Coupled fixed point theorems for nonlinear contractions without mixed monotone property

Fixed Point Theory and Applications20122012:170

DOI: 10.1186/1687-1812-2012-170

Received: 21 June 2012

Accepted: 19 September 2012

Published: 3 October 2012

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq1_HTML.gif is a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq2_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq3_HTML.gif . 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., [16]). 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq5_HTML.gif is a contraction mapping (i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq6_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif , where k is a non-negative number such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq8_HTML.gif ), then T has a unique fixed point. Several mathematicians have been dedicated to improvement and generalization of this principle (see [814]).

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 [1619] 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ1_HTML.gif
(1.1)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq12_HTML.gif for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq15_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equa_HTML.gif

then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif .

Theorem 1.2 (Bhaskar and Lakshmikantham [21])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif be a partially ordered set and suppose there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space. Suppose that X has the following property:
  1. (i)

    if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif is a non-decreasing sequence with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq19_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq20_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq21_HTML.gif ,

     
  2. (ii)

    if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif is a non-increasing sequence with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq23_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq24_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq21_HTML.gif .

     
Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be a mapping having the mixed monotone property on X. Assume that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ2_HTML.gif
(1.2)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq12_HTML.gif for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq15_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equb_HTML.gif

then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif .

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 [2231] 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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq25_HTML.gif denotes a collection of subsets of X, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif denotes a partially ordered set with the partial order ⪯. By http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq26_HTML.gif , we mean http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq27_HTML.gif . A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq28_HTML.gif is said to be non-decreasing (resp., non-increasing) if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq29_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq30_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq31_HTML.gif ).

Definition 2.1 (Bhaskar and Lakshmikantham [21])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif be a partially ordered set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ3_HTML.gif
(2.1)
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ4_HTML.gif
(2.2)

Definition 2.2 (Bhaskar and Lakshmikantham [21])

Let X be a nonempty set. An element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq32_HTML.gif is called a coupled fixed point of the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif .

Example 2.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equc_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif . It is easy to see that F has a unique coupled fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq34_HTML.gif .

Example 2.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equd_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq36_HTML.gif . We can see that a coupled fixed point of F is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq37_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq39_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif be a metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be a given mapping. Let M be a nonempty subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif . We say that M is an F-invariant subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif if and only if, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq41_HTML.gif ,
  1. (i)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq42_HTML.gif ;

     
  2. (ii)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq43_HTML.gif .

     

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

Definition 2.6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif be a metric space and M be a subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif . We say that M satisfies the transitive property if and only if, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq44_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Eque_HTML.gif

Remark 2.7 We can easily check that the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq45_HTML.gif is trivially F-invariant, which satisfies the transitive property.

Example 2.8 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq46_HTML.gif endowed with the usual metric and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq47_HTML.gif be defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equf_HTML.gif

It easy to see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq48_HTML.gif is F-invariant, which satisfies the transitive property.

Example 2.9 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq49_HTML.gif endowed with the usual metric and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq47_HTML.gif be defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equg_HTML.gif

It easy to see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq50_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq51_HTML.gif be a metric space endowed with a partial order ⪯. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq52_HTML.gif be a mapping satisfying the mixed monotone property, that is, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq53_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ5_HTML.gif
(2.3)
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ6_HTML.gif
(2.4)
Define a subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq54_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equh_HTML.gif

Then M is an F-invariant subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif , which satisfies the transitive property.

3 Coupled fixed point theorems without the mixed monotone property

Theorem 3.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif be a complete metric space and M be a nonempty subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif . Assume that there is a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq55_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq58_HTML.gif , and also suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif is a mapping such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ7_HTML.gif
(3.1)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq59_HTML.gif . Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)
    if for any two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq60_HTML.gif ,
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equi_HTML.gif
     

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq21_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq61_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq62_HTML.gif .

If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq63_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq64_HTML.gif and M is an F-invariant set which satisfies the transitive property, then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif , that is, F has a coupled fixed point.

Proof From http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq65_HTML.gif , we can construct two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif in X such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ8_HTML.gif
(3.2)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq67_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq69_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equj_HTML.gif

Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq70_HTML.gif is a coupled fixed point of F. This finishes the proof. Therefore, we may assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq71_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq72_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq73_HTML.gif and M is an F-invariant set, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equk_HTML.gif
Again, using the fact that M is an F-invariant set, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equl_HTML.gif
By repeating this argument, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equm_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . Denote http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq74_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif .

