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A generalization for the best proximity point of Geraghty-contractions

Nurcan Bilgili1, Erdal Karapınar2* and Kishin Sadarangani3

Author Affiliations

1 Department of Mathematics, Faculty of Science and Arts, Amasya University, Amasya, 06500, Turkey

2 Department of Mathematics, Atilim University, İncek, Ankara, 06836, Turkey

3 Department of Mathematics, University of Las Palmas de Gran Canaria, Campus Universitario de Tafira, Las Palmas de Gran Canaria, 35017, Spain

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Journal of Inequalities and Applications 2013, 2013:286  doi:10.1186/1029-242X-2013-286


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/286


Received:6 February 2013
Accepted:22 May 2013
Published:6 June 2013

© 2013 Bilgili 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

In this paper, we introduce the notion of Geraghty-contractions and consider the related best proximity point in the context of a metric space. We state an example to illustrate our result.

MSC: 47H10, 54H25, 46J10, 46J15.

1 Introduction and preliminaries

Fixed point theory and best proximity theory are very important tools in nonlinear functional analysis. These related research areas have wide application potential in various branches of mathematics and different disciplines such as economics, engineering. One of the most impressive results in this direction, known as the Banach contraction mapping principle, was given by Banach: Every contraction on a complete metric space has a unique fixed point. This celebrated result has been generalized in several ways in various abstract spaces. In particular, one of the interesting generalizations of the Banach contraction mapping principle was given by Geraghty [1].

Theorem 1 (Geraghty [1])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M1">View MathML</a>be a complete metric space and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M2">View MathML</a>be an operator. Suppose that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M3">View MathML</a>satisfying the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M4">View MathML</a>

IfTsatisfies the following inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M5">View MathML</a>

(1.1)

thenThas a unique fixed point.

It is clear that some mapping on a complete metric space has no fixed point, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M6">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M7">View MathML</a>. In this case, it is natural to ask the existence and uniqueness of the smallest value of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M8">View MathML</a>. This is the main motivation of a best proximity point. This research subject has attracted attention of a number of authors; see, e.g., [1-19].

First we recall fundamental definitions and basic results in this direction.

Let A and B be nonempty subsets of a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M1">View MathML</a>. A mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M10">View MathML</a> is called a k-contraction if there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M11">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M12">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M13">View MathML</a>. Notice that the k-contraction coincides with the Banach contraction mapping principle if one takes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M14">View MathML</a>, where A is a complete subset of X. A point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15">View MathML</a> is called the best proximity of T if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M17">View MathML</a>.

Let A and B be two nonempty subsets of a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a>. We denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M20">View MathML</a> the following sets:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M21">View MathML</a>

(1.2)

We denote by F the set of all functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M22">View MathML</a> satisfying the following property:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M23">View MathML</a>

(1.3)

Definition 2 (See [2])

Let A, B be two nonempty subsets of a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a>. A mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25">View MathML</a> is said to be a Geraghty-contraction if there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M26">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M27">View MathML</a>

(1.4)

Very recently Raj [10,11] introduced the notion of P-property as follows.

Definition 3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> be a pair of nonempty subsets of a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M30">View MathML</a>. Then the pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> is said to have the P-property if and only if for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M32">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M34">View MathML</a>

(1.5)

Example 4 (See, e.g., [11])

Let A be a nonempty subset of a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a>. It is evident that the pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M36">View MathML</a> has the P-property. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> be any pair of nonempty, closed, convex subsets of a real Hilbert space H. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> has the P-property.

Theorem 5 (See [2])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a>be a pair of nonempty closed subsets of a complete metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19">View MathML</a>is nonempty. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25">View MathML</a>be a continuous, Geraghty-contraction satisfying<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M43">View MathML</a>. Suppose that the pair<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a>has theP-property. Then there exists a unique<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15">View MathML</a>inAsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16">View MathML</a>.

The subject of this paper is to generalize, improve and extend the results of Caballero, Harjani and Sadarangani [2]. For this purpose, we first define the notion of generalized Geraghty-contraction as follows.

Definition 6 Let A, B be two nonempty subsets of a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a>. A mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25">View MathML</a> is said to be a generalized Geraghty-contraction if there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M26">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M50">View MathML</a>

(1.6)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M51">View MathML</a>.

Remark 7 Notice that since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M3">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M53">View MathML</a>

(1.7)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M51">View MathML</a>.

2 Main results

We start this section with our main result.

Theorem 8Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a>be a complete metric space. Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a>is a pair of nonempty closed subsets ofXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19">View MathML</a>is nonempty. Suppose also that the pair<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a>has theP-property. If a non-self-mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M25">View MathML</a>is a generalized Geraghty-contraction satisfying<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M43">View MathML</a>, then there exists a unique best proximity point, that is, there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15">View MathML</a>inAsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16">View MathML</a>.

