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On coupled fixed point results in asymmetric G-metric spaces

Ravi P Agarwal1, Zoran Kadelburg2* and Stojan Radenović3

Author Affiliations

1 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX, 78363-8202, USA

2 Faculty of Mathematics, University of Belgrade, Studentski trg 16, Beograd, 11000, Serbia

3 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Beograd, 11120, Serbia

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Journal of Inequalities and Applications 2013, 2013:528  doi:10.1186/1029-242X-2013-528

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/528


Received:13 March 2013
Accepted:2 September 2013
Published:11 November 2013

© 2013 Agarwal et al.; licensee Springer.

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

Abstract

Using a combination of techniques introduced by Jleli and Samet (Fixed Point Theory Appl. 2012:210, 2012) and Samet et al. (Int. J. Anal. 2013:917158, 2013) on the one hand, and by Kadelburg et al. (Bull. Math. Anal. Appl. 4:51-63, 2012) on the other hand, we show that several coupled fixed point results in (ordered) G-metric spaces obtained recently are simple consequences of the respective standard (ordered) metric results. The technique can be applied both in symmetric and asymmetric cases. Moreover, we show by an example that the results thus obtained are usually stronger than those presented in the literature.

MSC: 47H10, 54H25.

Keywords:
G-metric space; coupled fixed point

1 Introduction

As one of fruitful generalizations of metric spaces, G-metric spaces were introduced by Mustafa and Sims in [1]. In several subsequent papers, these and other authors obtained many fixed point and common fixed point results, thus extending the known theory from the standard metric case. It should be noted that there exist two kinds of G-metric spaces, symmetric and asymmetric ones, and while it was immediately clear that in the symmetric case these results can be easily reduced to their metric counterparts, in the asymmetric case, new proofs usually had to be found.

The notion of a coupled fixed point for mappings with two variables was introduced in the articles [2-4]. After that, a great number of mathematicians worked in this field and obtained a lot of results in metric and various abstract metric spaces (see, e.g., [5-10]). Coupled fixed points in G-metric spaces were investigated, e.g., in the papers [11-23].

Very recently, some new methods were presented for obtaining fixed point and coupled fixed point results. On the one hand, Jleli and Samet [24] and Samet et al.[25] showed that there is a very simple technique for reducing fixed point results in G-metric spaces, both in symmetric and in asymmetric cases, to their metric counterparts, avoiding complicated proofs from the known papers. On the other hand, in the papers [26-33], the authors presented another technique which reduces coupled fixed point results in metric and various abstract metric spaces to the results for mappings with one variable. This technique was used by Kadelburg et al.[34] and afterwards by Agarwal and Karapinar [35] to obtain coupled fixed point results in symmetric G-metric spaces.

By combining the mentioned techniques, we show in this paper that several coupled fixed point results in (ordered) G-metric spaces obtained in recent years and presented in the papers [11-23] simply follow from the well-known standard (ordered) metric results for mappings with one variable. The technique can be applied in both symmetric and asymmetric cases. Moreover, we will show by an example that the results obtained in this way are usually stronger and can be applied in a greater number of cases.

2 Preliminaries

For more details on the following definitions and results concerning G-metric spaces, we refer the reader to [1].

Definition 1 Let be a nonempty set, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M2">View MathML</a> be a function satisfying the following properties:

(G1) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M3">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M4">View MathML</a>;

(G2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M5">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M6">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M7">View MathML</a>;

(G3) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M8">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M9">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M10">View MathML</a>;

(G4) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M11">View MathML</a> (symmetry in all three variables);

(G5) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M12">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M13">View MathML</a> (rectangle inequality).

Then the function g is called a G-metric on and the pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is called a G-metric space.

Definition 2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> be a G-metric space, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17">View MathML</a> be a sequence of points in .

1. A point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M19">View MathML</a> is said to be the limit of a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M21">View MathML</a>, and one says that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17">View MathML</a> is g-convergent to x.

2. The sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M17">View MathML</a> is said to be a G-Cauchy sequence if, for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M24">View MathML</a>, there is a positive integer N such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M25">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M26">View MathML</a>; that is, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M27">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M28">View MathML</a>.

3. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is G-convergent in .

It was shown in [1] that a G-metric induces a Hausdorff topology and that the convergence, as described in the above definition, is relative to this topology. The topology being Hausdorff, a sequence can converge to at most one point.

Definition 3 A G-metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is called symmetric if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M33">View MathML</a>

holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M34">View MathML</a>.

The following are some simple examples of G-metric spaces.

Example 1 (1) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35">View MathML</a> be an ordinary metric space. Define g by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M36">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M37">View MathML</a>. Then it is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is a symmetric G-metric space.