Now, we show that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equn_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq75_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , from (3.1), it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ9_HTML.gif
(3.3)
Since M is an F-invariant set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq75_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq76_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . From (3.1) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq77_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ10_HTML.gif
(3.4)
Adding (3.3) and (3.4), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ11_HTML.gif
(3.5)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . From (3.5) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq78_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq58_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equo_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq79_HTML.gif is a monotone decreasing sequence. Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq80_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq81_HTML.gif .

Now, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq82_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq83_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq84_HTML.gif of both sides of (3.5), from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq85_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equp_HTML.gif
which is a contradiction. Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq82_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ12_HTML.gif
(3.6)
Next, we prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif are Cauchy sequences. Suppose that at least one, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif , is not a Cauchy sequence. Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq86_HTML.gif and two subsequences of integers http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq88_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq89_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ13_HTML.gif
(3.7)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq90_HTML.gif . Further, corresponding to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq88_HTML.gif , we can choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq87_HTML.gif in such a way that it is the smallest integer with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq89_HTML.gif satisfying (3.7). Then we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ14_HTML.gif
(3.8)
Using (3.7), (3.8) and the triangle inequality, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ15_HTML.gif
(3.9)

Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq91_HTML.gif and using (3.6), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq92_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq93_HTML.gif and M satisfies the transitive property, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ16_HTML.gif
(3.10)
From (3.1) and (3.10), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ17_HTML.gif
(3.11)
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ18_HTML.gif
(3.12)
Adding (3.11) and (3.12), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ19_HTML.gif
(3.13)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq90_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq94_HTML.gif of both sides of (3.13), from http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq85_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equq_HTML.gif
which is a contradiction. Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif are Cauchy sequences. Since X is complete, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ20_HTML.gif
(3.14)
Finally, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif . If the assumption (a) holds, then we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ21_HTML.gif
(3.15)
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ22_HTML.gif
(3.16)

Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif , that is, F has a coupled fixed point.

Suppose that (b) holds. We obtain that a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif converges to x and a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif converges to y for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif . By the assumption, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq61_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq61_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq95_HTML.gif , by the triangle inequality and (3.1), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ23_HTML.gif
(3.17)

Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq96_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq97_HTML.gif , and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif . Similarly, we can conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq98_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq49_HTML.gif endowed with the usual metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq99_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif and endowed with the usual partial order defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq100_HTML.gif . Define a continuous mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq101_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equr_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq102_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq104_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq105_HTML.gif , but http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq106_HTML.gif , and so the mapping F does not satisfy the mixed monotone property.

Now, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq55_HTML.gif be a function defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq107_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq108_HTML.gif . Then we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq58_HTML.gif . By simple calculation, we see that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq109_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equs_HTML.gif

Therefore, if we apply Theorem 3.1 with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq110_HTML.gif , we know that F has a unique coupled fixed point, that is, a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq111_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq112_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif in Theorem 3.1, then we get the following:

Corollary 3.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif be a complete metric space and M be a nonempty subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif is a mapping such that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ24_HTML.gif
(3.18)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq59_HTML.gif . Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)
    for any two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq60_HTML.gif , if
    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equt_HTML.gif
     

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq61_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif .

If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq63_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq64_HTML.gif and M is an F-invariant set which satisfies the transitive property, then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif , that is, F has 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 3.5 In addition to the hypotheses of Theorem 3.1, suppose that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq113_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq116_HTML.gif . Then F has a unique coupled fixed point.

Proof From Theorem 3.1, we know that F has a coupled fixed point. Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq118_HTML.gif are coupled fixed points of F, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq120_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq122_HTML.gif .

Now, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq124_HTML.gif . By the hypothesis, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq116_HTML.gif . We put http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq126_HTML.gif and construct two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq128_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equu_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif .

