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Some coincidence point results for generalized (ψ,φ)-weakly contractive mappings in ordered G-metric spaces

Zead Mustafa12, Vahid Parvaneh3*, Mujahid Abbas4 and Jamal Rezaei Roshan5

Author Affiliations

1 Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar

2 Department of Mathematics, The Hashemite University, P.O. Box 150459, Zarqa, 13115, Jordan

3 Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

4 Department of Mathematics and Applied Mathematics, University Pretoria, Lynnwood Road, Pretoria, 0002, South Africa

5 Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

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Fixed Point Theory and Applications 2013, 2013:326  doi:10.1186/1687-1812-2013-326

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/326


Received:7 July 2013
Accepted:28 October 2013
Published:2 December 2013

© 2013 Mustafa et al.; licensee Springer.

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

Abstract

The aim of this paper is to present some coincidence point results for six mappings satisfying the generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M1">View MathML</a>-weakly contractive condition in the framework of partially ordered G-metric spaces. To elucidate our results, we present two examples together with an application of a system of integral equations.

MSC: 47H10, 54H25.

Keywords:
coincidence point; common fixed point; generalized weak contraction; generalized metric space; partially weakly increasing mapping; altering distance function

1 Introduction and mathematical preliminaries

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M3">View MathML</a> be a metric space and f be a self-mapping on X. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M4">View MathML</a> for some x in X, then x is called a fixed point of f. The set of all fixed points of f is denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M5">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M6">View MathML</a>, and for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M7">View MathML</a> in a complete metric space X, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M8">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M9">View MathML</a> , converges to z, then f is called a Picard operator.

The function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M10">View MathML</a> is called an altering distance function if φ is continuous and nondecreasing and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M11">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M12">View MathML</a>[1].

A self-mapping f on X is a weak contraction if the following contractive condition holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M13">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>, where φ is an altering distance function.

The concept of a weakly contractive mapping was introduced by Alber and Guerre-Delabrere [2] in the setup of Hilbert spaces. Rhoades [3] considered this class of mappings in the setup of metric spaces and proved that a weakly contractive mapping is a Picard operator.

Let f and g be two self-mappings on a nonempty set X. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M15">View MathML</a> for some x in X, then x is called a common fixed point of f and g. Sessa [4] defined the concept of weakly commutative maps to obtain common fixed point for a pair of maps. Jungck generalized this idea, first to compatible mappings [5] and then to weakly compatible mappings [6]. There are examples which show that each of these generalizations of commutativity is a proper extension of the previous definition.

Zhang and Song [7] introduced the concept of a generalized φ-weak contractive mapping as follows.

Self-mappings f and g on a metric space X are called generalized φ-weak contractions if there exists a lower semicontinuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M16">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M17">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M19">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M21">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M22">View MathML</a>

Based on the above definition, they proved the following common fixed point result.

Theorem 1.1[7]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M3">View MathML</a>be a complete metric space. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M24">View MathML</a>are generalizedφ-weak contractive mappings, then there exists a unique point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M25">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M26">View MathML</a>.

For further works in this direction, we refer the reader to [8-20].

Recently, many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and studied fixed point theory in the so-called bistructural spaces. For more details on fixed point results, its applications, comparison of different contractive conditions and related results in ordered metric spaces, we refer the reader to [21-40] and the references mentioned therein.

Mustafa and Sims [41] generalized the concept of a metric, in which to every triplet of points of an abstract set, a real number is assigned. Based on the notion of generalized metric spaces, Mustafa et al.[42-49] obtained some fixed point theorems for mappings satisfying different contractive conditions. Chugh et al.[50] obtained some fixed point results for maps satisfying property P in G-metric spaces. Saadati et al.[51] studied fixed point of contractive mappings in partially ordered G-metric spaces. Shatanawi [52] obtained fixed points of Φ-maps in G-metric spaces. For more details, we refer to [21,53-65].

