Open Access Research

Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric

M Abbas1 and Basit Ali12*

Author Affiliations

1 Department of Mathematics, Lahore University of Management Sciences, Lahore, 54792, Pakistan

2 Department of Mathematics, University of Management and Technology, C-II, Johar Town, Lahore, Pakistan

For all author emails, please log on.

Fixed Point Theory and Applications 2013, 2013:21  doi:10.1186/1687-1812-2013-21


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/21


Received:9 October 2012
Accepted:12 January 2013
Published:31 January 2013

© 2013 Abbas and Ali; 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

Coincidence point theorems for hybrid pairs of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space using a partial Hausdorff metric have been proved. As an application of our main result, the existence and uniqueness of common and bounded solutions of functional equations arising in dynamic programming are discussed.

MSC: 47H10, 54H25, 54E50.

Keywords:
coincidence point; orbitally complete; common fixed point; partial metric space

1 Introduction and preliminaries

Fixed point theory plays a fundamental role in solving functional equations [1] arising in several areas of mathematics and other related disciplines as well. The Banach contraction principle is a key principle that made a remarkable progress towards the development of metric fixed point theory. Markin [2] and Nadler [3] proved a multi-valued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [4-9]) were obtained using different contractive conditions. The study of hybrid type contractive conditions involving single-valued and multi-valued mappings is a valuable addition to the metric fixed point theory and its applications (for details, see [10-14]). Among several generalizations of the Banach contraction principle, Suzuki’s work [[15], Theorem 2.1] led to a number of results (for details, see [13,16-21]).

On the other hand, Matthews [22] introduced the concept of a partial metric space as a part of the study of denotational semantics of dataflow networks. He obtained a modified version of the Banach contraction principle, more suitable in this context (see also [23,24]). Since then, results obtained in the framework of partial metric spaces have been used to constitute a suitable framework to model the problems related to the theory of computation (see [22,25-28]). Recently, Aydi et al.[29] initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem [3] in partial metric spaces.

The aim of this paper is to obtain some coincidence point theorems for a hybrid pair of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space. Our results extend, unify and generalize several known results in the existing literature (see [13,15,21,30]). As an application, we obtain the existence and uniqueness of a common and bounded solution for Suzuki-Zamfirescu class of functional equations under contractive conditions weaker than those given in [1,31-34].

Throughout this work, a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M1">View MathML</a> is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M2">View MathML</a>

(1.1)

In the sequel, the letters ℝ, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M3">View MathML</a> and ℕ will denote the set of all real numbers, the set of all non-negative real numbers and the set of all positive integers, respectively. Consistent with [22,29,35,36], the following definitions and results will be needed in the sequel.

Definition 1.1[22]

Let X be any non-empty set. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M4">View MathML</a> is said to be a partial metric if and only if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M5">View MathML</a> the following conditions are satisfied:

(P1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M6">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M7">View MathML</a>;

(P2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>;

(P3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>;

(P4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M10">View MathML</a>.

The pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> is called a partial metric space. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M12">View MathML</a>, then (P1) and (P2) imply that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M7">View MathML</a>. But the converse does not hold in general. A classical example of a partial metric space is the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M14">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M4">View MathML</a> is defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M16">View MathML</a> (see also [37]).

Example 1.2[22]

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M17">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M18">View MathML</a>

defines a partial metric p on X.

For more interesting examples, we refer to [23,27,28,35,38,39]. Each partial metric p on X generates a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M19">View MathML</a> topology <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20">View MathML</a> on X which has as a base the family of open balls (p-balls) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M21">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M22">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M24">View MathML</a>. A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25">View MathML</a> in a partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> is called convergent to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M29">View MathML</a> (for details, see [22]). If p is a partial metric on X, then the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M30">View MathML</a> given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M31">View MathML</a> defines a metric on X. Furthermore, a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25">View MathML</a> converges in a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a> to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> if and only if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M35">View MathML</a>

(1.2)

Definition 1.3[22]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> be a partial metric space, then

(a) A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25">View MathML</a> in X is called Cauchy if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M38">View MathML</a> exists and is finite.

(b) A partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> is said to be complete if every Cauchy sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25">View MathML</a> in X converges with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20">View MathML</a> to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M43">View MathML</a>.

Lemma A[22,35]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space, then

(c) A sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M25">View MathML</a>inXis Cauchy in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>if and only if it is Cauchy in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>.

(d) A partial metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>is complete if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>is complete.

