- Open Access
A generalization for the best proximity point of Geraghty-contractions
© Bilgili et al.; licensee Springer 2013
Received: 6 February 2013
Accepted: 22 May 2013
Published: 6 June 2013
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 .
Theorem 1 (Geraghty )
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., [1–19].
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 .
Definition 2 (See )
Example 4 (See, e.g., )
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 )
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 . For this purpose, we first define the notion of generalized Geraghty-contraction as follows.
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 .
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.
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.
Since , the pair has the P-property.
Notice that and and .
where is defined as .
Notice that β is nondecreasing since .
and it is easily seen that the function belongs to F.
The point is .
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