Open Access Research

Coupled fixed point theorems for nonlinear contractions without mixed monotone property

Wutiphol Sintunavarat1, Poom Kumam1* and Yeol Je Cho2*

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand

2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea

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Fixed Point Theory and Applications 2012, 2012:170  doi:10.1186/1687-1812-2012-170


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/170


Received:21 June 2012
Accepted:19 September 2012
Published:3 October 2012

© 2012 Sintunavarat 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 the existence of a coupled fixed point theorem of nonlinear contraction mappings in complete metric spaces without the mixed monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled fixed point by using the mixed monotone property. We also study the necessary condition for the uniqueness of a coupled fixed point of the given mapping. Further, we apply our results to the existence of a coupled fixed point of the given mapping in partially ordered metric spaces. Moreover, some applications to integral equations are presented.

MSC: 47H10, 54H25.

Keywords:
coupled fixed point; F-invariant set; transitive property; mixed monotone property; partially ordered set

1 Introduction

Let X be an arbitrary nonempty set. A fixed point for a self mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M1">View MathML</a> is a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M3">View MathML</a>. The applications of fixed point theorems are very important in diverse disciplines of mathematics, statistics, chemistry, biology, computer science, engineering and economics in dealing with problems arising in approximation theory, potential theory, game theory, mathematical economics, theory of differential equations, theory of integral equations, theory of matrix equations etc. (see, e.g., [1-6]). For example, fixed point theorems are incredibly useful when it comes to prove the existence of various types of Nash equilibria (see, e.g., [1]) in economics. Fixed point theorems are also helpful for proving the existence of weak periodic solutions for a model describing the electrical heating of a conductor taking into account the Joule-Thomson effect (see, e.g., [7]).

One of the very popular tools of a fixed point theory is the Banach contraction principle which first appeared in 1922. It states that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a> is a complete metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M5">View MathML</a> is a contraction mapping (i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M6">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>, where k is a non-negative number such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M8">View MathML</a>), then T has a unique fixed point. Several mathematicians have been dedicated to improvement and generalization of this principle (see [8-14]).

Especially, in 2004, Ran and Reurings [15] showed the existence of fixed points of nonlinear contraction mappings in metric spaces endowed with a partial ordering and presented applications of their results to matrix equations. Since 2004 some authors have studied fixed point theorems in partially ordered metric spaces (see [16-19] and references therein). Subsequently, Nieto and Rodríguez-López [18] extended the results in [15] for non-decreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions (see also [19]).

One of the interesting and crucial concepts, a coupled fixed point theorem, was introduced by Guo and Lakshmikantham [20]. In 2006 Bhaskar and Lakshmikantham [21] introduced the notion of the mixed monotone property of a given mapping. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. They also established the classical coupled fixed point theorems and gave some of their applications. The main results of Bhaskar and Lakshmikantham are as follows.

Theorem 1.1 (Bhaskar and Lakshmikantham [21])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>be a continuous mapping having the mixed monotone property onX. Assume that there exists a<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M13">View MathML</a>

(1.1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14">View MathML</a>for which<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M18">View MathML</a>

then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>.

Theorem 1.2 (Bhaskar and Lakshmikantham [21])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a>be a partially ordered set and suppose there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>is a complete metric space. Suppose thatXhas the following property:

(i) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a>is a non-decreasing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M25">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M26">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M27">View MathML</a>,

(ii) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a>is a non-increasing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M29">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M30">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M27">View MathML</a>.

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>be a mapping having the mixed monotone property onX. Assume that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M34">View MathML</a>

(1.2)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14">View MathML</a>for which<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M39">View MathML</a>

then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>.

Because of the important role of Theorems 1.1 and 1.2 in nonlinear differential equations, nonlinear integral equations and differential inclusions, many authors have studied the existence of coupled fixed points of the given mappings in several spaces and applications (see [22-31] and references therein).

In this paper, we establish the existence of a coupled fixed point of the given mapping in complete metric spaces without the mixed monotone property. We also give some illustrative examples to illustrate our main theorems. Furthermore, we find the necessary condition to guarantee the uniqueness of the coupled fixed point. Our results improve and extend some coupled fixed point theorems of Bhaskar and Lakshmikantham [21] and others. As an application, we apply the main results to the setting of partially ordered metric spaces and also present some applications to integral equations.

2 Preliminaries

In this section, we give some definitions, examples and remarks which are useful for main results in this paper.

Throughout this paper, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M43">View MathML</a> denotes a collection of subsets of X, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a> denotes a partially ordered set with the partial order ⪯. By <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M45">View MathML</a>, we mean <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M46">View MathML</a>. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M47">View MathML</a> is said to be non-decreasing (resp., non-increasing) if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M49">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M50">View MathML</a> (resp., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M51">View MathML</a>).