Since M is F-invariant and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq129_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equv_HTML.gif
that is,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equw_HTML.gif
From http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq130_HTML.gif , if we use again the property of F-invariant, then it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equx_HTML.gif
and so
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equy_HTML.gif
By repeating this process, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ25_HTML.gif
(3.19)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . From (3.1) and (3.19), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ26_HTML.gif
(3.20)
Since M is F-invariant and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq131_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq95_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ27_HTML.gif
(3.21)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . From (3.1) and (3.21), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ28_HTML.gif
(3.22)
Thus, from (3.20) and (3.22), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ29_HTML.gif
(3.23)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . By repeating this process, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ30_HTML.gif
(3.24)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . From http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq132_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq58_HTML.gif . Therefore, from (3.24), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ31_HTML.gif
(3.25)
Similarly, we can prove that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ32_HTML.gif
(3.26)
By the triangle inequality, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ33_HTML.gif
(3.27)

Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq96_HTML.gif in (3.27) and using (3.25) and (3.26), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq133_HTML.gif , and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq135_HTML.gif . 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 3.6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space. Assume that there is a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq136_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq58_HTML.gif and also suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif is a mapping such that F has the mixed monotone property and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ34_HTML.gif
(3.28)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq12_HTML.gif for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif . Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)
    X has the following property:
    1. (i)

      if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif is a non-decreasing sequence with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq19_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq20_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif ,

       
    2. (ii)

      if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif is a non-increasing sequence with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq23_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq137_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif .

       
     
If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq15_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equz_HTML.gif

then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif , that is, F has a coupled fixed point.

Proof First, we define a subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq54_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equaa_HTML.gif
From Example 2.10, we can conclude that M is an F-invariant set which satisfies the transitive property. By (3.28), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ35_HTML.gif
(3.29)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq12_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq138_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq15_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equab_HTML.gif
we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equac_HTML.gif
For the assumption (b), for any two sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif is a non-decreasing sequence in X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif is a non-increasing sequence in X with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq140_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equad_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equae_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif . Therefore, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq61_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif , 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 3.7 In addition to the hypotheses of Corollary 3.6, suppose that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq141_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq143_HTML.gif . Then F has a unique coupled fixed point.

Proof First, we define a subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq54_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equaf_HTML.gif

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq113_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq143_HTML.gif , we can conclude that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq116_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ36_HTML.gif
(3.30)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq12_HTML.gif for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq15_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equag_HTML.gif

then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif .

Proof Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq112_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif in Corollary 3.6(a), we can get the conclusion. □

Corollary 3.9 (Bhaskar and Lakshmikantham [21])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space. Suppose that X has the following property:
  1. (i)

    if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq18_HTML.gif is a non-decreasing sequence with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq19_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq20_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif ,

     
  2. (ii)

    if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq22_HTML.gif is a non-increasing sequence with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq23_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq24_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq66_HTML.gif .

     
Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq10_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ37_HTML.gif
(3.31)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq12_HTML.gif for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq14_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq15_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equah_HTML.gif

then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq17_HTML.gif .

Proof Taking http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq112_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq11_HTML.gif 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:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ38_HTML.gif
(4.1)

where T is a real number such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq145_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq146_HTML.gif denote the space of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq147_HTML.gif -valued continuous functions on the interval http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq148_HTML.gif . We endowed X with the metric http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq149_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equai_HTML.gif

It is clear that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq4_HTML.gif is a complete metric space.

Now, we consider the following assumptions:

Definition 4.1 An element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq150_HTML.gif is called a coupled lower and upper solution of the integral equation (4.1) if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq151_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equaj_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equak_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq152_HTML.gif .

(⋆1) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq153_HTML.gif is continuous;

(⋆2) for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq154_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq155_HTML.gif for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq157_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equal_HTML.gif

where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq55_HTML.gif is continuous, non-decreasing and satisfies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq57_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq58_HTML.gif .

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

Theorem 4.2 Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq159_HTML.gif hold. Then the integral equations (4.1) have the unique solution http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq160_HTML.gif if there exists a coupled lower and upper solution for (4.1).

Proof Define the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq161_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equam_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq162_HTML.gif . It is obvious that M is an F-invariant subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq40_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq160_HTML.gif .

Now, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq115_HTML.gif . Using http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq159_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq154_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equan_HTML.gif
which implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equ39_HTML.gif
(4.2)
Therefore, we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_Equao_HTML.gif

for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq115_HTML.gif . This implies that the condition (3.1) of Theorem 3.1 is satisfied. Moreover, it is easy to see that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq163_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq64_HTML.gif and all conditions in Theorem 3.1 are satisfied. Therefore, we apply Theorem 3.1 and then we get the solution http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_267_IEq164_HTML.gif . □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)
(2)
Department of Mathematics Education and the RINS, Gyeongsang National University

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