SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Highly Accessed Research

A generalized weak contraction principle with applications to coupled coincidence point problems

Binayak S Choudhury1, Nikhilesh Metiya2 and Mihai Postolache3*

Author Affiliations

1 Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, West Bengal, 711103, India

2 Department of Mathematics, Bengal Institute of Technology, Kolkata, West Bengal, 700150, India

3 Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independentei, Bucharest, 060042, Romania

For all author emails, please log on.

Fixed Point Theory and Applications 2013, 2013:152  doi:10.1186/1687-1812-2013-152

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/152


Received:12 March 2013
Accepted:24 May 2013
Published:11 June 2013

© 2013 Choudhury et al.; licensee Springer

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

Abstract

In this paper we establish some coincidence point results for generalized weak contractions with discontinuous control functions. The theorems are proved in metric spaces with a partial order. Our theorems extend several existing results in the current literature. We also discuss several corollaries and give illustrative examples. We apply our result to obtain some coupled coincidence point results which effectively generalize a number of established results.

MSC: 54H10, 54H25.

Keywords:
partially ordered set; control function; coincidence point; coupled coincidence point

1 Introduction

In this paper we prove certain coincidence point results in partially ordered metric spaces for functions which satisfy a certain inequality involving three control functions. Two of the control functions are discontinuous. Fixed point theory in partially ordered metric spaces is of relatively recent origin. An early result in this direction is due to Turinici [1], in which fixed point problems were studied in partially ordered uniform spaces. Later, this branch of fixed point theory has developed through a number of works, some of which are in [2-6].

Weak contraction was studied in partially ordered metric spaces by Harjani et al.[3]. In a recent result by Choudhury et al.[2], a generalization of the above result to a coincidence point theorem has been done using three control functions. Here we prove coincidence point results by assuming a weak contraction inequality with three control functions, two of which are not continuous. The results are obtained under two sets of additional conditions. A fixed point theorem is also established. There are several corollaries and two examples. One of the examples shows that the corollaries are properly contained in their respective theorem. The corollaries are generalizations of several existing works.

We apply our result to obtain some coupled coincidence point results. Coupled fixed theorems and coupled coincidence point theorems have appeared prominently in recent literature. Although the concept of coupled fixed points was introduced by Guo et al.[7], starting with the work of Gnana Bhaskar and Lakshmikantham [8], where they established a coupled contraction principle, this line of research has developed rapidly in partially ordered metric spaces. References [9-20] are some examples of these works. There is a viewpoint from which coupled fixed and coincidence point theorems can be considered as problems in product spaces [21]. We adopt this approach here. Specifically, we apply our theorem to a product of two metric spaces on which a metric is defined from the metric of the original spaces. We establish a generalization of several results. We also discuss an example which shows that our result is an actual improvement over the results it generalizes.

2 Mathematical preliminaries

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M2">View MathML</a>. The mapping T is said to be nondecreasing if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M4">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M5">View MathML</a> and nonincreasing if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M4">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M8">View MathML</a>.

Definition 2.1 ([22])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M10">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M11">View MathML</a>. The mapping T is said to be G-nondecreasing if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M13">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M14">View MathML</a> and G-nonincreasing if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M13">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M17">View MathML</a>.

Definition 2.2 Two self-mappings G and T of a nonempty set X are said to be commutative if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19">View MathML</a>.

Definition 2.3 ([23])

Two self-mappings G and T of a metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> are said to be compatible if the following relation holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M21">View MathML</a>

whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> is a sequence in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M23">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19">View MathML</a> is satisfied.

Definition 2.4 ([24])

Two self-mappings G and T of a nonempty set X are said to be weakly compatible if they commute at their coincidence points; that is, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M18">View MathML</a>.

Definition 2.5 ([8])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>. The mapping F is said to have the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument; that is, if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M30">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M31">View MathML</a>

Definition 2.6 ([17])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a> be a partially ordered set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>. We say that F has the mixedg-monotone property if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M35">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M36">View MathML</a>

Definition 2.7 ([8])

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a> is called a coupled fixed point of the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M39">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M40">View MathML</a>.

