SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

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

Open Access Research

Some coupled coincidence point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions

Saud M Alsulami

Author Affiliations

Department of Mathematics, King Abdulaziz University, P.O. Box 138381, Jeddah, 21323, Saudi Arabia

Fixed Point Theory and Applications 2013, 2013:194  doi:10.1186/1687-1812-2013-194

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/194


Received:1 April 2013
Accepted:4 July 2013
Published:22 July 2013

© 2013 Alsulami; licensee Springer

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

Abstract

In this paper, we present some coupled coincidence point results for mixed g-monotone mappings in partially ordered complete metric spaces involving altering distance functions. Moreover, we present an example to illustrate our main result. Our results extend some results in the field.

MSC: 47H09, 47H10, 49M05.

Keywords:
coupled coincidence points; partially metric spaces; contractive mappings; mixed g-monotone property

1 Introduction and preliminaries

The existence of a fixed point for contractive mappings in partially ordered metric spaces has attracted the attention of many mathematicians (cf.[1-11] and the references therein). In [3], Bhaskar and Lakshmikantham introduced the notion of a mixed monotone mapping and proved some coupled fixed point theorems for the mixed monotone mapping. Afterwards, Lakshmikantham and Ciric in [11] introduced the concept of a mixed g-monotone mapping and proved coupled coincidence point results for two mappings F and g, where F has the mixed g-monotone property and the functions F and g commute. It is well known that the concept of commuting has been weakened in various directions. One such notion which is weaker than commuting is the concept of compatibility introduced by Jungck [7]. In [5], Choudhury and Kundu defined the concept of compatibility of F and g. The purpose of this paper is to present some coupled coincidence point theorems for a mixed g-monotone mapping in the context of complete metric spaces endowed with a partial order by using altering distance functions which extend some results of [6]. We also present an example which illustrates the results.

Recall that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1">View MathML</a> is a partially ordered set, then f is said to be non-decreasing if for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M3">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M4">View MathML</a>. Similarly, f is said to be non-increasing if for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M3">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M7">View MathML</a>. We also recall the used definitions in the present work.

Definition 1.1[11] (Mixed g-monotone property)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1">View MathML</a> be a partially ordered set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M10">View MathML</a>. We say that the mapping F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument. That is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M12">View MathML</a>

(1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M13">View MathML</a>

(2)

Definition 1.2[11] (Coupled coincidence fixed point)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M14">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M15">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M16">View MathML</a>. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17">View MathML</a> is a coupled coincidence point of F and g if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M18">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M19">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2">View MathML</a>.

Definition 1.3[11]

Let X be a non-empty set and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M22">View MathML</a>. We say F and g are commutative if, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M23">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M24">View MathML</a>

Definition 1.4[5]

The mappings F and g, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M25">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M26">View MathML</a>, are said to be compatible if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M27">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M28">View MathML</a>

whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M29">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M30">View MathML</a> are sequences in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M31">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M32">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2">View MathML</a>.

Definition 1.5 (Altering distance function)

An altering distance function is a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M34">View MathML</a> satisfying

1. ψ is continuous and non-decreasing.

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M35">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M36">View MathML</a>.

2 Existence of coupled coincidence points

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1">View MathML</a> be a partially ordered set and suppose that there exists a metric d in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38">View MathML</a> is a complete metric space. Also, let φ and ϕ be altering distance functions. Now, we are in a position to state our main theorem.

Theorem 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21">View MathML</a>be a mapping having the mixedg-monotone property onXsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M40">View MathML</a>

(3)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M41">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M42">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M43">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M44">View MathML</a>, gis continuous, monotone increasing and suppose also thatFandgare compatible mappings. Moreover, suppose either

(a) Fis continuous, or

(b) Xhas the following properties:

(i) if a non-decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M45">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M46">View MathML</a>for alln,

(ii) if a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M47">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M48">View MathML</a>for alln.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M49">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M50">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M51">View MathML</a>, thenFandghave a coupled coincidence point.

