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 )
IfTsatisfies the following inequality:
thenThas 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., )
Theorem 5 (See )
Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty. Letbe a continuous, Geraghty-contraction satisfying. Suppose that the pairhas theP-property. Then there exists a uniqueinAsuch 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 8Letbe a complete metric space. Suppose thatis a pair of nonempty closed subsets ofXandis nonempty. Suppose also that the pairhas theP-property. If a non-self-mappingis a generalized Geraghty-contraction satisfying, then there exists a unique best proximity point, that is, there existsinAsuch that.
From (2.1), we get
On the other hand, by (2.1) and (2.2) we obtain that
Consequently, we have
Then, by (2.3) and (2.6), we deduce that
By (2.6), we get
Taking (2.9) into consideration, we find
Due to the triangular inequality, we have
Regarding (1.6) and (2.12), we have
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:
On the other hand, we obtain
Combining (1.6) and (2.14), we find
We claim that the best proximity point of T is unique.
By using the P-property, we find
Due to the fact that T is a generalized Geraghty-contraction, we have
a contradiction. This completes the proof. □
In order to illustrate our main result, we present the following example.
and consider the closed subsets
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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