Definition 2.8 ([17])

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a> is called a coupled coincidence point of the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M44">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M45">View MathML</a>.

Definition 2.9 ([17])

Let X be a nonempty set. The mappings g and F, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M47">View MathML</a>, are said to be commutative if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M48">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>.

Definition 2.10 ([12])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> be a metric space. The mappings g and F, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M51">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>, are said to be compatible if the following relations hold:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M53">View MathML</a>

whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M55">View MathML</a> are sequences in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M56">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M57">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a> are satisfied.

Definition 2.11 Let X be a nonempty set. The mappings g and F, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38">View MathML</a>, are said to be weakly compatible if they commute at their coupled coincidence points, that is, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M61">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M62">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M48">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M65">View MathML</a>.

Definition 2.12 ([25])

A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M66">View MathML</a> is called an altering distance function if the following properties are satisfied:

(i) ψ is monotone increasing and continuous;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M67">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M68">View MathML</a>.

In our results in the following sections, we use the following classes of functions.

We denote by Ψ the set of all functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M69">View MathML</a> satisfying

(iψ) ψ is continuous and monotone non-decreasing,

(iiψ) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M67">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M68">View MathML</a>;

and by Θ we denote the set of all functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M72">View MathML</a> such that

(iα) α is bounded on any bounded interval in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>,

(iiα) α is continuous at 0 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M74">View MathML</a>.

3 Main results

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M75">View MathML</a> be an ordered metric space. X is called regular if it has the following properties:

(i) if a nondecreasing sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M76">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M77">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M79">View MathML</a>;

(ii) if a nonincreasing sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M80">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M77">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M82">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M83">View MathML</a>.

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M86">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is compatible. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M91">View MathML</a>

(3.1)

for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M94">View MathML</a>,

(3.2)

and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M98">View MathML</a>

(3.3)

Also, suppose that

(a) Tis continuous, or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M105">View MathML</a>. Again, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M106">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M107">View MathML</a>. Continuing this process, we construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M109">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M105">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M112">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M113">View MathML</a>. Now, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M113">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M115">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M116">View MathML</a>. Again, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M116">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M118">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M119">View MathML</a>. Continuing this process, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M120">View MathML</a>

(3.5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M121">View MathML</a>

(3.6)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M122">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M83">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M124">View MathML</a>, from (3.3) and (3.4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M125">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M126">View MathML</a>

(3.7)

which, in view of the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M127">View MathML</a>, yields <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M128">View MathML</a>, which by (3.1) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M129">View MathML</a> for all positive integer n, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M130">View MathML</a> is a monotone decreasing sequence. Hence there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M131">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M132">View MathML</a>

(3.8)

Taking limit supremum on both sides of (3.7), using (3.8), the property (iα) of φ and θ, and the continuity of ψ, we obtain

Since , it follows that

that is,

which by (3.2) is a contradiction unless <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M137">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M138">View MathML</a>

(3.9)

Next we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139">View MathML</a> is a Cauchy sequence.

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139">View MathML</a> is not a Cauchy sequence. Then there exists an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M141">View MathML</a> for which we can find two sequences of positive integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M142">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M143">View MathML</a> such that for all positive integers k, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M144">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M145">View MathML</a>. Assuming that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M146">View MathML</a> is the smallest such positive integer, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M147">View MathML</a>

Now,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M148">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M149">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M150">View MathML</a> in the above inequality and using (3.9), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M151">View MathML</a>

(3.10)

Again,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M152">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M153">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M150">View MathML</a> in the above inequalities, using (3.9) and (3.10), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M155">View MathML</a>

(3.11)

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M156">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M157">View MathML</a>, from (3.3) and (3.4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M158">View MathML</a>

Taking limit supremum on both sides of the above inequality, using (3.10), (3.11), the property (iα) of φ and θ, and the continuity of ψ, we obtain

Since , it follows that

that is,

which is a contradiction by (3.2). Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139">View MathML</a> is a Cauchy sequence in X. From the completeness of X, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M164">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M165">View MathML</a>

(3.12)

Since the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a> is compatible, from (3.12), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M167">View MathML</a>

(3.13)

Let the condition (a) hold.

