Open Access Research

A best proximity point theorem for Geraghty-contractions

J Caballero, J Harjani and K Sadarangani*

Author Affiliations

Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, Las Palmas de Gran Canaria, 35017, Spain

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Fixed Point Theory and Applications 2012, 2012:231  doi:10.1186/1687-1812-2012-231


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/231


Received:16 May 2012
Accepted:10 December 2012
Published:27 December 2012

© 2012 Caballero et al.; 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

The purpose of this paper is to provide sufficient conditions for the existence of a unique best proximity point for Geraghty-contractions.

Our paper provides an extension of a result due to Geraghty (Proc. Am. Math. Soc. 40:604-608, 1973).

Keywords:
fixed point; Geraghty-contraction; P-property; best proximity point

1 Introduction

Let A and B be nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M1">View MathML</a>.

An operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a> is said to be a k-contraction if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M3">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M4">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M5">View MathML</a>. Banach’s contraction principle states that when A is a complete subset of X and T is a k-contraction which maps A into itself, then T has a unique fixed point in A.

A huge number of generalizations of this principle appear in the literature. Particularly, the following generalization of Banach’s contraction principle is due to Geraghty [1].

First, we introduce the class ℱ of those functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M6">View MathML</a> satisfying the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M7">View MathML</a>

Theorem 1.1 ([1])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>be a complete metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M9">View MathML</a>be an operator. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10">View MathML</a>such that for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M11">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M12">View MathML</a>

(1)

ThenThas a unique fixed point.

Since the constant functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M13">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M3">View MathML</a>, belong to ℱ, Theorem 1.1 extends Banach’s contraction principle.

Remark 1.1 Since the functions belonging to ℱ are strictly smaller than one, condition (1) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M15">View MathML</a>

Therefore, any operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M9">View MathML</a> satisfying (1) is a continuous operator.

The aim of this paper is to give a generalization of Theorem 1.1 by considering a non-self map T.

First, we present a brief discussion about a best proximity point.

Let A be a nonempty subset of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M18">View MathML</a> be a mapping. The solutions of the equation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M19">View MathML</a> are fixed points of T. Consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M20">View MathML</a> is a necessary condition for the existence of a fixed point for the operator T. If this necessary condition does not hold, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M21">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M22">View MathML</a> and the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M23">View MathML</a> does not have any fixed point. In this setting, our aim is to find an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M22">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M25">View MathML</a> is minimum in some sense. The best approximation theory and best proximity point analysis have been developed in this direction.

In our context, we consider two nonempty subsets A and B of a complete metric space and a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a>.

A natural question is whether one can find an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M27">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M28">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M29">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M22">View MathML</a>, the optimal solution to this problem will be the one for which the value <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M31">View MathML</a> is attained by the real valued function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M32">View MathML</a> given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M33">View MathML</a>.

Some results about best proximity points can be found in [2-9].

2 Notations and basic facts

Let A and B be two nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>.

We denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M36">View MathML</a> the following sets:

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M38">View MathML</a>.

In [8], the authors present sufficient conditions which determine when the sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M36">View MathML</a> are nonempty.

Now, we present the following definition.

Definition 2.1 Let A, B be two nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a> is said to be a Geraghty-contraction if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M44">View MathML</a>

Notice that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M45">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M46">View MathML</a>

Therefore, every Geraghty-contraction is a contractive mapping.

In [10], the author introduces the following definition.

Definition 2.2 ([10])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> be a pair of nonempty subsets of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M49">View MathML</a>. Then the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> is said to have the P-property if and only if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M51">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M52">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M53">View MathML</a>

It is easily seen that for any nonempty subset A of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M55">View MathML</a> has the P-property.

In [10], the author proves that any pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.

3 Main results

We start this section presenting our main result.

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a>be a pair of nonempty closed subsets of a complete metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35">View MathML</a>is nonempty. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a>be a Geraghty-contraction satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M61">View MathML</a>. Suppose that the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a>has theP-property. Then there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M63">View MathML</a>inAsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M64">View MathML</a>.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35">View MathML</a> is nonempty, we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M27">View MathML</a>.

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M67">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M69">View MathML</a>. Similarly, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M70">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M71">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M72">View MathML</a>. Repeating this process, we can get a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M73">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M35">View MathML</a> satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M75">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> has the P-property, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M77">View MathML</a>

Taking into account that T is a Geraghty-contraction, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M78">View MathML</a>, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M79">View MathML</a>

(2)

Suppose that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M80">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M81">View MathML</a>.

In this case,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M82">View MathML</a>

and consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M83">View MathML</a>.

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M84">View MathML</a>

and this is the desired result.

In the contrary case, suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M85">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M78">View MathML</a>.

By (2), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M87">View MathML</a> is a decreasing sequence of nonnegative real numbers, and hence there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M89">View MathML</a>

In the sequel, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M90">View MathML</a>.

Assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M91">View MathML</a>, then from (2) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M92">View MathML</a>

The last inequality implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M93">View MathML</a> and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M90">View MathML</a> and this contradicts our assumption.

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M96">View MathML</a>

(3)

Notice that since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M97">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M98">View MathML</a>, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M99">View MathML</a> fixed, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M100">View MathML</a>, and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> satisfies the P-property, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M102">View MathML</a>.

In what follows, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M73">View MathML</a> is a Cauchy sequence.

In the contrary case, we have that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M104">View MathML</a>

By using the triangular inequality,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M105">View MathML</a>

By (2) and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M106">View MathML</a>, by the above mentioned comment, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M107">View MathML</a>

which gives us

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M108">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M109">View MathML</a> and by (3), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M110">View MathML</a>, from the last inequality it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M111">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M112">View MathML</a>.