(2) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M39">View MathML</a>. Define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M40">View MathML</a>

and extend g to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M41">View MathML</a> by using the symmetry in the variables. Then it is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is an asymmetric G-metric space.

Remark 1 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is a G-metric space, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M44">View MathML</a>

defines a standard metric on . If the G-metric g is symmetric, this reduces to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M46">View MathML</a>. This simple fact implies that most of the fixed point results in symmetric G-metric spaces can be easily reduced to their metric counterparts. In the asymmetric case, another approach is needed.

Definition 4[3,4,36]

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M47">View MathML</a> be a partially ordered set, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a>.

1. f is said to have the h-mixed monotone property if the following two conditions are satisfied:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M50">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M51">View MathML</a> (the identity map), we say that f has the mixed monotone property.

2. A point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M52">View MathML</a> is said to be a coupled coincidence point of f and h if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M53">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M54">View MathML</a>, and their common coupled fixed point if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M55">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M56">View MathML</a>.

3. The mappings f and h are called w-compatible if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M57">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M58">View MathML</a> whenever <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M53">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M54">View MathML</a>.

If is a nonempty set, then the triple <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62">View MathML</a> is called an ordered G-metric space if:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is a G-metric space, and

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M47">View MathML</a> is a partially ordered set.

3 Main results

We will use the following simple lemma for obtaining our results.

Lemma 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62">View MathML</a>be an orderedG-metric space.

(a) If a relationis defined on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a>by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M67">View MathML</a>

and mappings<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M68">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M69">View MathML</a>are given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M70">View MathML</a>

for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M71">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M72">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M73">View MathML</a>are orderedG-metric spaces. The spaces<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75">View MathML</a>are complete iff<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>is complete.

(b) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a>is a self-map, and a mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>has theh-mixed monotone property, then the mapping<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M79">View MathML</a>given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M80">View MathML</a>

isH-nondecreasing w.r.t. ⊑, i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M81">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M82">View MathML</a>is defined by<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M83">View MathML</a>.

(c) Ifhis continuous in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>, thenHis continuous in both<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75">View MathML</a>. Iffis continuous from<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74">View MathML</a>to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> (resp. from<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75">View MathML</a>to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>), thenFis continuous in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74">View MathML</a> (resp. in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75">View MathML</a>).

Proof We will only check the second part of assertion (a); the proofs of all other parts are straightforward.

It was stated already in [1] that if g is a G-metric on , then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M94">View MathML</a>

define standard metrics on , and that topologies thus generated are the same as the topology of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>. In particular, the completeness is satisfied simultaneously.

On the other hand, it is well known that for each (standard) metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35">View MathML</a>, the mappings

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M98">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M99">View MathML</a> are metrics on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a>, also preserving the completeness property. Combining these two facts, we obtain that the mappings

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M101">View MathML</a>

satisfy all the stated properties. □

Remark 2 If the given G-metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is symmetric, then, using the construction given in [[34], Lemma 3.1] and afterwards in [[35], Sections 4 and 5], we can obtain the previous result in a slightly different way. Namely, in this case, one can consider the G-metric G on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M104">View MathML</a>

(3.1)

and then associate the metrics

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M105">View MathML</a>

It is easy to see that, in fact, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M106">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M107">View MathML</a>.

However, this approach cannot be used in the asymmetric case, since in this case, (3.1) does not define a G-metric on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a> (see [[1], Section 4] and, further, Example 2).

Now we are ready to state some of our main results. We start, as a sample, with the following theorem.

Theorem 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62">View MathML</a>be a complete partially orderedG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a>be mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113">View MathML</a>is closed andfhas the mixedh-monotone property. Suppose that there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M114">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M115">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M116">View MathML</a>. Suppose also that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M118">View MathML</a>

(3.2)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M119">View MathML</a>satisfying (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M120">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M121">View MathML</a>) or (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M122">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M123">View MathML</a>). Let us assume also that if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M124">View MathML</a>is a nondecreasing sequence inconverging to some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M126">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M127">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M128">View MathML</a>. Then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharew-compatible, thenfandhhave a common coupled fixed point.

Proof Consider the complete partially ordered metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M72">View MathML</a> and the mappings <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M79">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M82">View MathML</a> defined in Lemma 1. Then, obviously, the following conditions hold:

1. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M132">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M133">View MathML</a> is closed;

2. F is H-nondecreasing;

3. There exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M134">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M135">View MathML</a>.

