# A Note on Some Best Proximity Point Theorems Proved underP-Property

Authors: Ali Abkar; Moosa Gabeleh
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We show that some recent results concerning the existence of best proximity points can be obtained from the same results in fixed point theory.


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## Body

## 1. Introduction

LetA and B be two nonempty subsets of a metric space (X,d). In this paper, we adopt the following notations and definitions:
(1)D(x,B)∶=inf{d(x,y):y∈B},∀x∈X,A0∶={x∈A:d(x,y)=dist(A,B),forsomey∈B},B0∶={y∈B:d(x,y)=dist(A,B),forsomex∈A}.The notion ofbest proximity point is defined as follows.Definition 1.
LetA and B be nonempty subsets of a metric space (X,d) and T:A→B a non-self-mapping. A point x*∈A is called a best proximity point of T if d(x*,Tx*)=dist(A,B), where
(2)dist(A,B)∶=inf{d(x,y):(x,y)∈A×B}.Similarly, for a multivalued non-self-mappingT:A→2B, where (A,B) is a nonempty pair of subsets of a metric space (X,d), a point x*∈A is a best proximity point of T provided that D(x*,Tx*)=dist(A,B).Recently, the notion ofP-property was introduced in [1] as follows.Definition 2 (see [1]).
Let(A,B) be a pair of nonempty subsets of a metric space (X,d) with A0≠∅. The pair (A,B) is said to have P-property if and only if
(3)d(x1,y1)=dist(A,B)d(x2,y2)=dist(A,B)⟹d(x1,x2)=d(y1,y2),
where x1,x2∈A0 and y1,y2∈B0.By using this notion, some best proximity point results were proved for various classes of non-self-mappings. Here, we state some of them.Theorem 3 (see [1]).
Let(A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:A→B be a weakly contractive non-self-mapping; that is,
(4)d(Tx,Ty)≤d(x,y)-ϕ(d(x,y))∀x,y∈A,
where ϕ:[0,∞)→[0,∞) is a continuous and nondecreasing function such that ϕ is positive on (0,∞), ϕ(0)=0, and limt→∞ϕ(t)=∞. Assume that the pair (A,B) has the P-property and T(A0)⊆B0. Then, T has a unique best proximity point.Theorem 4 (see [2]).
Let(A,B) be a pair of nonempty closed subsets of a Banach space X such that A is compact and A0 is nonempty. Let T:A→B be a nonexpansive mapping; that is,
(5)∥Tx-Ty∥≤∥x-y∥,∀x,y∈A.
Assume that the pair (A,B) has the P-property and T(A0)⊆B0. Then, T has a best proximity point.Theorem 5 (see [3]).
Let(A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:A→B be a Meir-Keeler non-self-mapping; that is, for all x,y∈A and for any ε>0, there exists δ(ε)>0 such that
(6)ε≤d(x,y)<ε+δimpliesd(Tx,Ty)≤ε.
Assume that the pair (A,B) has the P-property and T(A0)⊆B0. Then, T has a unique best proximity point.Theorem 6 (see [4]).
Let(A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A0≠∅ and (A,B) satisfies the P-property. Let T:A→2B be a multivalued contraction non-self-mapping; that is,
(7)H(Tx,Ty)≤αd(x,y),
for some α∈(0,1) and for all x,y∈A. If  Tx is bounded and closed in B for all x∈A and Tx0 is included in B0 for each x0∈A0, then T has a best proximity point in A.Theorem 7 (see [5]).
Let(A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:A→B be a Geraghty-contraction non-self-mapping; that is,
(8)d(Tx,Ty)≤β(d(x,y)),d(x,y),∀x,y∈A,
where β:[0,∞)→[0,1) is a function which satisfies the following condition:
(9)β(tn)⟶1⟹tn⟶0.
Suppose that the pair (A,B) has the P-property and T(A0)⊆B0. Then, T has a unique best proximity point.


## 2. Main Result

In this section, we show that the existence of a best proximity point in the main theorems of [1–5] can be obtained from the existence of the fixed point for a self-map. We begin our argument with the following lemmas.Lemma 8 (see [6]).
Let(A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that A0 is nonempty and (A,B) has the P-property. Then, (A0,B0) is a closed pair of subsets of X.Lemma 9.
Let(A,B) be a pair of nonempty closed subsets of a metric space (X,d) such that A0 is nonempty. Assume that the pair (A,B) has the P-property. Then there exists a bijective isometry g:A0→B0 such that d(x,gx)=dist(A,B).Proof.
Letx∈A0; then there exists an element y∈B0 such that
(10)d(x,y)=dist(A,B).
Assume that there exists another point y´∈B0 such that
(11)d(x,y´)=dist(A,B).
By the fact that (A,B) has the P-property, we conclude that y=y´. Consider the non-self-mapping g:A0→B0 such that d(x,gx)=dist(A,B). Clearly, g is well defined. Moreover, g is an isometry. Indeed, if x1,x2∈A0, then
(12)d(x1,gx1)=dist(A,B),d(x2,gx2)=dist(A,B).
Again, since (A,B) has the P-property,
(13)d(x1,x2)=d(gx1,gx2);
that is, g is an isometry.Here, we prove that the existence and uniqueness of the best proximity point in Theorem3 are a sample result of the existence of fixed point for a weakly contractive self-mapping.Theorem 10.
Let(A,B) be a pair of nonempty closed subsets of a complete metric space X such that A0 is nonempty. Let T:A→B be a weakly contractive mapping. Assume that the pair (A,B) has the P-property and T(A0)⊆B0. Then, T has a unique best proximity point.Proof.
Consider the bijective isometryg:A0→B0 as in Lemma 9. Since T(A0)⊆B0, for the self-mapping g-1T:A0→A0, we have
(14)d(g-1(Tx),g-1(Ty))=d(Tx,Ty)≤φ(d(x,y)),
for all x,y∈A0 which implies that the self-mapping g-1T is weakly contractive. Note that A0 is closed by Lemma 8. Thus, g-1T has a unique fixed point [7]. Suppose that x*∈A0 is a unique fixed point of the self-mapping g-1T; that is, g-1T(x*)=x*. So, Tx*=gx*, and then
(15)d(x*,Tx*)=d(x*,gx*)=dist(A,B),
from which it follows that x*∈A0 is a unique best proximity point of the non-self weakly contractive mapping T.Remark 11.
By a similar argument, using the fact that every nonexpansive self-mapping defined on a nonempty compact and convex subset of a Banach space has a fixed point, we conclude Theorem4. Also, the existence and uniqueness of best proximity point for Meir-Keeler non-self-mapping T (Theorem 5) follow from the Meir-Keeler's fixed point theorem ([8]). Moreover, in Theorem 6, Nadler's fixed point theorem ([9]) ensures the existence of a best proximity point for multivalued non-self mapping T. Finally, Theorem 7 due to Caballero et al., is obtained from Geraghty's fixed point theorem ([10]).



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*Source: 189567-2013-11-04.xml*