Abstract
In this paper, we introduce the notion of Geraghtycontractions 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
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,
First we recall fundamental definitions and basic results in this direction.
Let A and B be nonempty subsets of a metric space
Let A and B be two nonempty subsets of a metric space
We denote by F the set of all functions
Definition 2 (See [2])
Let A, B be two nonempty subsets of a metric space
Very recently Raj [10,11] introduced the notion of Pproperty as follows.
Definition 3 Let
Example 4 (See, e.g., [11])
Let A be a nonempty subset of a metric space
Theorem 5 (See [2])
Let
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 Geraghtycontraction as follows.
Definition 6 Let A, B be two nonempty subsets of a metric space
where
Remark 7 Notice that since
where
2 Main results
We start this section with our main result.
Theorem 8Let
Proof Let us fix an element
Due to the fact that the pair
From (2.1), we get
On the other hand, by (2.1) and (2.2) we obtain that
Consequently, we have
If there exists
which yields that
For the rest of the proof, we suppose that
Then, by (2.3) and (2.6), we deduce that
Suppose that
a contradiction. As a result, we conclude that
By (2.6), we get
for all
for each
On the other hand, since
Since,
for all
Taking (2.9) into consideration, we find
We shall show that
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
Letting
On the other hand, we obtain
Taking limit as
So, we deduce that
and hence
Combining (1.6) and (2.14), we find
Since
which yields
As a result, we deduce that
We claim that the best proximity point of T is unique.
Suppose, on the contrary, that
By using the Pproperty, we find
and
Due to the fact that T is a generalized Geraghtycontraction, we have
a contradiction. This completes the proof. □
Remark 9 Let
Corollary 10Suppose that
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
and consider the closed subsets
and let
Since
Notice that
Moreover,
and, as
where
Notice that β is nondecreasing since
Therefore,
and it is easily seen that the function
Therefore, since the assumptions of Theorem 8 are satisfied, by Theorem 8 there exists
a unique
The point
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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