Open Access

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 f : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq1_HTML.gif is a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq2_HTML.gif such that f x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space and T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq5_HTML.gif is a contraction mapping (i.e., d ( T x , T y ) k d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq6_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif, where k is a non-negative number such that k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists a k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif with
d ( F ( x , y ) , F ( u , v ) ) k ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ1_HTML.gif
(1.1)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq12_HTML.gif for which x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif. If there exists x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq15_HTML.gif such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equa_HTML.gif

then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif.

Theorem 1.2 (Bhaskar and Lakshmikantham [21])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif be a partially ordered set and suppose there is a metric d on X such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space. Suppose that X has the following property:
  1. (i)

    if { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif is a non-decreasing sequence with { x n } x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq19_HTML.gif, then x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq20_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq21_HTML.gif,

     
  2. (ii)

    if { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif is a non-increasing sequence with { y n } y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq23_HTML.gif, then y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq24_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq21_HTML.gif.

     
Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be a mapping having the mixed monotone property on X. Assume that there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif with
d ( F ( x , y ) , F ( u , v ) ) k ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ2_HTML.gif
(1.2)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq12_HTML.gif for which x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif. If there exists x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq15_HTML.gif such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equb_HTML.gif

then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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, P ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq25_HTML.gif denotes a collection of subsets of X, and ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif denotes a partially ordered set with the partial order . By x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq26_HTML.gif, we mean y x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq27_HTML.gif. A mapping f : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq28_HTML.gif is said to be non-decreasing (resp., non-increasing) if for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif, x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq29_HTML.gif implies f ( x ) f ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq30_HTML.gif (resp., f ( y ) f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq31_HTML.gif).

Definition 2.1 (Bhaskar and Lakshmikantham [21])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif be a partially ordered set and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif,
x 1 , x 2 X , x 1 x 2 F ( x 1 , y ) F ( x 2 , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ3_HTML.gif
(2.1)
and
y 1 , y 2 X , y 1 y 2 F ( x , y 1 ) F ( x , y 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ4_HTML.gif
(2.2)

Definition 2.2 (Bhaskar and Lakshmikantham [21])

Let X be a nonempty set. An element ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq32_HTML.gif is called a coupled fixed point of the mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif if x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif.

Example 2.3 Let X = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq33_HTML.gif and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be defined by
F ( x , y ) = x + y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equc_HTML.gif

for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif. It is easy to see that F has a unique coupled fixed point ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq34_HTML.gif.

Example 2.4 Let X = P ( [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq35_HTML.gif and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be defined by
F ( A , B ) = A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equd_HTML.gif

for all A , B X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq36_HTML.gif. We can see that a coupled fixed point of F is ( A ˜ , B ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq37_HTML.gif, where A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq38_HTML.gif and B ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif be a metric space and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be a given mapping. Let M be a nonempty subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq40_HTML.gif. We say that M is an F-invariant subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq40_HTML.gif if and only if, for all x , y , z , w X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq41_HTML.gif,
  1. (i)

    ( x , y , z , w ) M ( w , z , y , x ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq42_HTML.gif;

     
  2. (ii)

    ( x , y , z , w ) M ( F ( x , y ) , F ( y , x ) , F ( z , w ) , F ( w , z ) ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq43_HTML.gif.

     

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

Definition 2.6 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif be a metric space and M be a subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq40_HTML.gif. We say that M satisfies the transitive property if and only if, for all x , y , z , w , a , b X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq44_HTML.gif,
( x , y , z , w ) M and ( z , w , a , b ) M ( x , y , a , b ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Eque_HTML.gif

Remark 2.7 We can easily check that the set M = X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq45_HTML.gif is trivially F-invariant, which satisfies the transitive property.

Example 2.8 Let X = { 0 , 1 , 2 , 3 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq46_HTML.gif endowed with the usual metric and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq47_HTML.gif be defined by
F ( x , y ) = { 1 , x , y { 1 , 2 } , 3 , otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equf_HTML.gif

It easy to see that M = { 1 , 2 } 4 X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq48_HTML.gif is F-invariant, which satisfies the transitive property.