Consistent with [29], let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M50">View MathML</a> be the family of all non-empty, closed and bounded subsets of the partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>, induced by the partial metric p. Note that closedness is taken from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M52">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20">View MathML</a> is the topology induced by p) and boundedness is given as follows: A is a bounded subset in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> if there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M55">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M56">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M57">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M58">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M59">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M60">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a>, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M62">View MathML</a> and

It can be verified that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M64">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M65">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M66">View MathML</a>.

Lemma B[35]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space andAbe a non-empty subset ofX, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M68">View MathML</a>if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M69">View MathML</a>.

Proposition 1.4[29]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space. For any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M71">View MathML</a>, we have the following:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M72">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M73">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M74">View MathML</a>implies<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M75">View MathML</a>;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M76">View MathML</a>.

Proposition 1.5[29]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space. For any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M71">View MathML</a>, we have the following:

(h1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M79">View MathML</a>;

(h2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M80">View MathML</a>;

(h3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M81">View MathML</a>;

(h4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M82">View MathML</a>implies that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M83">View MathML</a>.

The mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M84">View MathML</a> is called a partial Hausdorff metric induced by a partial metric p. Every Hausdorff metric is a partial Hausdorff metric, but the converse is not true (see Example 2.6 in [29]).

Lemma C[29]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M60">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M87">View MathML</a>, then for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M57">View MathML</a>, there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M89">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M90">View MathML</a>.

Theorem 1.6[29]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M92">View MathML</a>is a multi-valued mapping such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M93">View MathML</a>, we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M94">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M95">View MathML</a>. ThenThas a fixed point.

Definition 1.7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> be a partial metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M97">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M98">View MathML</a>. A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> is said to be (i) a fixed point off if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M100">View MathML</a>, (ii) a fixed point ofT if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M101">View MathML</a>, (iii) a coincidence point of a pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M103">View MathML</a>, (iv) a common fixed point of the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M105">View MathML</a>.

We denote the set of all fixed points of f, the set of all coincidence points of the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> and the set of all common fixed points of the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M108">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M109">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M110">View MathML</a>, respectively. Motivated by the work of [4,13], we give the following definitions in partial metric spaces.

Definition 1.8 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> be a partial metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M97">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M98">View MathML</a>. The pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> is called (i) commuting if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a>, (ii) weakly compatible if the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> commutes at their coincidence points, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M118">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M119">View MathML</a>, (iii) IT-commuting [11] at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M121">View MathML</a>.

Definition 1.9 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> be a partial metric space and Y be any non-empty set. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124">View MathML</a> be single-valued and multi-valued mappings, respectively. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M125">View MathML</a>, then the set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M126">View MathML</a>

(1.3)

is called an orbit for the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M128">View MathML</a>. A partial metric space X is called <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete if and only if every Cauchy sequence in the orbit for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M128">View MathML</a> converges with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M20">View MathML</a> to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M134">View MathML</a>.

Singh and Mishra [13] introduced Suzuki-Zamfirescu type hybrid contractive condition in complete metric spaces. In the context of partial metric spaces, the condition is given as follows.

Definition 1.10 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> be a partial metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124">View MathML</a> be single-valued and multi-valued mappings, respectively. The hybrid pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> is said to satisfy Suzuki-Zamfirescu hybrid contraction condition if there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M140">View MathML</a> implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M141">View MathML</a>

(1.4)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M143">View MathML</a>

(1.5)

Lemma DLet<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124">View MathML</a>be single-valued and multi-valued mappings, respectively. Then the partial metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete.

Proof Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M128">View MathML</a> is an arbitrary element of X. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M155">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>, then it is also Cauchy in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>. Therefore, by (1.2) we deduce that there exists y in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M158">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154">View MathML</a> converges to y in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>. Conversely, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a> be <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M164">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>, then it is also a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M167">View MathML</a>

For given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M24">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M169">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M170">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M171">View MathML</a>. Consequently, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M172">View MathML</a>

whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M171">View MathML</a>. The result follows. □

2 Coincidence points of a hybrid pair of mappings

In the following theorem, the existence of coincidence points of a hybrid pair of single-valued and multi-valued mappings that satisfy Suzuki-Zamfirescu hybrid contraction condition in partial metric spaces is established.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space andYbe any non-empty set. Assume that a pair of mappings<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124">View MathML</a>satisfies Suzuki-Zamfirescu hybrid contraction condition with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M177">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete at<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M182">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>isIT-commuting at coincidence points of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M186">View MathML</a>provided thatfzis a fixed point offfor some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M187">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M188">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M190">View MathML</a>. By the given assumption, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M191">View MathML</a>. So, there exists a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M192">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M193">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M87">View MathML</a>, so by Lemma C, there exists a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M195">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M196">View MathML</a>

Using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M197">View MathML</a>, we obtain a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M198">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M199">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M200">View MathML</a>

