Open Access Research

Discussion on some coupled fixed point theorems

Bessem Samet1*, Erdal Karapınar2, Hassen Aydi3 and Vesna Ćojbas̆ić Rajić4

Author Affiliations

1 Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

2 Department of Mathematics, Atılım University, İncek, Ankara, 06836, Turkey

3 Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, H. Sousse, Tunisie

4 Faculty of Economics, University of Belgrade, Kamenička 6, Beograd, 11000, Serbia

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Fixed Point Theory and Applications 2013, 2013:50  doi:10.1186/1687-1812-2013-50


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/50


Received:29 August 2012
Accepted:18 February 2013
Published:10 March 2013

© 2013 Samet 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

In this paper, we show that, unexpectedly, most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature.

MSC: 47H10, 54H25.

Keywords:
coupled fixed point; fixed point; ordered set; metric space

1 Introduction

In recent years, there has been recent interest in establishing fixed point theorems on ordered metric spaces with a contractivity condition which holds for all points that are related by partial ordering.

In [1], Ran and Reurings established the following fixed point theorem that extends the Banach contraction principle to the setting of ordered metric spaces.

Theorem 1.1 (Ran and Reurings [1])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Tis continuous nondecreasing (with respect to ⪯);

(iii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(iv) there exists a constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M9">View MathML</a>

ThenThas a fixed point. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13">View MathML</a>, we obtain uniqueness of the fixed point.

Nieto and López [2] extended the above result for a mapping T not necessarily continuous by assuming an additional hypothesis on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M14">View MathML</a>.

Theorem 1.2 (Nieto and López [2])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln;

(iii) Tis nondecreasing;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(v) there exists a constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M26">View MathML</a>

ThenThas a fixed point. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13">View MathML</a>, we obtain uniqueness of the fixed point.

Theorems 1.1 and 1.2 are extended and generalized by many authors. Before presenting some of theses results, we need to introduce some functional sets.

Denote by Φ the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M31">View MathML</a> satisfying the following conditions:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M32">View MathML</a>) φ is continuous nondecreasing;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M33">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M34">View MathML</a>.

Denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M35">View MathML</a> the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M36">View MathML</a> satisfying the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M37">View MathML</a>

Denote by Θ the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M38">View MathML</a> which satisfy the condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M39">View MathML</a>

Denote by Ψ the set of functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M40">View MathML</a> satisfying the following conditions:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M41">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M42">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M43">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M44">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M45">View MathML</a>.

In [3], Harjani and Sadarangani established the following results.

Theorem 1.3 (Harjani and Sadarangani [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Tis continuous nondecreasing;

(iii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M51">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M54">View MathML</a>

ThenThas a fixed point. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13">View MathML</a>, we obtain uniqueness of the fixed point.

Theorem 1.4 (Harjani and Sadarangani [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln;

(iii) Tis nondecreasing;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(v) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M51">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M70">View MathML</a>

ThenThas a fixed point. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13">View MathML</a>, we obtain uniqueness of the fixed point.

In [4], Amini-Harandi and Emami established the following results.

Theorem 1.5 (Amini-Harandi and Emami [4])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Tis continuous nondecreasing;

(iii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M80">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M83">View MathML</a>

ThenThas a fixed point. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13">View MathML</a>, we obtain uniqueness of the fixed point.

Theorem 1.6 (Amini-Harandi and Emami [4])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln;

(iii) Tis nondecreasing;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(v) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M80">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M99">View MathML</a>

ThenThas a fixed point. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M11">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M13">View MathML</a>, we obtain uniqueness of the fixed point.

Remark 1.1 Jachymski [5] established that Theorem 1.5 (resp. Theorem 1.6) follows from Theorem 1.3 (resp. Theorem 1.4).

Remark 1.2 Theorems 1.3 and 1.4 hold if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M104">View MathML</a> satisfies only the following conditions: φ is lower semi-continuous and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M34">View MathML</a> (see, for example, [6]).

The following results are special cases of Theorem 2.2 in [7].

Theorem 1.7 (Ćirić et al.[7])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Tis continuous nondecreasing;

(iii) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(iv) there exists a continuous function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M111">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M112">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M43">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M116">View MathML</a>

ThenThas a fixed point.

Theorem 1.8 (Ćirić et al.[7])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be an ordered set endowed with a metricdand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M2">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln;

(iii) Tis nondecreasing;

(iv) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M4">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M5">View MathML</a>;

(v) there exists a continuous function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M111">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M112">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M43">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M130">View MathML</a>

ThenThas a fixed point.

Remark 1.3 Theorems 1.7 and 1.8 hold if we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M131">View MathML</a> (see, for example, [8]).