Very recently, Jleli and Samet [66] and Samet et al.[67] noticed that some fixed point theorems in the context of a G-metric space can be concluded by some existing results in the setting of a (quasi-)metric space. In fact, if the contraction condition of the fixed point theorem on a G-metric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setting of a usual metric space. More precisely, in [66,67], the authors noticed that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M27">View MathML</a> forms a quasi-metric. Therefore, if one can transform the contraction condition of existence results in a G-metric space in such terms, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M28">View MathML</a>, then the related fixed point results become the known fixed point results in the context of a quasi-metric space.

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

Definition 1.2[41]

Let X be a nonempty set. Suppose that a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M29">View MathML</a> satisfies:

(G1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M30">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M31">View MathML</a>;

(G2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M32">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M34">View MathML</a>;

(G3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M35">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M37">View MathML</a>;

(G4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M38">View MathML</a> (symmetry in all three variables); and

(G5) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M39">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M40">View MathML</a>.

Then G is called a G-metric on X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41">View MathML</a> is called a G-metric space.

Definition 1.3[41]

A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a> in a G-metric space X is:

(i) a G-convergent sequence if there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a> such that for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M45">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M46">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M47">View MathML</a>.

(ii) a G-Cauchy sequence if, for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44">View MathML</a>, there is a natural number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M49">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M50">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M51">View MathML</a>.

A G-metric space on X is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X. It is known that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a>G-converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M54">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M55">View MathML</a>.

Lemma 1.4[41]

LetXbe aG-metric space. Then the following are equivalent:

(1) The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a>isG-convergent tox.

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M57">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M58">View MathML</a>.

(3) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M59">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M58">View MathML</a>.

Lemma 1.5[68]

LetXbe aG-metric space. Then the following are equivalent:

The sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a>isG-Cauchy.

For every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M45">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M46">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M65">View MathML</a>; that is, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M66">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M55">View MathML</a>.

Definition 1.6[41]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M69">View MathML</a> be two G-metric spaces. Then a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M70">View MathML</a> is G-continuous at a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a> if and only if it is G-sequentially continuous at x; that is, whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a> is G-convergent to x, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M73">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M74">View MathML</a>-convergent to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M75">View MathML</a>.

Definition 1.7 A G-metric on X is said to be symmetric if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M76">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>.

Proposition 1.8EveryG-metric onXdefines a metric<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M78">View MathML</a>onXby

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M79">View MathML</a>

(1.1)

For a symmetricG-metric space, one obtains

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M80">View MathML</a>

(1.2)

However, ifGis not symmetric, then the following inequality holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M81">View MathML</a>

(1.3)

Definition 1.9 A partially ordered G-metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a> is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a> in X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M84">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M85">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M86">View MathML</a>).

Definition 1.10 Let f and g be two self-maps on a partially ordered set X. A pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a> is said to be

(i) weakly increasing if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M88">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M89">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M90">View MathML</a>[69],

(ii) partially weakly increasing if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M88">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>[22].

Let X be a nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M93">View MathML</a> be a given mapping. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M95">View MathML</a>.

Definition 1.11 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M96">View MathML</a> be a partially ordered set, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M97">View MathML</a> be mappings such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M98">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M99">View MathML</a>. The ordered pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a> is said to be: (a) weakly increasing with respect to h if and only if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M103">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M104">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M105">View MathML</a>[34], (b) partially weakly increasing with respect to h if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M106">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M103">View MathML</a>[32].

Remark 1.12 In the above definition: (i) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M108">View MathML</a>, we say that f is weakly increasing (partially weakly increasing) with respect to h, (ii) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M109">View MathML</a> (the identity mapping on X), then the above definition reduces to a weakly increasing (partially weakly increasing) mapping (see [34,40]).

The following is an example of mappings f, g, h, R, S and T for which all ordered pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a> are partially weakly increasing with respect to R, S and T but not weakly increasing with respect to them.

Example 1.13 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M113">View MathML</a>. We define functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M114">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M115">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M116">View MathML</a>

Definition 1.14[60,62]

Let X be a G-metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M117">View MathML</a>. The pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a> is said to be compatible if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M119">View MathML</a>, whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a> is a sequence in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M121">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M122">View MathML</a>.

Definition 1.15 (see, e.g., [67])

A quasi-metric on a nonempty set X is a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M123">View MathML</a> such that (p1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M124">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M125">View MathML</a>, (p2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M126">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M127">View MathML</a>. A pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M128">View MathML</a> is said to be a quasi-metric space.