By the triangular inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M168">View MathML</a>

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M169">View MathML</a> in the above inequality, using (3.12), (3.13) and the continuities of T and G, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M170">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25">View MathML</a>, that is, x is a coincidence point of the mappings G and T.

Next we suppose that the condition (b) holds.

By (3.5) and (3.12), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M172">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M173">View MathML</a>. Using the monotone property of G, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M174">View MathML</a>

(3.14)

As G is continuous and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a> is compatible, by (3.12) and (3.13), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M176">View MathML</a>

(3.15)

Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M177">View MathML</a>

Since ψ is continuous, from the above inequality, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M178">View MathML</a>

which, by (3.3) and (3.14), implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M179">View MathML</a>

Using (3.15) and the property (iiα) of φ and θ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M180">View MathML</a>

which, by the property of ψ, implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M181">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25">View MathML</a>, that is, x is a coincidence point of the mappings G and T. □

Next we discuss some corollaries of Theorem 3.1. By an example, we show that Theorem 3.1 properly contains all its corollaries.

Every commuting pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a> is also a compatible pair. Then considering <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a> to be the commuting pair in Theorem 3.1, we have the following corollary.

Corollary 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is commutative. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90">View MathML</a>such that (3.1), (3.2) and (3.3) are satisfied. Also, suppose that

(a) Tis continuous, or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

Considering ψ to be the identity mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M194">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M195">View MathML</a> in Theorem 3.1, we have the following corollary.

Corollary 3.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is compatible. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M201">View MathML</a>such that for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M94">View MathML</a>,

(3.16)

and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M208">View MathML</a>

(3.17)

Also, suppose that

(a) Tis continuous, or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

Considering <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M194">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M195">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M213">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214">View MathML</a> in Theorem 3.1, we have the following corollary.

Corollary 3.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is compatible. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M222">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M224">View MathML</a>

(3.18)

Also, suppose that

(a) Tis continuous, or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

Considering φ to be the function ψ in Theorem 3.1, we have the following corollary.

Corollary 3.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is compatible. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233">View MathML</a>such that for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M236">View MathML</a>,

(3.19)

and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M240">View MathML</a>

(3.20)

Also, suppose that

(a) Tis continuous or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

If ψ and φ are the identity mappings in Theorem 3.1, we have the following corollary.

Corollary 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is compatible. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233">View MathML</a>such that for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M94">View MathML</a>, and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M255">View MathML</a>

(3.21)

Also, suppose that

(a) Tis continuous, or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

Considering ψ and φ to be the identity mappings and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M258">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M259">View MathML</a> in Theorem 3.1, we have the following corollary.

Corollary 3.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M187">View MathML</a>be two mappings such thatGis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>, TisG-nondecreasing with respect toand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is compatible. Assume that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M267">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M268">View MathML</a>

(3.22)

Also, suppose that

(a) Tis continuous, or

(b) Xis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

The condition (i), the continuity and the monotone property of the function G, and (ii), the compatibility condition of the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>, which were considered in Theorem 3.1, are relaxed in our next theorem by taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272">View MathML</a> to be closed in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>.

Theorem 3.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M86">View MathML</a>be two mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M87">View MathML</a>andTisG-nondecreasing with respect toand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272">View MathML</a>is closed inX. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M280">View MathML</a>such that (3.1), (3.2) and (3.3) are satisfied. Also, suppose thatXis regular.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M99">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M100">View MathML</a>, thenGandThave a coincidence point inX.