Moreover, it follows from (3.2) that there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M137">View MathML</a>

(3.3)

holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M138">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M139">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M140">View MathML</a>. Now, all the conditions of a special case of [[37], Theorem 2.2] are fulfilled and it follows that the mappings F and H have a coincidence point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M141">View MathML</a> which is, obviously, a coupled coincidence point of f and h. The last assertion also follows easily. □

Corollary 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M62">View MathML</a>be a complete partially orderedG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a>be mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113">View MathML</a>is closed andfhas the mixedh-monotone property. Suppose that there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M114">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M115">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M116">View MathML</a>. Suppose also that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M151">View MathML</a>

(3.4)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M152">View MathML</a>satisfying (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M153">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M154">View MathML</a>) or (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M155">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M156">View MathML</a>). Let us assume also that if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M124">View MathML</a>is a nondecreasing sequence inconverging to some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M126">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M127">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M128">View MathML</a>. Then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharew-compatible, thenfandhhave a common coupled fixed point.

Proof We have only to prove that condition (3.4) implies condition (3.2). Indeed, putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M162">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M163">View MathML</a> in (3.4), we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M164">View MathML</a>

and putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M165">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M166">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M167">View MathML</a>

Adding up, and taking into account the definitions of mappings F and H as well as the definition of metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M168">View MathML</a>, we obtain condition (3.2) (i.e., condition (3.3)). □

In an even easier way, one can obtain the following versions of the previous results in the space without order. In this case, the given technique reduces the problem simply to the Banach contraction principle.

Theorem 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>be a completeG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a>be mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113">View MathML</a>is closed. Suppose also that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a>such that condition (3.2) holds for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M119">View MathML</a>. Ifhis continuous, then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharew-compatible, thenfandhhave a common coupled fixed point.

Corollary 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>be a completeG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a>be mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M112">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M113">View MathML</a>is closed. Suppose also that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a>such that condition (3.4) holds for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M152">View MathML</a>. Ifhis continuous, then the mappingsfandhhave a coupled coincidence point. Moreover, iffandharew-compatible, thenfandhhave a common coupled fixed point.

Remark 3 The last result was obtained (in the special case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M183">View MathML</a>) in [[34], Corollary 4.1] and afterwards (implicitly) in the course of proof of [[35], Theorem 5.3], but in the case when the given G-metric g is symmetric. The proof from these articles cannot be applied in the asymmetric case (see Remark 2 and, further, Example 2).

Remark 4 The obtained results are strict improvements of some results obtained earlier. In the symmetric case, this was shown by Kadelburg et al. (see [[34], Example 4.1]). We present an example in an asymmetric case.

Example 2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> be the G-metric space considered in [[38], Example 3.4], i.e., let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M185">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M2">View MathML</a> be given as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M187">View MathML</a>

and extended by symmetry in the variables. Then it is easy to check that g is a G-metric which is asymmetric since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M188">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M189">View MathML</a>.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M49">View MathML</a> be given as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M191">View MathML</a>, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M193">View MathML</a>

We will show first that f and h satisfy neither the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M194">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a> (which is condition (3.1) from [15]) nor (weaker) condition (3.4). Indeed, take, e.g., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M196">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M197">View MathML</a> and condition (3.4) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M198">View MathML</a>

which is a contradiction for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a>. Hence, neither [[15], Theorem 3.1] nor Corollary 2 can be used to conclude that f and h have a coupled coincidence point (i.e., that f has a coupled fixed point).

In order to show that f and h satisfy the conditions of Theorem 2, we first note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M200">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M201">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M202">View MathML</a>

Take now <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M203">View MathML</a>. By a careful calculation (there are 21 nontrivial cases) it can be checked that condition (3.2) (i.e., condition (3.3)) is satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M138">View MathML</a>.

Hence, Theorem 2 can be applied to conclude that f has a coupled fixed point (which is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M205">View MathML</a>).

We note also that the approach from the papers [34,35] cannot be used in this example since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M206">View MathML</a>

does not define a G-metric on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a>. Indeed, e.g.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M208">View MathML</a>

although <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M209">View MathML</a>, and the property (G3) of G-metrics is not satisfied.

For the sake of simplicity, we shall consider in the rest of the paper ‘unordered’ versions of coupled fixed point results. ‘Ordered’ versions can be formulated and proved using usual variations.

The proof of our next result uses the metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M210">View MathML</a>.

Theorem 3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>be a completeG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>. Suppose that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M117">View MathML</a>such that for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M214">View MathML</a>the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M215">View MathML</a>

(3.5)

Thenfhas a unique coupled fixed point in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a>and it is of the form<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M217">View MathML</a>for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M19">View MathML</a>.

