Open Access Research

Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces

Hassen Aydi1, Calogero Vetro2, Wutiphol Sintunavarat3 and Poom Kumam3*

Author Affiliations

1 Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, H. Sousse, Tunisia

2 Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, Palermo, 90123, Italy

3 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand

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Fixed Point Theory and Applications 2012, 2012:124 doi:10.1186/1687-1812-2012-124


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/124


Received:23 February 2012
Accepted:10 July 2012
Published:25 July 2012

© 2012 Aydi et al.; licensee Springer

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

Abstract

We prove some coincidence and common fixed point results for three mappings satisfying a generalized weak contractive condition in ordered partial metric spaces. As application of the presented results, we give a unique fixed point result for a mapping satisfying a weak cyclical contractive condition. We also provide some illustrative examples.

MSC: 47H10, 54H25.

Keywords:
coincidence point; common fixed point; compatible mappings; cyclic weak <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1">View MathML</a>-contraction; partial metric space; weakly increasing mappings

1 Introduction and preliminaries

In the last decades, several authors have worked on domain theory in order to equip semantics domain with a notion of distance. In 1994, Matthews [29] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle [16] can be generalized to the partial metric context for applications in program verification. Later on, many researchers studied fixed point theorems in partial metric spaces as well as ordered partial metric spaces. For more details, see [5,6,9-15,19,20,33,34,36].

Recently, there have been so many exciting developments in the field of existence of fixed points in partially ordered sets. For instance, Ran and Reurings [38] extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. For more details on fixed point theory in partially ordered sets, we refer the reader to [1-4,7,8,17,18,24,28,30-32,39,41] and the references cited therein.

In this paper, we establish some coincidence and common fixed point results for three self-mappings on an ordered partial metric space satisfying a generalized weak contractive condition. The presented theorems extend some recent results in the literature. Moreover, as application, we give a unique fixed point theorem for a mapping satisfying a weak cyclical contractive condition.

Throughout this paper, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M2">View MathML</a> will denote the set of all non-negative real numbers. First, we start by recalling some known definitions and properties of partial metric spaces.

Definition 1.1 ([29])

A partial metric on a nonempty set X is a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M3">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M4">View MathML</a>:

(p1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M5">View MathML</a>,

(p2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M6">View MathML</a>,

(p3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M7">View MathML</a>,

(p4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M8">View MathML</a>.

A partial metric space is a pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a> such that X is a nonempty set and p is a partial metric on X.

It is clear that, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M10">View MathML</a>, then from (p1) and (p2), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M11">View MathML</a>; but if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M12">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M13">View MathML</a> may not be 0. A basic example of a partial metric space is the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M14">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M15">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M16">View MathML</a>.

Other examples of partial metric spaces which are interesting from a computational point of view may be found in [22,29].

Each partial metric p on X generates a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M17">View MathML</a> topology <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M18">View MathML</a> on X which has as a base the family of open p-balls <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M19">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M20">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M22">View MathML</a>.

If p is a partial metric on X, then the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M23">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M24">View MathML</a>

(1.1)

is a metric on X.

Definition 1.2 ([29])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> be a sequence in X. Then

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> converges to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M28">View MathML</a>. We may write this as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M29">View MathML</a>.

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> is called a Cauchy sequence if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M31">View MathML</a> exists and is finite.

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a> is said to be complete if every Cauchy sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> in X converges, with respect to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M18">View MathML</a>, to a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a>, such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M36">View MathML</a>.

Lemma 1.3 ([29])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a>be a partial metric space. Then

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a>is a Cauchy sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a>if and only if it is a Cauchy sequence in the metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M40">View MathML</a>.

(b) A partial metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a>is complete if and only if the metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M40">View MathML</a>is complete. Furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M43">View MathML</a>if and only if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M44">View MathML</a>

Definition 1.4 ([5])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a> be a partial metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M46">View MathML</a> be a given mapping. We say that T is continuous at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M47">View MathML</a>, if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M49">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M50">View MathML</a>.