The study of unique common fixed points of mappings satisfying strict contractive conditions has been at the center of vigorous research activity. The study of common fixed point theorems in generalized metric spaces was initiated by Abbas and Rhoades [56] (see also [21,53,54]). Motivated by the work in [8,13,16,17,22,32] and [40], we prove some coincidence point results for nonlinear generalized <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M1">View MathML</a>-weakly contractive mappings in partially ordered G-metric spaces.

2 Main results

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a> be an ordered G-metric space, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M131">View MathML</a> be six self-mappings. Throughout this paper, unless otherwise stated, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33">View MathML</a>, let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M133">View MathML</a>

Let X be any nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M131">View MathML</a> be six mappings such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M138">View MathML</a> be an arbitrary point of X. Choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M140">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M142">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M143">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M144">View MathML</a>. This can be done as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137">View MathML</a>.

Continuing in this way, we construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148">View MathML</a> defined by: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M149">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M150">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152">View MathML</a>. The sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M153">View MathML</a> in X is said to be a Jungck-type iterative sequence with initial guess <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M138">View MathML</a>.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a>be a partially orderedG-completeG-metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M156">View MathML</a>be six mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137">View MathML</a>. Suppose that for every three comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M160">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M161">View MathML</a>

(2.1)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a>are altering distance functions. Letf, g, h, R, SandTbe continuous, the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a>be compatible and the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a>be partially weakly increasing with respect toR, SandT, respectively. Then the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a>have a coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172">View MathML</a>inX. Moreover, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M173">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M174">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M175">View MathML</a>are comparable, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172">View MathML</a>is a coincidence point off, g, h, R, SandT.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148">View MathML</a> be a Jungck-type iterative sequence with initial guess <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M7">View MathML</a> in X; that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M149">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M150">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152">View MathML</a>.

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M183">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M184">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M185">View MathML</a>, and the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a> are partially weakly increasing with respect to R, S and T, so we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M189">View MathML</a>

Continuing this process, we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M190">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152">View MathML</a>.

We will complete the proof in three steps.

Step I. We will prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M192">View MathML</a>.

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M193">View MathML</a>. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M194">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M195">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M196">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M197">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M198">View MathML</a> gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M199">View MathML</a>. Indeed,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M200">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M201">View MathML</a>

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M202">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M203">View MathML</a>; that is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M204">View MathML</a>. Similarly, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M205">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M199">View MathML</a> gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M207">View MathML</a>. Also, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M208">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M207">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M210">View MathML</a>. Consequently, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M211">View MathML</a> becomes constant for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M212">View MathML</a>.

Suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M213">View MathML</a>

(2.2)

for each k. We now claim that the following inequality holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M214">View MathML</a>

(2.3)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M215">View MathML</a> .

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M216">View MathML</a> and for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M152">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M218">View MathML</a>. Then, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M219">View MathML</a>, using (2.1), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M220">View MathML</a>

(2.4)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M221">View MathML</a>

Hence, (2.4) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M222">View MathML</a>

which is possible only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M223">View MathML</a>; that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M224">View MathML</a>, a contradiction to (2.2). Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M225">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M226">View MathML</a>

Therefore, (2.3) is proved for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M216">View MathML</a>.

Similarly, it can be shown that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M228">View MathML</a>

(2.5)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M229">View MathML</a>

(2.6)

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M230">View MathML</a> is a nondecreasing sequence of nonnegative real numbers. Therefore, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M231">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M232">View MathML</a>

(2.7)

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M233">View MathML</a>

(2.8)

taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M234">View MathML</a> in (2.8), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M235">View MathML</a>

(2.9)

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M236">View MathML</a> in (2.4), using (2.7), (2.9) and the continuity of ψ and φ, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M237">View MathML</a>. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M238">View MathML</a>. Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M239">View MathML</a>

(2.10)

from our assumptions about φ. Also, from Definition 1.2, part (G3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M240">View MathML</a>

(2.11)

and, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M241">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M243">View MathML</a>

(2.12)

Step II. We now show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148">View MathML</a> is a G-Cauchy sequence in X. Because of (2.10), it is sufficient to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M245">View MathML</a> is G-Cauchy.

