Abstract and Applied Analysis

Volume 2013 (2013), Article ID 189567, 3 pages

http://dx.doi.org/10.1155/2013/189567

Research Article

A Note on Some Best Proximity Point Theorems Proved under P-Property

Ali Abkar¹ and Moosa Gabeleh²

¹Department of Mathematics, Imam Khomeini International University, Qazvin 34149, Iran

²Department of Mathematics, Ayatollah Boroujerdi University, Borujerd, Iran

Received 20 July 2013; Accepted 29 September 2013

Academic Editor: Mohamed Jleli

Copyright © 2013 Ali Abkar and Moosa Gabeleh. 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 show that some recent results concerning the existence of best proximity points can be obtained from the same results in fixed point theory.

1. Introduction

Let and be two nonempty subsets of a metric space . In this paper, we adopt the following notations and definitions:

The notion of best proximity point is defined as follows.

Definition 1. Let and be nonempty subsets of a metric space and a non-self-mapping. A point is called a best proximity point of if , where

Similarly, for a multivalued non-self-mapping , where is a nonempty pair of subsets of a metric space , a point is a best proximity point of provided that .

Recently, the notion of -property was introduced in [1] as follows.

Definition 2 (see [1]). Let be a pair of nonempty subsets of a metric space with . The pair is said to have -property if and only if

where and .

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 be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a weakly contractive non-self-mapping; that is,

where is a continuous and nondecreasing function such that is positive on , , and . Assume that the pair has the P-property and . Then, has a unique best proximity point.

Theorem 4 (see [2]). Let be a pair of nonempty closed subsets of a Banach space such that is compact and is nonempty. Let be a nonexpansive mapping; that is,

Assume that the pair has the P-property and . Then, has a best proximity point.

Theorem 5 (see [3]). Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a Meir-Keeler non-self-mapping; that is, for all and for any , there exists such that

Assume that the pair has the P-property and . Then, has a unique best proximity point.

Theorem 6 (see [4]). Let be a pair of nonempty closed subsets of a complete metric space such that and satisfies the P-property. Let be a multivalued contraction non-self-mapping; that is,

for some and for all . If   is bounded and closed in for all and is included in for each , then has a best proximity point in .

Theorem 7 (see [5]). Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a Geraghty-contraction non-self-mapping; that is,

where is a function which satisfies the following condition:

Suppose that the pair has the P-property and . Then, 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 be a pair of nonempty closed subsets of a complete metric space such that is nonempty and has the P-property. Then, is a closed pair of subsets of .

Lemma 9. Let be a pair of nonempty closed subsets of a metric space such that is nonempty. Assume that the pair has the -property. Then there exists a bijective isometry such that .

Proof. Let ; then there exists an element such that

Assume that there exists another point such that

By the fact that has the -property, we conclude that . Consider the non-self-mapping such that . Clearly, is well defined. Moreover, is an isometry. Indeed, if , then

Again, since has the -property,

that is, is an isometry.

Here, we prove that the existence and uniqueness of the best proximity point in Theorem 3 are a sample result of the existence of fixed point for a weakly contractive self-mapping.

Theorem 10. Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a weakly contractive mapping. Assume that the pair has the P-property and . Then, has a unique best proximity point.

Proof. Consider the bijective isometry as in Lemma 9. Since , for the self-mapping , we have

for all which implies that the self-mapping is weakly contractive. Note that is closed by Lemma 8. Thus, has a unique fixed point [7]. Suppose that is a unique fixed point of the self-mapping ; that is, . So, , and then

from which it follows that is a unique best proximity point of the non-self weakly contractive mapping .

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 Theorem 4. Also, the existence and uniqueness of best proximity point for Meir-Keeler non-self-mapping (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 . Finally, Theorem 7 due to Caballero et al., is obtained from Geraghty's fixed point theorem ([10]).

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