## Journal of Inequalities and Applications

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# A generalization for the best proximity point of Geraghty-contractions

Journal of Inequalities and Applications20132013:286

DOI: 10.1186/1029-242X-2013-286

Accepted: 22 May 2013

Published: 6 June 2013

## Abstract

In this paper, we introduce the notion of Geraghty-contractions and consider the related best proximity point in the context of a metric space. We state an example to illustrate our result.

MSC: 47H10, 54H25, 46J10, 46J15.

## 1 Introduction and preliminaries

Fixed point theory and best proximity theory are very important tools in nonlinear functional analysis. These related research areas have wide application potential in various branches of mathematics and different disciplines such as economics, engineering. One of the most impressive results in this direction, known as the Banach contraction mapping principle, was given by Banach: Every contraction on a complete metric space has a unique fixed point. This celebrated result has been generalized in several ways in various abstract spaces. In particular, one of the interesting generalizations of the Banach contraction mapping principle was given by Geraghty [1].

Theorem 1 (Geraghty [1])

Let be a complete metric space and let be an operator. Suppose that there exists satisfying the condition
If T satisfies the following inequality:
(1.1)

then T has a unique fixed point.

It is clear that some mapping on a complete metric space has no fixed point, that is, for all . In this case, it is natural to ask the existence and uniqueness of the smallest value of . This is the main motivation of a best proximity point. This research subject has attracted attention of a number of authors; see, e.g., [119].

First we recall fundamental definitions and basic results in this direction.

Let A and B be nonempty subsets of a metric space . A mapping is called a k-contraction if there exists such that for any . Notice that the k-contraction coincides with the Banach contraction mapping principle if one takes , where A is a complete subset of X. A point is called the best proximity of T if , where .

Let A and B be two nonempty subsets of a metric space . We denote by and the following sets:
(1.2)
We denote by F the set of all functions satisfying the following property:
(1.3)

Definition 2 (See [2])

Let A, B be two nonempty subsets of a metric space . A mapping is said to be a Geraghty-contraction if there exists such that
(1.4)

Very recently Raj [10, 11] introduced the notion of P-property as follows.

Definition 3 Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the P-property if and only if for any and ,
(1.5)

Example 4 (See, e.g., [11])

Let A be a nonempty subset of a metric space . It is evident that the pair has the P-property. Let be any pair of nonempty, closed, convex subsets of a real Hilbert space H. Then has the P-property.

Theorem 5 (See [2])

Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a continuous, Geraghty-contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .

The subject of this paper is to generalize, improve and extend the results of Caballero, Harjani and Sadarangani [2]. For this purpose, we first define the notion of generalized Geraghty-contraction as follows.

Definition 6 Let A, B be two nonempty subsets of a metric space . A mapping is said to be a generalized Geraghty-contraction if there exists such that
(1.6)

where .

Remark 7 Notice that since , we have
(1.7)

where .

## 2 Main results

We start this section with our main result.

Theorem 8 Let be a complete metric space. Suppose that is a pair of nonempty closed subsets of X and is nonempty. Suppose also that the pair has the P-property. If a non-self-mapping is a generalized Geraghty-contraction satisfying , then there exists a unique best proximity point, that is, there exists in A such that .

Proof Let us fix an element in . Since , we can find such that . Further, as , there is an element in such that . Recursively, we obtain a sequence in with the following property:
(2.1)
Due to the fact that the pair has the P-property, we derive that
(2.2)
From (2.1), we get
On the other hand, by (2.1) and (2.2) we obtain that
Consequently, we have
(2.3)
If there exists such that , then the proof is completed. In fact, due to (2.2), we have
(2.4)
which yields that . Hence, equation (2.1) implies that
(2.5)
For the rest of the proof, we suppose that for any . Owing to the fact T is a generalized Geraghty-contraction, we derive that
(2.6)
Then, by (2.3) and (2.6), we deduce that
Suppose that . Then we get that
a contradiction. As a result, we conclude that and hence
(2.7)
By (2.6), we get
(2.8)
for all . Consequently, is a nonincreasing sequence and bounded below. Thus, there exists such that . We shall show that . Suppose, on the contrary, . Then, by (2.8), we have
for each . In what follows,
On the other hand, since , we conclude , that is,
(2.9)
Since, holds for all and satisfies the P-property, then, for all , we can write, . We also have
for all . It follows that
Taking (2.9) into consideration, we find
(2.10)
We shall show that is a Cauchy sequence. Suppose, on the contrary, that we have
(2.11)
Due to the triangular inequality, we have
(2.12)
Regarding (1.6) and (2.12), we have
(2.13)
Taking (2.10), (2.13) and (2.9) into account, we derive that
Owing to (2.11), we get
which implies . By the property of β, we have . Consequently, we have , a contradiction. Hence, we conclude that the sequence is Cauchy. Since A is a closed subset of the complete metric space and , and we can find such that as . We assert that . Suppose, on the contrary, that . First, we obtain the following inequalities:
Letting in the inequalities above, we conclude that
On the other hand, we obtain
Taking limit as in the inequality above, we find
So, we deduce that . As a consequence, we derive
and hence
(2.14)
Combining (1.6) and (2.14), we find
(2.15)
Since together with (2.15), we get . Hence, we have
which yields

As a result, we deduce that , a contradiction. So, and hence , is a best proximity point of T. Hence, we conclude that T has a best proximity point.

We claim that the best proximity point of T is unique.

Suppose, on the contrary, that and are two distinct best proximity points of T. Thus, we have
(2.16)
By using the P-property, we find
(2.17)
and
Due to the fact that T is a generalized Geraghty-contraction, we have

a contradiction. This completes the proof. □

Remark 9 Let be a metric space and A be any nonempty subset of X. It is evident that a pair satisfies the P-property.

Corollary 10 Suppose that is a complete metric space and A is a nonempty closed subset of X. If a self-mapping is a generalized Geraghty-contraction, then it has a unique fixed point.

Proof Taking Remark 9 into consideration, we conclude the desired result by applying Theorem 8 with . □

In order to illustrate our main result, we present the following example.

Example 11 Suppose that with the metric
and consider the closed subsets
and let be the mapping defined by

Since , the pair has the P-property.

Notice that and and .

Moreover,
and, as , we have

where is defined as .

Notice that β is nondecreasing since .

Therefore,

and it is easily seen that the function belongs to F.

Therefore, since the assumptions of Theorem 8 are satisfied, by Theorem 8 there exists a unique such that

The point is .

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Arts, Amasya University
(2)
Department of Mathematics, Atilim University
(3)
Department of Mathematics, University of Las Palmas de Gran Canaria

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