Lemma 1.5 (Sequential characterization of continuity)

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a>be a partial metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M52">View MathML</a>be a given mapping. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M53">View MathML</a>is continuous at<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M47">View MathML</a>if it is sequentially continuous at<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M55">View MathML</a>, that is, if and only if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M56">View MathML</a>

Let X be a nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M57">View MathML</a> be a given mapping. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a>, we denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M59">View MathML</a> the subset of X defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M60">View MathML</a>

Definition 1.6 Let X be a nonempty set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M61">View MathML</a> is called an ordered partial metric space if and only if

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a> is a partial metric space,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63">View MathML</a> is a partially ordered set.

Definition 1.7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63">View MathML</a> be a partially ordered set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M65">View MathML</a> are called comparable if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M66">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M67">View MathML</a> holds.

Definition 1.8 ([30])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M68">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M69">View MathML</a> be given mappings such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M70">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M71">View MathML</a>. We say that S and T are weakly increasing with respect to R if and only if, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M73">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M74">View MathML</a>

Remark 1.9 If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M75">View MathML</a> is the identity mapping (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M76">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a>, shortly <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M78">View MathML</a>), then the fact that S and T are weakly increasing with respect to R implies that S and T are weakly increasing mappings, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M79">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M80">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81">View MathML</a>. Finally, a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M82">View MathML</a> is weakly increasing if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M83">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81">View MathML</a>.

Example 1.10 Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M85">View MathML</a> endowed with the usual ordering of real numbers and define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M86">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M87">View MathML</a>

Now, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M88">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M89">View MathML</a>, then S and T are weakly increasing with respect to R.

Definition 1.11 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M90">View MathML</a> be an ordered partial metric space. We say that X is regular if and only if the following hypothesis holds: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M91">View MathML</a> is a non-decreasing sequence in X with respect to ⪯ such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M92">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M94">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M95">View MathML</a>.

Finally, we recall the following definition of partial-compatibility introduced by Samet et al.[40].

Definition 1.12 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a> be a partial metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M97">View MathML</a> be given mappings. We say that the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M98">View MathML</a> is partial-compatible if the following conditions hold:

(b1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M99">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M100">View MathML</a>.

(b2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M101">View MathML</a>, whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M102">View MathML</a> is a sequence in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M103">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M104">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M105">View MathML</a>.

Note that Definition 1.12 extends and generalizes the notion of compatibility introduced by Jungck [25].

2 Main results

We start this section with some auxiliary results (see also [37]).

Lemma 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M106">View MathML</a>be a metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a>be a sequence inXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M108">View MathML</a>is non-increasing and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M109">View MathML</a>

If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M110">View MathML</a>is not a Cauchy sequence, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48">View MathML</a>and two sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M112">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M113">View MathML</a>of positive integers such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M114">View MathML</a>and the following four sequences tend toεwhen<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M115">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M116">View MathML</a>

As a corollary, applying Lemma 2.1 to the associated metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M117">View MathML</a> of a partial metric p, and using Lemma 1.3, we obtain the following lemma (see also [21]).

Lemma 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M32">View MathML</a>be a partial metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a>be a sequence inXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M120">View MathML</a>is non-increasing and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M121">View MathML</a>

If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M110">View MathML</a>is not a Cauchy sequence, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48">View MathML</a>and two sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M112">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M113">View MathML</a>of positive integers such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M114">View MathML</a>and the following four sequences tend toεwhen<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M115">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M128">View MathML</a>

In the sequel, let Ψ be the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M129">View MathML</a> such that ψ is continuous, strictly increasing and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M130">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M131">View MathML</a>. Also, let Φ be the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M132">View MathML</a> such that φ is lower semi-continuous and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M133">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M131">View MathML</a>. Such ψ and φ are called control functions.

Our first main result is the following.

Theorem 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63">View MathML</a>be a partially ordered set. Suppose that there exists a partial metricponXsuch that the partial metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>is complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M137">View MathML</a>be given mappings satisfying

(a) T, SandRare continuous,

(b) the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139">View MathML</a>are partial-compatible,

(c) TandSare weakly increasing with respect toR.

Suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140">View MathML</a>such thatRxandRyare comparable, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M141">View MathML</a>

(2.1)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143">View MathML</a>. ThenT, SandRhave a coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M145">View MathML</a>.