Proof By using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M44">View MathML</a>, we construct sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M53">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M54">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M55">View MathML</a>

(4)

We are going to divide the proof into several steps in order to make it easy to read.

Step 1. We will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M56">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M57">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M58">View MathML</a>.

We use the mathematical induction to show that. From the assumption of the theorem, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M59">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M60">View MathML</a>, so our claim is satisfied for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M61">View MathML</a>. Now, suppose that our claim holds for some fixed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M62">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M63">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M64">View MathML</a> and F has the mixed g-monotone property, then we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M65">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M66">View MathML</a>

Thus the claim holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M67">View MathML</a> and by the mathematical induction our claim is proved.

Step 2. We will show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M68">View MathML</a>.

In fact, using (3), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M69">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M64">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M71">View MathML</a>

(5)

Since ϕ is non-negative, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M72">View MathML</a>

and since φ is non-decreasing, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M73">View MathML</a>

(6)

In the same way, we get the following:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M74">View MathML</a>

(7)

and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M75">View MathML</a>

(8)

Using (6) and (8), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M76">View MathML</a>

From the last inequality, we notice that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M77">View MathML</a> is non-negative decreasing. This implies that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M78">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M79">View MathML</a>

(9)

It is easily seen that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M80">View MathML</a> is non-decreasing, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M81">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M82">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M83">View MathML</a>. Using this, (5) and (7), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M84">View MathML</a>

(10)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85">View MathML</a> in the last inequality and using (6), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M86">View MathML</a>

and this implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M87">View MathML</a>. Thus, using the fact that ϕ is an altering distance function, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M88">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M89">View MathML</a>

(11)

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M90">View MathML</a> and this completes the proof of our claim.

Step 3. We will prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M91">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M92">View MathML</a> are Cauchy sequences.

Suppose that one of the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M93">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M92">View MathML</a> is not a Cauchy sequence. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M95">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M96">View MathML</a>, and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M97">View MathML</a>

This means that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M98">View MathML</a>, for which we can find subsequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M99">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M100">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M101">View MathML</a>, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M102">View MathML</a>

(12)

Further, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M103">View MathML</a> corresponding to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M104">View MathML</a> in such a way that it is the smallest integer with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M105">View MathML</a> and satisfying (12). Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M106">View MathML</a>

(13)

Using (3), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M107">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M108">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M109">View MathML</a>

(14)

and also we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M110">View MathML</a>

(15)

Combining (14) and (15), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M111">View MathML</a>

(16)

Using the triangular inequality and (13), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M112">View MathML</a>

(17)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M113">View MathML</a>

(18)

Using (12), (17) and (18), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M114">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M115">View MathML</a> in the last inequality and using (11), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M116">View MathML</a>

(19)

Similarly, using the triangular inequality and (13), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M117">View MathML</a>

(20)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M118">View MathML</a>

(21)

Combining (20) and (21), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M119">View MathML</a>

(22)

Using the triangular inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M120">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M121">View MathML</a>

Using the two last inequalities and (12), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M122">View MathML</a>

(23)

Using (22) and (23), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M123">View MathML</a>

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M115">View MathML</a> in the last inequality and using (11), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M125">View MathML</a>

(24)

Finally, letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M115">View MathML</a> in (15) and using (18), (23) and the continuity of φ and ϕ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M127">View MathML</a>

and, consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M128">View MathML</a>. Since ϕ is an altering distance function, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M129">View MathML</a>, and this is a contradiction. This proves our claim.

Since X is a complete metric space, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M2">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M131">View MathML</a>

(25)

Since F and g are compatible mappings, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M132">View MathML</a>

(26)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M133">View MathML</a>

(27)

We now show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M134">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M135">View MathML</a>. Suppose that assumption (a) holds. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M136">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M137">View MathML</a>

Taking the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85">View MathML</a>, using (3), (25), (26) and the fact that F and g are continuous, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M139">View MathML</a>. Similarly, using (3), (25), (27) and the fact that F and g are continuous, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M140">View MathML</a>. Hence, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M141">View MathML</a>

Finally, suppose that (b) holds. In fact, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M93">View MathML</a> is non-decreasing and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M143">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M92">View MathML</a> is non-increasing and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M145">View MathML</a>, by our assumption, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M146">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M147">View MathML</a> for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M148">View MathML</a>.