Proof We take the same sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> as in the proof of Theorem 3.1. Then we have (3.12), that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M284">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139">View MathML</a> is a sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272">View MathML</a> is closed in X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M288">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M288">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M290">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M291">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M292">View MathML</a>

(3.23)

Now, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M139">View MathML</a> is nondecreasing and converges to Gz. So, by the order condition of the metric space X, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M294">View MathML</a>

(3.24)

Putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M295">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M296">View MathML</a> in (3.3), by the virtue of (3.24), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M297">View MathML</a>

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M298">View MathML</a> in the above inequality, using (3.23), the property (iiα) of φ and θ and the continuity of ψ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M299">View MathML</a>

which, by the property of ψ, implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M300">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M301">View MathML</a>, that is, z is a coincidence point of the mappings G and T. □

In the following, our aim is to prove the existence and uniqueness of the common fixed point in Theorems 3.1 and 3.2.

Theorem 3.3In addition to the hypotheses of Theorems 3.1 and 3.2, in both of the theorems, suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a>there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M303">View MathML</a>such thatTuis comparable toTxandTy, and also the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a>is weakly compatible. ThenGandThave a unique common fixed point.

Proof From Theorem 3.1 or Theorem 3.2, the set of coincidence points of G and T is non-empty. Suppose x and y are coincidence points of G and T, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M306">View MathML</a>. Now, we show

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M307">View MathML</a>

(3.25)

By the assumption, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M303">View MathML</a> such that Tu is comparable with Tx and Ty.

Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M309">View MathML</a> and choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M310">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M311">View MathML</a>. Then, similarly to the proof of Theorem 3.1, we can inductively define sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M312">View MathML</a> where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M313">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M314">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M315">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M316">View MathML</a> are comparable.

Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M317">View MathML</a> (the proof is similar to that in the other case).

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M318">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M319">View MathML</a>.

In fact, we will use mathematical induction. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M317">View MathML</a>, our claim is true for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M321">View MathML</a>. We presume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M318">View MathML</a> holds for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M323">View MathML</a>. Since T is G-nondecreasing with respect to ⪯, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M324">View MathML</a>

and this proves our claim.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M325">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M318">View MathML</a>, using the contractive condition (3.3), for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M327">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M328">View MathML</a>

that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M329">View MathML</a>, which, in view of the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M127">View MathML</a>, yields <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M331">View MathML</a>, which by (3.1) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M129">View MathML</a> for all positive integer n, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M130">View MathML</a> is a monotone decreasing sequence.

Then, as in the proof of Theorem 3.1, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M334">View MathML</a>

(3.26)

Similarly, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M335">View MathML</a>

(3.27)

By the triangle inequality, using (3.26) and (3.27), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M336">View MathML</a>

Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M337">View MathML</a>. Thus (3.25) is proved.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M25">View MathML</a>, by weak compatibility of G and T, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M339">View MathML</a>

(3.28)

Denote

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M340">View MathML</a>

(3.29)

Then from (3.28) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M341">View MathML</a>

Thus z is a coincidence point of G and T. Then from (3.25) with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M342">View MathML</a> it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M343">View MathML</a>

By (3.29) it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M344">View MathML</a>

(3.30)

From (3.29) and (3.30), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M345">View MathML</a>.

Therefore, z is a common fixed point of G and T.

To prove the uniqueness, assume that r is another common fixed point of G and T. Then by (3.25) we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M346">View MathML</a>. Hence the common fixed point of G and T is unique. □

Example 3.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M347">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348">View MathML</a> is a partially ordered set with the natural ordering of real numbers. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> is a complete metric space.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M352">View MathML</a> be given respectively by the formulas <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M353">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M354">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19">View MathML</a>. Then T and G satisfy all the properties mentioned in Theorem 3.1.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356">View MathML</a> be given respectively by the formulas

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M357">View MathML</a>

Then ψ, φ and θ have the properties mentioned in Theorem 3.1.

It can be verified that (3.3) is satisfied for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M58">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M97">View MathML</a>. Hence the required conditions of Theorem 3.1 are satisfied and it is seen that 0 is a coincidence point of G and T. Also, the conditions of Theorem 3.3 are satisfied and it is seen that 0 is the unique common fixed point of G and T.