Proof Let us fix an element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M63">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M65">View MathML</a>, we can find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M66">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M67">View MathML</a>. Further, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M68">View MathML</a>, there is an element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M69">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M71">View MathML</a>. Recursively, we obtain a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M72">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M19">View MathML</a> with the following property:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M74">View MathML</a>

(2.1)

Due to the fact that the pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> has the P-property, we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M76">View MathML</a>

(2.2)

From (2.1), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M77">View MathML</a>

On the other hand, by (2.1) and (2.2) we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M78">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M79">View MathML</a>

(2.3)

If there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M80">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M81">View MathML</a>, then the proof is completed. In fact, due to (2.2), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M82">View MathML</a>

(2.4)

which yields that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M83">View MathML</a>. Hence, equation (2.1) implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M84">View MathML</a>

(2.5)

For the rest of the proof, we suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M85">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M86">View MathML</a>. Owing to the fact T is a generalized Geraghty-contraction, we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M87">View MathML</a>

(2.6)

Then, by (2.3) and (2.6), we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M88">View MathML</a>

Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M89">View MathML</a>. Then we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M90">View MathML</a>

a contradiction. As a result, we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M91">View MathML</a> and hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M92">View MathML</a>

(2.7)

By (2.6), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M93">View MathML</a>

(2.8)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M86">View MathML</a>. Consequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M95">View MathML</a> is a nonincreasing sequence and bounded below. Thus, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M96">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M97">View MathML</a>. We shall show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M98">View MathML</a>. Suppose, on the contrary, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M99">View MathML</a>. Then, by (2.8), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M100">View MathML</a>

for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M101">View MathML</a>. In what follows,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M102">View MathML</a>

On the other hand, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M103">View MathML</a>, we conclude <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M104">View MathML</a>, that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M105">View MathML</a>

(2.9)

Since, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M106">View MathML</a> holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M86">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> satisfies the P-property, then, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M109">View MathML</a>, we can write, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M110">View MathML</a>. We also have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M111">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M112">View MathML</a>. It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M113">View MathML</a>

Taking (2.9) into consideration, we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M114">View MathML</a>

(2.10)

We shall show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M72">View MathML</a> is a Cauchy sequence. Suppose, on the contrary, that we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M116">View MathML</a>

(2.11)

Due to the triangular inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M117">View MathML</a>

(2.12)

Regarding (1.6) and (2.12), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M118">View MathML</a>

(2.13)

Taking (2.10), (2.13) and (2.9) into account, we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M119">View MathML</a>

Owing to (2.11), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M120">View MathML</a>

which implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M121">View MathML</a>. By the property of β, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M122">View MathML</a>. Consequently, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M123">View MathML</a>, a contradiction. Hence, we conclude that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M72">View MathML</a> is Cauchy. Since A is a closed subset of the complete metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M126">View MathML</a>, and we can find <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M127">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M128">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M129">View MathML</a>. We assert that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M16">View MathML</a>. Suppose, on the contrary, that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M131">View MathML</a>. First, we obtain the following inequalities:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M132">View MathML</a>

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M129">View MathML</a> in the inequalities above, we conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M134">View MathML</a>

On the other hand, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M135">View MathML</a>

Taking limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M129">View MathML</a> in the inequality above, we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M137">View MathML</a>

So, we deduce that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M138">View MathML</a>. As a consequence, we derive

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M139">View MathML</a>

and hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M140">View MathML</a>

(2.14)

Combining (1.6) and (2.14), we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M141">View MathML</a>

(2.15)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M142">View MathML</a> together with (2.15), we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M143">View MathML</a>. Hence, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M144">View MathML</a>

which yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M145">View MathML</a>

As a result, we deduce that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M146">View MathML</a>, a contradiction. So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M147">View MathML</a> and hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M148">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15">View MathML</a> is a best proximity point of T. Hence, we conclude that T has a best proximity point.

We claim that the best proximity point of T is unique.

Suppose, on the contrary, that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M151">View MathML</a> are two distinct best proximity points of T. Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M152">View MathML</a>

(2.16)

By using the P-property, we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M153">View MathML</a>

(2.17)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M154">View MathML</a>

Due to the fact that T is a generalized Geraghty-contraction, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M155">View MathML</a>

a contradiction. This completes the proof. □

Remark 9 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M18">View MathML</a> be a metric space and A be any nonempty subset of X. It is evident that a pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M36">View MathML</a> satisfies the P-property.

Corollary 10Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M1">View MathML</a>is a complete metric space andAis a nonempty closed subset ofX. If a self-mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M159">View MathML</a>is a generalized Geraghty-contraction, then it has a unique fixed point.

Proof Taking Remark 9 into consideration, we conclude the desired result by applying Theorem 8 with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M160">View MathML</a>. □

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

Example 11 Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M161">View MathML</a> with the metric

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M162">View MathML</a>

and consider the closed subsets

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M163">View MathML</a>

and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M164">View MathML</a> be the mapping defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M165">View MathML</a>

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M166">View MathML</a>, the pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M28">View MathML</a> has the P-property.

Notice that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M168">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M169">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M43">View MathML</a>.

Moreover,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M171">View MathML</a>

and, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M172">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M173">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M174">View MathML</a> is defined as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M175">View MathML</a>.

Notice that β is nondecreasing since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M176">View MathML</a>.

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M177">View MathML</a>

and it is easily seen that the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M178">View MathML</a> belongs to F.

Therefore, since the assumptions of Theorem 8 are satisfied, by Theorem 8 there exists a unique <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M179">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M180">View MathML</a>

The point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M179">View MathML</a> is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/286/mathml/M182">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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