Definition 2.1 (Bhaskar and Lakshmikantham [21])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a> be a partially ordered set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>. The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M55">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M56">View MathML</a>

(2.2)

Definition 2.2 (Bhaskar and Lakshmikantham [21])

Let X be a nonempty set. An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M57">View MathML</a> is called a coupled fixed point of the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>.

Example 2.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M61">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M63">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>. It is easy to see that F has a unique coupled fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M65">View MathML</a>.

Example 2.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M66">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M68">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M69">View MathML</a>. We can see that a coupled fixed point of F is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M70">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M71">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M72">View MathML</a> are disjoint sets.

Next, we give the notion of an F-invariant set which is due to Samet and Vetro [32].

Definition 2.5 (Samet and Vetro [32])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a> be a metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a> be a given mapping. Let M be a nonempty subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a>. We say that M is an F-invariant subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a> if and only if, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M77">View MathML</a>,

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M78">View MathML</a>;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M79">View MathML</a>.

Here, we introduce the new property which is useful for our main results.

Definition 2.6 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a> be a metric space and M be a subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a>. We say that M satisfies the transitive property if and only if, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M83">View MathML</a>

Remark 2.7 We can easily check that the set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M84">View MathML</a> is trivially F-invariant, which satisfies the transitive property.

Example 2.8 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M85">View MathML</a> endowed with the usual metric and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M86">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M87">View MathML</a>

It easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M88">View MathML</a> is F-invariant, which satisfies the transitive property.

Example 2.9 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M89">View MathML</a> endowed with the usual metric and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M86">View MathML</a> be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M91">View MathML</a>

It easy to see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M92">View MathML</a> is F-invariant, which satisfies the transitive property.

The following example plays a key role in the proof of our main results in a partially ordered set.

Example 2.10 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M93">View MathML</a> be a metric space endowed with a partial order ⪯. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M94">View MathML</a> be a mapping satisfying the mixed monotone property, that is, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M95">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M96">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M97">View MathML</a>

(2.4)

Define a subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M98">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M99">View MathML</a>

Then M is an F-invariant subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a>, which satisfies the transitive property.

3 Coupled fixed point theorems without the mixed monotone property

Theorem 3.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>be a complete metric space andMbe a nonempty subset of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a>. Assume that there is a function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M103">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a>for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106">View MathML</a>, and also suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>is a mapping such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M108">View MathML</a>

(3.1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M109">View MathML</a>. Suppose that either

(a) Fis continuous or

(b) if for any two sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M113">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M27">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M116">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M118">View MathML</a>andMis anF-invariant set which satisfies the transitive property, then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>, that is, Fhas a coupled fixed point.

Proof From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M122">View MathML</a>, we can construct two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a> in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M125">View MathML</a>

(3.2)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. If there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M127">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M128">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M129">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M130">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M131">View MathML</a> is a coupled fixed point of F. This finishes the proof. Therefore, we may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M132">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M133">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M135">View MathML</a> and M is an F-invariant set, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M136">View MathML</a>

Again, using the fact that M is an F-invariant set, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M137">View MathML</a>

By repeating this argument, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M138">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M140">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>.

Now, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M142">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M144">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, from (3.1), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M146">View MathML</a>

(3.3)

Since M is an F-invariant set and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M144">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M149">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. From (3.1) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M153">View MathML</a>

(3.4)

Adding (3.3) and (3.4), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M154">View MathML</a>

(3.5)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. From (3.5) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M156">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M158">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M160">View MathML</a> is a monotone decreasing sequence. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M161">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M162">View MathML</a>.

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M163">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M164">View MathML</a>. Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M165">View MathML</a> of both sides of (3.5), from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M167">View MathML</a>, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M168">View MathML</a>

which is a contradiction. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M163">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M170">View MathML</a>

(3.6)

Next, we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a> are Cauchy sequences. Suppose that at least one, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a>, is not a Cauchy sequence. Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M175">View MathML</a> and two subsequences of integers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M176">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M177">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M179">View MathML</a>

(3.7)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M180">View MathML</a>. Further, corresponding to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M177">View MathML</a>, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M176">View MathML</a> in such a way that it is the smallest integer with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M178">View MathML</a> satisfying (3.7). Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M184">View MathML</a>

(3.8)

Using (3.7), (3.8) and the triangle inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M185">View MathML</a>

(3.9)

Letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M186">View MathML</a> and using (3.6), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M187">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M188">View MathML</a> and M satisfies the transitive property, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M189">View MathML</a>

(3.10)

From (3.1) and (3.10), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M190">View MathML</a>

(3.11)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M191">View MathML</a>

(3.12)

Adding (3.11) and (3.12), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M192">View MathML</a>

(3.13)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M180">View MathML</a>. Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M194">View MathML</a> of both sides of (3.13), from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M167">View MathML</a>, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M197">View MathML</a>

which is a contradiction. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a> are Cauchy sequences. Since X is complete, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M201">View MathML</a>

(3.14)

Finally, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>. If the assumption (a) holds, then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M204">View MathML</a>

(3.15)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M205">View MathML</a>

(3.16)

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>, that is, F has a coupled fixed point.