Taking into account that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M114">View MathML</a> and this contradicts our assumption.

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M73">View MathML</a> is a Cauchy sequence.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M116">View MathML</a> and A is a closed subset of the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M119">View MathML</a>.

Since any Geraghty-contraction is a contractive mapping and hence continuous, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M120">View MathML</a>.

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M121">View MathML</a>.

Taking into account that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M122">View MathML</a> is a constant sequence with value <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M31">View MathML</a>, we deduce

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M124">View MathML</a>

This means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M63">View MathML</a> is a best proximity point of T.

This proves the part of existence of our theorem.

For the uniqueness, suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M126">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M127">View MathML</a> are two best proximity points of T with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M128">View MathML</a>.

This means that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M129">View MathML</a>

Using the P-property, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M130">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M131">View MathML</a>

which is a contradiction.

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M132">View MathML</a>.

This finishes the proof. □

4 Examples

In order to illustrate our results, we present some examples.

Example 4.1 Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M133">View MathML</a> with the usual metric.

Let A and B be the subsets of X defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M134">View MathML</a>

Obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M135">View MathML</a> and A, B are nonempty closed subsets of X.

Moreover, it is easily seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M137">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a> be the mapping defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M139">View MathML</a>

In the sequel, we check that T is a Geraghty-contraction.

In fact, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M140">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M141">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M142">View MathML</a>

(4)

Now, we prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M143">View MathML</a>

(5)

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M144">View MathML</a> (the same reasoning works for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M145">View MathML</a>).

Then, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M146">View MathML</a> is strictly increasing in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M148">View MathML</a>

This proves (5).

Taking into account (4) and (5), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M149">View MathML</a>

(6)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M146">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M151">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M152">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M153">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M154">View MathML</a>.

Obviously, when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M155">View MathML</a>, the inequality (6) is satisfied.

It is easily seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M156">View MathML</a> by using elemental calculus.

Therefore, T is a Geraghty-contraction.

Notice that the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> satisfies the P-property.

Indeed, if

then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M159">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M160">View MathML</a> and consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M161">View MathML</a>

By Theorem 3.1, T has a unique best proximity point.

Obviously, this point is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M162">View MathML</a>.

The condition A and B are nonempty closed subsets of the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a> is not a necessary condition for the existence of a unique best proximity point for a Geraghty-contraction <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a> as it is proved with the following example.

Example 4.2 Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M133">View MathML</a> with the usual metric and the subsets of X given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M166">View MathML</a>

Obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M135">View MathML</a> and B is not a closed subset of X.

Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M168">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M137">View MathML</a>.

We consider the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M2">View MathML</a> defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M171">View MathML</a>

Now, we check that T is a Geraghty-contraction.

In fact, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M140">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M141">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M174">View MathML</a>

(7)

In what follows, we need to prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M175">View MathML</a>

(8)

In fact, suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M144">View MathML</a> (the same argument works for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M145">View MathML</a>).

Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M178">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M179">View MathML</a> (notice that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M180">View MathML</a> since the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M181">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M151">View MathML</a> is strictly increasing).

Taking into account that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M183">View MathML</a>

and since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M184">View MathML</a>, we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M185">View MathML</a>, and consequently, from the last inequality it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M186">View MathML</a>

Applying ϕ (notice that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M181">View MathML</a>) to the last inequality and taking into account the increasing character of ϕ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M188">View MathML</a>

or equivalently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M189">View MathML</a>

and this proves (8).

By (7) and (8), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M190">View MathML</a>

(9)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M191">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M153">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M154">View MathML</a>. Obviously, the inequality (9) is satisfied for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M194">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M155">View MathML</a>.

Now, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10">View MathML</a>.

In fact, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M197">View MathML</a>, then the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M198">View MathML</a> is a bounded sequence since in the contrary case, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M199">View MathML</a> and thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M200">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M201">View MathML</a>. This means that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M202">View MathML</a> such that, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M98">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M204">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M205">View MathML</a>. The bounded character of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M198">View MathML</a> gives us the existence of a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M207">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M208">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M207">View MathML</a> convergent. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M210">View MathML</a>. From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M211">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M212">View MathML</a> and, as the unique solution of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M213">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M214">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M215">View MathML</a>.

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M216">View MathML</a> and this contradicts the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M217">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M98">View MathML</a>.

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M219">View MathML</a> and this proves that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M10">View MathML</a>.

A similar argument to the one used in Example 4.1 proves that the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M47">View MathML</a> has the P-property.

On the other hand, the point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M162">View MathML</a> is a best proximity point for T since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M223">View MathML</a>

Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M224">View MathML</a> is the unique best proximity point for T.

Indeed, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M225">View MathML</a> is a best proximity point for T, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M226">View MathML</a>

and this gives us

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M227">View MathML</a>

Taking into account that the unique solution of this equation is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M228">View MathML</a>, we have proved that T has a unique best proximity point which is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M224">View MathML</a>.

Notice that in this case B is not closed.

Since for any nonempty subset A of X, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M55">View MathML</a> satisfies the P-property, we have the following corollary.

Corollary 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M8">View MathML</a>be a complete metric space andAbe a nonempty closed subset of X. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M232">View MathML</a>be a Geraghty-contraction. ThenThas a unique fixed point.

Proof Using Theorem 3.1 when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M233">View MathML</a>, the desired result follows. □

Notice that when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/231/mathml/M234">View MathML</a>, Corollary 4.1 is Theorem 1.1 due to Gerahty [1].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The three authors have contributed equally in this paper. They read and approval the final manuscript.

Acknowledgements

This research was partially supported by ‘Universidad de Las Palmas de Gran Canaria’, Project ULPGC 2010-006.

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