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Coupled fixed point theorems for α-ψ-contractive type mappings in partially ordered metric spaces

Mohammad Mursaleen1, Syed Abdul Mohiuddine2* and Ravi P Agarwal23

Author Affiliations

1 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

3 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX, 78363, USA

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Fixed Point Theory and Applications 2012, 2012:228  doi:10.1186/1687-1812-2012-228

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/228


Received:17 May 2012
Accepted:29 November 2012
Published:18 December 2012

© 2012 Mursaleen et al.; licensee Springer

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

Abstract

The object of this paper is to determine some coupled fixed point theorems for nonlinear contractive mappings in the framework of a metric space endowed with partial order. We also prove the uniqueness of a coupled fixed point for such mappings in this setup.

MSC: 47H10, 54H25, 34B15.

Keywords:
coupled fixed point; contractive mapping; partially ordered set; metric space

1 Introduction

Fixed point theory is a very useful tool in solving a variety of problems in control theory, economic theory, nonlinear analysis and global analysis. The Banach contraction principle [1] is the most famous, simplest and one of the most versatile elementary results in fixed point theory. A huge amount of literature is witnessed on applications, generalizations and extensions of this principle carried out by several authors in different directions, e.g., by weakening the hypothesis, using different setups, considering different mappings.

Many authors obtained important fixed point theorems, e.g., Abbas et al.[2], Agarwal et al.[3,4], Bhaskar and Lakshmikantham [5], Choudhury and Kundu [6], Choudhury and Maity [7], Ćirić et al.[8], Luong and Thuan [9], Nieto and López [10,11], Ran and Reurings [12] and Samet [13] presented some new results for contractions in partially ordered metric spaces. In [14], Ilić and Rakočević determined some common fixed point theorems by considering the maps on cone metric spaces. Recently, Haghi et al.[15] have shown that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations. For more detail on fixed point theory and related concepts, we refer to [16-34] and the references therein.

In [5], Bhaskar and Lakshmikantham introduced the notions of mixed monotone property and coupled fixed point for the contractive mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M1">View MathML</a>, where X is a partially ordered metric space, and proved some coupled fixed point theorems for a mixed monotone operator. As an application of the coupled fixed point theorems, they determined the existence and uniqueness of the solution of a periodic boundary value problem. Recently, Lakshmikantham and Ćirić [35] have proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Most recently, Samet et al.[36] have defined α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces.

The aim of this paper is to determine some coupled fixed point theorems for generalized contractive mappings in the framework of partially ordered metric spaces.

2 Definitions and preliminary results

We start with the definition of a mixed monotone property and a coupled fixed point and state the related results.

Definition 2.1 ([5])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a> be a mapping. Then a map F is said to have the mixed monotone property if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M4">View MathML</a> is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M6">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M7">View MathML</a>

Definition 2.2 ([5])

An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M8">View MathML</a> is said to be a coupled fixed point of the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a> if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M10">View MathML</a>

Theorem 2.3 ([5])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a>be a continuous mapping having the mixed monotone property onX. Assume that there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M14">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M15">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17">View MathML</a>. If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M19">View MathML</a>

then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22">View MathML</a>.

Theorem 2.4 ([5])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>is a complete metric space. Assume thatXhas the following property:

(i) if a non-decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M25">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M26">View MathML</a>for alln;

(ii) if a non-increasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M27">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M28">View MathML</a>for alln.

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a>be a mapping having the mixed monotone property onX. Assume that there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M14">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M31">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17">View MathML</a>. If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M35">View MathML</a>

then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22">View MathML</a>.

3 Main results

In this section, we establish some coupled fixed point results by considering maps on metric spaces endowed with partial order.

Denote by Ψ the family of non-decreasing functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M40">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M42">View MathML</a> is the nth iterate of ψ satisfying (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M43">View MathML</a>, (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M44">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a> and (iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M46">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a>.

Lemma 3.1If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M48">View MathML</a>is non-decreasing and right continuous, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M49">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M51">View MathML</a>if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M44">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a>.