Proof Consider the complete metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M75">View MathML</a> and the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M79">View MathML</a> defined in Lemma 1. Putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M162">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M163">View MathML</a> in (3.5), one gets

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M223">View MathML</a>

interchanging the places of x and y, as well as of s and t, this gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M224">View MathML</a>

Putting now <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M165">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M166">View MathML</a> in (3.5), one gets

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M227">View MathML</a>

interchanging the places of x and y, as well as of s and t, this gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M228">View MathML</a>

Taking into account the definition of metric <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M210">View MathML</a> in Lemma 1, it follows from the last four inequalities that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M230">View MathML</a>

holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M138">View MathML</a>. By a well-known result from the theory of standard metric spaces (see, e.g., [39]), it follows that there exists a unique point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M233">View MathML</a>. Obviously, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M234">View MathML</a> is a coupled fixed point of the mapping f. Since the coupled fixed point is unique, it must be of the form <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M217">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M19">View MathML</a>. □

When considering situations where contractive conditions are of ‘weak’ kind, or they use the so-called comparison functions, a variation of the previous approach is needed. One possibility is to use quasi-metrics. Recall that a pair <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35">View MathML</a> is called a quasi-metric space if the mapping d has all the properties of a metric except, possibly, the symmetry <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M238">View MathML</a>. For some properties of quasi-metric spaces, we refer to [24]. In particular, the following fixed point result was proved in that paper.

Theorem 4 [[24], Theorem 3.2]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M35">View MathML</a>be a complete quasi-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M240">View MathML</a>be a mapping satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M241">View MathML</a>

(3.6)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M242">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M243">View MathML</a>is continuous with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M244">View MathML</a>. ThenThas a unique fixed point.

Remark 5 An additional nondecreasing function ψ could be added in (3.6) to become

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M245">View MathML</a>

However, it was shown in [40] that it is redundant, hence we will not use it here.

Obviously, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is a G-metric space, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M247">View MathML</a>

defines a quasi-metric on . Moreover,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M249">View MathML</a>

(3.7)

defines a quasi-metric on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a>. The space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M251">View MathML</a> is complete iff <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a> is G-complete. Now we can easily prove the following theorem.

Theorem 5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>be a completeG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>satisfy

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M255">View MathML</a>

(3.8)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M256">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M243">View MathML</a>is continuous with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M244">View MathML</a>. Thenfhas a unique coupled fixed point.

Proof Consider the (complete) quasi-metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M251">View MathML</a> given by (3.7) and the self-mapping F given on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M66">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M261">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M141">View MathML</a>. Then contractive condition (3.8) gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M263">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M264">View MathML</a>. Hence, Theorem 4 can be applied to conclude that F has a unique fixed point, which is then a coupled fixed point of f. □

Another possibility is to impose an additional condition on the function φ, or on the comparison function Φ. This is illustrated in the next result.

Theorem 6Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M15">View MathML</a>be a completeG-metric space, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M48">View MathML</a>satisfy the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M267">View MathML</a>

(3.9)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M152">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M269">View MathML</a>is right-continuous, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M270">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M271">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M272">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M273">View MathML</a>. Thenfhas a unique coupled fixed point.

Proof Consider again the space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M74">View MathML</a> and the mapping F given in Lemma 1. Putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M162">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M163">View MathML</a> in (3.9), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M277">View MathML</a>

interchanging places of x and y, as well as s and t, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M278">View MathML</a>

On the other hand, putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M165">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M166">View MathML</a> in (3.9), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M281">View MathML</a>

interchanging places of x and y, as well as s and t, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M282">View MathML</a>

Adding up the last four inequalities, and using the assumed properties of function Φ, we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M283">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M284">View MathML</a>. Hence, by a classical result of Boyd and Wong [41], it follows that F has a unique fixed point, which is then a coupled fixed point of f. □

It is clear that a lot of other known coupled fixed point results in G-metric spaces (both symmetric and asymmetric) can be easily obtained in this way.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors are indebted to the referees whose suggestions helped them to improve the exposition. The second and third authors are thankful to the Ministry of Education, Science and Technological Development of Serbia.

