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Coupled fixed point theorems on partially ordered G-metric spaces

Erdal Karapınar1, Poom Kumam23 and Inci M Erhan1*

Author Affiliations

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

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

3 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada

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Fixed Point Theory and Applications 2012, 2012:174  doi:10.1186/1687-1812-2012-174

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/174


Received:7 July 2012
Accepted:26 September 2012
Published:11 October 2012

© 2012 Karapınar 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

The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered G-metric spaces in a virtually different and more natural way.

MSC: 46N40, 47H10, 54H25, 46T99.

Keywords:
coupled fixed point; coupled coincidence point; mixed g-monotone property; ordered set; G-metric space

1 Introduction and preliminaries

The notion of metric space was introduced by Fréchet [1] in 1906. In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, e.g., [2]) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasi-metric spaces and b-metric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences.

Inspired by this motivation Mustafa and Sims [3] introduced the notion of a G-metric space in 2004 (see also [4-7]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [8] from the point of view of G-metrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [9] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, e.g., [10-20]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], Gnana-Bhaskar and Lakshmikantham [15] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, many authors conducted research on the coupled fixed point theory and many results in this direction were published (see, for example, [21-35]).

In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called G-metrics, coupled fixed points and partially ordered spaces.

We shall start with some necessary definitions and a detailed overview of the fundamental results developed in the remarkable works mentioned above. Throughout this paper, ℕ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M1">View MathML</a> denote the set of non-negative integers and the set of positive integers respectively.

Definition 1 (See [3])

Let X be a non-empty set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M2">View MathML</a> be a function satisfying the following properties:

(G1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M3">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M4">View MathML</a>,

(G2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M5">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M6">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M7">View MathML</a>,

(G3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M8">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M9">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M10">View MathML</a>,

(G4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M11">View MathML</a> (symmetry in all three variables),

(G5) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M12">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M13">View MathML</a> (rectangle inequality).

Then the function G is called a generalized metric or, more specially, a G-metric on X, and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a> is called a G-metric space.

It can be easily verified that every G-metric on X induces a metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M15">View MathML</a> on X given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M16">View MathML</a>

(1.1)

Trivial examples of G-metric are as follows.

Example 2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M17">View MathML</a> be a metric space. The function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M18">View MathML</a>, defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M19">View MathML</a>

or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M20">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M21">View MathML</a>, is a G-metric on X.

The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].

Definition 3 (See [3])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a> be a G-metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> be a sequence of points of X. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> is G-convergent to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M25">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M26">View MathML</a>, that is, if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M27">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M28">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M29">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M30">View MathML</a>. We call x the limit of the sequence and write <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M31">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M32">View MathML</a>.

Proposition 4 (See [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space. The following statements are equivalent:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a>isG-convergent tox,

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M35">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M36">View MathML</a>,

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M37">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M36">View MathML</a>,

(4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M39">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M40">View MathML</a>.

Definition 5 (See [3])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a> be a G-metric space. A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> is called G-Cauchy sequence if for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M27">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M45">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M46">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M47">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M48">View MathML</a>.

Proposition 6 (See [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space. The following statements are equivalent:

(1) The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a>isG-Cauchy.

(2) For any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M27">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M44">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M53">View MathML</a>, for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M54">View MathML</a>.

Definition 7 (See [3])

A G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a> is called G-complete if every G-Cauchy sequence is G-convergent in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a>.

Definition 8 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a> be a G-metric space. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M58">View MathML</a> is said to be continuous if for any three G-convergent sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M60">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M61">View MathML</a> converging to x, y and z respectively, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M62">View MathML</a> is G-convergent to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M63">View MathML</a>.

We define below g-ordered complete G-metric spaces.

Definition 9 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a> be a partially ordered set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a> be a G-metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66">View MathML</a> be a mapping. A partially ordered G-metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M67">View MathML</a>, is called g-ordered complete if for each G-convergent sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M68">View MathML</a>, the following conditions hold:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M69">View MathML</a>) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> is a non-increasing sequence in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M71">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M72">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M73">View MathML</a>.

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M74">View MathML</a>) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> is a non-decreasing sequence in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M71">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M77">View MathML</a><a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M73">View MathML</a>.

In particular, if g is the identity mapping in (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M69">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M74">View MathML</a>), the partially ordered G-metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M67">View MathML</a>, is called ordered complete.

We next recall some basic notions from the coupled fixed point theory. In 1987 Guo and Lakshmikantham [14] defined the concept of a coupled fixed point. In 2006, in order to prove the existence and uniqueness of the coupled fixed point of an operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M82">View MathML</a> on a partially ordered metric space, Gnana-Bhaskar and Lakshmikantham [15] reconsidered the notion of a coupled fixed point via the mixed monotone property.

Definition 10 ([15])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M83">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M84">View MathML</a>. The mapping F is said to have the mixed monotone property if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M85">View MathML</a> is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M6">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M87">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M88">View MathML</a>

Definition 11 ([15])

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M89">View MathML</a> is called a coupled fixed point of the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M90">View MathML</a> if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M91">View MathML</a>

The results in [15] were extended by Lakshmikantham and Ćirić in [16] by defining the mixed g-monotone property.

