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Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces

Chi-Ming Chen1 and Erdal Karapınar2*

Author Affiliations

1 Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu, Taiwan

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

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Journal of Inequalities and Applications 2013, 2013:410  doi:10.1186/1029-242X-2013-410

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/410


Received:4 March 2013
Accepted:8 August 2013
Published:23 August 2013

© 2013 Chen and Karapınar; licensee Springer

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

Abstract

The purpose of this paper is to study fixed point theorems for a multi-valued mapping satisfying the α-Meir-Keeler contraction with respect to the partial Hausdorff metric ℋ in complete partial metric spaces. Our result generalizes and extends some results in the literature.

MSC: 47H10, 54C60, 54H25, 55M20.

Keywords:
fixed point; α-Meir-Keeler contraction; partial metric space; partial Hausdorff metric space

1 Introduction and preliminaries

Fixed point theory is one of the most crucial tools in nonlinear functional analysis, and it has application in distinct branches of mathematics and in various sciences, such as economics, engineering and computer science. The most impressed fixed point result was given by Banach [1] in 1922. He concluded that each contraction has a unique fixed point in the complete metric space. Since then, this pioneer work has been generalized and extended in different abstract spaces. One of the interesting generalization of Banach fixed point theorem was given by Matthews [2] in 1994. In this remarkable paper, the author introduced the following notion of partial metric spaces and proved the Banach fixed point theorem in the context of complete partial metric space.

For the sake of completeness, we recall basic definitions and fundamental results from the literature.

Throughout this paper, by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M1">View MathML</a>, we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers.

Definition 1[2]

A partial metric on a nonempty set X is a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M2">View MathML</a> such that for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M3">View MathML</a>

(p1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M4">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M5">View MathML</a>;

(p2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M6">View MathML</a>;

(p3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M7">View MathML</a>;

(p4) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M8">View MathML</a>.

A partial metric space is a pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> such that X is a nonempty set, and p is a partial metric on X.

Remark 1 It is clear that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M10">View MathML</a>, then from (p1) and (p2), we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M4">View MathML</a>. But if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M4">View MathML</a>, the expression <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M13">View MathML</a> may not be 0.

Each partial metric p on X generates a <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M14">View MathML</a> topology <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M15">View MathML</a> on X, which has as a base the family of open p-balls <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M16">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M17">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M19">View MathML</a>. If p is a partial metric on X, then the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M20">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M21">View MathML</a>

is a metric on X.

We recall some definitions of a partial metric space, as follows.

Definition 2[2]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> be a partial metric space. Then

(1) a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> in a partial metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> converges to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M26">View MathML</a>;

(2) a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> in a partial metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> is called a Cauchy sequence if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M29">View MathML</a> exists (and is finite);

(3) a partial metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> is said to be complete if every Cauchy sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> in X converges, with respect to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M32">View MathML</a>, to a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M34">View MathML</a>;

(4) a subset A of a partial metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> is closed if whenever <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> is a sequence in A such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> converges to some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M39">View MathML</a>.

Remark 2 The limit in a partial metric space is not unique.

Lemma 1[2,3]

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a>is a Cauchy sequence in a partial metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>if and only if it is a Cauchy sequence in the metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42">View MathML</a>;

(2) a partial metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>is complete if and only if the metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42">View MathML</a>is complete. Furthermore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M45">View MathML</a>if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M46">View MathML</a>.

Recently, fixed point theory has developed rapidly on partial metric spaces, see, e.g., [3-12] and the reference therein. Very recently, Haghi et al.[13] proved that some fixed point results in partial metric space results are equivalent to the results in the context of a usual metric space. On the other hand, this case is not valid for our main results, that is, the recent result of Haghi et al.[13] is not applicable to the main theorems.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M47">View MathML</a> be a metric space, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M48">View MathML</a> denote the collection of all nonempty, closed and bounded subsets of X. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M49">View MathML</a>, we define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M50">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M51">View MathML</a>, and it is well known that ℋ is called the Hausdorff metric induced the metric d. A multi-valued mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M52">View MathML</a> is called a contraction if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M53">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M54">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M55">View MathML</a>. The study of fixed points for multi-valued contractions using the Hausdorff metric was introduced in Nadler [14].