We assume on contrary that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44">View MathML</a> for which we can find subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M247">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M248">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M245">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M250">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M251">View MathML</a>

(2.13)

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M252">View MathML</a> is the smallest number such that the above statement holds; i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M253">View MathML</a>

(2.14)

From the rectangle inequality and (2.14), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M254">View MathML</a>

(2.15)

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M234">View MathML</a> in (2.15), from (2.11) and (2.13) we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M256">View MathML</a>

(2.16)

Using the rectangle inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M257">View MathML</a>

(2.17)

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M234">View MathML</a> in (2.17), from (2.16), (2.10), (2.11) and (2.12) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M259">View MathML</a>

(2.18)

Finally,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M260">View MathML</a>

(2.19)

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M261">View MathML</a> in (2.19) and using (2.16), (2.10), (2.11) and (2.12), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M262">View MathML</a>

Consider

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M263">View MathML</a>

(2.20)

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M261">View MathML</a> and using (2.10), (2.11) and (2.12), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M265">View MathML</a>

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M266">View MathML</a>

(2.21)

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M267">View MathML</a>, so from (2.1) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M268">View MathML</a>

(2.22)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M269">View MathML</a>

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M261">View MathML</a> and using (2.11), (2.12), (2.18) and (2.21) in (2.22), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M271">View MathML</a>

a contradiction. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148">View MathML</a> is a G-Cauchy sequence.

Step III. We will show that f, g, h, R, S and T have a coincidence point.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M148">View MathML</a> is a G-Cauchy sequence in the G-complete G-metric space X, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M274">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M275">View MathML</a>

(2.23)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M276">View MathML</a>

(2.24)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M277">View MathML</a>

(2.25)

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M278">View MathML</a>

(2.26)

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a> is compatible, so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M280">View MathML</a>

(2.27)

Moreover, from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M281">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M282">View MathML</a>, and the continuity of T and f, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M283">View MathML</a>

(2.28)

By the rectangle inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M284">View MathML</a>

(2.29)

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M58">View MathML</a> in (2.29), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M286">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M287">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172">View MathML</a> is a coincidence point of f and T.

Similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M289">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M290">View MathML</a>. Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M173">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M174">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M175">View MathML</a> be comparable. By (2.1) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M294">View MathML</a>

(2.30)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M295">View MathML</a>

Hence (2.30) gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M296">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M297">View MathML</a>. □

In the following theorem, the continuity assumption on the mappings f, g, h, R, S and T is in fact replaced by the sequential limit comparison property of the space, and the compatibility of the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a> is in fact replaced by weak compatibility of the pairs.

Theorem 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a>be a partially orderedG-completeG-metric space with the sequential limit comparison property, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M131">View MathML</a>be six mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M136">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M137">View MathML</a>, RX, SXandTXbeG-complete subsets of X. Suppose that for comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M306">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M307">View MathML</a>

(2.31)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a>are altering distance functions. Then the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a>have a coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172">View MathML</a>inXprovided that the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a>are weakly compatible and the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a>are partially weakly increasing with respect toR, SandT, respectively. Moreover, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M173">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M174">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M175">View MathML</a>are comparable, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M274">View MathML</a>is a coincidence point off, g, h, R, SandT.

Proof Following the proof of Theorem 2.1, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M274">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M324">View MathML</a>

(2.32)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M325">View MathML</a> is G-complete and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M326">View MathML</a>, therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M327">View MathML</a>. Hence, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M25">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M329">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M330">View MathML</a>

(2.33)

Similarly, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M331">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M332">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M333">View MathML</a>

(2.34)

Now, we prove that w is a coincidence point of f and T. For this purpose, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M334">View MathML</a>. Suppose opposite <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M335">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M336">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M236">View MathML</a>, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M338">View MathML</a>.

Therefore, from (2.31), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M339">View MathML</a>

(2.35)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M340">View MathML</a>

Taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M236">View MathML</a> in (2.35), as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M342">View MathML</a> and from (G2) and the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M343">View MathML</a> , we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M344">View MathML</a>

which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M345">View MathML</a>, a contradiction, so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M334">View MathML</a>. Again from the above inequality it is easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M347">View MathML</a>. So, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M348">View MathML</a>.