Proof By Definition 1.8, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M146">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M55">View MathML</a> be an arbitrary point in X. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M148">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M149">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M150">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M151">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M152">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M153">View MathML</a>. Continuing this process, we can construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> in X defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M155">View MathML</a>

(2.2)

By construction, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M156">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M157">View MathML</a>. Then using the fact that S and T are weakly increasing with respect to R, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M158">View MathML</a>

We continue this process to get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M159">View MathML</a>

(2.3)

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160">View MathML</a> is a Cauchy sequence in the partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>. To this aim, we distinguish the following two cases.

Case 1. We suppose that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M162">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M163">View MathML</a>, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M164">View MathML</a>. By (2.3), applying (2.1) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M165">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M166">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M167">View MathML</a>

Since ψ is strictly increasing, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M168">View MathML</a>

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M169">View MathML</a>. Continuing this process, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M170">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M171">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M172">View MathML</a>, therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160">View MathML</a> is Cauchy in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>. The same conclusion holds if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M175">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M176">View MathML</a>.

Case 2. Now, we suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M177">View MathML</a>

(2.4)

Here, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M178">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M179">View MathML</a>. Thanks to (2.3), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M180">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M181">View MathML</a> are comparable, then using (2.2) and taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M182">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M183">View MathML</a> in (2.1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M184">View MathML</a>

Since ψ is strictly increasing, the above inequality implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M185">View MathML</a>

(2.5)

Now, taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M186">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M183">View MathML</a> in (2.1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M188">View MathML</a>

(2.6)

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M189">View MathML</a>

(2.7)

Combining (2.5) and (2.7), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M190">View MathML</a>

(2.8)

It follows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M191">View MathML</a> is non-increasing and bounded below by 0. Hence, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M193">View MathML</a>

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M195">View MathML</a>. Taking the lim sup as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93">View MathML</a> in (2.6) and using the properties of the functions ψ and φ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M197">View MathML</a>

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M198">View MathML</a>, and by a property of the function φ, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194">View MathML</a>, that is a contradiction. We deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194">View MathML</a>, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M201">View MathML</a>

(2.9)

We shall show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160">View MathML</a> is a Cauchy sequence in the partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>. For this, it is sufficient to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M204">View MathML</a> is Cauchy in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>. Suppose to the contrary that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M204">View MathML</a> is not a Cauchy sequence. Then, having in mind that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M191">View MathML</a> is non-increasing and (2.9), it follows by Lemma 2.2 that there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48">View MathML</a> and two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M112">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M113">View MathML</a> of positive integers such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M114">View MathML</a> and the following four sequences tend to ε when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M115">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M213">View MathML</a>

Applying (2.1) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M214">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M215">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M216">View MathML</a>

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M217">View MathML</a> in the above inequality and using the continuity of ψ and the lower semi-continuity of φ, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M218">View MathML</a>

(2.10)

from which a contradiction follows since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M48">View MathML</a>. Then, we deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160">View MathML</a> is a Cauchy sequence in the partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>, which is complete, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M160">View MathML</a> converges to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144">View MathML</a>, that is, from (p3) and Definition 1.2,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M224">View MathML</a>

But from (2.9) and condition (p2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M225">View MathML</a>

therefore, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M226">View MathML</a>

(2.11)

From (2.11) and the continuity of R, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M227">View MathML</a>

(2.12)

The triangular inequality yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M228">View MathML</a>

(2.13)

By (2.2) and (2.11), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M229">View MathML</a>

(2.14)

Having in mind that the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138">View MathML</a> is partial-compatible, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M231">View MathML</a>

(2.15)

Also, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M232">View MathML</a>, then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M233">View MathML</a>. The continuity of T together with (2.11) give us

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M234">View MathML</a>

(2.16)

Combining (2.12) and (2.15) together with (2.16) and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93">View MathML</a> in (2.13), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M236">View MathML</a>

(2.17)

By condition (p2) and (2.17), one can write

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M237">View MathML</a>

(2.18)

Similarly, by triangular inequality, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M238">View MathML</a>

(2.19)

By (2.2) and (2.11), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M239">View MathML</a>

(2.20)

Since the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139">View MathML</a> is partial-compatible, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M241">View MathML</a>

(2.21)

Also, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M232">View MathML</a>, it follows <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M243">View MathML</a>. Thus, from (2.18), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M244">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M245">View MathML</a>.

