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Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces

Vasile Berinde1* and Mădălina Păcurar2

Author Affiliations

1 Department of Mathematics and Computer Science, North University of Baia Mare, Victoriei 76, Baia Mare, 430122, Romania

2 Department of Statistics, Analysis, Forecast and Mathematics, Faculty of Economics and Bussiness Administration, Babeş-Bolyai University of Cluj-Napoca, 56-60 T. Mihali St., Cluj-Napoca, 400591, Romania

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Fixed Point Theory and Applications 2012, 2012:115  doi:10.1186/1687-1812-2012-115

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/115


Received:14 January 2012
Accepted:2 July 2012
Published:20 July 2012

© 2012 Berinde and Păcurar; 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 introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M1">View MathML</a> in partially ordered metric spaces. The obtained results extend, complement and unify some recent coupled fixed point theorems due to Samet (Nonlinear Anal. 72:4508-4517, 2010), Bhaskar and Lakshmikantham (Nonlinear Anal. 65:1379-1393, 2006) and some other very recent papers. An example to show that our generalizations are effective is also presented.

1 Introduction

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2">View MathML</a> be a metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M3">View MathML</a> a self mapping. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2">View MathML</a> is complete and T is a contraction, i.e., there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M5">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M6">View MathML</a>

(1.1)

then, by Banach contraction mapping principle, which is a classical and powerful tool in nonlinear analysis, we know that T has a unique fixed point p and, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M7">View MathML</a>, the Picard iteration <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M8">View MathML</a> converges to p.

The Banach contraction mapping principle has been generalized in several directions, see for example [2] and [21] for recent surveys. One of these generalizations, known as the Meir-Keeler fixed point theorem [13], has been obtained by replacing the contraction condition (1.1) by the following more general assumption: for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M10">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M11">View MathML</a>

(1.2)

Recently, Ran and Reurings [20] have initiated another important direction in generalizing the Banach contraction mapping principle by considering a partial ordering on the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2">View MathML</a> and by requiring that the contraction condition (1.1) is satisfied only for comparable elements, that is, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M13">View MathML</a>

(1.3)

In compensation, the authors in [20] assumed that T satisfies a certain monotonicity condition.

This new approach has been then followed by several authors: Agarwal et al.[1], Nieto and Lopez [16,17], O’Regan and Petruşel [18], who obtained fixed point theorems, and also by Bhaskar and Lakshmikantham [9], Lakshmikantham and Ćirić [12], Luong and Thuan [15], Samet [22] and many others, who obtained coupled fixed point theorems or coincidence point theorems. These results also found important applications to the existence of solutions for matrix equations or ordinary differential equations and integral equations, see [9,10,15-20] and some of the references therein.

In order to state the main result in [22], we recall the following notions. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M14">View MathML</a> be a partially ordered set and endow the product space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M15">View MathML</a> with the following partial order:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M16">View MathML</a>

We say that a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M1">View MathML</a> has the mixed monotone property if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M18">View MathML</a> is monotone nondecreasing in x and is monotone non-increasing in y, that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M20">View MathML</a>

(1.4)

and, respectively,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M21">View MathML</a>

(1.5)

We say F has the strict mixed monotone property if the strict inequality in the left-hand side of (1.4) and (1.5) implies the strict inequality in the right-hand side, respectively.

A pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M22">View MathML</a> is called a coupled fixed point of F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M23">View MathML</a>

The next theorem is the main existence result in [22].

Theorem 1 (Samet [22])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M14">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M25">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M26">View MathML</a>be a continuous mapping having the strict mixed monotone property onX. Assume also thatFis a generalized Meir-Keeler operator, that is, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M10">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M29">View MathML</a>

(1.6)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M30">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M32">View MathML</a>.

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M33">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M34">View MathML</a>

then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M19">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M36">View MathML</a>

In the same paper [22] the author also established other existence as well as existence and uniqueness results for coupled fixed points of mixed strict monotone generalized Meir-Keeler operators.

Starting from the results in [22], our main aim in this paper is to obtain more general coupled fixed point theorems for mixed monotone operators <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M26">View MathML</a> satisfying a generalized Meir-Keeler contractive condition which is significantly weaker than (1.6). Our technique of proof is different and slightly simpler than the ones used in [22] and [9]. We thus extend, unify, generalize and complement several related results in literature, amongst which we mention the ones in [1,9,13,20] and [22].