References

  1. Mustafa, Z, Sims, B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal.. 7(2), 289–297 (2006)

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

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

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

  5. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl.. 2011, (2011) Article ID 81

  6. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled fixed point theorems for weak contraction mapping under F-invariant set. Abstr. Appl. Anal.. 2012, (2012) Article ID 324874

  7. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled fixed point theorem for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 128

  8. Sintunavarat, W, Petruşel, A, Kumam, P: Common coupled fixed point theorems for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M285">View MathML</a>-compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo. 61, 361–382 (2012). Publisher Full Text OpenURL

  9. Sintunavarat, W, Kumam, P, Cho, YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl.. 2012, (2012) Article ID 170

  10. Karapınar, E, Kumam, P, Sintunavarat, W: Coupled fixed point theorems in cone metric spaces with a c-distance and applications. Fixed Point Theory Appl.. 2012, (2012) Article ID 194

  11. Abbas, M, Sintunavarat, W, Kumam, P: Coupled fixed point of generalized contractive mappings on partially ordered G-metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 31

  12. Aydi, H, Damjanović, B, Samet, B, Shatanawi, W: Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math. Comput. Model.. 54, 2443–2450 (2011). Publisher Full Text OpenURL

  13. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/528/mathml/M287">View MathML</a>-weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl.. 63, 298–309 (2012). Publisher Full Text OpenURL

  14. Cho, YJ, Rhoades, BE, Saadati, R, Samet, B, Shatanawi, W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl.. 2012, (2012) Article ID 8

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

  16. Karapinar, E, Kaymakcalan, B, Taş, K: On coupled fixed point theorems on partially ordered G-metric spaces. J. Inequal. Appl.. 2012, (2012) Article ID 200

  17. Karapinar, E, Kumam, P, Erhan, IM: Coupled fixed points on partially ordered G-metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 174

  18. Luong, NV, Thuan, NX: Coupled fixed point theorems in partially ordered G-metric space. Math. Comput. Model.. 55, 1601–1609 (2012). Publisher Full Text OpenURL

  19. Mohiuddine, SM, Alotaibi, A: On coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Abstr. Appl. Anal.. 2012, (2012) Article ID 897198

  20. Nashine, HK: Coupled common fixed point results in ordered G-metric spaces. J. Nonlinear Sci. Appl.. 1, 1–13 (2012)

  21. Shatanawi, W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat.. 40, 441–447 (2011)

  22. Shatanawi, W, Abbas, M, Nazir, T: Common coupled coincidence and coupled fixed point results in two generalized metric space. Fixed Point Theory Appl.. 2011, (2011) Article ID 80

  23. Wangkeeree, R: Coupled fixed point theorems for generalized contractive mappings in partially ordered G-metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 172

  24. Jleli, M, Samet, B: Remarks on G-metric spaces and fixed point theorems. Fixed Point Theory Appl.. 2012, (2012) Article ID 210

  25. Samet, B, Vetro, C, Vetro, F: Remarks on G-metric spaces. Int. J. Anal.. 2013, (2013) Article ID 917158

  26. Amini-Harandi, A: Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem. Math. Comput. Model.. 57, 2343–2348 (2013). Publisher Full Text OpenURL

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

  28. Berinde, V: Coupled fixed point theorems for ϕ-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.. 75, 3218–3228 (2012). Publisher Full Text OpenURL

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

  30. Golubović, Z, Kadelburg, Z, Radenović, S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal.. 2012, (2012) Article ID 192581

  31. Kadelburg, Z, Radenović, S: Coupled fixed point results under tvs-cone metric and w-cone-distance. Adv. Fixed Point Theory. 2, 29–46 (2012)

  32. Jleli, M, Rajić, VĆ, Samet, B, Vetro, C: Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations. J. Fixed Point Theory Appl.. 12, 175–192 (2012). Publisher Full Text OpenURL

  33. Kadelburg, Z, Nashine, HK, Radenović, S: Coupled fixed point results in 0-complete ordered partial metric spaces. J. Adv. Math. Stud.. 6, 159–172 (2013)

  34. Kadelburg, Z, Nashine, HK, Radenović, S: Common coupled fixed point results in partially ordered G-metric spaces. Bull. Math. Anal. Appl.. 4, 51–63 (2012)

  35. Agarwal, RP, Karapinar, E: Remarks on some coupled fixed point theorems in G-metric spaces. Fixed Point Theory Appl.. 2013, (2013) Article ID 2

  36. Abbas, M, Ilić, D, Khan, MA: Coupled coincidence point and coupled common fixed point theorems in partially ordered metric spaces with w-distance. Fixed Point Theory Appl.. 2010, (2010) Article ID 134897

  37. Ćirić, LB, Cakić, N, Rajović, M, Ume, JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl.. 2008, (2008) Article ID 131294

  38. Nashine, HK, Kadelburg, Z, Radenović, S: Coincidence and fixed point results under generalized weakly contractive condition in partially ordered G-metric spaces. Filomat (to appear)

  39. Ćirić, LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc.. 45, 267–273 (1974)

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

  41. Boyd, DW, Wong, JS: On nonlinear contractions. Proc. Am. Math. Soc.. 20, 458–464 (1969). Publisher Full Text OpenURL