Example 2.9 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq49_HTML.gif endowed with the usual metric and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq47_HTML.gif be defined by
F ( x , y ) = { x , x , y ( , 1 ) ( 1 , ) , cos ( x + y ) sin ( x y ) , otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equg_HTML.gif

It easy to see that M = [ ( , 1 ) ( 1 , ) ] 4 X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq51_HTML.gif be a metric space endowed with a partial order . Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq52_HTML.gif be a mapping satisfying the mixed monotone property, that is, for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq53_HTML.gif, we have
x 1 , x 2 X , x 1 x 2 F ( x 1 , y ) F ( x 2 , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ5_HTML.gif
(2.3)
and
y 1 , y 2 X , y 1 y 2 F ( x , y 2 ) F ( x , y 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ6_HTML.gif
(2.4)
Define a subset M X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq54_HTML.gif by
M = { ( a , b , c , d ) X 4 : a c , b d } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equh_HTML.gif

Then M is an F-invariant subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq40_HTML.gif, which satisfies the transitive property.

3 Coupled fixed point theorems without the mixed monotone property

Theorem 3.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif be a complete metric space and M be a nonempty subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq40_HTML.gif. Assume that there is a function φ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq55_HTML.gif with 0 = φ ( 0 ) < φ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq56_HTML.gif and lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif for each t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq58_HTML.gif, and also suppose that F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif is a mapping such that
d ( F ( x , y ) , F ( u , v ) ) φ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ7_HTML.gif
(3.1)
for all ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq59_HTML.gif. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)
    if for any two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif with ( x n + 1 , y n + 1 , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq60_HTML.gif,
    { x n } x , { y n } y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equi_HTML.gif
     

for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq21_HTML.gif, then ( x , y , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq61_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq62_HTML.gif.

If there exists ( x 0 , y 0 ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq63_HTML.gif such that ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , x 0 , y 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq64_HTML.gif and M is an F-invariant set which satisfies the transitive property, then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif, that is, F has a coupled fixed point.

Proof From F ( X × X ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq65_HTML.gif, we can construct two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif in X such that
x n = F ( x n 1 , y n 1 ) , y n = F ( y n 1 , x n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ8_HTML.gif
(3.2)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. If there exists n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq67_HTML.gif such that x n 1 = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq68_HTML.gif and y n 1 = y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq69_HTML.gif, then
x n 1 = F ( x n 1 , y n 1 ) , y n 1 = F ( y n 1 , x n 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equj_HTML.gif

Thus, ( x n 1 , y n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq70_HTML.gif is a coupled fixed point of F. This finishes the proof. Therefore, we may assume that x n 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq71_HTML.gif or y n 1 y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq72_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif.

Since ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , x 0 , y 0 ) = ( x 1 , y 1 , x 0 , y 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq73_HTML.gif and M is an F-invariant set, we get
( F ( x 1 , y 1 ) , F ( y 1 , x 1 ) , F ( x 0 , y 0 ) , F ( y 0 , x 0 ) ) = ( x 2 , y 2 , x 1 , y 1 ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equk_HTML.gif
Again, using the fact that M is an F-invariant set, we have
( F ( x 2 , y 2 ) , F ( y 2 , x 2 ) , F ( x 1 , y 1 ) , F ( y 1 , x 1 ) ) = ( x 3 , y 3 , x 2 , y 2 ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equl_HTML.gif
By repeating this argument, we get
( F ( x n 1 , y n 1 ) , F ( y n 1 , x n 1 ) , x n 1 , y n 1 ) = ( x n , y n , x n 1 , y n 1 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equm_HTML.gif

for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. Denote δ n 1 : = d ( x n , x n 1 ) + d ( y n , y n 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq74_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif.