2 Main results

The first main result in this paper is the following coupled fixed point result which generalizes Theorem 1 (Theorem 2.1 in [22]) and Theorem 2.1 in [9] and some other related results.

Theorem 2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M14">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M25">View MathML</a>is a complete metric space. Assume<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M26">View MathML</a>is continuous and has the mixed monotone property and is also a generalized symmetric Meir-Keeler operator, that is, for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M41">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M10">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M30">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M31">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M32">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M46">View MathML</a>

implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M47">View MathML</a>

(2.1)

If there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M33">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M49">View MathML</a>

(2.2)
or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M50">View MathML</a>

(2.3)
then there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M51">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M52">View MathML</a>

Proof Consider the functional <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M53">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M54">View MathML</a>

It is a simple task to check that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M55">View MathML</a> is a metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56">View MathML</a> and, moreover, that, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M2">View MathML</a> is complete, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M58">View MathML</a> is a complete metric space, too. Now consider the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M59">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M60">View MathML</a>

Clearly, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M61">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M62">View MathML</a>, in view of the definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M55">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M64">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M65">View MathML</a>

Hence, by the contractive condition (2.1) we obtain a usual Meir-Keeler type condition: for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M67">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M68">View MathML</a>

(2.4)

Assume (2.2) holds (the case (2.3) is similar). Then, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M69">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M70">View MathML</a>

Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M71">View MathML</a> and consider the Picard iteration associated to T and to the initial approximation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M72">View MathML</a>, that is, the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M73">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M74">View MathML</a>

(2.5)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M75">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M76">View MathML</a>.

Since F is mixed monotone, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M77">View MathML</a>

and, by induction,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M78">View MathML</a>

which shows that T is monotone and the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M79">View MathML</a> is nondecreasing.

Note that (2.4) implies the strict contractive condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M80">View MathML</a>

(2.6)

Take now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M81">View MathML</a> in (2.6) to obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M82">View MathML</a>

which shows that the sequence of nonnegative numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M84">View MathML</a>

(2.7)

is non-increasing, hence convergent to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M85">View MathML</a>.

We now prove that necessarily <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86">View MathML</a>. Suppose, to the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9">View MathML</a>. Then, there exist a positive integer p such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M88">View MathML</a>

which, by the Meir-Keeler condition (2.4), yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M89">View MathML</a>

a contradiction, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83">View MathML</a> converges non-increasingly to ϵ. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M92">View MathML</a>

(2.8)

Let now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9">View MathML</a> be arbitrary and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M94">View MathML</a> the corresponding value from the hypothesis of our theorem. By (2.8), there exists a positive integer k such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M95">View MathML</a>

(2.9)

For this fixed number k, consider now the set

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M96">View MathML</a>

By (2.9), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M97">View MathML</a>. We claim that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M98">View MathML</a>

(2.10)

Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M99">View MathML</a>. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M100">View MathML</a> and hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M101">View MathML</a>

which, by (2.10) and Meir-Keeler type condition (2.1), is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M102">View MathML</a>. Thus, by (2.10) we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M103">View MathML</a> and, by induction,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M104">View MathML</a>

This implies that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M105">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M106">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M79">View MathML</a> is a Cauchy sequence in the complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M58">View MathML</a>, and hence there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M109">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M110">View MathML</a>

By hypothesis, T is continuous in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M58">View MathML</a>, and hence by (2.5) it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112">View MathML</a> is a fixed point of T, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M113">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M114">View MathML</a>. Then, by the definition of T, this means

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M115">View MathML</a>

that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116">View MathML</a> is a coupled fixed point of F. □

Remark 1 Theorem 2 is more general than Theorem 1 (i.e., Theorem 2.1 in [22]), since the contractive condition (2.1) is weaker than (1.6), a fact which is clearly illustrated by Example 1.

Apart from these improvements, we note that our proof is significantly simpler and shorter than the one in [22].

Example 1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M117">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M118">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M119">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M120">View MathML</a>

Then F is mixed monotone and satisfies condition (2.1) but does not satisfy condition (1.6).

Assume, to the contrary, that (1.6) holds. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M121">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M122">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M123">View MathML</a>, such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M124">View MathML</a>

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M125">View MathML</a>, this gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M126">View MathML</a>

(2.11)

which by (1.6) would imply

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M127">View MathML</a>

and this in turn, by (2.11), would imply

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M128">View MathML</a>

a contradiction. Hence F does not satisfy condition (1.6).