Theorem 1[14]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M47">View MathML</a>be a complete metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M52">View MathML</a>be a multi-valued contraction. Then there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M59">View MathML</a>.

Very recently, Aydi et al.[15] established the notion of partial Hausdorff metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60">View MathML</a> induced by the partial metric p. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> be a partial metric space, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M62">View MathML</a> be the collection of all nonempty, closed and bounded subset of the partial metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>. Note that closedness is taken from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M64">View MathML</a>, and boundedness is given as follows: A is a bounded subset in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> if there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M66">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M67">View MathML</a> such that for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M68">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M69">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M70">View MathML</a>. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M18">View MathML</a>, they define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M73">View MathML</a>

It is immediate to get that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M74">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M75">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M76">View MathML</a>.

Remark 3[15]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> be a partial metric space, and let A be a nonempty subset of X. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M78">View MathML</a>

Aydi et al.[15] also introduced the following properties of mappings <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M79">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M80">View MathML</a>.

Proposition 1[15]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>be a partial metric space. For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71">View MathML</a>, the following properties hold:

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M83">View MathML</a>;

(2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M84">View MathML</a>;

(3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M85">View MathML</a>implies that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M86">View MathML</a>;

(4) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M87">View MathML</a>.

Proposition 2[15]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>be a partial metric space. For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71">View MathML</a>, the following properties hold:

(1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M90">View MathML</a>;

(2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M91">View MathML</a>;

(3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M92">View MathML</a>;

(4) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M93">View MathML</a>implies that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M94">View MathML</a>.

Aydi et al.[15] proved the following important result.

Lemma 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>be a partial metric space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M71">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M97">View MathML</a>. For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M68">View MathML</a>, there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M99">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M100">View MathML</a>

In this study, we also recall the Meir-Keeler-type contraction [16] and α-admissible [17]. In 1969, Meir and Keeler [16] introduced the following notion of Meir-Keeler-type contraction in a metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M47">View MathML</a>.

Definition 3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> be a metric space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M103">View MathML</a>. Then f is called a Meir-Keeler-type contraction whenever for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M104">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M105">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M106">View MathML</a>

The following definition was introduced in [17].

Definition 4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M103">View MathML</a> be a self-mapping of a set X and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M108">View MathML</a>. Then f is called an α-admissible if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M109">View MathML</a>

2 Main results

We first introduce the following notions of a strictly α-admissible and and an α-Meir-Keeler contraction with respect to the partial Hausdorff metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60">View MathML</a>.

Definition 5 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> be a partial metric space, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M113">View MathML</a>. We say that T is strictly α-admissible if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M114">View MathML</a>

Definition 6 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> be a partial metric space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M116">View MathML</a>. We call <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a> an α-Meir-Keeler contraction with respect to the partial Hausdorff metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60">View MathML</a> if the following conditions hold:

(c1) T is strictly α-admissible;

(c2) for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M104">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M19">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M121">View MathML</a>

Remark 4 Note that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a> is a α-Meir-Keeler contraction with respect to the partial Hausdorff metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M123">View MathML</a>, then we have that for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M54">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M125">View MathML</a>

Further, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M10">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M127">View MathML</a>. On the other hand, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M10">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M129">View MathML</a>.

We now state and prove our main result.

Theorem 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>be a complete partial metric space. Suppose that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a>is anα-Meir-Keeler contraction with respect to the partial Hausdorff metricand that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M66">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M133">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M134">View MathML</a>. ThenThas a fixed point inX (that is, there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M135">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M136">View MathML</a>).