The continuity of S and (2.20) give us

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M246">View MathML</a>

(2.22)

Combining (2.12) and (2.21) together with (2.22) and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93">View MathML</a> in (2.19), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M248">View MathML</a>

(2.23)

By condition (p2) and (2.23), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M249">View MathML</a>

(2.24)

Applying (2.1) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M250">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M251">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M252">View MathML</a>

and so it follows <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M253">View MathML</a>, that is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M254">View MathML</a>. Thus, we have obtained

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M255">View MathML</a>

that is, u is a coincidence point of T, S and R. □

Remark 2.4 We point out that the order in which the mappings in condition (b) of Theorem 2.3 are considered is crucial. Trivially, Theorem 2.3 remains true if we assume that the partial-compatible pairs are <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M256">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M257">View MathML</a>.

Example 2.5 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M258">View MathML</a> be endowed with the partial metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M259">View MathML</a> and the order given as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M260">View MathML</a>

Consider the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M86">View MathML</a> defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M262">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M263">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81">View MathML</a>. Also, define the functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M265">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M266">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M267">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M268">View MathML</a>. Clearly, condition (2.1) is satisfied. In fact, for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M270">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M271">View MathML</a>

All the other hypotheses of Theorem 2.3 are satisfied and T, S and R have a coincidence point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M272">View MathML</a>. (Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M272">View MathML</a> is the unique common fixed point of T, S and R.)

Note that Theorem 2.3 is not applicable in respect of the usual order of real numbers because T is not weakly increasing. It follows that the partial order may be fundamental.

Under different hypotheses, the conclusion of Theorem 2.3 remains true without assuming the continuity of T, S and R, and the partial-compatibility of the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M256">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M257">View MathML</a>. This is the purpose of the next theorem.

Theorem 2.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63">View MathML</a>be a partially ordered set. Suppose that there exists a partial metricponXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>is complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M137">View MathML</a>be given mappings satisfying

(a) RXis a closed subspace of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>,

(b) TandSare weakly increasing with respect toR,

(c) Xis regular.

Suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140">View MathML</a>such thatRxandRyare comparable, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M281">View MathML</a>

(2.25)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143">View MathML</a>. Then, T, SandRhave a coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M145">View MathML</a>.

Proof Following the proof of Theorem 2.3, we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M286">View MathML</a> is a Cauchy sequence in the closed subspace RX, then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M287">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144">View MathML</a>, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M289">View MathML</a>

(2.26)

Thanks to (2.3), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M290">View MathML</a> is a non-decreasing sequence, and so, since it converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M291">View MathML</a>, from the regularity of X, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M292">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M293">View MathML</a> and Ru are comparable. Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M186">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M295">View MathML</a> in (2.25) and using (2.2), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M296">View MathML</a>

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M297">View MathML</a> in the above inequality, using (2.26) and the properties of φ and ψ, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M298">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M299">View MathML</a>

which is true if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M300">View MathML</a>. This means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M301">View MathML</a>.

Analogously, putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M302">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M183">View MathML</a> in (2.25), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M304">View MathML</a>

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M297">View MathML</a> in the above inequality, using (2.26) and the properties of φ and ψ, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M306">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M307">View MathML</a>

We conclude that u is a coincidence point of T, S and R. □

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M75">View MathML</a> is the identity mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M309">View MathML</a>, by Theorem 2.6, we obtain the following common fixed point result involving two mappings.

Corollary 2.7Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63">View MathML</a>be a partially ordered set. Suppose that there exists a partial metricponXsuch that the partial metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>is complete. LetXbe regular and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M312">View MathML</a>be given mappings such thatTandSare weakly increasing. Suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140">View MathML</a>such thatxandyare comparable, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M314">View MathML</a>

(2.27)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143">View MathML</a>. Then, TandShave a common fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M318">View MathML</a>.

The following example shows that the hypothesis ‘T and S are weakly increasing (with respect to R)’ has a key role for the validity of our results.