Definition 12 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a> be a partially ordered set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M93">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M94">View MathML</a>. The function F is said to have mixed g-monotone property if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M85">View MathML</a> is monotone g-non-decreasing in x and is monotone g-non-increasing in y, that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M96">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M97">View MathML</a>

(1.2)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M98">View MathML</a>

(1.3)

It is clear that Definition 12 reduces to Definition 10 when g is the identity mapping.

Definition 13 An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M99">View MathML</a> is called a coupled coincidence point of the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M100">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M94">View MathML</a> if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M102">View MathML</a>

and a common coupled fixed point of F and g if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M103">View MathML</a>

Definition 14 The mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M100">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M94">View MathML</a> are said to commute if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M106">View MathML</a>

Throughout the rest of the paper, we shall use the notation gx instead of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M107">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M108">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M25">View MathML</a>, for brevity. In [35], Nashine proved the following theorems.

Theorem 15Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>be a partially orderedG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M111">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66">View MathML</a>be mappings such thatFhas the mixedg-monotone property, and let there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M114">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M115">View MathML</a>. Suppose that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M116">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M117">View MathML</a>the following holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M118">View MathML</a>

(1.4)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M119">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M120">View MathML</a>, where either<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M121">View MathML</a>or<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M122">View MathML</a>. Assume the following hypotheses:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M123">View MathML</a>,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M124">View MathML</a>isG-complete,

(iii) gisG-continuous and commutes withF.

ThenFandghave a coupled coincidence point, that is, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M125">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M126">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M127">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M128">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M129">View MathML</a>, thenFandghave a common fixed point, that is, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M25">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M131">View MathML</a>.

Theorem 16If in the above theorem, we replace the condition (ii) by the assumption thatXisg-ordered complete, then we have the conclusions of Theorem 15.

We next give the definition of G-compatible mappings inspired by the definition of compatible mappings in [13].

Definition 17 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a> be a G-metric space. The mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M133">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M108">View MathML</a> are said to be G-compatible if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M136">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M138">View MathML</a> are sequences in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M139">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M140">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M6">View MathML</a> are satisfied.

In this paper, we aim to extend the results on coupled fixed points mentioned above. Our results improve, enrich and extend some existing theorems in the literature. We also give examples to illustrate our results. This paper can also be considered as a continuation of the works of Berinde [11,12].

2 Main results

We start with an example which shows the weakness of Theorem 15.

Example 18 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M142">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M143">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M144">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M145">View MathML</a>. Let ⪯ be usual order. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a> is a G-metric space. Define a map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M90">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M148">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M150">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M96">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M152">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M153">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M154">View MathML</a>

(2.2)

It is clear that there is no <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M116">View MathML</a> for which the statement (1.4) of Theorem 15 holds. Notice, however, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M156">View MathML</a> is the unique coupled coincidence point of F and g. In fact, it is a common fixed point of F and g, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M157">View MathML</a>.

We now state our first result which successively guarantees the existence of a coupled coincidence point.

Theorem 19Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a>be a partially ordered set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-completeG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161">View MathML</a>be two mappings such thatFhas the mixedg-monotone property onXand

(2.3)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M163">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M164">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M120">View MathML</a>. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166">View MathML</a>, gisG-continuous and thatFandgareG-compatible mappings. Suppose further that either

(a) Fis continuous or

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>isg-ordered complete.

Suppose also that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M170">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171">View MathML</a>, thenFandghave a coupled coincidence point, that is, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M172">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M173">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M174">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M175">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M114">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M177">View MathML</a>. Using the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166">View MathML</a>, we can construct two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M23">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M138">View MathML</a> in X in the following way:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M181">View MathML</a>

(2.4)

We shall prove that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M182">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M183">View MathML</a>

(2.5)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M114">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M177">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M186">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M187">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M188">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M189">View MathML</a>, that is, (2.5) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M190">View MathML</a>. Assume that (2.5) holds for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M191">View MathML</a>. Since F has the mixed g-monotone property, from (2.4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M192">View MathML</a>

(2.6)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M193">View MathML</a>

(2.7)

By mathematical induction, it follows that (2.5) holds for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M194">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M195">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M196">View MathML</a>

(2.9)

If there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M197">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M198">View MathML</a>, then F and g have a coupled coincidence point. Indeed, in that case we would have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M199">View MathML</a>

We suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M200">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M201">View MathML</a>. More precisely, we assume that either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M202">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M203">View MathML</a>.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M204">View MathML</a>, we set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M205">View MathML</a>

Then by using (2.3) and (2.6), for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M204">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M207">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M208">View MathML</a>

(2.10)

Now, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M209">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M210">View MathML</a>, by using rectangle inequality (G5) of G-metric and (2.10), we get

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M212">View MathML</a>

Then by Proposition 6, we conclude that the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M213">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M214">View MathML</a> are G-Cauchy.