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M137">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a> is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60">View MathML</a>, by Remark 4, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M140">View MathML</a>

(1)

Put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M141">View MathML</a>, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M142">View MathML</a>. From Lemma 2 with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M143">View MathML</a>, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M144">View MathML</a>

(2)

Using (1) and (2), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M145">View MathML</a>

(3)

So, we can obtain a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M146">View MathML</a> recursively as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M147">View MathML</a>

Since T is strictly α-admissible, we deduce that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M148">View MathML</a>. Continuing this process, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M149">View MathML</a>

(4)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a> is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M60">View MathML</a>, by Remark 4, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M152">View MathML</a>

(5)

From Lemma 2 with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M153">View MathML</a>, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M154">View MathML</a>

(6)

Using (5) and (6), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M155">View MathML</a>

(7)

Now, from (7) and by the mathematical induction, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M156">View MathML</a>

(8)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M157">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M158">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M159">View MathML</a>

Put

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M160">View MathML</a>

(9)

Using (8) and (9), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M161">View MathML</a>

(10)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M162">View MathML</a> in (10). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M163">View MathML</a>

(11)

By the property (p2) of a partial metric and using (11), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M164">View MathML</a>

(12)

Using (10) and the property (p4) of a partial metric, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M165">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M166">View MathML</a>

(13)

Using (12) and (13), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M167">View MathML</a>

By the definition of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M168">View MathML</a>, we get that for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M165">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M170">View MathML</a>

(14)

This yields that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a> is complete, from Lemma 1, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M42">View MathML</a> is a complete metric space. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M23">View MathML</a> converges to some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M135">View MathML</a> with respect to the metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M168">View MathML</a>, and we also have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M178">View MathML</a>

(15)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a> is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric ℋ, by Remark 4, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M180">View MathML</a>

By the definition of the mapping α, we have that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M181">View MathML</a>. Using (15), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M182">View MathML</a>

(16)

Now <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M183">View MathML</a> gives that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M184">View MathML</a>

Using (16), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M185">View MathML</a>

(17)

By the property (p4) of a partial metric, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M186">View MathML</a>

Taking limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M162">View MathML</a>, and using (12), (15) and (17), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M188">View MathML</a>

Therefore, from (15), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M189">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M190">View MathML</a>

which implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M136">View MathML</a> by Remark 3. □

The following theorem, the main result of [15], is a consequence of Theorem 2 by taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M192">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M193">View MathML</a>.

Theorem 3[15]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M9">View MathML</a>be a complete partial metric space. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M112">View MathML</a>is a multi-valued mapping such that for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M54">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M197">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/410/mathml/M193">View MathML</a>. ThenThas a fixed point.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

References

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

  2. Matthews, SG: Partial metric topology. Proc. 8th Summer of Conference on General Topology and Applications. 183–197 (1994)

  3. Oltra, S, Valero, O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste. 36, 17–26 (2004)

  4. Abdeljawad, T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model.. 54, 2923–2927 (2011). Publisher Full Text OpenURL

  5. Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 40

  6. Altun, I, Erduran, A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl.. 2011, (2011) Article ID 508730

  7. Aydi, H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud.. 4(2), 1–12 (2011)

  8. Aydi, H, Vetro, C, Sintunavarat, W, Kumam, P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 124

  9. Chi, KP, Karapinar, E, Thanh, TD: A generalized contraction principle in partial metric spaces. Math. Comput. Model.. 55, 1673–1681 (2012). Publisher Full Text OpenURL

  10. Karapinar, E: Weak ϕ-contraction on partial metric spaces. J. Comput. Anal. Appl.. 14(2), 206–210 (2012)

  11. Karapinar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl.. 2011, (2011) Article ID 4

  12. Karapinar, E, Erhan, IM, Yıldız, UA: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci.. 6, 239–244 (2012)

  13. Haghi, RH, Rezapour, S, Shahzad, N: Be careful on partial metric fixed point results. Topol. Appl.. 160(3), 450–454 (2013). Publisher Full Text OpenURL

  14. Nadler, SB: Multi-valued contraction mappings. Pac. J. Math.. 30, 475–488 (1969). Publisher Full Text OpenURL

  15. Aydi, H, Abbas, M, Vetro, C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl.. 159, 3234–3242 (2012). Publisher Full Text OpenURL

  16. Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl.. 28, 326–329 (1969). Publisher Full Text OpenURL

  17. Samet, B, Vetro, C, Vetro, P: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal.. 75, 2154–2165 (2012). Publisher Full Text OpenURL