Definition 3.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a> be a partially ordered metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a> be a mapping. Then a map F is said to be <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M56">View MathML</a>-contractive if there exist two functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M58">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M59">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17">View MathML</a>.

Definition 3.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57">View MathML</a> be two mappings. Then F is said to be <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65">View MathML</a>-admissible if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M66">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60">View MathML</a>.

Theorem 3.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a>be a mapping having the mixed monotone property ofX. Suppose that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M58">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57">View MathML</a>such that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60">View MathML</a>, the following holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M74">View MathML</a>

(3.1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17">View MathML</a>. Suppose also that

(i) Fis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65">View MathML</a>-admissible,

(ii) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M79">View MathML</a>

(iii) Fis continuous.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M81">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M82">View MathML</a>, thenFhas a coupled fixed point; that is, there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M84">View MathML</a>

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M86">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M87">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M88">View MathML</a> (say) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M89">View MathML</a> (say). Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M90">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M91">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M92">View MathML</a>. Continuing this process, we can construct two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94">View MathML</a> in X as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M95">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96">View MathML</a>. We will show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M97">View MathML</a>

(3.2)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96">View MathML</a>. We will use the mathematical induction. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M99">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M81">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M82">View MathML</a> and as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M102">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M103">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M104">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M105">View MathML</a>. Thus, (3.2) hold for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M99">View MathML</a>. Now suppose that (3.2) hold for some fixed n, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96">View MathML</a>. Then, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M108">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M109">View MathML</a> and by the mixed monotone property of F, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M110">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M111">View MathML</a>

From above, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M112">View MathML</a>

Thus, by the mathematical induction, we conclude that (3.2) hold for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96">View MathML</a>. If for some n we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M114">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M115">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M116">View MathML</a>; that is, F has a coupled fixed point. Now, we assumed that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M117">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M96">View MathML</a>. Since F is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65">View MathML</a>-admissible, we have

Thus, by the mathematical induction, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M121">View MathML</a>

(3.3)

and similarly,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M122">View MathML</a>

(3.4)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M123">View MathML</a>. Using (3.1) and (3.3), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M124">View MathML</a>

(3.5)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M125">View MathML</a>

(3.6)

Adding (3.5) and (3.6), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M126">View MathML</a>

Repeating the above process, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M127">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M123">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M129">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M131">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M132">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M133">View MathML</a>. Then, by using the triangle inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M134">View MathML</a>

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M135">View MathML</a>. Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M136">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M137">View MathML</a>

and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94">View MathML</a> are Cauchy sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a> is a complete metric space and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94">View MathML</a> are convergent in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>. Then there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M146">View MathML</a>

Since F is continuous and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M147">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M148">View MathML</a>, taking limit <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M150">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M151">View MathML</a>

that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22">View MathML</a> and hence F has a coupled fixed point. □

In the next theorem, we omit the continuity hypothesis of F.

Theorem 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a>be a mapping such thatFhas the mixed monotone property. Assume that there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M58">View MathML</a>and a mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M159">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M16">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M17">View MathML</a>. Suppose that

(i) conditions (i) and (ii) of Theorem 3.4 hold,

(ii) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94">View MathML</a>are sequences inXsuch that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M165">View MathML</a>

for allnand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M166">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M167">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M168">View MathML</a>

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M18">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M81">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M82">View MathML</a>, then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22">View MathML</a>; that is, Fhas a coupled fixed point inX.

Proof Proceeding along the same lines as in the proof of Theorem 3.4, we know that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M93">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M94">View MathML</a> are Cauchy sequences in the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M12">View MathML</a>. Then there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M179">View MathML</a>

(3.7)

On the other hand, from (3.3) and hypothesis (ii), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M180">View MathML</a>

(3.8)

and similarly,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M181">View MathML</a>

(3.9)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M123">View MathML</a>. Using the triangle inequality, (3.8) and the property of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M44">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M185">View MathML</a>

Similarly, using (3.9), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M186">View MathML</a>

Taking the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50">View MathML</a> in the above two inequalities, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M188">View MathML</a>

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M21">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M22">View MathML</a>. Thus, F has a coupled fixed point. □

In the following theorem, we will prove the uniqueness of the coupled fixed point. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M2">View MathML</a> is a partially ordered set, then we endow the product <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192">View MathML</a> with the following partial order relation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M193">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M194">View MathML</a>.