Remark 3.1 In the above example, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a> is compatible but not commuting so that Corollary 3.1 is not applicable to this example and hence Theorem 3.1 properly contains its Corollary 3.1.

Remark 3.2 In the above example, ψ is not the identity mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M361">View MathML</a> for all t in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>. Let us consider the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M364">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M365">View MathML</a> for all n. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M366">View MathML</a> for all n. Now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M367">View MathML</a>, but . Therefore, Corollary 3.2 is not applicable to this example, and hence Theorem 3.1 properly contains its Corollary 3.2.

Remark 3.3 The above example <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M361">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M370">View MathML</a>, and hence Corollary 3.3 is not applicable to the example, and so Theorem 3.1 properly contains its Corollary 3.3.

Remark 3.4 In the above example, φ is not identical to the function ψ, and also for any sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M374">View MathML</a>. Therefore, Corollaries 3.4 and 3.5 are not applicable to this example, and hence Theorem 3.1 properly contains its Corollaries 3.4 and 3.5.

Remark 3.5 In the above example, ψ and φ are not the identity functions and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M375">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M259">View MathML</a>. Therefore, Corollary 3.6 is not applicable to the above example. Hence Theorem 3.1 properly contains its Corollary 3.6.

Remark 3.6 Theorem 3.1 generalizes the results in [2-4,6,25-29].

Example 3.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M377">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348">View MathML</a> is a partially ordered set with the natural ordering of real numbers. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> is a metric space with the required properties of Theorem 3.2.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M382">View MathML</a> be given respectively by the formulas

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M383">View MathML</a>

Then T and G have the properties mentioned in Theorem 3.2.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356">View MathML</a> be given respectively by the formulas

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M385">View MathML</a>

Then ψ, φ and θ have the properties mentioned in Theorem 3.2.

All the required conditions of Theorem 3.2 are satisfied. It is seen that every rational number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M386">View MathML</a> is a coincidence point of G and T. Also, the conditions of Theorem 3.3 are satisfied and it is seen that 1 is the unique common fixed point of G and T.

Remark 3.7 In the above example, the function g is not continuous. Therefore, Theorem 3.1 is not applicable to the above example.

4 Applications to coupled coincidence point results

In this section, we use the results of the previous section to establish new coupled coincidence point results in partially ordered metric spaces. Our results are extensions of some existing results.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a> be a partially ordered set. Now, we endow the product space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M388">View MathML</a> with the following partial order:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M389">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> be a metric space. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M391">View MathML</a> given by the law

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M392">View MathML</a>

is a metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M393">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38">View MathML</a> be two mappings. Then we define two functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M396">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M397">View MathML</a> respectively as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M398">View MathML</a>

Lemma 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>. IfFhas the mixedg-monotone property, thenTisG-nondecreasing.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M402">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M403">View MathML</a>. Then, by the definition of G, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M404">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M405">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M406">View MathML</a>. Since F has the mixed g-monotone property, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M407">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M408">View MathML</a>

It follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M409">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M410">View MathML</a>. Therefore, T is G-nondecreasing. □

Lemma 4.2LetXbe a nonempty set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>. IfgandFare commutative, then the mappingsGandTare also commutative.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a>. Since g and F are commutative, by the definition of G and T, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M414">View MathML</a>

which shows that G and T are commutative. □

Lemma 4.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>be metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>. IfgandFare compatible, then the mappingsGandTare also compatible.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M418">View MathML</a> be a sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M393">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M420">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a>. By the definition of G and T, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M422">View MathML</a>, which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M423">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M424">View MathML</a>

Since g and F are compatible, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M425">View MathML</a>

