SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

On best proximity point of ψ-Geraghty contractions

Erdal Karapınar

Author Affiliations

Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey

Fixed Point Theory and Applications 2013, 2013:200  doi:10.1186/1687-1812-2013-200


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/200


Received:12 March 2013
Accepted:17 May 2013
Published:24 July 2013

© 2013 Karapınar; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [1] 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 [2].

Theorem 1.1 (Geraghty [2])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M2">View MathML</a>be an operator. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M3">View MathML</a>satisfying the condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M4">View MathML</a>

IfTsatisfies the following inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M5">View MathML</a>

(1)

thenThas a unique fixed point.

It is very natural that some mappings, especially non-self-mappings defined on a complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>, do not necessarily possess a fixed point, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M7">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M8">View MathML</a>. In such situations, it is reasonable to search for the existence (and uniqueness) of a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M9">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M10">View MathML</a> is an approximation of an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M12">View MathML</a>. In other words, one speculates to determine an approximate solution <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M13">View MathML</a> that is optimal in the sense that the distance between <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M13">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M15">View MathML</a> is minimum. Here, the point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M16">View MathML</a> 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 [6]. 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1">View MathML</a> be a metric space and A and B be nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1">View MathML</a>. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M19">View MathML</a> is called a k-contraction if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M20">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M21">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M22">View MathML</a>. It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M23">View MathML</a>, where A is a complete subset of X.

Let A and B be two nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>. We denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M26">View MathML</a> the following sets:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M27">View MathML</a>

(2)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M28">View MathML</a>.

In [13], the authors presented sufficient conditions which determine when the sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M26">View MathML</a> are nonempty. In [19], the author introduced the following definition.

Definition 1.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> be a pair of nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M33">View MathML</a>. Then the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> is said to have the P-property if and only if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M35">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M36">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M37">View MathML</a>

(3)

It can be easily seen that for any nonempty subset A of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M39">View MathML</a> has the P-property. In [19], the author proved that any pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property. Now, we introduce the class F of those functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M3">View MathML</a> satisfying the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M42">View MathML</a>

(4)

Definition 1.2 (See [6])

Let A, B be two nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44">View MathML</a> is said to be a Geraghty-contraction if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M46">View MathML</a>

(5)

Theorem 1.2 (See [6])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a>be a pair of nonempty closed subsets of a complete metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25">View MathML</a>is nonempty. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44">View MathML</a>be a continuous Geraghty-contraction satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M51">View MathML</a>. Suppose that the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a>has theP-property. Then there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53">View MathML</a>inAsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M54">View MathML</a>.

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

We start this section with the definition of the following auxiliary function. Let Ψ denote the class of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M55">View MathML</a> which satisfy the following conditions:

(a) ψ is nondecreasing;

(b) ψ is subadditive, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M56">View MathML</a>;

(c) ψ is continuous;

(d) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M57">View MathML</a>.

We introduce the following contraction.

Definition 2.1 Let A, B be two nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44">View MathML</a> is said to be a ψ-Geraghty contraction if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M61">View MathML</a>

(6)

Remark 2.1 Notice that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M3">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M63">View MathML</a>

(7)

We are now ready to state and prove our main theorem.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a>be a pair of nonempty closed subsets of a complete metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25">View MathML</a>is nonempty. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M44">View MathML</a>be aψ-Geraghty contraction satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M51">View MathML</a>. Suppose that the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a>has theP-property. Then there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53">View MathML</a>inAsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M54">View MathML</a>.

Proof Regarding that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25">View MathML</a> is nonempty, we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M73">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M74">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M75">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M76">View MathML</a>. Analogously, regarding the assumption <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M77">View MathML</a>, we determine <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M79">View MathML</a>. Recursively, we obtain a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M80">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M25">View MathML</a> satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M82">View MathML</a>

(8)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> has the P-property, we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M84">View MathML</a>

(9)

If there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M85">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M86">View MathML</a>, then the proof is completed. Indeed,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M87">View MathML</a>

(10)

and consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M88">View MathML</a>. On the other hand, due to (8) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M89">View MathML</a>

Therefore, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M90">View MathML</a>

(11)

For the rest of the proof, we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M91">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M92">View MathML</a>. Since T is a ψ-Geraghty contraction, for any ℕ, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M93">View MathML</a>

(12)

Consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M94">View MathML</a> is a nonincreasing sequence and bounded below, and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M95">View MathML</a> exists. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M96">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M97">View MathML</a>. Then, from (6), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M98">View MathML</a>

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M99">View MathML</a>, which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M100">View MathML</a>

On the other hand, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45">View MathML</a>, we conclude <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M102">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M103">View MathML</a>

(13)

Notice that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M104">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M105">View MathML</a>, for fixed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M106">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M107">View MathML</a>, and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> satisfies the P-property, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M109">View MathML</a>. In what follows, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M80">View MathML</a> is a Cauchy sequence. On the contrary, assume that we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M111">View MathML</a>

(14)

By using the triangular inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M112">View MathML</a>

(15)

By (12) and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M113">View MathML</a>, by the comment mentioned above, regarding the discussion on the P-property above together with (12), (15) and the property of the function ψ, we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M114">View MathML</a>

(16)

By a simple manipulation, (16) yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M115">View MathML</a>

(17)

By taking the properties of the function ψ into account, together with (13) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M116">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M117">View MathML</a>, the last inequality yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M118">View MathML</a>

(18)

Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M119">View MathML</a>. By taking the fact <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M45">View MathML</a> into account, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M121">View MathML</a>

Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M80">View MathML</a> is a Cauchy sequence.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M123">View MathML</a> and A is a closed subset of the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M6">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M126">View MathML</a>.

