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A new common coupled fixed point theorem in generalized metric space and applications to integral equations

Feng Gu* and Yun Yin

Author Affiliations

Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, 310036, China

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Fixed Point Theory and Applications 2013, 2013:266  doi:10.1186/1687-1812-2013-266


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/266


Received:8 June 2013
Accepted:19 September 2013
Published:7 November 2013

© 2013 Gu and Yin; licensee Springer.

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

Abstract

In the present paper, we prove a common coupled fixed point theorem in the setting of a generalized metric space in the sense of Mustafa and Sims. Our results improve and extend the corresponding results of Shatanawi. We also present an application to integral equations.

Keywords:
G-metric space; common coupled coincidence fixed point; common fixed point; integral equation

1 Introduction and preliminaries

The study of fixed points of mappings satisfying certain contractive conditions has been in the center of rigorous research activity. For a survey of common fixed point theory in metric and cone metric spaces, we refer the reader to [1-9]. In 2006, Bhaskar and Lakshmikantham [10] initiated the study of a coupled fixed point in ordered metric spaces and applied their results to prove the existence and uniqueness of solutions for a periodic boundary value problem. For more works in coupled and coincidence point theorems, we refer the reader to [11-13].

Some authors generalized the concept of metric spaces in different ways. Mustafa and Sims [14] introduced the notion of G-metric space, in which the real number is assigned to every triplet of an arbitrary set as a generalization of the notion of metric spaces. Based on the notion of G-metric spaces, many authors (for example, [15-33]) obtained some fixed point and common fixed point theorems for mappings satisfying various contractive conditions. Fixed point problems have also been considered in partially ordered G-metric spaces [34-39].

The purpose of this paper is to obtain some common coupled coincidence point theorems in G-metric spaces satisfying some contractive conditions.

The following definitions and results will be needed in the sequel.

Definition 1.1[14]

Let X be a nonempty set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M1">View MathML</a> be a function satisfying the following axioms:

(G1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M2">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M3">View MathML</a>;

(G2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M4">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M5">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M6">View MathML</a>;

(G3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M7">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M8">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M9">View MathML</a>;

(G4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M10">View MathML</a> (symmetry in all three variables);

(G5) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M12">View MathML</a> (rectangle inequality),

then the function G is called a generalized metric, or more specifically, a G-metric on X, and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a> is called a G-metric space.

Definition 1.2[14]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a> be a G-metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a> be a sequence of points in X, a point x in X is said to be the limit of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M17">View MathML</a>, and one says that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a> is G-convergent to x.

Thus, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M19">View MathML</a> in a G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>, then for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M21">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M22">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M23">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M24">View MathML</a>.

Proposition 1.3[14]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space, then the following are equivalent:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a>isG-convergent tox.

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M27">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M28">View MathML</a>.

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M29">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M28">View MathML</a>.

(4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M31">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M32">View MathML</a>.

Definition 1.4[14]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> be a G-metric space. A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a> is called G-Cauchy sequence if for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M35">View MathML</a>, there exists a positive integer <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M22">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M37">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M38">View MathML</a>; i.e., if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M39">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M40">View MathML</a>.

Definition 1.5[14]

A G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> is said to be G-complete if every G-Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> is G-convergent in X.

Proposition 1.6[14]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a>be aG-metric space, then the following are equivalent:

(1) The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a>isG-Cauchy.

(2) For every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M21">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M46">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M47">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M48">View MathML</a>.

Proposition 1.7[14]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a>be aG-metric space. Then the function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M50">View MathML</a>is jointly continuous in all three of its variables.

Definition 1.8[14]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M52">View MathML</a> be G-metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M53">View MathML</a> be a function. Then f is said to be G-continuous at a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M54">View MathML</a> if and only if for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M21">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M56">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M58">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M59">View MathML</a>. A function f is G-continuous at X if and only if it is G-continuous at all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M54">View MathML</a>.

Proposition 1.9[14]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M52">View MathML</a>beG-metric spaces, then a function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M63">View MathML</a>isG-continuous at a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>if and only if it isG-sequentially continuous atx; that is, whenever<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M65">View MathML</a>isG-convergent tox, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M66">View MathML</a>isG-convergent to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M67">View MathML</a>.

Proposition 1.10[14]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a>be aG-metric space. Then for anyx, y, z, ainX, it follows that

(i) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M2">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M3">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M71">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M72">View MathML</a>;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M73">View MathML</a>;

(v) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M74">View MathML</a>;

(vi) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M75">View MathML</a>.

Definition 1.11[10]

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M76">View MathML</a> is called a coupled fixed point of a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M78">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M79">View MathML</a>.

Definition 1.12[11]

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M76">View MathML</a> is called a coupled coincidence point of the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M83">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M84">View MathML</a>.

Definition 1.13[11]

Let X be a nonempty set. Then we say that the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a> are commutative if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M87">View MathML</a>.

2 Main results

We start our work by proving the following crucial lemma.