Suppose that (b) holds. We obtain that a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a> converges to x and a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a> converges to y for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>. By the assumption, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M214">View MathML</a>, by the triangle inequality and (3.1), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M215">View MathML</a>

(3.17)

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M216">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M217">View MathML</a>, and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>. Similarly, we can conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M219">View MathML</a>. Therefore, F has a coupled fixed point. This completes the proof. □

Now, we give an example to validate Theorem 3.1.

Example 3.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M89">View MathML</a> endowed with the usual metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M221">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a> and endowed with the usual partial order defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M223">View MathML</a>. Define a continuous mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M224">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M225">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M226">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M227">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M228">View MathML</a>. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M229">View MathML</a>, but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M230">View MathML</a>, and so the mapping F does not satisfy the mixed monotone property.

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M103">View MathML</a> be a function defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M232">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M233">View MathML</a>. Then we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106">View MathML</a>. By simple calculation, we see that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M237">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M238">View MathML</a>

Therefore, if we apply Theorem 3.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M239">View MathML</a>, we know that F has a unique coupled fixed point, that is, a point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M240">View MathML</a> is a unique coupled fixed point.

Remark 3.3 Although the mixed monotone property is an essential tool in the partially ordered metric spaces to show the existence of coupled fixed points, the mappings do not have the mixed monotone property in a general case as in the above example. Therefore, Theorem 3.1 is interesting, as a new auxiliary tool, in showing the existence of a coupled fixed point.

If we take the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M241">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a> in Theorem 3.1, then we get the following:

Corollary 3.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>be a complete metric space andMbe a nonempty subset of<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a>. Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>is a mapping such that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M247">View MathML</a>

(3.18)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M109">View MathML</a>. Suppose that either

(a) Fis continuous or

(b) for any two sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M112">View MathML</a>, if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M252">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M118">View MathML</a>andMis anF-invariant set which satisfies the transitive property, then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>, that is, Fhas a coupled fixed point.

Now, from Theorem 3.1, we have the following question:

(Q1) Is it possible to guarantee the uniqueness of the coupled fixed point of F?

Now, we give positive answers to this question.

Theorem 3.5In addition to the hypotheses of Theorem 3.1, suppose that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M261">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M264">View MathML</a>. ThenFhas a unique coupled fixed point.

Proof From Theorem 3.1, we know that F has a coupled fixed point. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M265">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M266">View MathML</a> are coupled fixed points of F, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M267">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M268">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M269">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M270">View MathML</a>.

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M271">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M272">View MathML</a>. By the hypothesis, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M264">View MathML</a>. We put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M276">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M277">View MathML</a> and construct two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M278">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M279">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M280">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>.

Since M is F-invariant and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M282">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M283">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M284">View MathML</a>

From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M285">View MathML</a>, if we use again the property of F-invariant, then it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M286">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M287">View MathML</a>

By repeating this process, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M288">View MathML</a>

(3.19)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. From (3.1) and (3.19), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M290">View MathML</a>

(3.20)

Since M is F-invariant and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M291">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M214">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M293">View MathML</a>

(3.21)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. From (3.1) and (3.21), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M295">View MathML</a>

(3.22)

Thus, from (3.20) and (3.22), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M296">View MathML</a>

(3.23)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. By repeating this process, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M298">View MathML</a>

(3.24)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M156">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M302">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106">View MathML</a>. Therefore, from (3.24), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M304">View MathML</a>

(3.25)

Similarly, we can prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M305">View MathML</a>

(3.26)

By the triangle inequality, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M307">View MathML</a>

(3.27)

Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M216">View MathML</a> in (3.27) and using (3.25) and (3.26), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M309">View MathML</a>, and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M310">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M311">View MathML</a>. Therefore, F has a unique coupled fixed point. This completes the proof. □

Next, we give a simple application of our results to coupled fixed point theorems in partially ordered metric spaces.

Corollary 3.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>is a complete metric space. Assume that there is a function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M314">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a>for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106">View MathML</a>and also suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>is a mapping such thatFhas the mixed monotone property and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M319">View MathML</a>

(3.28)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14">View MathML</a>for which<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a>. Suppose that either

(a) Fis continuous or

(b) Xhas the following property:

(i) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a>is a non-decreasing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M25">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M26">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>,

(ii) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a>is a non-increasing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M29">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M329">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M332">View MathML</a>

then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>, that is, Fhas a coupled fixed point.

