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])
Letbe a complete metric space and letbe an operator. Suppose that there existssatisfying 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., [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 kcontraction if there exists such that for any . Notice that the kcontraction 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 [2])
Let A, B be two nonempty subsets of a metric space . A mapping is said to be a Geraghtycontraction if there exists such that
Very recently Raj [10,11] introduced the notion of Pproperty as follows.
Definition 3 Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the Pproperty if and only if for any and ,
Example 4 (See, e.g., [11])
Let A be a nonempty subset of a metric space . It is evident that the pair has the Pproperty. Let be any pair of nonempty, closed, convex subsets of a real Hilbert space H. Then has the Pproperty.
Theorem 5 (See [2])
Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty. Letbe a continuous, Geraghtycontraction satisfying. Suppose that the pairhas thePproperty. Then there exists a uniqueinAsuch 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 Geraghtycontraction as follows.
Definition 6 Let A, B be two nonempty subsets of a metric space . A mapping is said to be a generalized Geraghtycontraction if there exists such that
Remark 7 Notice that since , we have
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 thePproperty. If a nonselfmappingis a generalized Geraghtycontraction satisfying, then there exists a unique best proximity point, that is, there existsinAsuch 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 Pproperty, 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 Geraghtycontraction, 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
On the other hand, since , we conclude , that is,
Since, holds for all and satisfies the Pproperty, then, for all , we can write, . We also have
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
and hence
Combining (1.6) and (2.14), we find
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
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 be a metric space and A be any nonempty subset of X. It is evident that a pair satisfies the Pproperty.
Corollary 10Suppose thatis a complete metric space andAis a nonempty closed subset ofX. If a selfmappingis a generalized Geraghtycontraction, 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 Pproperty.
Moreover,
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
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
References

Geraghty, M: On contractive mappings . Proc. Am. Math. Soc.. 40, 604–608 (1973). Publisher Full Text

Caballero, J, Harjani, J, Sadarangani, K: A best proximity point theorem for Geraghtycontractions . Fixed Point Theory Appl.. 2012, (2012) Article ID 231

Eldred, AA, Veeramani, P: Existence and convergence of best proximity points . J. Math. Anal. Appl.. 323, 1001–1006 (2006). Publisher Full Text

Anuradha, J, Veeramani, P: Proximal pointwise contraction . Topol. Appl.. 156, 2942–2948 (2009). Publisher Full Text

Markin, J, Shahzad, N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces . Nonlinear Anal.. 70, 2435–2441 (2009). Publisher Full Text

Basha, SS, Veeramani, P: Best proximity pair theorems for multifunctions with open fibres . J. Approx. Theory. 103, 119–129 (2000). Publisher Full Text

Raj, VS, Veeramani, P: Best proximity pair theorems for relatively nonexpansive mappings . Appl. Gen. Topol.. 10, 21–28 (2009)

AlThagafi, MA, Shahzad, N: Convergence and existence results for best proximity points . Nonlinear Anal.. 70, 3665–3671 (2009). Publisher Full Text

Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems . Numer. Funct. Anal. Optim.. 24, 851–862 (2003). Publisher Full Text

Raj, VS: A best proximity theorem for weakly contractive nonself mappings . Nonlinear Anal.. 74, 4804–4808 (2011). Publisher Full Text

Raj, VS: Banach’s contraction principle for nonself mappings. Preprint

Karapınar, E: Best proximity points of cyclic mappings . Appl. Math. Lett.. 25(11), 1761–1766 (2012). Publisher Full Text

Karapınar, E, Erhan, ÝM: Best proximity point on different type contractions . Appl. Math. Inf. Sci.. 3(3), 342–353 (2011)

Karapınar, E: Best proximity points of Kannan type cyclic weak ϕcontractions in ordered metric spaces . An. Stiint. Univ. Ovidius Constanta. 20(3), 51–64 (2012)

Mongkolkeha, C, Kumam, P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces . J. Optim. Theory Appl.. 155, 215–226 (2012). Publisher Full Text

Mongkolkeha, C, Kumam, P: Some common best proximity points for proximity commuting mappings . Optim. Lett. (2012) doi:10.1007/s1159001205251 in press

Jleli, M, Samet, B: Best proximity points for αψproximal contractive type mappings and applications . Bull. Sci. Math. (2013) doi:10.1016/j.bulsci.2013.02.003

Mongkolkeha, C, Cho, YJ, Kumam, P: Best proximity points for generalized proximal Ccontraction mappings in metric spaces with partial orders . J. Inequal. Appl.. 2013, (2013) Article ID 94

De la Sen, M: Fixed point and best proximity theorems under two classes of integraltype contractive conditions in uniform metric spaces . Fixed Point Theory Appl.. 2010, (2010) Article ID 510974