Now, we show that
δ n 2 φ ( δ n 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equn_HTML.gif
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. Since ( x n , y n , x n 1 , y n 1 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq75_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif, from (3.1), it follows that
d ( x n + 1 , x n ) = d ( F ( x n , y n ) , F ( x n 1 , y n 1 ) ) φ ( d ( x n , x n 1 ) + d ( y n , y n 1 ) 2 ) = φ ( δ n 1 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ9_HTML.gif
(3.3)
Since M is an F-invariant set and ( x n , y n , x n 1 , y n 1 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq75_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif, we get ( y n 1 , x n 1 , y n , x n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq76_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. From (3.1) and ( y n 1 , x n 1 , y n , x n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq77_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif, we get
d ( y n + 1 , y n ) = d ( F ( y n , x n ) , F ( y n 1 , x n 1 ) ) = d ( F ( y n 1 , x n 1 ) , F ( y n , x n ) ) φ ( d ( y n 1 , y n ) + d ( x n 1 , x n ) 2 ) = φ ( δ n 1 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ10_HTML.gif
(3.4)
Adding (3.3) and (3.4), we get
δ n 2 φ ( δ n 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ11_HTML.gif
(3.5)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. From (3.5) and φ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq78_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq58_HTML.gif, we have
δ n 2 φ ( δ n 1 2 ) < δ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equo_HTML.gif

for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif, that is, { δ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq79_HTML.gif is a monotone decreasing sequence. Therefore, lim n δ n = δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq80_HTML.gif for some δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq81_HTML.gif.

Now, we show that δ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq82_HTML.gif. Suppose that δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq83_HTML.gif. Taking n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq84_HTML.gif of both sides of (3.5), from lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif for all r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq85_HTML.gif, it follows that
δ = lim n δ n 2 lim n φ ( δ n 1 2 ) = 2 lim δ n 1 δ + φ ( δ n 1 2 ) < 2 ( δ 2 ) = δ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equp_HTML.gif
which is a contradiction. Thus, δ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq82_HTML.gif and
lim n δ n = lim n [ d ( x n + 1 , x n ) + d ( y n + 1 , y n ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ12_HTML.gif
(3.6)
Next, we prove that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif are Cauchy sequences. Suppose that at least one, { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif or { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif, is not a Cauchy sequence. Then there exists ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq86_HTML.gif and two subsequences of integers n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq87_HTML.gif and m k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq88_HTML.gif with n k > m k k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq89_HTML.gif such that
r k : = d ( x m k , x n k ) + d ( y m k , y n k ) ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ13_HTML.gif
(3.7)
for all k { 1 , 2 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq90_HTML.gif. Further, corresponding to m k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq88_HTML.gif, we can choose n k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq87_HTML.gif in such a way that it is the smallest integer with n k > m k k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq89_HTML.gif satisfying (3.7). Then we have
d ( x m k , x n k 1 ) + d ( y m k , y n k 1 ) < ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ14_HTML.gif
(3.8)
Using (3.7), (3.8) and the triangle inequality, we have
ϵ r k = d ( x m k , x n k ) + d ( y m k , y n k ) d ( x m k , x n k 1 ) + d ( x n k 1 , x n k ) + d ( y m k , y n k 1 ) + d ( y n k 1 , y n k ) = [ d ( x m k , x n k 1 ) + d ( y m k , y n k 1 ) ] + [ d ( x n k , x n k 1 ) + d ( y n k , y n k 1 ) ] < ϵ + δ n k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ15_HTML.gif
(3.9)

Letting k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq91_HTML.gif and using (3.6), we have lim k r k = ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq92_HTML.gif.

Since n k > m k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq93_HTML.gif and M satisfies the transitive property, we get
( x n k , y n k , x m k , y m k ) M and ( y m k , x m k , y n k , x n k ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ16_HTML.gif
(3.10)
From (3.1) and (3.10), we get
d ( x m k + 1 , x n k + 1 ) = d ( F ( x m k , y m k ) , F ( x n k , y n k ) ) = d ( F ( x n k , y n k ) , F ( x m k , y m k ) ) φ ( d ( x n k , x m k ) + d ( y n k , y m k ) 2 ) = φ ( r k 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ17_HTML.gif
(3.11)
and
d ( y m k + 1 , y n k + 1 ) = d ( F ( y m k , x m k ) , F ( y n k , x n k ) ) φ ( d ( y m k , y n k ) + d ( x m k , x n k ) 2 ) = φ ( r k 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ18_HTML.gif
(3.12)
Adding (3.11) and (3.12), we get
r k + 1 2 φ ( r k 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ19_HTML.gif
(3.13)
for all k { 1 , 2 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq90_HTML.gif. Taking k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq94_HTML.gif of both sides of (3.13), from lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif for all r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq85_HTML.gif, it follows that
ϵ = lim k r k + 1 2 lim k φ ( r k 2 ) = 2 lim r k ϵ + φ ( r k 2 ) < 2 ( ϵ 2 ) = ϵ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equq_HTML.gif
which is a contradiction. Therefore, { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif are Cauchy sequences. Since X is complete, there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that
lim n x n = x , lim n y n = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ20_HTML.gif
(3.14)
Finally, we show that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif. If the assumption (a) holds, then we have
x = lim n x n + 1 = lim n F ( x n , y n ) = F ( lim n x n , lim n y n ) = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ21_HTML.gif
(3.15)
and
y = lim n y n + 1 = lim n F ( y n , x n ) = F ( lim n y n , lim n x n ) = F ( y , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ22_HTML.gif
(3.16)