Lemma 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90">View MathML</a>be four mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M91">View MathML</a>

(2.1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M94">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M95">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M96">View MathML</a>is a common coupled coincidence point of the mappings pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M97">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M98">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M99">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M100">View MathML</a>

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M96">View MathML</a> is a common coupled coincidence point of the mappings pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M97">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M98">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M99">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M105">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M106">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M107">View MathML</a>. Then by (2.1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M108">View MathML</a>

Also by (2.1), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M109">View MathML</a>

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M110">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M111">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M112">View MathML</a>

which is a contradiction. So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M113">View MathML</a>, and hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M114">View MathML</a>

 □

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a>be four mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M118">View MathML</a>

(2.2)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M94">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M122">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>andgsatisfy the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M128">View MathML</a>;

(ii) gXisG-complete;

(iii) gisG-continuous and commutes with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>.

Then there exist unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M133">View MathML</a>

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M134">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M128">View MathML</a>, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M138">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M139">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M140">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M141">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M142">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M143">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M144">View MathML</a>. Combining this process, we can construct two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M15">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M146">View MathML</a> in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M147">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M148">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M149">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M150">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M151">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M152">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M153">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M154">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M155">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M156">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M157">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M158">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M159">View MathML</a>. On the other hand, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M160">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M161">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M151">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M150">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M164">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M165">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M155">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M154">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M168">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M169">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M159">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M158">View MathML</a>. Without loss of generality, we can assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M172">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M173">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M174">View MathML</a> .

By (2.2) and (G3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M175">View MathML</a>

(2.3)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M176">View MathML</a>

(2.4)

By combining (2.3) and (2.4), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M177">View MathML</a>

(2.5)

In the same way, we can show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M178">View MathML</a>

(2.6)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M179">View MathML</a>

(2.7)

It follows from (2.5), (2.6) and (2.7) that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M180">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M181">View MathML</a>

(2.8)

Where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M182">View MathML</a>. From (G3), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M183">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M184">View MathML</a>. Hence, by the (G3) and (2.8), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M185">View MathML</a>

(2.9)

Therefore, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M186">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M187">View MathML</a>, by (G5) and (2.9), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M188">View MathML</a>

(2.10)

Which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M189">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M190">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M191">View MathML</a> are all G-Cauchy in gX. Since gX is G-complete, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M190">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M191">View MathML</a> are G-convergent to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M194">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M195">View MathML</a>, respectively. Since g is G-continuous, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M196">View MathML</a> is G-convergent to gx and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M197">View MathML</a> is G-convergent to gy. That is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M198">View MathML</a>

(2.11)

Also, since g commutes with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>, respectively, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M202">View MathML</a>

Thus, from condition (2.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M203">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M204">View MathML</a>, using (2.11) and the fact that G is continuous on its variables, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M205">View MathML</a>

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M153">View MathML</a>. Similarly, we may show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M165">View MathML</a>. Also for the same reason, we may show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M149">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M161">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M157">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M169">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M96">View MathML</a> is a common coupled coincidence point of the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M97">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M98">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M99">View MathML</a>. By Lemma 2.1, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M216">View MathML</a>

(2.12)

Since the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M217">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M218">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M219">View MathML</a> are all a subsequence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M190">View MathML</a>, then they are all G-convergent to x. Similarly, we may show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M221">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M222">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M223">View MathML</a> are all G-convergent to y. From (2.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M224">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M204">View MathML</a>, and using the fact that G is continuous on its variables, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M226">View MathML</a>

Similarly, we may show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M227">View MathML</a>

Thus, using the Proposition 1.10(iii), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M228">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M229">View MathML</a>, so the last inequality happens only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M230">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M231">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M232">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M233">View MathML</a>. From (2.12), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M234">View MathML</a>, thus, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M235">View MathML</a>

To prove the uniqueness, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M236">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M237">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M238">View MathML</a>

Again using condition (2.2) and Proposition 1.10(iii), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M239">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M229">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M241">View MathML</a>, which is a contradiction. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a> and g have a unique common fixed point. □

Remark 2.1 Theorem 2.1 extends and improves Theorem 3.2 of Shatanawi [26].

The following corollary can be obtained from Theorem 2.1 immediately.

Corollary 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a>be mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M248">View MathML</a>

(2.13)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M251">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M252">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>andgsatisfy the following conditions:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127">View MathML</a>;

(2) gXisG-complete;

(3) gisG-continuous and commutes with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>.

Then there exist unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M263">View MathML</a>

Remark 2.2 If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M264">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M265">View MathML</a>, then Corollary 2.1 is reduced to Theorem 3.2 of Shatanawi [26].

Now, we give an example to support Corollary 2.1.

Example 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M266">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M267">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M268">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> is a complete G-metric space. Define a map

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M271">View MathML</a>

by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M272">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57">View MathML</a>. Also, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M275">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277">View MathML</a>. Through calculation, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M278">View MathML</a>

Then the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a> and g are satisfying condition (2.13) of Corollary 2.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M282">View MathML</a>. So that all the conditions of Corollary 2.1 are satisfied. By Corollary 2.4, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a> and g have a unique common fixed point. Moreover, 0 is the unique common fixed point for all of the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a> and g.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M289">View MathML</a>, then Theorem 2.1 is reduced to the following.