Let X be a nonempty set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a> be a given mapping. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a> is a coupled fixed point of F if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M134">View MathML</a>

In [9], Bhaskar and Lakshmikantham established some coupled fixed point theorems on ordered metric spaces and applied the obtained results to the study of existence and uniqueness of solutions to a class of periodic boundary value problems. The obtained results in [9] have been extended and generalized by many authors (see, for example, [8,10-23]).

In this paper, we will prove that most of the coupled fixed point theorems are in fact immediate consequences of well-known fixed point theorems in the literature.

2 Main results

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a> be a partially ordered set endowed with a metric d and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M136">View MathML</a> be a given mapping. We endow the product set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M137">View MathML</a> with the partial order:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M138">View MathML</a>

Definition 2.1F is said to have the mixed monotone property if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M139">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/2013/1/50/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M140">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M141">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M142">View MathML</a>. It is easy to show that the mappings <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M143">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M144">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a>, are metrics on Y.

Now, define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M146">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M147">View MathML</a>

It is easy to show the following.

Lemma 2.1The following properties hold:

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete if and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M149">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M150">View MathML</a>are complete;

(b) Fhas the mixed monotone property if and only ifTis monotone nondecreasing with respect to2;

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M10">View MathML</a>is a coupled fixed point ofFif and only if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M152">View MathML</a>is a fixed point ofT.

2.1 Bhaskar and Lakshmikantham’s coupled fixed point results

We present the obtained results in [9] in the following theorem.

Theorem 2.1 (see Bhaskar and Lakshmikantham [9])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be a partially ordered set endowed with a metricd. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Fhas the mixed monotone property;

(iii) Fis continuous orXhas the following properties:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) if a decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163">View MathML</a>for alln;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166">View MathML</a>;

(v) there exists a constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M171">View MathML</a>

ThenFhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172">View MathML</a>. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176">View MathML</a>, we have uniqueness of the coupled fixed point and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177">View MathML</a>.

We will prove the following result.

Theorem 2.2Theorem 2.1 follows from Theorems 1.1 and 1.2.

Proof From (v), for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M181">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M182">View MathML</a>

This implies that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M186">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M187">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189">View MathML</a>. From Lemma 2.1, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> is complete, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M149">View MathML</a> is also complete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯2. From (iv), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M192">View MathML</a>. Now, if F is continuous, then T is continuous. In this case, applying Theorem 1.1, we get that T has a fixed point, which implies from Lemma 2.1 that F has a coupled fixed point. If conditions (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) are satisfied, then Y satisfies the following property: if a nondecreasing (with respect to ⪯2) sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M195">View MathML</a> in Y converges to some point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M196">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M197">View MathML</a> for all n. Applying Theorem 1.2, we get that T has a fixed point, which implies that F has a coupled fixed point. If, in addition, we suppose that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176">View MathML</a>, from the last part of Theorems 1.1 and 1.2, we obtain the uniqueness of the fixed point of T, which implies the uniqueness of the coupled fixed point of F. Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172">View MathML</a> be a unique coupled fixed point of F. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M203">View MathML</a> is also a coupled fixed point of F, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177">View MathML</a>. □

2.2 Harjani, López and Sadarangani’s coupled fixed point results

We present the results obtained in [16] in the following theorem.

Theorem 2.3 (see Harjani et al.[16])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be a partially ordered set endowed with a metricd. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Fhas the mixed monotone property;

(iii) Fis continuous orXhas the following properties:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) if a decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163">View MathML</a>for alln;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166">View MathML</a>;

(v) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M219">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M223">View MathML</a>

ThenFhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172">View MathML</a>. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176">View MathML</a>, we have uniqueness of the coupled fixed point and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177">View MathML</a>.

We will prove the following result.

Theorem 2.4Theorem 2.3 follows from Theorems 1.3 and 1.4.

Proof From (v), for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M233">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M234">View MathML</a>

This implies (since ψ is nondecreasing) that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M238">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M239">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189">View MathML</a>. Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.3 (resp. Theorem 1.4). The rest of the proof is similar to the above proof. □

2.3 Lakshmikantham and Ćirić’s coupled fixed point results

In [8], putting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M242">View MathML</a> (the identity mapping), we obtain the following result.

Theorem 2.5 (see Lakshmikantham and Ćirić’s [8])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be a partially ordered set endowed with a metricd. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Fhas the mixed monotone property;

(iii) Fis continuous orXhas the following properties:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) if a decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163">View MathML</a>for alln;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166">View MathML</a>;

(v) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M257">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M261">View MathML</a>

ThenFhas a coupled fixed point.

We will prove the following result.

Theorem 2.6Theorem 2.5 follows from Theorems 1.7 and 1.8.

