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 )
Let be a complete metric space and let be an operator. Suppose that there exists satisfying the condition
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 .
Let A and B be two nonempty subsets of a metric space . We denote by and the following sets:
We denote by F the set of all functions satisfying the following property:
Definition 2 (See )
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
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 ,
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 theP-property. Then there exists a unique inAsuch 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.
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
Remark 7 Notice that since , we have
2 Main results
We start this section with our main result.
Theorem 8Let be a complete metric space. Suppose that is a pair of nonempty closed subsets ofXand is nonempty. Suppose also that the pair has theP-property. If a non-self-mapping is a generalized Geraghty-contraction satisfying , then there exists a unique best proximity point, that is, there exists inAsuch 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:
Due to the fact that the pair has the P-property, we derive that
From (2.1), we get
On the other hand, by (2.1) and (2.2) we obtain that
Consequently, we have
If there exists such that , then the proof is completed. In fact, due to (2.2), we have
which yields that . Hence, equation (2.1) implies that
For the rest of the proof, we suppose that for any . Owing to the fact T is a generalized Geraghty-contraction, we derive that
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
By (2.6), we get
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,
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
We shall show that is a Cauchy sequence. Suppose, on the contrary, that we have
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:
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
Combining (1.6) and (2.14), we find
Since together with (2.15), we get . Hence, we have
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
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. □
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 10Suppose that is a complete metric space andAis a nonempty closed subset ofX. 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 .
and, as , we have
where is defined as .
Notice that β is nondecreasing since .
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 .
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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