As f and T are weakly compatible, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M349">View MathML</a>. Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172">View MathML</a> is a coincidence point of f and T.

Similarly it can be shown that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M172">View MathML</a> is a coincidence point of the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a>.

The rest of the proof can be obtained from the same arguments as those in the proof of Theorem 2.1. □

Remark 2.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41">View MathML</a> be a G-metric space. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M355">View MathML</a> be mappings. If we define functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M356">View MathML</a> in the following way:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M357">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M358">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>, it is easy to see that both mappings p and q do not satisfy the conditions of Definition 1.15. Hence, Theorem 2.1 and Theorem 2.2 cannot be characterized in the context of quasi-metric as it is suggested in [66,67].

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M360">View MathML</a> in Theorem 2.1, we obtain the following result.

Corollary 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a>be a partially orderedG-completeG-metric space, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M362">View MathML</a>be four mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M363">View MathML</a>. Suppose that for every three comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M364">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M365">View MathML</a>

(2.36)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M366">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a>are altering distance functions. Thenf, g, handRhave a coincidence point inXprovided that the pairs<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a>are partially weakly increasing with respect toRand either

a. fis continuous and the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M371">View MathML</a>is compatible, or

b. gis continuous and the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a>is compatible, or

c. his continuous and the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M373">View MathML</a>is compatible.

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M374">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M375">View MathML</a> in Theorem 2.1, we obtain the following coincidence point result.

Corollary 2.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a>be a partially orderedG-completeG-metric space, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M377">View MathML</a>be two mappings such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M135">View MathML</a>. Suppose that for every three comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M364">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M380">View MathML</a>

(2.37)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M381">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a>are altering distance functions. Then the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M371">View MathML</a>has a coincidence point inXprovided thatfandRare continuous, the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M371">View MathML</a>is compatible andfis weakly increasing with respect toR.

Example 2.6 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M385">View MathML</a> and G on X be given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M386">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33">View MathML</a>. We define an ordering ‘⪯’ on X as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M388">View MathML</a>

Define self-maps f, g, h, S, T and R on X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M389">View MathML</a>

To prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a> are partially weakly increasing with respect to R, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M392">View MathML</a>; that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M393">View MathML</a>. By the definition of f and R, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M394">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M395">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M396">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M397">View MathML</a>, therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M398">View MathML</a>, or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M399">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M106">View MathML</a>. Hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a> is partially weakly increasing with respect to R. Similarly, one can show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a> are partially weakly increasing with respect to S and T, respectively.

Furthermore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M404">View MathML</a> and the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a> are compatible. Indeed, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a> is a sequence in X such that for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M122">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M410">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M411">View MathML</a>

Continuity of sinh−1 and sinh implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M412">View MathML</a>

and the uniqueness of the limit gives that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M413">View MathML</a>. But

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M414">View MathML</a>

So, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M12">View MathML</a>. Since f and T are continuous, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M416">View MathML</a>

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M418">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M419">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M420">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M421">View MathML</a>.

Using the mean value theorem simultaneously for the functions sinh−1 and sinh, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M422">View MathML</a>

Thus, (2.1) is satisfied for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33">View MathML</a>. Therefore, all the conditions of Theorem 2.1 are satisfied. Moreover, 0 is a coincidence point of f, g, h, R, S and T.

The following example supports the usability of our results for non-symmetric G-metric spaces.

Example 2.7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M424">View MathML</a> be endowed with the usual order. Let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M425">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M426">View MathML</a>

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M427">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M428">View MathML</a>

It is easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41">View MathML</a> is a non-symmetric G-metric space.

Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41">View MathML</a> has the sequential limit comparison property. Indeed, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M42">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M59">View MathML</a> for an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>, there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M434">View MathML</a> such that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M435">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M436">View MathML</a>.

Define the self-maps f, g, h, R, S and T by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M437">View MathML</a>

We see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M438">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M439">View MathML</a>

Also, RX, SX and TX are G-complete. The pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M165">View MathML</a> are weakly compatible.