Proof From (v), for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M265">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M266">View MathML</a>

This implies that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M270">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M271">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189">View MathML</a>. Here, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M274">View MathML</a> is the metric on Y given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M275">View MathML</a>

Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.7 (resp. Theorem 1.8). Then T has a fixed point, which implies that F has a coupled fixed point. □

2.4 Luong and Thuan’s coupled fixed point results

Luong and Thuan [18] presented a coupled fixed point result involving an ICS mapping.

Definition 2.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> be a metric space. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M277">View MathML</a> is said to be ICS if S is injective, continuous and has the property: for every sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a> in X, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279">View MathML</a> is convergent, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a> is also convergent.

We have the following result.

Lemma 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>be a metric space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M277">View MathML</a>be an ICS mapping. Then the mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M283">View MathML</a>defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M284">View MathML</a>

is a metric onX. Moreover, if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286">View MathML</a>is also complete.

Proof Let us prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M287">View MathML</a> is a metric on X. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M289">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M290">View MathML</a>. Since S is injective, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M291">View MathML</a>. Other properties of the metric can be easily checked. Now, suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> is complete and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a> be a Cauchy sequence in the metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279">View MathML</a> is Cauchy in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> is complete, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279">View MathML</a> is convergent in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> to some point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>. Since S is an ICS mapping, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a> is also convergent in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> to some point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>. On the other hand, the continuity of S implies the convergence of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M279">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a> to Sx. By the uniqueness of the limit in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M307">View MathML</a>, which implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M308">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M309">View MathML</a>. Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a> is a convergent sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286">View MathML</a>. This proves that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M286">View MathML</a> is complete. □

The obtained result in [18] is the following.

Theorem 2.7 (see Luong and Thuan [18])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be a partially ordered set endowed with a metricd. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M277">View MathML</a>be an ICS mapping. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Fhas the mixed monotone property;

(iii) Fis continuous orXhas the following properties:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) if a decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163">View MathML</a>for alln;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166">View MathML</a>;

(v) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M328">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M220">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M332">View MathML</a>

ThenFhas a coupled fixed point.

We will prove the following result.

Theorem 2.8Theorem 2.7 follows from Theorems 1.7 and 1.8.

Proof The condition (v) implies that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M336">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M337">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M189">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M340">View MathML</a> is the metric (see Lemma 2.2) on Y defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M341">View MathML</a>

Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.7 (resp. Theorem 1.8). Then T has a fixed point, which implies that F has a coupled fixed point. □

2.5 Berind’s coupled fixed point results

The following result was established in [11].

Theorem 2.9 (see Berinde [11])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be a partially ordered set endowed with a metric d. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Fhas the mixed monotone property;

(iii) Fis continuous orXhas the following properties:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) if a decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163">View MathML</a>for alln;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166">View MathML</a>;

(v) there exists a constant<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M6">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M360">View MathML</a>

ThenFhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172">View MathML</a>. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176">View MathML</a>, we have uniqueness of the coupled fixed point and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177">View MathML</a>.

We have the following result.

Theorem 2.10Theorem 2.9 follows from Theorems 1.1 and 1.2.

Proof From the condition (v), the mapping T satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M367">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M369">View MathML</a>. Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.1 (resp. Theorem 1.2). Then T has a fixed point, which implies that F has a coupled fixed point. The rest of the proof is similar to the above proofs. □

2.6 Rasouli and Bahrampour’s coupled fixed point results

Theorem 2.11 (see Rasouli and Bahrampour [20])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M1">View MathML</a>be a partially ordered set endowed with a metricd. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M132">View MathML</a>be a given mapping. Suppose that the following conditions hold:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M3">View MathML</a>is complete;

(ii) Fhas the mixed monotone property;

(iii) Fis continuous orXhas the following properties:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M156">View MathML</a>) if a nondecreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M18">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M19">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M20">View MathML</a>for alln,

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M160">View MathML</a>) if a decreasing sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M161">View MathML</a>inXconverges to some point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M162">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M163">View MathML</a>for alln;

(iv) there exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M164">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M165">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M166">View MathML</a>;

(v) there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M80">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M169">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M388">View MathML</a>

ThenFhas a coupled fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M172">View MathML</a>. Moreover, if for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M168">View MathML</a>there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M175">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M176">View MathML</a>, we have uniqueness of the coupled fixed point and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M177">View MathML</a>.

We have the following result.

Theorem 2.12Theorem 2.11 follows from Theorems 1.5 and 1.6.

Proof From the condition (v), the mapping T satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M395">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M145">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/50/mathml/M369">View MathML</a>. Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.5 (resp. Theorem 1.6). Then T has a fixed point, which implies that F has a coupled fixed point. The rest of the proof is similar to the above proofs. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This work is supported by the Research Center, College of Science, King Saud University.

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