Applying (3), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M149">View MathML</a>

and as φ is non-decreasing, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M150">View MathML</a>

(28)

Using the triangular inequality and (28), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M151">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M152">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M153">View MathML</a>, taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85">View MathML</a> in the last inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M155">View MathML</a>

and, consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M18">View MathML</a>.

Using a similar argument, it can be proved that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M135">View MathML</a> and this completes the proof. □

Corollary 2.1[6]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1">View MathML</a>be a partially ordered set and suppose that there exists a metricdinXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21">View MathML</a>be a mapping having the mixed monotone property onXsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M161">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M41">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M163">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M164">View MathML</a>, whereφandϕare altering distance functions. Moreover, suppose either

(a) Fis continuous, or

(b) Xhas the following properties:

(i) if a non-decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M45">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M46">View MathML</a>for alln,

(ii) if a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M47">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M48">View MathML</a>for alln.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M49">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M170">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M171">View MathML</a>, thenFhas a coupled fixed point.

Corollary 2.2[3]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1">View MathML</a>be a partially ordered set and suppose that there exists a metricdinXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M21">View MathML</a>be a mapping having the mixed monotone property onXsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M175">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M41">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M163">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M164">View MathML</a>. Moreover, suppose either

(a) Fis continuous, or

(b) Xhas the following properties:

(i) if a non-decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M45">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M46">View MathML</a>for alln,

(ii) if a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M47">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M48">View MathML</a>for alln.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M49">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M170">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M171">View MathML</a>, thenFhas a coupled fixed point.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M186">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M187">View MathML</a> and g is the identity function. Then applying Theorem 2.1, we get Corollary 2.2. □

3 Uniqueness of the coupled coincidence point

In this section, we prove the uniqueness of the coupled coincidence point. Note that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M1">View MathML</a> is a partially ordered set, then we endow the product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189">View MathML</a> with the following partial order relation, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M190">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M191">View MathML</a>

Theorem 3.1In addition to the hypotheses of Theorem 2.1, suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189">View MathML</a>, there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M195">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189">View MathML</a>that is comparable to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193">View MathML</a>, thenFandghave a unique coupled coincidence point.

Proof Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M17">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193">View MathML</a> are coupled coincidence points of F, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M134">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M202">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M203">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M204">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M195">View MathML</a> be an element of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M189">View MathML</a> comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M209">View MathML</a> (the proof is similar in the other case).

We construct the sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M210">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M211">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M212">View MathML</a>

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M213">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M148">View MathML</a>. In fact, we will use mathematical induction.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M61">View MathML</a>, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M209">View MathML</a>, this means <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M217">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M218">View MathML</a> and, consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M219">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M213">View MathML</a>, then since F has the mixed g-monotone property and since g is monotone increasing, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M221">View MathML</a>

and this proves our claim.

Now, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M222">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M223">View MathML</a>, using (3), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M224">View MathML</a>

(29)

In the same way, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M225">View MathML</a>

(30)

Using (29) and (30) and the fact that ϕ is non-decreasing, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M226">View MathML</a>

(31)

Using the last inequality and the fact that φ is non-decreasing, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M227">View MathML</a>

Thus the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M228">View MathML</a> is decreasing and non-negative, and hence, for certain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M78">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M230">View MathML</a>

(32)

Using (32) and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M85">View MathML</a> in (31), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M232">View MathML</a>

This gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M87">View MathML</a> and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M88">View MathML</a>.

Finally, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M235">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M236">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M237">View MathML</a>. Using a similar argument for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M193">View MathML</a>, we can get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M239">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M240">View MathML</a>, and the uniqueness of the limit gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M241">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M242">View MathML</a>. This completes the proof. □

Theorem 3.2Under the assumptions of Theorem 2.1, suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M243">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M244">View MathML</a>are comparable, then the coupled coincidence point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M14">View MathML</a>satisfies<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M246">View MathML</a>.