Noting that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M124">View MathML</a> is G-complete, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M216">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M217">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M214">View MathML</a> are G-convergent to x and y respectively, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M219">View MathML</a>

(2.11)

Since F and g are G-compatible mappings, by (2.11), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M220">View MathML</a>

(2.12)

Suppose that the condition (a) holds. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M191">View MathML</a>, we have

(2.13)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M223">View MathML</a> in the above inequality, using (2.11), (2.12) and the continuities of F and g, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M224">View MathML</a>

Hence, we derive that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M126">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M127">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M227">View MathML</a> is a coupled coincidence point of F and g. Suppose that the condition (b) holds. By (2.8), (2.9) and (2.11), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M228">View MathML</a>

(2.14)

Due to the fact that F and g are G-compatible mappings and g is continuous, by (2.11) and (2.12), we have

(2.15)

(2.16)

Keeping (2.15) and (2.16) in mind, we consider now

(2.17)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M223">View MathML</a> in the above inequality, by using (2.15), (2.16) and the continuity of g, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M233">View MathML</a>

(2.18)

By (G1), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M126">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M127">View MathML</a>. Consequently, the element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M236">View MathML</a> is a coupled coincidence point of the mappings F and g. □

Corollary 20Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a>be a partially ordered set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a>isG-complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161">View MathML</a>be two mappings such thatFhas the mixedg-monotone property onXand

(2.19)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M244">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M245">View MathML</a>. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166">View MathML</a>, the self-mappinggisG-continuous andFandgareG-compatible mappings. Suppose that either

(a) Fis continuous or

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>isg-ordered complete.

Suppose also that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M250">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171">View MathML</a>, thenFandghave a coupled coincidence point.

Proof It is sufficient to take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M252">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M253">View MathML</a> in Theorem 19. □

Corollary 21Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a>be a partially ordered set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a>isG-complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161">View MathML</a>be two mappings such thatFhas the mixedg-monotone property onXand

(2.20)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M164">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M262">View MathML</a>. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166">View MathML</a>and that the self-mappinggisG-continuous and commutes withF. Suppose that either

(a) Fis continuous or

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>isg-ordered complete.

Suppose further that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M250">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171">View MathML</a>, thenFandghave a coupled coincidence point.

Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Theorem 19. □

Corollary 22Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a>be a partially ordered set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a>isG-complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M161">View MathML</a>be two mappings such thatFhas the mixedg-monotone property onXand

(2.21)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M244">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M245">View MathML</a>. Assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M166">View MathML</a>and thatgisG-continuous and commutes withF. Suppose that either

(a) Fis continuous or

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>isg-ordered complete.

Assume also that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M250">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171">View MathML</a>, thenFandghave a coupled coincidence point.

Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Corollary 20. □

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M284">View MathML</a> in Theorem 19 and in Corollary 20, we get the following results.

Corollary 23Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a>be a partially ordered set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a>isG-complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160">View MathML</a>be a mapping having the mixed monotone property onXand

(2.22)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M163">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M291">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M292">View MathML</a>. Suppose that either

(a) Fis continuous or

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>is ordered complete.

Suppose also that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M295">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M296">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171">View MathML</a>, thenFhas a coupled fixed point.

Corollary 24Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M64">View MathML</a>be a partially ordered set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M22">View MathML</a>be aG-metric space such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a>isG-complete. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M160">View MathML</a>be a mapping having the mixed monotone property onXand

(2.23)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M243">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M304">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M305">View MathML</a>. Suppose that either

(a) Fis continuous or

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M110">View MathML</a>is ordered complete.

Suppose further that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M168">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M295">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M296">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M171">View MathML</a>, thenFhas a coupled fixed point.

Example 25 Let us recall Example 18. We have

(2.24)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M312">View MathML</a>

(2.25)

It is clear that there any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M313">View MathML</a> provides the statement (2.3) of Theorem 19.

Notice that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M156">View MathML</a> is the unique coupled coincidence point of F and g which is also common coupled fixed point, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M157">View MathML</a>.

Example 26 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M142">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M143">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M318">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M145">View MathML</a>. Let ⪯ be usual order. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M14">View MathML</a> is a G-metric space.

Define a map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M90">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M322">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M66">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M324">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M96">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M326">View MathML</a>. We observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M327">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M328">View MathML</a>

(2.26)

then the statement (2.3) of Theorem 19 is satisfied for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M329">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M156">View MathML</a>.

Notice that if we replace the condition (2.3) of Theorem 19 with the condition (1.4) of Theorem 15 [21], that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M331">View MathML</a>

(2.27)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M332">View MathML</a>, then the coupled coincidence point exists even though the contractive condition is not satisfied.

More precisely, consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M152">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M334">View MathML</a>

(2.28)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M335">View MathML</a>

(2.29)

It is clear that the condition (2.27) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/174/mathml/M336">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The second author gratefully acknowledges the support provided by the Department of Mathematics and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta as a visitor for the short term research.

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