Now we prove that (2.1) holds. Indeed, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M129">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M130">View MathML</a>

and, by summing up the two inequalities above, we get for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M122">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M123">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M133">View MathML</a>

which holds if we simply take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M134">View MathML</a>. Thus, condition (2.1) holds. Note also that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M135">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M136">View MathML</a> satisfy (2.2).

So Theorem 2 can be applied to F in this example to conclude that F has a (unique) coupled fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M137">View MathML</a>, while Theorem 1 cannot be applied since (1.6) is not satisfied.

Remark 2 One can prove that the coupled fixed point ensured by Theorem 2 is in fact unique, like in Example 1, provided that: every pair of elements in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56">View MathML</a> has either a lower bound or an upper bound, which is known, see [9], to be equivalent to the following condition: for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M139">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M140">View MathML</a>

(2.12)

Theorem 3Adding condition (2.12) to the hypotheses of Theorem 2, we obtain the uniqueness of the coupled fixed point ofF.

Proof By Theorem 2 there exists a coupled fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116">View MathML</a>. In search for a contradiction, assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M142">View MathML</a> is a coupled fixed point of F, different from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M143">View MathML</a>. This means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M144">View MathML</a>. We discuss two cases:

Case 1. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145">View MathML</a> is comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112">View MathML</a>.

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145">View MathML</a> is comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112">View MathML</a> with respect to the ordering in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56">View MathML</a>, by taking in (2.4) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M150">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M151">View MathML</a> (or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M152">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M153">View MathML</a>), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M154">View MathML</a>

a contradiction.

Case 2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112">View MathML</a> are not comparable.

In this case, there exists an upper bound or a lower bound <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M157">View MathML</a> of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M145">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M112">View MathML</a>. Then, in view of the monotonicity of T, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M160">View MathML</a> is comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M161">View MathML</a> and to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M162">View MathML</a>. Assume, without any loss of generality, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M163">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M164">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M165">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M166">View MathML</a>, which means <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M167">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M168">View MathML</a>. By the monotonicity of T, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M169">View MathML</a>

Note that, like in the proof of Theorem 2, condition (2.4) implies the strict contractive condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M170">View MathML</a>

(2.13)

Take now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M171">View MathML</a> in (2.13) to obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M172">View MathML</a>

(2.14)

which shows that the sequence of nonnegative numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M174">View MathML</a>

(2.15)

is non-increasing, hence convergent to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M85">View MathML</a>.

We now prove that necessarily <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86">View MathML</a>. Suppose, to the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M9">View MathML</a>. Then, there exists a positive integer p such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M178">View MathML</a>

which, by the Meir-Keeler condition (2.4), yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M179">View MathML</a>

a contradiction, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M83">View MathML</a> converges non-increasingly to ϵ. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M86">View MathML</a>, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M182">View MathML</a>

(2.16)

Similarly, one obtains

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M183">View MathML</a>

(2.17)

Now, by (2.16) and (2.17), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M184">View MathML</a>

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M185">View MathML</a>, which leads to the contradiction <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M186">View MathML</a>. □

Similarly to [9] and [22], by assuming a similar condition to (2.12), but this time with respect to the ordered set X, that is, by assuming that every pair of elements of X have either an upper bound or a lower bound in X, one can show that even the components of the coupled fixed points are equal.

Theorem 4In addition to the hypotheses of Theorem 3, suppose that every pair of elements ofXhas an upper bound or a lower bound inX. Then for the coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116">View MathML</a>we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M188">View MathML</a>, that is, Fhas a fixed point

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M189">View MathML</a>

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116">View MathML</a> be a coupled fixed point of F (ensured by Theorem 2). Suppose, to the contrary, that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M191">View MathML</a>. Without any loss of generality, we can assume <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M192">View MathML</a>. We consider again two cases.

Case 1. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M193">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M194">View MathML</a> are comparable, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M195">View MathML</a> is comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M196">View MathML</a> and hence, by taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M197">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M198">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M199">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M200">View MathML</a>, in (2.13) one obtains

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M201">View MathML</a>

(2.18)

a contradiction.

Case 2. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M193">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M194">View MathML</a> are not comparable, then there exists a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M204">View MathML</a> comparable to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M193">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M194">View MathML</a>. Suppose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M208">View MathML</a> (the other case is similar). Then in view of the order on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56">View MathML</a>, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M210">View MathML</a>

that is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M212">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M213">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M214">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M214">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M216">View MathML</a> are comparable in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56">View MathML</a>. Now, similarly to the proof of Theorem 3, we obtain that, for any two comparable elements Y, V in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M56">View MathML</a>, one has

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M219">View MathML</a>

(2.19)

where T was defined in the proof of Theorem 3.