It follows that G and T are compatible. □

Lemma 4.4LetXbe a nonempty set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>. IfgandFare weak compatible, then the mappingsGandTare also weak compatible.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a> be a coincidence point G and T. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M429">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M430">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>. Since g and F are weak compatible, by the definition of G and T, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M433">View MathML</a>

which shows that G and T commute at their coincidence point, that is, G and T are weak compatible. □

Theorem 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is compatible. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90">View MathML</a>such that (3.1) and (3.2) are satisfied and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M444">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M445">View MathML</a>

(4.1)

Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

Proof We consider the product space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M452">View MathML</a>, the metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M391">View MathML</a> on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M393">View MathML</a> and the functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M455">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M397">View MathML</a> as mentioned above. Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M457">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M458">View MathML</a> is a complete metric space. By the definition of G and T, we have that

(i) G is continuous and nondecreasing; and T is continuous,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M459">View MathML</a>,

(iii) T is G-nondecreasing with respect to ⪯,

(iv) the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M88">View MathML</a> is compatible.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M461">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M462">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M463">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M464">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M465">View MathML</a>. Then (4.1) reduces to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M466">View MathML</a>

Now, the existence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a> implies the existence of a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M470">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M471">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M472">View MathML</a>. Therefore, the theorem reduces to Theorem 3.1, and hence there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M473">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M474">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M429">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M430">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M478">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M479">View MathML</a> is a coupled coincidence point of g and F. □

The following corollary is a consequence of Corollary 3.1.

Corollary 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M483">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is commutative. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90">View MathML</a>such that (3.1), (3.2) and (4.1) are satisfied. Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

The following corollary is a consequence of Corollary 3.2.

Corollary 4.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M496">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M498">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is compatible. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M201">View MathML</a>such that for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M503">View MathML</a>,

and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M508">View MathML</a>

Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

The following corollary is a consequence of Corollary 3.3.

Corollary 4.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M496">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M498">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is compatible. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M526">View MathML</a>

Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

The following corollary is a consequence of Corollary 3.4.

Corollary 4.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M496">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M411">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M498">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is compatible. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233">View MathML</a>such that for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M541">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M503">View MathML</a>, and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M548">View MathML</a>

(4.2)

Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

Remark 4.1 The above result is also true if the arguments of ψ and θ in (4.2) are replaced by their half values, that is, when (4.2) is replaced by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M555">View MathML</a>

In this case, we can write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M556">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M557">View MathML</a> and proceed with the same proof by replacing ψ, θ by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M558">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M559">View MathML</a> respectively. Then we obtain a generalization of Theorem 2 of Berinde in [10].

The following corollary is a consequence of Corollary 3.5.

Corollary 4.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is compatible. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M233">View MathML</a>such that for any sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M73">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M503">View MathML</a>, and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M443">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M507">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M574">View MathML</a>

Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

The following corollary is a consequence of Corollary 3.6.

Corollary 4.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M29">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>be two mappings such thatgis continuous and nondecreasing, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M438">View MathML</a>, Fhas the mixedg-monotone property onXand the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is compatible. Assume that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M214">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M442">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M589">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M590">View MathML</a>

Also, suppose that

(a) Fis continuous, or

(b) Xis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

The following theorem is a consequence of Theorem 3.2.

Theorem 4.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M1">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a>is a complete metric space. Consider the mappings<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M38">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M34">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M601">View MathML</a>, Fhas the mixedg-monotone property onXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M272">View MathML</a>is closed inX. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M603">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M90">View MathML</a>such that (3.1), (3.2) and (4.1) are satisfied. Also, suppose thatXis regular.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M446">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M447">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M448">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M431">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M432">View MathML</a>; that is, gandFhave a coupled coincidence point inX.

The following theorem is a consequence of Theorem 3.3.

Theorem 4.3In addition to the hypotheses of Theorems 4.1 and 4.2, in both of the theorems, suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M611">View MathML</a>, there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M612">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M613">View MathML</a>is comparable to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M614">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M615">View MathML</a>, and also the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a>is weakly compatible. ThengandFhave a unique coupled common fixed point; that is, there exists a unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M618">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M619">View MathML</a>.