Example 2.8 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M319">View MathML</a> be endowed with the partial metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M259">View MathML</a> and the order ⪯ given as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M321">View MathML</a>

Consider the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M322">View MathML</a> defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M323">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M324">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M81">View MathML</a>. Also, define the functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M326">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M266">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M328">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M268">View MathML</a>. It is easy to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M330">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M331">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M21">View MathML</a>, that is, T and S are weakly increasing. Now, take x and y comparable and, without loss of generality, assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M333">View MathML</a>, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M334">View MathML</a>. It is easy to show that (2.27) holds and all the other hypotheses of Corollary 2.7 are satisfied. Then, T and S have a unique common fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M272">View MathML</a>.

Note that Corollary 2.7 is not applicable in respect of the usual order of real numbers because T and S are not weakly increasing.

Now, we shall prove the existence and uniqueness of a common fixed point for three mappings.

Theorem 2.9In addition to the hypotheses of Theorem 2.3, suppose that for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M337">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M338">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M339">View MathML</a>. Then, T, SandRhave a unique common fixed point, that is, there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M144">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M341">View MathML</a>.

Proof Referring to Theorem 2.3, the set of coincidence points of T, S and R is nonempty. Now, we shall show that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M342">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M343">View MathML</a> are coincidence points of T, S and R, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M344">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M345">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M346">View MathML</a>

(2.28)

For the coincidence points <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M342">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M343">View MathML</a>, Theorem 2.3 gives us that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M349">View MathML</a>

By assumption, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M350">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M351">View MathML</a>

(2.29)

Now, proceeding similarly to the proof of Theorem 2.3, we can immediately define a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M352">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M353">View MathML</a>

(2.30)

Since T and S are weakly increasing with respect to R, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M354">View MathML</a>

(2.31)

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M355">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M356">View MathML</a> in (2.1) and using (2.31), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M357">View MathML</a>

Since ψ is strictly increasing, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M358">View MathML</a>

This gives us

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M359">View MathML</a>

(2.32)

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M360">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M361">View MathML</a> in (2.1), then similarly to the above, one can find

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M362">View MathML</a>

(2.33)

We combine (2.32) and (2.33) to remark that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M363">View MathML</a>

(2.34)

Then, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M364">View MathML</a> is non-increasing and bounded below, so there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M366">View MathML</a>

Adopting the strategy used in the proof of Theorem 2.3, one can show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M194">View MathML</a>, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M368">View MathML</a>

(2.35)

The same idea yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M369">View MathML</a>

(2.36)

Now, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M370">View MathML</a> and from (2.35), (2.36), we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M371">View MathML</a>, and so (2.28) holds.

Thanks to (2.30) and (2.35), one can write

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M372">View MathML</a>

(2.37)

From partial-compatibility of the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139">View MathML</a>, using (2.35) and (2.37), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M375">View MathML</a>

(2.38)

Denote

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M376">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M377">View MathML</a>, so again by partial-compatibility of the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M138">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M139">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M380">View MathML</a>

(2.39)

By triangular inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M381">View MathML</a>

Using (2.37), (2.38), (2.39), the continuity of T and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93">View MathML</a> in the above inequality, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M383">View MathML</a>

that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M245">View MathML</a> and u is a coincidence point of T and R.

Analogously, the triangular inequality gives us

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M385">View MathML</a>

Using (2.37), (2.38), (2.39), the continuity of S and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M93">View MathML</a> in the above inequality, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M387">View MathML</a>

By condition (p2), it follows immediately

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M388">View MathML</a>

Now, applying (2.1) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M250">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M390">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M391">View MathML</a>

then we deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M253">View MathML</a>, and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M254">View MathML</a>. Until now, we have obtained

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M394">View MathML</a>

With <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M395">View MathML</a> and from (2.28), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M396">View MathML</a>

This proves that u is a common fixed point of the mappings T, S and R.

Now our purpose is to check that such a point is unique. Suppose to the contrary that there is another common fixed point of T, S and R, say q. Then, applying (2.1) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M397">View MathML</a>, we obtain easily that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M398">View MathML</a>. It is immediate that q is a coincidence point of T, S and R. From (2.28), this implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M399">View MathML</a>

Hence, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M400">View MathML</a>

which yields the uniqueness of the common fixed point of T, S and R. This completes the proof. □

Remark 2.10 We leave, as exercise for the reader, to verify that our results hold even if we replace condition (2.1) by the following

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M401">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M402">View MathML</a> such that Rx and Ry are comparable.