Theorem 3.6In addition to the hypothesis of Theorem 3.4, suppose that for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M198">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M200">View MathML</a>

and also assume that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201">View MathML</a>is comparable to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196">View MathML</a>. ThenFhas a unique coupled fixed point.

Proof From Theorem 3.4, the set of coupled fixed points is nonempty. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196">View MathML</a> are coupled fixed points of the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a>; that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M207">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M208">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M209">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M210">View MathML</a>. By assumption, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M192">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201">View MathML</a> is comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M196">View MathML</a>. Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M216">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M217">View MathML</a> and choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M218">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M219">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M220">View MathML</a>. Thus, we can define two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M221">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M222">View MathML</a> as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M223">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M201">View MathML</a> is comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M195">View MathML</a>, it is easy to show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M226">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M227">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M228">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M229">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M230">View MathML</a>. Since for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M231">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M232">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M233">View MathML</a>

(3.10)

Since F is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M65">View MathML</a>-admissible, so from (3.10), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M235">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M216">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M217">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M238">View MathML</a>

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M239">View MathML</a>

Therefore, by the mathematical induction, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M240">View MathML</a>

(3.11)

for all n∈ and similarly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M241">View MathML</a>. From (3.10) and (3.11), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M242">View MathML</a>

(3.12)

Similarly, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M243">View MathML</a>

(3.13)

Adding (3.12) and (3.13), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M244">View MathML</a>

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M245">View MathML</a>

(3.14)

for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M230">View MathML</a>. Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M50">View MathML</a> in (3.14) and using Lemma 3.1, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M248">View MathML</a>

This implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M249">View MathML</a>

(3.15)

Similarly, one can show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M250">View MathML</a>

(3.16)

From (3.15) and (3.16), we conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M251">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M252">View MathML</a>. Hence, F has a unique coupled fixed point. □

Example 3.7 (Linear case)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M253">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M254">View MathML</a> be a standard metric. Define a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M3">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M256">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>. Consider a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M57">View MathML</a> be such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M259">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M260">View MathML</a> holds for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M60">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M262">View MathML</a>

It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M263">View MathML</a>

Thus (3.1) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M264">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a>, and we also see that all the hypotheses of Theorem 3.4 are fulfilled. Then there exists a coupled fixed point of F. In this case, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M266">View MathML</a> is a coupled fixed point of F.

Example 3.8 (Nonlinear case)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M267">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M268">View MathML</a> be a standard metric. Define a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M1">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M270">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M5">View MathML</a>. Consider a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M272">View MathML</a> be such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M273">View MathML</a>

Then we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M274">View MathML</a>

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M275">View MathML</a>

Therefore (3.1) holds for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M276">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M41">View MathML</a>, and also the hypothesis of Theorem 3.4 is fulfilled. Then there exists a coupled fixed point of F. In this case, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M266">View MathML</a> is a coupled fixed point of F.

4 Concluding remark

The author of [33] recently established some coupled fixed point theorems in partially ordered metric spaces shortly by using some usual corresponding fixed point theorems on the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M279">View MathML</a>. Note that if the right-hand side of the α-ψ-contractive type condition (3.1) is replaced by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M280">View MathML</a>, then a very short proof similar to what followed in [33] can be provided for a coupled fixed point theorem of Theorem 3.4 type by making just use of the results in [36]. However, since the right-hand side of (3.1) is not of the form <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M280">View MathML</a>, specially for nonlinear functions ψ, then it is not possible to apply the method [33]. In this connection, notice that Example 3.7 works for both when the right-hand side is either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/228/mathml/M280">View MathML</a> or as in (3.1), but Example 3.8 works only for (3.1). Hence, our results are more interesting and different from the existing results of [33] and [36].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The work of the second author was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. He acknowledges with thanks DSR technical and financial support.

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