On the other hand, one can easily check that the pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a> are partially weakly increasing with respect to R, S and T, respectively.

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M447">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M448">View MathML</a>.

According to the definition of f, g, h and G for each three elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M33">View MathML</a>, we see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M450">View MathML</a>

But

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M451">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M452">View MathML</a>

Hence, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M453">View MathML</a>

Therefore, all the conditions of Theorem 2.2 are satisfied. Moreover, 0 is a coincidence point of f, g, h, R, S and T.

Let Λ be the set of all functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M454">View MathML</a> satisfying the following conditions:

(I) μ is a positive Lebesgue integrable mapping on each compact subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M455">View MathML</a>.

(II) For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M44">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M457">View MathML</a>.

Remark 2.8 Suppose that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M458">View MathML</a> such that mappings f, g, h, R, S and T satisfy the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M459">View MathML</a>

(2.38)

Then f, g, h, R, S and T have a coincidence point if the other conditions of Theorem 2.1 are satisfied.

For this, define the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M460">View MathML</a>. Then (2.38) becomes

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M461">View MathML</a>

Take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M462">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M463">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M464">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M465">View MathML</a> are altering distance functions.

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M466">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M467">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M468">View MathML</a> in Theorems 2.1 and 2.2, we obtain the following common fixed point result.

Theorem 2.9Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a>be a partially orderedG-completeG-metric space, and letfandgbe two self-mappings onX. Suppose that for every comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M471">View MathML</a>

(2.39)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M472">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a>are altering distance functions. Then the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>has a common fixed pointzinXprovided that the pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>is weakly increasing and either

a. forgis continuous, or

b. Xhas the sequential limit comparison property.

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M108">View MathML</a> in the above, we obtain the following common fixed point result.

Theorem 2.10Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M82">View MathML</a>be a partially ordered completeG-metric space, and letfbe a self-mapping onX. Suppose that for every comparable elements<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M479">View MathML</a>

(2.40)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M480">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M162">View MathML</a>are altering distance functions. Thenfhas a fixed pointzinXprovided thatfis weakly increasing and either

a. fis continuous, or

b. Xhas the sequential limit comparison property.

3 Existence of a common solution for a system of integral equations

Motivated by the work in [21] and [32], we consider the following system of integral equations:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M482">View MathML</a>

(3.1)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M483">View MathML</a>. The aim of this section is to prove the existence of a solution for (3.1) which belongs to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M484">View MathML</a> (the set of all continuous real-valued functions defined on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M485">View MathML</a>) as an application of Corollary 2.4.

The considered problem can be reformulated as follows.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M97">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M487">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M488">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a> and for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M490">View MathML</a>. Obviously, the existence of a solution for (3.1) is equivalent to the existence of a common fixed point of f, g and h.

Let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M491">View MathML</a>

Equip X with the G-metric given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M492">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M493">View MathML</a>. Evidently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M41">View MathML</a> is a complete G-metric space. We endow X with the partial ordered ⪯ given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M495">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M490">View MathML</a>. It is known that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M96">View MathML</a> has the sequential limit comparison property [37].

Now, we will prove the following result.

Theorem 3.1Suppose that the following hypotheses hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M498">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M499">View MathML</a>are continuous;

(ii) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M500">View MathML</a>and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M502">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M503">View MathML</a>

(iii) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M504">View MathML</a>and for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M506">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M507">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M508">View MathML</a>is a continuous function satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M509">View MathML</a>

Then system (3.1) has a solution<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>.

Proof From condition (ii), the ordered pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M87">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M112">View MathML</a> are partially weakly increasing.

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M14">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M515">View MathML</a>. From condition (iii), for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M490">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M517">View MathML</a>

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M518">View MathML</a>

(3.2)

Similarly, we can show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M519">View MathML</a>

(3.3)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M520">View MathML</a>

(3.4)

Therefore, from (3.2), (3.3) and (3.4), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M521">View MathML</a>

Putting, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M522">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M523">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M524">View MathML</a> in Corollary 2.4, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/326/mathml/M43">View MathML</a>, a common fixed point of f and g and h, which is a solution of (3.1). □

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.

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