Now use (2.19) for the comparable pairs <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M220">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M221">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M222">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M223">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M224">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M225">View MathML</a>, respectively, to get

(2.20)

(2.21)

(2.22)

Now, by using the triangle inequality and (2.20), (2.21), (2.22), one has

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M229">View MathML</a>

which shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M230">View MathML</a>, that is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M231">View MathML</a>. □

Similarly, one can obtain the same conclusion under the following alternative assumption.

Theorem 5In addition to the hypotheses of Theorem 3, suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M232">View MathML</a>are comparable. Then for the coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M116">View MathML</a>we have<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M234">View MathML</a>, that is, Fhas a fixed point:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M235">View MathML</a>

Remark 3 Note that our contractive condition (2.1) is symmetric, while the contractive condition (1.6) used in [22] is not. Our generalization is based in fact on the idea of making the last one symmetric, which is very natural, as the great majority of contractive conditions in metrical fixed point theory are symmetric, see [2] and [21].

Remark 4 Note also that if F satisfies the contractive condition in [9], that is, there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M236">View MathML</a> with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M237">View MathML</a>

then, as pointed out by Proposition 2.1 in [22], F also satisfies the contractive condition (1.6) and hence (2.1).

This follows by simply taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/115/mathml/M238">View MathML</a>.

In view of the results in [14] and [24], the coupled fixed point theorems established in the present paper are also generalizations of all results in [1,13,20] and [22]. See also [3-8] and [11,23] for other recent results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the writing of this paper. They read and approved the final manuscript.

Acknowledgement

The research was supported by the Grant PN-II-RU-TE-2011-3-0239 of the Romanian Ministry of Education and Research.

References

  1. Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.. 87, 1–8 (2008). Publisher Full Text OpenURL

  2. Berinde, V: Iterative Approximation of Fixed Points, Springer, Berlin (2007)

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

  4. Berinde, V, Borcut, M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal.. 74, 4889–4897 (2011). Publisher Full Text OpenURL

  5. Borcut, M, Berinde, V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput.. 218, 5929–5936 (2012). Publisher Full Text OpenURL

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

  7. Berinde, V: Stability of Picard iteration for contractive mappings satisfying an implicit relation. Carpath. J. Math.. 27(1), 13–23 (2011)

  8. Berzig, M, Samet, B: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl.. 63(8), 1319–1334 (2012). Publisher Full Text OpenURL

  9. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal.. 65(7), 1379–1393 (2006). Publisher Full Text OpenURL

  10. Jachymski, J: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl.. 194, 293–303 (1995). Publisher Full Text OpenURL

  11. Karapinar, E, Berinde, V: Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. Banach J. Math. Anal.. 6(1), 74–89 (2012)

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

  13. Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl.. 28, 326–329 (1969). Publisher Full Text OpenURL

  14. Lim, TC: On characterizations of Meir-Keeler contractive maps. Nonlinear Anal.. 46, 113–120 (2001). Publisher Full Text OpenURL

  15. Luong, NV, Thuan, NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal.. 74, 983–992 (2011). Publisher Full Text OpenURL

  16. Nieto, JJ, Rodriguez-Lopez, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22(3), 223–239 (2005). Publisher Full Text OpenURL

  17. Nieto, JJ, Rodriguez-Lopez, R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser.. 23(12), 2205–2212 (2007). Publisher Full Text OpenURL

  18. O’Regan, D, Petruşel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl.. 341, 1241–1252 (2008). Publisher Full Text OpenURL

  19. Păcurar, M: Common fixed points for almost Prešić type operators. Carpath. J. Math.. 28(1), 117–126 (2012)

  20. Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.. 132(5), 1435–1443 (2004). Publisher Full Text OpenURL

  21. Rus, IA, Petruşel, A, Petruşel, G: Fixed Point Theory, Cluj University Press, Cluj-Napoca (2008)

  22. Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.. 72, 4508–4517 (2010). Publisher Full Text OpenURL

  23. Samet, B, Vetro, C, Vetro, P: Fixed point theorems for α-ψ contractive type mappings. Nonlinear Anal.. 75(4), 2154–2165 (2012). Publisher Full Text OpenURL

  24. Suzuki, T: Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Anal.. 64, 971–978 (2006). Publisher Full Text OpenURL