Example 4.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M347">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348">View MathML</a> is a partially ordered set with the natural ordering of real numbers. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> is a complete metric space.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M625">View MathML</a> be given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M626">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M19">View MathML</a>. Also, consider

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M628">View MathML</a>

which obeys the mixed g-monotone property.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M22">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M55">View MathML</a> be two sequences in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M631">View MathML</a>

Then, obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M632">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M633">View MathML</a>.

Now, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M79">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M635">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M636">View MathML</a>, while

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M637">View MathML</a>

Then it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M638">View MathML</a>

Hence, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a> is compatible in X.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M640">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M641">View MathML</a> be two points in X. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M642">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356">View MathML</a> be given respectively by the formulas

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M644">View MathML</a>

Then ψ, φ and θ have the properties mentioned in Theorem 4.1.

We now verify inequality (4.1) of Theorem 4.1.

We take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M645">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M646">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M647">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M648">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M649">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M650">View MathML</a>.

The following are the four possible cases.

Case-1: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M651">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M652">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M653">View MathML</a>

Case-2: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M654">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M655">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M656">View MathML</a>

Case-3: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M651">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M658">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M659">View MathML</a>

Case-4: The case ‘<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M660">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M661">View MathML</a>’ is not possible. Under this condition, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M662">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M663">View MathML</a>. Then by the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M649">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M665">View MathML</a>, which contradicts that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M648">View MathML</a>.

In all the above cases, it can be verified that (4.1) is satisfied. Hence the required conditions of Theorem 4.1 are satisfied, and it is seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M667">View MathML</a> is a coupled coincidence point of g and F in X. Also, the conditions of Theorem 4.3 are satisfied, and it is seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M667">View MathML</a> is the unique coupled common fixed point of g and F in X.

Remark 4.2 In the above example, the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M439">View MathML</a> is compatible but not commuting so that Corollary 4.1 is not applicable to this example, and hence Theorem 4.1 properly contains its Corollary 4.1.

Remark 4.3 As discussed in Remarks 3.2-3.5, Theorem 4.1 properly contains its Corollaries 4.2-4.6.

Remark 4.4 Theorem 4.1 properly contains its Corollary 4.6, which is an extension of Theorem 3 of Berinde [9], and Theorems 2.1 and 2.2 of Bhaskar and Lakshmikantham [8]. Therefore, Theorem 4.1 is an actual extension over Theorem 3 of Berinde [9] and Theorems 2.1 and 2.2 of Bhaskar and Lakshmikantham [8].

Example 4.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M377">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M348">View MathML</a> is a partially ordered set with the natural ordering of real numbers. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M349">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M12">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M20">View MathML</a> is a metric space with the required properties of Theorem 4.2.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M675">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M676">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M677">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M625">View MathML</a> be given by the formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M679">View MathML</a>

Then F and g have the properties mentioned in Theorems 4.2.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M356">View MathML</a> be given respectively by the formulas

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M681">View MathML</a>

Then ψ, φ and θ have the properties mentioned in Theorem 4.2.

All the required conditions of Theorem 4.2 are satisfied. It is seen that every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M37">View MathML</a>, where both x and y are rational, is a coupled coincidence point of g and F in X. Also, the conditions of Theorem 4.3 are satisfied and it is seen that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M683">View MathML</a> is the unique coupled common fixed point of g and F in X.

Remark 4.5 In the above example, the function g is not continuous. Therefore, Theorem 4.1 is not applicable to the above example.

Remark 4.6 In some recent papers [30,31] it has been proved that some of the contractive conditions involving continuous control functions are equivalent. Here two of our control functions are discontinuous. Therefore, the contraction we use here is not included in the class of contractions addressed by Aydi et al.[30] and Jachymski [31].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgements

The work is partially supported by the Council of Scientific and Industrial Research, India (No. 25(0168)/09/EMR-II). Professor BS Choudhury gratefully acknowledges the support.