Therefore, x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif, that is, F has a coupled fixed point.

Suppose that (b) holds. We obtain that a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif converges to x and a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif converges to y for some x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif. By the assumption, we have ( x , y , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. Since ( x , y , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq95_HTML.gif, by the triangle inequality and (3.1), we get
d ( F ( x , y ) , x ) d ( F ( x , y ) , x n + 1 ) + d ( x n + 1 , x ) = d ( F ( x , y ) , F ( x n , y n ) ) + d ( x n + 1 , x ) φ ( d ( x , x n ) + d ( y , y n ) 2 ) + d ( x n + 1 , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ23_HTML.gif
(3.17)

Taking n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq96_HTML.gif, we have d ( F ( x , y ) , x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq97_HTML.gif, and so x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif. Similarly, we can conclude that y = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq49_HTML.gif endowed with the usual metric d ( x , y ) = | x y | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq99_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif and endowed with the usual partial order defined by x y y x [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq100_HTML.gif. Define a continuous mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq101_HTML.gif by
F ( x , y ) = x + y + 2 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equr_HTML.gif

for all ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq102_HTML.gif. Let y 1 = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq103_HTML.gif and y 2 = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq104_HTML.gif. Then we have y 1 y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq105_HTML.gif, but F ( x , y 1 ) F ( x , y 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq106_HTML.gif, and so the mapping F does not satisfy the mixed monotone property.

Now, let φ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq55_HTML.gif be a function defined by φ ( t ) = 2 3 t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq107_HTML.gif for all t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq108_HTML.gif. Then we obtain 0 = φ ( 0 ) < φ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq56_HTML.gif and lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif for any t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq58_HTML.gif. By simple calculation, we see that for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq109_HTML.gif,
d ( F ( x , y ) , F ( u , v ) ) = | x + y + 2 3 u + v + 2 3 | 1 3 ( d ( x , u ) + d ( y , v ) ) = 2 3 ( d ( x , u ) + d ( y , v ) 2 ) = φ ( d ( x , u ) + d ( y , v ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equs_HTML.gif

Therefore, if we apply Theorem 3.1 with M = X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq110_HTML.gif, we know that F has a unique coupled fixed point, that is, a point ( 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 φ ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq112_HTML.gif for some k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif in Theorem 3.1, then we get the following:

Corollary 3.4 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif be a complete metric space and M be a nonempty subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq40_HTML.gif. Suppose that F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif is a mapping such that there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif such that
d ( F ( x , y ) , F ( u , v ) ) k ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ24_HTML.gif
(3.18)
for all ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq59_HTML.gif. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)
    for any two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif with ( x n + 1 , y n + 1 , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq60_HTML.gif, if
    { x n } x , { y n } y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equt_HTML.gif
     

for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif, then ( x , y , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif.

If there exists ( x 0 , y 0 ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq63_HTML.gif such that ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , x 0 , y 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq64_HTML.gif and M is an F-invariant set which satisfies the transitive property, then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x , y ) , ( z , t ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq113_HTML.gif, there exists ( u , v ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq114_HTML.gif such that ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq115_HTML.gif and ( z , t , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq117_HTML.gif and ( z , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq118_HTML.gif are coupled fixed points of F, that is, x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq119_HTML.gif, y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq120_HTML.gif, z = F ( z , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq121_HTML.gif and t = F ( t , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq122_HTML.gif.

