection; that is, there is a continuous fn:A0→B0 such that dist(fn(v),Fn(v))≤α, for v∈A0. Choosing α=1/n, by the definition of Fn the selection fn satisfies
(20)∥T(v)-fn(v)∥≤(2n)+dist(A,B),
for v∈A0.
Consider the metric projection operatorPA:M→A. Since fn(A0)⊂B0 and PA(B0)⊂A0, the map PA○fn sends A0 into A0. Since PA○fn is continuous and A0 is compact and convex, by the Schauder fixed point theorem there is a fixed point xn=PA○fn(xn)∈A0. Let yn=fn(xn)∈B0, and assume by compactness that xn, yn converge to x0∈A0, y0∈B0, respectively. By continuity of PA, x0=PA(y0).
By definition of the mapfn, ∥T(xn)-yn∥≤(2/n)+dist(A,B), and since yn→y0 we have
(21)∥T(xn)-y0∥≤∥T(xn)-yn∥+∥yn-y0∥⟶dist(A,B).
Therefore, by Proposition 13,
(22)T(xn)⟶PA(y0).
Byu-continuity of T, for any ϵ>0 there is a δ>0 such that
(23)∥T(xn)-T(y0)∥<ϵ+dist(A,B)provided∥xn-y0∥<δ+dist(A,B).
Since xn→x0, choose n sufficiently large that ∥xn-x0∥<δ. Then
(24)∥xn-y0∥≤∥xn-x0∥+∥x0-y0∥<δ+dist(A,B),
which implies that
(25)dist(A,B)≤∥T(xn)-T(y0)∥<ϵ+dist(A,B).
Since ϵ is arbitrary,
(26)∥T(xn)-T(y0)∥⟶dist(A,B).
Therefore, by Proposition 13,
(27)T(xn)⟶PA(T(y0)).
By the relations (22) and (27), T(xn) converges to both PA(y0) and PA(T(y0)). Therefore, x0=PA(y0)=PA(T(y0)). Since y0,T(y0)∈B0, ∥x0-y0∥=∥x0-T(y0)∥=dist(A,B), and by strict convexity of M, y0=T(y0).
Since∥x0-y0∥=dist(A,B), we have by u-continuity of T that ∥T(x0)-T(y0)∥=dist(A,B). Therefore, T(x0)=PA(T(y0)), and since x0=PA(T(y0)), this implies that x0=T(x0).


## 4. Best Proximity Points in Hyperconvex Spaces

The following is a best proximity point result for relativelyu-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.
LetA, B be admissible subsets of a hyperconvex metric space (M,d), let A0 be a compact subset of M and let T:A∪B→A∪B be a relatively u-continuous mapping such that T(A)⊂B, and T(B)⊂A. Then there is an x0∈A0 such that d(x0,T(x0))=dist(A,B).Proof.
By a result of Kirk et al. [18], the sets A0 and B0 are nonempty and hyperconvex. For x∈A0, choose y∈B0 such that d(x,y)=dist(A,B). Then, by u-continuity of T, for any ε>0 there is a δ>0 such that for u∈A, v∈B,
(28)d(u,v)<δ+dist(A,B)impliesthatd(T(u),T(v))<ε+dist(A,B).
It follows that d(T(x),T(y))=dist(A,B). This implies that T(x)∈B0 for x∈A0.
Define an open neighborhood ofx in A0 by U(x)={u∈A0:d(u,x)<δ}.
Thenu∈U(x) implies that
(29)d(u,y)≤d(u,x)+d(x,y)<δ+dist(A,B),
and therefore, by u-continuity of T,
(30)d(T(u),T(y))<ε+dist(A,B).
Define a multivaluedF:A0→2A0 by
(31)F(v)=B(T(v);dist(A,B))∩A,
for v∈A0. Since T(v)∈B0 for v∈A0, F(v) is a nonempty subset of A0, and since A is admissible, F(v) is also admissible.
We show thatF is almost lower semicontinuous by establishing that B(T(y);ε)∩F(u)≠∅ for u∈U(x). By (30) and the hyperconvexity of M, for u∈U(x),
(32)B(T(y);ε)∩B(T(u);dist(A,B))≠∅.
Since T(u)∈B0, we have
(33)B(T(u);dist(A,B))∩A≠∅.
Any point p in the intersection (33) is in A0 since d(p,T(u))=dist(A,B). Therefore,
(34)B(T(u);dist(A,B))∩A⊂A0.
By (32), (33), and the fact that T(y)∈A0, the sets B(T(y);ε), B(T(u);dist(A,B)), and A have pairwise nonempty intersection. Since all of these sets are ball intersections, the hyperconvexity of the space M implies that
(35)B(T(y);ε)∩B(T(u);dist(A,B))∩A≠∅.
Further, by (34), the intersection in (35) is contained in A0. It follows from (35) that B(T(y);ε)∩F(u)≠∅ for u∈U(x). This implies that the mapping F 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 f:A0→A0 such that f(x)∈F(x) for x∈A0. By Theorem 11, a continuous self-mapping on a compact hyperconvex space has a fixed point. Therefore, there is a w∈A0 such that w=f(w)∈F(w). By the definition of F,
(36)d(w,T(w))=dist(A,B).



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