Very recently, Caballero, Harjani and Sadarangani (Fixed Point Theory Appl. 2012:231, 2012) observed some best proximity point results for Geraghty contractions by using the P-property. In this paper, we introduce the notion of ψ-Geraghty contractions and show the existence and uniqueness of the best proximity point of such contractions in the setting of a metric space. We state examples to illustrate our result.
MSC: 41A65, 90C30, 47H10.
Keywords:best proximity point; non-self mapping; partial order; metric space; fixed point
1 Introduction and preliminaries
In nonlinear functional analysis, fixed point theory and best proximity point theory play a crucial role in the establishment of the existence of certain differential and integral equations. As a consequence, fixed point theory is very useful for various quantitative sciences that involve such equations. To list a few, certain branches of computer sciences, engineering and economics are well-known examples in which fixed point theory is used.
The most remarkable paper in this field was reported by Banach  in 1922. In this paper, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this outstanding paper, many authors have extended, generalized and improved this remarkable fixed point theorem of Banach by changing either the conditions of the mappings or the construction of the space. In particular, one of the notable generalizations of Banach fixed point theorem was reported by Geraghty .
Theorem 1.1 (Geraghty )
IfTsatisfies the following inequality:
thenThas a unique fixed point.
It is very natural that some mappings, especially non-self-mappings defined on a complete metric space , do not necessarily possess a fixed point, that is, for all . In such situations, it is reasonable to search for the existence (and uniqueness) of a point such that is an approximation of an such that . In other words, one speculates to determine an approximate solution that is optimal in the sense that the distance between and is minimum. Here, the point is called a best proximity point.
This research subject has attracted attention of a number of authors; for example, see [2-23]. In this paper we generalize and improve certain results of Caballero et al. in . Notice also that in the best proximity point theory, we usually consider a non-self-mapping. In fixed point theory, almost all maps are self-mappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.
Let be a metric space and A and B be nonempty subsets of a metric space . A mapping is called a k-contraction if there exists such that for any . It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes , where A is a complete subset of X.
It can be easily seen that for any nonempty subset A of , the pair has the P-property. In , the author proved that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property. Now, we introduce the class F of those functions satisfying the following condition:
Definition 1.2 (See )
Theorem 1.2 (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.
In the following section, we improve the theorem above by using a distance function ψ in Definition 1.2. In particular, we introduce Definition 2.1 and broaden the scope of Theorem 1.2 to ψ-Geraghty-contractions.
2 Main results
(a) ψ is nondecreasing;
(c) ψ is continuous;
We introduce the following contraction.
We are now ready to state and prove our main theorem.
Theorem 2.1Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty. Letbe aψ-Geraghty contraction satisfying. Suppose that the pairhas theP-property. Then there exists a uniqueinAsuch that.
Therefore, we conclude that
By using the triangular inequality,
By a simple manipulation, (16) yields that
Due to the properties of ψ, we get
Using the fact that T is a ψ-Geraghty contraction, we have
Using the P-property, we have
Using the fact that T is a ψ-Geraghty contraction, we have
a contradiction. This completes the proof. □
In order to illustrate our results, we present the following example.
and consider the closed subsets
Also, . Further, clearly, the pair has the P-property. Let . Note that, if , then condition (6) holds. Hence, we assume that . We shall show that (6) holds. Suppose, on the contrary, there exist such that
The author declares that he has no competing interests.
The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.
Geraghty, M: On contractive mappings. Proc. Am. Math. Soc.. 40, 604–608 (1973). Publisher Full Text
Al-Thagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal.. 70, 3665–3671 (2009). Publisher Full Text
Anuradha, J, Veeramani, P: Proximal pointwise contraction. Topol. Appl.. 156, 2942–2948 (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
Jleli, M, Samet, B: Best proximity points for α-ψ-proximal contractive type mappings and applications. Bull. Sci. Math. (in press). doi:10.1016/j.bulsci.2013.02.003 Publisher Full Text
Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl.. 323, 1001–1006 (2006). Publisher Full Text
Karapınar, E: Best proximity points of cyclic mappings. Appl. Math. Lett.. 25(11), 1761–1766 (2012). 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
Markin, J, Shahzad, N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal.. 70, 2435–2441 (2009). Publisher Full Text
Pragadeeswarar, V, Marudai, M: Best proximity points: approximation and optimization in partially ordered metric spaces. Optim. Lett. (2012). Publisher Full Text
Raj, VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal.. 74, 4804–4808 (2011). Publisher Full Text
Samet, B: Some results on best proximity points. J. Optim. Theory Appl. (2013). Publisher Full Text
Shahzad, N, Basha, SS, Jeyaraj, R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl.. 148, 69–78 (2011). Publisher Full Text