Corollary 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a>be four mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M293">View MathML</a>

(2.14)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M295">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M296">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M297">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>andgsatisfy the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M126">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M127">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M128">View MathML</a>;

(ii) gXisG-complete;

(iii) gisG-continuous and commutes with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a>.

Then there exist unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M308">View MathML</a>

If we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M264">View MathML</a> in Corollary 2.2, then the following corollary is obtained.

Corollary 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a>be four mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M313">View MathML</a>

(2.15)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M295">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M296">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M297">View MathML</a>. Suppose thatFandgsatisfy the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277">View MathML</a>;

(ii) gXisG-complete;

(iii) gisG-continuous and commutes withF.

Then there exist unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M320">View MathML</a>

Now, we give an example to support Corollary 2.3.

Example 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M266">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M267">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M323">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> is a complete G-metric space. Define a map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M326">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M327">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57">View MathML</a>. Also, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M330">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277">View MathML</a>. Through calculation, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M333">View MathML</a>

Then the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a> and g are satisfying condition (2.15) of Corollary 2.3 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M337">View MathML</a>. So that all the conditions of Corollary 2.3 are satisfied. By Corollary 2.3, F and g have a unique common fixed point. Moreover, 0 is the unique common fixed point for all of the mappings F and g.

If we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M264">View MathML</a> in Theorem 2.1, then the following corollary is obtained.

Corollary 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M13">View MathML</a>be aG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M77">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M82">View MathML</a>be mappings such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M342">View MathML</a>

(2.16)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M93">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M94">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M122">View MathML</a>. Suppose thatFandgsatisfy the following conditions:

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M277">View MathML</a>;

(2) gXisG-complete;

(3) gisG-continuous and commutes withF.

Then there exist unique<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M349">View MathML</a>.

Now, we introduce an example to support Corollary 2.4.

Example 2.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M350">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M267">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M352">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> is a complete G-metric space. Define a map

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M355">View MathML</a>

by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M356">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57">View MathML</a>. Also, define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M90">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M330">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>.

Clearly, we can get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M361">View MathML</a>, and g is G-continuous and commutes with F.

By the definition of the mappings of F and g, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M362">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M363">View MathML</a>

Then the mappings F and g are satisfying condition (2.16) of Corollary 2.4 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M289">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M365">View MathML</a>. So that all the conditions of Corollary 2.4 are satisfied. By Corollary 2.4, F and g have a unique common fixed point. Here <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M366">View MathML</a> is the unique common fixed point of mappings F and g; that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M367">View MathML</a>.

3 Application to integral equations

Throughout this section, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M368">View MathML</a> is the set of all continuous functions defined on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M369">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M370">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M371">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M269">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M33">View MathML</a> is a G-complete metric space.

Consider the following integral equations:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M374">View MathML</a>

(3.1)

Next, we will analyze (3.1) under the following conditions:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M375">View MathML</a> is continuous.

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M376">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M377">View MathML</a>) are continuous functions.

(iii) There exist constants <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M378">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M379">View MathML</a>) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M380">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M381">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M382">View MathML</a>.

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M383">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M384">View MathML</a>

The aim of this section is to give an existence theorem for a solution of the above integral equations by using the obtained result given by Theorem 2.1.

Theorem 3.1Under conditions (i)-(iv), integral equation (3.1) has a unique common solution in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M385">View MathML</a>.

Proof First, we consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M386">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M387">View MathML</a>). By virtue of our assumptions, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M388">View MathML</a> is well defined (this means that for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M57">View MathML</a> then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M390">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M387">View MathML</a>)). Then we can get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M392">View MathML</a>

(3.2)

By conditions (iii),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M393">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M394">View MathML</a>

Taking these inequalities into (3.2), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M395">View MathML</a>

(3.3)

Using the Cauchy-Schwartz inequality in (3.3), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M396">View MathML</a>

(3.4)

Similarly, we can obtain the following estimate

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M397">View MathML</a>

(3.5)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M398">View MathML</a>

(3.6)

Substituting (3.4), (3.5) and (3.6) into (3.3), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M399">View MathML</a>

(3.7)

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M330">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M64">View MathML</a>, and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M402">View MathML</a>

then inequality (3.7) becomes

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M403">View MathML</a>

(3.8)

By condition (iv), we know that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M404">View MathML</a>

This proves that the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M388">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M387">View MathML</a>) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M407">View MathML</a> satisfy contractive condition (2.2) appearing in Theorem 2.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M407">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M123">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M124">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M125">View MathML</a> have a unique common coupled fixed point, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M412">View MathML</a>, and so, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/266/mathml/M413">View MathML</a> is the unique solution of equation (3.1). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this work. Both authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the editor and the reviewer for suggestions which improved the contents of the article. This work is supported by the National Natural Science Foundation of China (11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030) and the Physical Experiment Center of Hangzhou Normal University.

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