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M201">View MathML</a>

we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M202">View MathML</a>

If

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M203">View MathML</a>

then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M204">View MathML</a>

If

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M205">View MathML</a>

then we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M206">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M207">View MathML</a>, we choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M208">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M209">View MathML</a>. Using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M210">View MathML</a>, we obtain a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M211">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M212">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M213">View MathML</a>

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M214">View MathML</a>

so we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M215">View MathML</a>

Following the arguments similar to those given above, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M216">View MathML</a>

which further implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M217">View MathML</a>

Continuing this process, we obtain a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M218">View MathML</a> such that for any integer <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M219">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M220">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M221">View MathML</a>

for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M222">View MathML</a>. This shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M223">View MathML</a>. Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M224">View MathML</a>

so we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M225">View MathML</a>

Now, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M226">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M227">View MathML</a>

It follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M229">View MathML</a>. By Lemma A, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M154">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M231">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M232">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181">View MathML</a>, so again by Lemma D, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M235">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181">View MathML</a>. Hence, there exists an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M238">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M239">View MathML</a>. This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M240">View MathML</a>

(2.1)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M241">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M242">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M243">View MathML</a>. Now,

give

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M245">View MathML</a>

Similarly, we can show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M246">View MathML</a>

Now, we will claim that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M247">View MathML</a>

(2.2)

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M248">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M249">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M250">View MathML</a>. This gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M251">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M103">View MathML</a> and we are done. Now from (2.1), there exists a positive integer <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M253">View MathML</a> such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M254">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M255">View MathML</a>

So, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M256">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M257">View MathML</a>

Hence, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M256">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M259">View MathML</a>

This implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M260">View MathML</a>

On taking limit as n tends to ∞, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M261">View MathML</a>

If

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M262">View MathML</a>

then we are done. If

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M263">View MathML</a>

then we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M264">View MathML</a>

and hence (2.2) holds. Next, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M265">View MathML</a>

(2.3)

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M266">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M248">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M249">View MathML</a>, and the claim follows from (2.2). Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M269">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M270">View MathML</a>. As f is a non-constant single-valued mapping, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M271">View MathML</a>

This implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M272">View MathML</a>

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M273">View MathML</a>

Hence, (2.3) holds for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M266">View MathML</a>. Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M275">View MathML</a>

On taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M276">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M277">View MathML</a>

We obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M278">View MathML</a>, which further implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M279">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M280">View MathML</a>. Further if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M282">View MathML</a>, then due to IT-commutativity of the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M284">View MathML</a>. This shows that fz is a common fixed point of the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>. □

Corollary ALet<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space andYbe any non-empty set. Assume that here exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M139">View MathML</a>such that the mappings<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M123">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124">View MathML</a>satisfy

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M290">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142">View MathML</a>, with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M177">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete at<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M297">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>isIT-commuting at coincidence points of the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M186">View MathML</a>provided thatfzis a fixed point offfor some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M187">View MathML</a>.

Example 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M303">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M304">View MathML</a>. Define a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M4">View MathML</a> as follows:

Then p is a partial metric on X. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M307">View MathML</a> be as given in Theorem 2.1 and the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M124">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M309">View MathML</a> be given as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M310">View MathML</a>

Note that

If we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M312">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M313">View MathML</a>, then for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M315">View MathML</a>

holds. If we consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M316">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M317">View MathML</a>. Then, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M318">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M319">View MathML</a>, hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M320">View MathML</a> is satisfied trivially. Now consider

Hence, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M323">View MathML</a>

implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M324">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M325">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M326">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M327">View MathML</a>, there exists a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M328">View MathML</a> in Y such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M329">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M330">View MathML</a>, we obtain a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M331">View MathML</a> in Y such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M332">View MathML</a>. Continuing this way, we construct an orbit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M333">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M325">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M338">View MathML</a>. So, all the conditions of Corollary A are satisfied. Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M339">View MathML</a>.

On the other hand, the metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M340">View MathML</a> induced by the partial metric p is given by

Now, we show that Corollary A is not applicable (in the case of a metric induced by a partial metric p) in this case. Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M342">View MathML</a>

is satisfied for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M139">View MathML</a>, x and y in X, so it must imply <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M344">View MathML</a>. But

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M345">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M346">View MathML</a>

Hence, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M347">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M348">View MathML</a>

Corollary BLet<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be a partial metric space, Ybe any non-empty set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M350">View MathML</a>be such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M177">View MathML</a>. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M178">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M179">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>-orbitally complete at<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M181">View MathML</a>. Assume further that there exists an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M347">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M357">View MathML</a>

implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M358">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M142">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M182">View MathML</a>. Further, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183">View MathML</a>and the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M362">View MathML</a>is commuting atxwhere<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M119">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M110">View MathML</a>is a singleton.