Proof Assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M247">View MathML</a> (a similar argument applies to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M248">View MathML</a>).

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M249">View MathML</a> for all n, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M250">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M251">View MathML</a>.

Obviously, the inequality is satisfied for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M61">View MathML</a>. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M253">View MathML</a>. Using the mixed g-monotone property of F, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M254">View MathML</a>

and since g is non-decreasing, this proves our claim.

Now, using (3) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M249">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M256">View MathML</a>

(33)

and since φ is non-decreasing, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M257">View MathML</a>

We notice that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M258">View MathML</a> is decreasing. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M259">View MathML</a> for certain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M260">View MathML</a>. Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M261">View MathML</a>

and this gives us <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M88">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M143">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M145">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M265">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M266">View MathML</a>

and thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M246">View MathML</a>. This completes the proof. □

4 Example

The following example illustrates our main result.

Example 4.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M268">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M269">View MathML</a> is a partially ordered set with the natural ordering of real numbers. Let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M270">View MathML</a>

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M38">View MathML</a> is a complete metric space. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M16">View MathML</a> be defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M273">View MathML</a>

and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M274">View MathML</a> be defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M275">View MathML</a>

Then, F satisfies the mixed g-monotone property.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M276">View MathML</a> be defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M277">View MathML</a>

and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M278">View MathML</a> be defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M279">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M29">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M30">View MathML</a> be two sequences in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M282">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M283">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M284">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M285">View MathML</a>. Then, obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M286">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M287">View MathML</a>. Now, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M136">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M289">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M290">View MathML</a>

Then it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M291">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M292">View MathML</a>

Hence, the mappings F and g are compatible in X. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M293">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M294">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M295">View MathML</a>) are two points in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M296">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M297">View MathML</a>

We next verify the contraction of Theorem 2.1. We take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M298">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M299">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M300">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M301">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M302">View MathML</a>.

We consider the following cases.

Case 1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M303">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M304">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M305">View MathML</a>

Case 2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M303">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M307">View MathML</a> Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M308">View MathML</a>

Case 3. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M309">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M304">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M311">View MathML</a>

Case 4. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M309">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M307">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M314">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M315">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M316">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M317">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/194/mathml/M318">View MathML</a>

Obviously, the contraction of Theorem 2.1 is satisfied.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-037-D1433). The author, therefore, acknowledges with thanks DSR technical and financial support. Also, the author would like to thank Prof. Abdullah Alotaibi for useful discussion on this paper. Moreover, many thanks to the referees and the editor for their helpful comments.

References

  1. Alotaibi, A, Alsulami, SM: Coupled coincidence points for monotone operators in partially ordered metric spaces. Fixed Point Theory Appl.. 2011, Article ID 44 (2011)

  2. Alsulami, SM, Nawab, H, Alotaibi, A: Coupled fixed and coincidence points for monotone operators in partial metric spaces. Fixed Point Theory Appl.. 2012, Article ID 173 (2012)

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

  4. Yeol, JC, Shah, MH, Nawab, H: Coupled fixed points of weakly F-contractive mappings in topological spaces. Appl. Math. Lett. (2011, in press)

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

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

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

  8. Khamsi, M, Kirk, W: An Introduction to Metric Spaces and Fixed Point Theory, Wiley-Interscience, New York (2001)

  9. Nguyen, V, Nguyen, X: Coupled fixed point in partially ordered metric spaces and applications. Nonlinear Anal., Theory Methods Appl.. 74, 983–992 (2011). Publisher Full Text OpenURL

  10. Zhang, X: Fixed point theorems of multivalued monotone mappings in ordered metric spaces. Appl. Math. Lett.. 23, 235–240 (2010). Publisher Full Text OpenURL

  11. Lakshmikantham, V, Ciric, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal., Theory Methods Appl.. 70(12), 4341–4349 (2009). Publisher Full Text OpenURL