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## Optimization Problems via Best Proximity Point Analysis

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Research Article | Open Access

Volume 2013 |Article ID 189567 | https://doi.org/10.1155/2013/189567

Ali Abkar, Moosa Gabeleh, "A Note on Some Best Proximity Point Theorems Proved under P-Property", Abstract and Applied Analysis, vol. 2013, Article ID 189567, 3 pages, 2013. https://doi.org/10.1155/2013/189567

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

Accepted29 Sep 2013
Published04 Nov 2013

#### 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  as follows.

Definition 2 (see ). 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 ). 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 ). 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 ). 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 ). 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 ). 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  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 ). 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 . 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 (). Moreover, in Theorem 6, Nadler's fixed point theorem () 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 ().

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