Proof First, we define a subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M98">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M337">View MathML</a>

From Example 2.10, we can conclude that M is an F-invariant set which satisfies the transitive property. By (3.28), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M338">View MathML</a>

(3.29)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M340">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M342">View MathML</a>

we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M343">View MathML</a>

For the assumption (b), for any two sequences <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a> is a non-decreasing sequence in X with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M347">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a> is a non-increasing sequence in X with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M349">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M350">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M351">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>. Therefore, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>, and so the assumption (b) of Theorem 3.1 holds.

Now, since all the hypotheses of Theorem 3.1 hold, F has a coupled fixed point. This completes the proof. □

Corollary 3.7In addition to the hypotheses of Corollary 3.6, suppose that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M355">View MathML</a>, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M359">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M360">View MathML</a>. ThenFhas a unique coupled fixed point.

Proof First, we define a subset <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M98">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M362">View MathML</a>

From Example 2.10, we can conclude that M is an F-invariant set which satisfies the transitive property. Thus, the proof of the existence of a coupled fixed point is straightforward by following the same lines as in the proof of Corollary 3.6.

Next, we show the uniqueness of a coupled fixed point of F. Since for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M261">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M262">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M359">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M360">View MathML</a>, we can conclude that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M264">View MathML</a>. Therefore, since all the hypotheses of Theorem 3.5 hold, F has a unique coupled fixed point. This completes the proof. □

Corollary 3.8 (Bhaskar and Lakshmikantham [21])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>be a continuous mapping having the mixed monotone property onX. Assume that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M375">View MathML</a>

(3.30)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14">View MathML</a>for which<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M380">View MathML</a>

then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>.

Proof Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M241">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a> in Corollary 3.6(a), we can get the conclusion. □

Corollary 3.9 (Bhaskar and Lakshmikantham [21])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M9">View MathML</a>be a partially ordered set and suppose that there is a metricdonXsuch that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a>is a complete metric space. Suppose thatXhas the following property:

(i) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M24">View MathML</a>is a non-decreasing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M25">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M26">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>,

(ii) if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M28">View MathML</a>is a non-increasing sequence with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M29">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M30">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M126">View MathML</a>.

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M11">View MathML</a>be a continuous mapping having the mixed monotone property onX. Assume that there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M398">View MathML</a>

(3.31)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M14">View MathML</a>for which<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M15">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M16">View MathML</a>. If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M17">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M403">View MathML</a>

then there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M7">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M20">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M21">View MathML</a>.

Proof Taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M241">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M12">View MathML</a> in Corollary 3.6(b), we can get the conclusion. □

4 Applications

In this section, we apply our theorem to the existence theorem for a solution of the following nonlinear integral equations:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M409">View MathML</a>

(4.1)

where T is a real number such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M410">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M411">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M412">View MathML</a> denote the space of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M413">View MathML</a>-valued continuous functions on the interval <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M414">View MathML</a>. We endowed X with the metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M415">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M416">View MathML</a>

It is clear that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M4">View MathML</a> is a complete metric space.

Now, we consider the following assumptions:

Definition 4.1 An element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M418">View MathML</a> is called a coupled lower and upper solution of the integral equation (4.1) if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M419">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M420">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M421">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M422">View MathML</a>.

(⋆1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M423">View MathML</a> is continuous;

(⋆2) for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M424">View MathML</a> and for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M425">View MathML</a> for which <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M426">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M427">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M428">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M103">View MathML</a> is continuous, non-decreasing and satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M104">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M105">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M106">View MathML</a>.

Next, we give the existence theorem for a unique solution of the integral equations (4.1).

Theorem 4.2Suppose that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M433">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M434">View MathML</a>hold. Then the integral equations (4.1) have the unique solution<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M435">View MathML</a>if there exists a coupled lower and upper solution for (4.1).

Proof Define the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M436">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M437">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M438">View MathML</a>. It is obvious that M is an F-invariant subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M75">View MathML</a> which satisfies the transitive property. It is easy to see that (b) given in Theorem 3.1 is satisfied.

Next, we prove that F has a coupled fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M435">View MathML</a>.

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263">View MathML</a>. Using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M434">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M424">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M444">View MathML</a>

which implies that

(4.2)

Therefore, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M446">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M263">View MathML</a>. This implies that the condition (3.1) of Theorem 3.1 is satisfied. Moreover, it is easy to see that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M448">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M118">View MathML</a> and all conditions in Theorem 3.1 are satisfied. Therefore, we apply Theorem 3.1 and then we get the solution <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/170/mathml/M450">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

This project was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No.55000613). The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST), the third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).

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