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M127">View MathML</a>. Suppose, on the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M128">View MathML</a>. This means that we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M129">View MathML</a> such that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M130">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M131">View MathML</a> with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M132">View MathML</a>

(19)

Due to the properties of ψ, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M133">View MathML</a>

(20)

Using the fact that T is a ψ-Geraghty contraction, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M134">View MathML</a>

(21)

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M130">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M137">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M138">View MathML</a> such that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M139">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M140">View MathML</a>

(22)

Consequently, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M139">View MathML</a> we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M142">View MathML</a>

(23)

a contradiction. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M127">View MathML</a>.

Regarding the fact that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M144">View MathML</a> is a constant sequence with value <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M145">View MathML</a>, we derive

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M146">View MathML</a>

(24)

which is equivalent to saying that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53">View MathML</a> is the best proximity point of T. This completes the proof of the existence of a best proximity point.

We shall show the uniqueness of the best proximity point of T. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M53">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M149">View MathML</a> are two distinct best proximity points of T, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M150">View MathML</a>. This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M151">View MathML</a>

(25)

Using the P-property, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M152">View MathML</a>

(26)

Using the fact that T is a ψ-Geraghty contraction, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M153">View MathML</a>

(27)

a contradiction. This completes the proof. □

Notice that the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M39">View MathML</a> satisfies the P-property for any nonempty subset A of X. Consequently, we have the following corollary.

Corollary 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1">View MathML</a>be a complete metric space andAbe a nonempty closed subset of X. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M156">View MathML</a>be aψ-Geraghty-contraction. ThenThas a unique fixed point.

Proof Apply Theorem 2.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M157">View MathML</a>. □

If we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M158">View MathML</a> we obtain Theorem 1.2 as a corollary of Theorem 2.1.

Corollary 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M1">View MathML</a>be a complete metric space andAbe a nonempty closed subset of X. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M156">View MathML</a>be a Geraghty-contraction. ThenThas a unique fixed point.

Proof Apply Theorem 2.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M157">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M158">View MathML</a>. □

In order to illustrate our results, we present the following example.

Example 2.1 Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M163">View MathML</a> with the metric

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M164">View MathML</a>

and consider the closed subsets

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M165">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M166">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M167">View MathML</a>.

Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M168">View MathML</a> to be the mapping defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M169">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M170">View MathML</a>, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> has the P-property.

Notice that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M172">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M173">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M51">View MathML</a>.

Without loss of generality, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M175">View MathML</a>. Moreover,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M176">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M177">View MathML</a>

We have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M178">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M179">View MathML</a> is defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M180">View MathML</a>.

Therefore, T is a ψ-Geraghty-contraction. Notice that the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M181">View MathML</a> satisfies the P-property. Indeed, if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M182">View MathML</a>

then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M183">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M184">View MathML</a> and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M185">View MathML</a>

Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there exists a unique <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M186">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M187">View MathML</a>

More precisely, the point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M188">View MathML</a> is the best proximity point of T.

Example 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M189">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M190">View MathML</a> be a metric on X. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M191">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M192">View MathML</a> are two closed subsets of ℝ. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M168">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M194">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M195">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M196">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M197">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M198">View MathML</a>. Clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M199">View MathML</a>. Now we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M200">View MathML</a>

Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M201">View MathML</a>. Further, clearly, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M31">View MathML</a> has the P-property. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M203">View MathML</a>. Note that, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M204">View MathML</a>, then condition (6) holds. Hence, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M205">View MathML</a>. We shall show that (6) holds. Suppose, on the contrary, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M206">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M207">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M208">View MathML</a>

which yields that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M209">View MathML</a>, a contradiction. Therefore condition (6) holds for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M203">View MathML</a>. Hence, the conditions of Theorem 2.1 hold and T has a unique best proximity point. Here, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/200/mathml/M211">View MathML</a> is the best proximity point of T.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.

References

  1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math.. 3, 133–181 (1922)

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

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

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

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

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

  7. 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 OpenURL

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

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

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

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

  12. Karapınar, E: Best proximity points of Kannan type cyclic weak ϕ-contractions in ordered metric spaces. An. Univ. Ovidius Constanţa. 20(3), 51–64 (2012)

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

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

  15. Pragadeeswarar, V, Marudai, M: Best proximity points: approximation and optimization in partially ordered metric spaces. Optim. Lett. (2012). Publisher Full Text OpenURL

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

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

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

  19. Raj, VS: Banach’s contraction principle for non-self mappings. Preprint

  20. Samet, B: Some results on best proximity points. J. Optim. Theory Appl. (2013). Publisher Full Text OpenURL

  21. Sintunavarat, W, Kumam, P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl.. 2012, Article ID 93 (2012)

  22. Shahzad, N, Basha, SS, Jeyaraj, R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl.. 148, 69–78 (2011). Publisher Full Text OpenURL

  23. Srinivasan, PS: Best proximity pair theorems. Acta Sci. Math.. 67, 421–429 (2001)