Proof It follows from Theorem 2.1, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M182">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M366">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M367">View MathML</a>. Further, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M183">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M102">View MathML</a> is commuting at u, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M370">View MathML</a>. Now,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M371">View MathML</a>

implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M372">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M373">View MathML</a>, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M374">View MathML</a>, which further implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M375">View MathML</a>. Hence, fu is a common fixed point of f and T.

For uniqueness, assume there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M376">View MathML</a>, such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M377">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M378">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M379">View MathML</a>

which implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M380">View MathML</a>

We obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M381">View MathML</a>, which further implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M382">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M383">View MathML</a>. □

3 An application

In this section, we assume that U and V are Banach spaces, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M384">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M385">View MathML</a>. Suppose that

Considering W and D as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:

(3.1)

(3.2)

Then equations (3.1) and (3.2) can be reformulated as

(3.3)

(3.4)

For more on the multistage process involving such functional equations, we refer to [23,31-34]. Now, we study the existence and uniqueness of a common and bounded solution of the functional equations (3.3)-(3.4) arising in dynamic programming in the setup of partial metric spaces.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391">View MathML</a> denote the set of all bounded real-valued functions on W. For an arbitrary <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392">View MathML</a>, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M393">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M394">View MathML</a> is a Banach space endowed with the metric d defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M395">View MathML</a>. Now, consider

(3.5)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M397">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M398">View MathML</a> and is a partial metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M307">View MathML</a> be defined as in Section 1. Suppose that the following conditions hold:

(C1): G, F, g, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M402">View MathML</a> are bounded.

(C2): For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M398">View MathML</a>, define

(3.6)

(3.7)

Moreover, assume that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M347">View MathML</a> such that for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M409">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M397">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M411">View MathML</a>,

(3.8)

implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M413">View MathML</a>

(3.9)

where

(C3): For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M416">View MathML</a> such that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M418">View MathML</a>

(C4): There exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M392">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M420">View MathML</a>

Theorem 3.1Assume that the conditions (C1)-(C4) are satisfied. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M421">View MathML</a>is a closed convex subspace of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391">View MathML</a>, then the functional equations (3.3) and (3.4) have a unique, common and bounded solution.

Proof Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M423">View MathML</a> is a complete partial metric space. By (C1), J, K are self-maps of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M391">View MathML</a>. The condition (C3) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M425">View MathML</a>. It follows from (C4) that J and K commute at their coincidence points. Let λ be an arbitrary positive number and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M426">View MathML</a>. Choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M428">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M429">View MathML</a>

(3.10)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M430">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M431">View MathML</a>. Further, from (3.5) and (3.6), we have

(3.11)

(3.12)

Therefore, (3.8) in (C2) becomes

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M434">View MathML</a>

(3.13)

Then (3.13) together with (3.10) and (3.12) implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M435">View MathML</a>

(3.14)

Now, (3.10), (3.11) and (3.13) imply

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M436">View MathML</a>

(3.15)

From (3.14) and (3.15), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M437">View MathML</a>

(3.16)

As the above inequality is true for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M403">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M439">View MathML</a> is taken arbitrarily, so from (3.13) we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M440">View MathML</a>

(3.17)

implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M441">View MathML</a>

(3.18)

Therefore, by Corollary B, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M442">View MathML</a> has a common fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M443">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/21/mathml/M444">View MathML</a> is a unique, bounded and common solution of (3.3) and (3.4). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their useful comments that helped to improve the presentation of this paper.

References

  1. Baskaran, R, Subrahmanyam, PV: A note on the solution of a class of functional equations. Appl. Anal.. 22(3-4), 235–241 (1986). Publisher Full Text OpenURL

  2. Markin, J: A fixed point theorem for set valued mappings. Bull. Am. Math. Soc.. 74, 639–640 (1968). Publisher Full Text OpenURL

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

  4. Ćirić, L: Fixed points for generalized multi-valued contractions. Mat. Vesn.. 9, 265–272 (1972)

  5. Ćirić, L: Multi-valued nonlinear contraction mappings. Nonlinear Anal.. 71, 2716–2723 (2009). Publisher Full Text OpenURL

  6. Covitz, H, Nadler, SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math.. 8, 5–11 (1970). Publisher Full Text OpenURL

  7. Daffer, PZ, Kaneko, H: Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl.. 192, 655–666 (1995). Publisher Full Text OpenURL