References

  1. Turinici, M: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl.. 117, 100–127 (1986). Publisher Full Text OpenURL

  2. Choudhury, BS, Kundu, A: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/152/mathml/M684">View MathML</a>-weak contractions in partially ordered metric spaces. Appl. Math. Lett.. 25(1), 6–10 (2012). Publisher Full Text OpenURL

  3. Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal.. 71, 3403–3410 (2009). Publisher Full Text OpenURL

  4. Harjani, J, Sadarangani, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal.. 72, 1188–1197 (2010). Publisher Full Text OpenURL

  5. Nieto, JJ, Lopez, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22, 223–239 (2005). Publisher Full Text OpenURL

  6. Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.. 132, 1435–1443 (2004). Publisher Full Text OpenURL

  7. Guo, D, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal.. 11(5), 623–632 (1987). Publisher Full Text OpenURL

  8. Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal.. 65, 1379–1393 (2006). Publisher Full Text OpenURL

  9. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.. 74(18), 7347–7355 (2011). Publisher Full Text OpenURL

  10. Berinde, V: Coupled fixed point theorems for ϕ-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.. 75, 3218–3228 (2011)

  11. Berinde, V: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl.. 64, 1770–1777 (2012). Publisher Full Text OpenURL

  12. Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal.. 73, 2524–2531 (2010). Publisher Full Text OpenURL

  13. Choudhury, BS, Maity, P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model.. 54, 73–79 (2011). Publisher Full Text OpenURL

  14. Choudhury, BS, Metiya, N, Kundu, A: Coupled coincidence point theorems in ordered metric spaces. Ann. Univ. Ferrara. 57, 1–16 (2011). Publisher Full Text OpenURL

  15. Harjani, J, López, B, Sadarangani, K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal.. 74, 1749–1760 (2011). Publisher Full Text OpenURL

  16. Karapınar, E, Luong, NV, Thuan, NX: Coupled coincidence points for mixed monotone operators in partially ordered metric spaces. Arab. J. Math.. 1, 329–339 (2012). Publisher Full Text OpenURL

  17. Lakshmikantham, V, Ćirić, Lj: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.. 70, 4341–4349 (2009). Publisher Full Text OpenURL

  18. Luong, NV, Thuan, NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal.. 74, 983–992 (2011). Publisher Full Text OpenURL

  19. Samet, B: Coupled fixed point theorems for generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.. 72, 4508–4517 (2010). Publisher Full Text OpenURL

  20. Shatanawi, W, Samet, B, Abbas, M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model.. 55(3-4), 680–687 (2012). Publisher Full Text OpenURL

  21. Chen, YZ: Existence theorems of coupled fixed points. J. Math. Anal. Appl.. 154, 142–150 (1991). Publisher Full Text OpenURL

  22. Kadelburg, Z, Pavlović, M, Radenović, S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl.. 59, 3148–3159 (2010). Publisher Full Text OpenURL

  23. Jungck, G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci.. 9, 771–779 (1986). Publisher Full Text OpenURL

  24. Jungck, G, Rhoades, BE: Fixed point for set-valued functions without continuity. Indian J. Pure Appl. Math.. 29(3), 227–238 (1998)

  25. Khan, MS, Swaleh, M, Sessa, S: Fixed points theorems by altering distances between the points. Bull. Aust. Math. Soc.. 30, 1–9 (1984). Publisher Full Text OpenURL

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

  27. Dutta, PN, Choudhury, BS: A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl.. 2008, (2008) Article ID 406368

  28. Jungck, G: Commuting maps and fixed points. Am. Math. Mon.. 83, 261–263 (1976). Publisher Full Text OpenURL

  29. Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal.. 47(4), 2683–2693 (2001). Publisher Full Text OpenURL

  30. Aydi, H, Karapınar, E, Samet, B: Remarks on some recent fixed point theorems. Fixed Point Theory Appl.. 76, (2012) Article ID 2012

  31. Jachymski, J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal.. 74, 768–774 (2011). Publisher Full Text OpenURL