3 Application to cyclical contractions

In this section we use the previous results to prove a fixed point theorem for a mapping satisfying a weak cyclical contractive condition. In 2003, Kirk et al.[27] studied existence and uniqueness of a fixed point for mappings satisfying cyclical contractive conditions in complete metric spaces.

Definition 3.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M403">View MathML</a> be a metric space, m a positive integer and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M404">View MathML</a> nonempty subsets of X. A mapping T on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M405">View MathML</a> is called a m-cyclic mapping if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M406">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M407">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M408">View MathML</a>.

Later on, Pacurar and Rus [35] introduced the following notion, suggested by the considerations in [27].

Definition 3.2 Let Y be a nonempty set, m a positive integer and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M409">View MathML</a> an operator. By definition, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M410">View MathML</a> is a cyclic representation of Y with respect to T if T is a m-cyclic mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411">View MathML</a> are nonempty sets.

Example 3.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M412">View MathML</a>. Assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M413">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M414">View MathML</a>, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M415">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M409">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M417">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M418">View MathML</a>. It is clear that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M419">View MathML</a> is a cyclic representation of Y.

Inspired by Karapinar [26] and Gopal et al.[23], we present the notion of a cyclic weak <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1">View MathML</a>-contraction in partial metric spaces.

Definition 3.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M421">View MathML</a> be an ordered partial metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M422">View MathML</a> be closed subsets of X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M410">View MathML</a>. An operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M409">View MathML</a> is called a cyclic weak <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1">View MathML</a>-contraction if the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M410">View MathML</a> is a cyclic representation of Y with respect to T,

(ii) there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M142">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M143">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M429">View MathML</a>

(3.1)

for every comparable <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M430">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M431">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M432">View MathML</a>).

Now, we state and prove the following result.

Theorem 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M63">View MathML</a>be a partially ordered set. Suppose that there exists a partial metricponXsuch that the partial metric space<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M9">View MathML</a>is complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M435">View MathML</a>be a given mapping satisfying

(a) Tis a cyclic weak<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M1">View MathML</a>-contraction,

(b) Tis weakly increasing and continuous,

(c) the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M437">View MathML</a>is partial-compatible,

(d) for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M140">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M337">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M338">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M441">View MathML</a>.

Then, Thas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M442">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M443">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M444">View MathML</a> and set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M445">View MathML</a>

(3.2)

For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M446">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M447">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M448">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M449">View MathML</a>. Then, following the lines of the proof of Theorem 2.3, it is easy to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> is a Cauchy sequence in the partial metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M451">View MathML</a>, which is complete, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> converges to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M453">View MathML</a>. On the other hand, by condition (i) of Definition 3.4, it follows that the iterative sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> has an infinite number of terms in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M456">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M451">View MathML</a> is complete, from each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M456">View MathML</a>, one can extract a subsequence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M25">View MathML</a> that converges to y. In virtue of the fact that each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M411">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M456">View MathML</a>, is closed, we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M463">View MathML</a> and thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M464">View MathML</a>. Obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M465">View MathML</a> is closed and complete. Now, consider the restriction of T on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M465">View MathML</a>, that is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M467">View MathML</a> which satisfies the assumptions of Theorem 2.3 and thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M468">View MathML</a> has a unique fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M465">View MathML</a>, say u, which is obtained by iteration from the starting point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M470">View MathML</a>. To conclude, we have to show that, for any initial value <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M418">View MathML</a>, we get the same limit point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M472">View MathML</a>. Due to condition (c) and using the analogous ideas of the proof of Theorem 2.9, it can be obtained that, for any initial value <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M418">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M474">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/124/mathml/M475">View MathML</a>. This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors are really thankful to the anonymous referee for his/her precious suggestions useful to improve the quality of the paper. The third author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the forth author would like to thank the Commission on Higher Education and the Thailand Research Fund under Grant MRG no. 5380044 for financial support during the preparation of this manuscript.

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