  8. Reich, S: Fixed points of contractive functions. Boll. Unione Mat. Ital.. 5, 26–42 (1972)

  9. Semenov, PV: Fixed points of multi-valued contractions. Funct. Anal. Appl.. 36, 159–161 (2002). Publisher Full Text OpenURL

  10. Naimpally, SA, Singh, SL, Whitfield, JHM: Coincidence theorems for hybrid contractions. Math. Nachr.. 127, 177–180 (1986). Publisher Full Text OpenURL

  11. Singh, SL, Mishra, SN: Nonlinear hybrid contractions. J. Natur. Phys. Sci.. 5/8, 191–206 (1991/1994)

  12. Singh, SL, Mishra, SN: On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications. Indian J. Pure Appl. Math.. 33, 531–542 (2002)

  13. Singh, SL, Mishra, SN: Coincidence theorems for certain classes of hybrid contractions. Fixed Point Theory Appl.. 2010, Article ID 898109 (2010)

  14. Singh, SL, Mishra, SN: Remarks on recent fixed point theorems. Fixed Point Theory Appl. doi:10.1155/2010/452905 (2010)

  15. Suzuki, T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc.. 136, 1861–1869 (2008)

  16. Ali, B, Abbas, M: Suzuki type fixed point theorem for fuzzy mappings in ordered metric spaces. Fixed Point Theory Appl.. 2013, Article ID 9 (2013)

  17. Ćirić, L, Abbas, M, Rajović, M, Ali, B: Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metrics. Appl. Math. Comput.. 219, 1712–1723 (2012). Publisher Full Text OpenURL

  18. Dhompongsa, S, Yingtaweesittikul, H: Fixed points for multi-valued mappings and the metric completeness. Fixed Point Theory Appl.. 2009, Article ID 972395 (2009)

  19. Kikkawa, M, Suzuki, T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal.. 69, 2942–2949 (2008). Publisher Full Text OpenURL

  20. Kikkawa, M, Suzuki, T: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl.. 2008, Article ID 649749 (2008)

  21. Moţ, G, Petruşel, A: Fixed point theory for a new type of contractive multi-valued operators. Nonlinear Anal.. 70, 3371–3377 (2009). Publisher Full Text OpenURL

  22. Matthews, SG: Partial metric topology. Proc. 8th Summer Conference on General Topology Appl.. 183–197 (1994)

  23. Bari, CD, Vetro, P: Fixed points for weak φ-contractions on partial metric spaces. Int. J. Eng. Contemp. Math. Sci.. 1, 5–13 (2011)

  24. Paesano, D, Vetro, P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl.. 159, 911–920 (2012). Publisher Full Text OpenURL

  25. Ćirić, L, Samet, B, Aydi, H, Vetro, C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput.. 218, 2398–2406 (2011). Publisher Full Text OpenURL

  26. Heckmann, R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct.. 7, 71–83 (1999). Publisher Full Text OpenURL

  27. Romaguera, S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl.. 2010, Article ID 493298 (2010)

  28. Schellekens, MP: The correspondence between partial metrics and semivaluations. Theor. Comput. Sci.. 315, 135–149 (2004). Publisher Full Text OpenURL

  29. 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

  30. Zamfirescu, T: Fixed point theorems in metric spaces. Arch. Math.. 23, 292–298 (1972). Publisher Full Text OpenURL

  31. Bellman, R: Methods of Nonlinear Analysis. Vol. II, Academic Press, New York (1973)

  32. Bellman, R, Lee, ES: Functional equations in dynamic programming. Aequ. Math.. 17, 1–18 (1978). Publisher Full Text OpenURL

  33. Bhakta, PC, Mitra, S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl.. 98, 348–362 (1984). Publisher Full Text OpenURL

  34. Pathak, HK, Cho, YJ, Kang, SM, Lee, BS: Fixed point theorems for compatible mappings of type P and applications to dynamic programming. Matematiche. 50, 15–33 (1995)

  35. Altun, I, Simsek, H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud.. 1, 1–8 (2008)

  36. Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces. Topol. Appl.. 157, 2778–2785 (2010). Publisher Full Text OpenURL

  37. Abbas, M, Nazir, T: Fixed point of generalized weakly contractive mappings in ordered partial metric spaces. Fixed Point Theory Appl.. 2012, Article ID 1 (2012)

  38. Bukatin, MA, Shorina, SY: Partial metrics and co-continuous valuations. In: Nivat M (ed.) Foundations of Software Science and Computation Structure, pp. 125–139. Springer, Berlin (1998)

  39. Romaguera, S, Valero, O: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci.. 19, 541–563 (2009). Publisher Full Text OpenURL