Now, we show that x = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq123_HTML.gif and y = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq124_HTML.gif. By the hypothesis, there exists ( u , v ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq114_HTML.gif such that ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq115_HTML.gif and ( z , t , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq116_HTML.gif. We put u 0 = u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq125_HTML.gif and v 0 = v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq126_HTML.gif and construct two sequences { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq127_HTML.gif and { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq128_HTML.gif by
u n = F ( u n 1 , v n 1 ) , v n = F ( v n 1 , u n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equu_HTML.gif

for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif.

Since M is F-invariant and ( x , y , u 0 , v 0 ) = ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq129_HTML.gif, we have
( F ( x , y ) , F ( y , x ) , F ( u 0 , v 0 ) , F ( v 0 , u 0 ) ) M , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equv_HTML.gif
that is,
( x , y , u 1 , v 1 ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equw_HTML.gif
From ( x , y , u 1 , v 1 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq130_HTML.gif, if we use again the property of F-invariant, then it follows that
( F ( x , y ) , F ( y , x ) , F ( u 1 , v 1 ) , F ( v 1 , u 1 ) ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equx_HTML.gif
and so
( x , y , u 2 , v 2 ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equy_HTML.gif
By repeating this process, we get
( x , y , u n , v n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ25_HTML.gif
(3.19)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. From (3.1) and (3.19), we have
d ( x , u n + 1 ) = d ( F ( x , y ) , F ( u n , v n ) ) φ ( d ( x , u n ) + d ( y , v n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ26_HTML.gif
(3.20)
Since M is F-invariant and ( x , y , u n , v n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq131_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq95_HTML.gif, we have
( v n , u n , y , x ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ27_HTML.gif
(3.21)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. From (3.1) and (3.21), we get
d ( v n + 1 , y ) = d ( F ( v n , u n ) , F ( y , x ) ) φ ( d ( v n , y ) + d ( u n , x ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ28_HTML.gif
(3.22)
Thus, from (3.20) and (3.22), we have
d ( x , u n + 1 ) + d ( y , v n + 1 ) 2 φ ( d ( x , u n ) + d ( y , v n ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ29_HTML.gif
(3.23)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. By repeating this process, we get
d ( x , u n + 1 ) + d ( y , v n + 1 ) 2 φ n ( d ( x , u 1 ) + d ( y , v 1 ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ30_HTML.gif
(3.24)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. From φ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq78_HTML.gif and lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif, it follows that lim n φ n ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq132_HTML.gif for each t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq58_HTML.gif. Therefore, from (3.24), we have
lim n [ d ( x , u n + 1 ) + d ( y , v n + 1 ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ31_HTML.gif
(3.25)
Similarly, we can prove that
lim n [ d ( z , u n + 1 ) + d ( t , v n + 1 ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ32_HTML.gif
(3.26)
By the triangle inequality, for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif, we have
d ( x , z ) + d ( y , t ) [ d ( x , u n + 1 ) + d ( u n + 1 , z ) ] + [ d ( y , v n + 1 ) + d ( v n + 1 , t ) ] = [ d ( x , u n + 1 ) + d ( y , v n + 1 ) ] + [ d ( z , u n + 1 ) + d ( t , v n + 1 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ33_HTML.gif
(3.27)

Taking n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq96_HTML.gif in (3.27) and using (3.25) and (3.26), we have d ( x , z ) + d ( y , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq133_HTML.gif, and so x = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq134_HTML.gif and y = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space. Assume that there is a function φ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq136_HTML.gif with 0 = φ ( 0 ) < φ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq56_HTML.gif and lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif for each t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq58_HTML.gif and also suppose that F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif is a mapping such that F has the mixed monotone property and
d ( F ( x , y ) , F ( u , v ) ) φ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ34_HTML.gif
(3.28)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq12_HTML.gif for which x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    X has the following property:

     
  3. (i)

    if { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif is a non-decreasing sequence with { x n } x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq19_HTML.gif, then x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq20_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif,

     
  4. (ii)

    if { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif is a non-increasing sequence with { y n } y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq23_HTML.gif, then y y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq137_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif.

     
If there exists x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq15_HTML.gif such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equz_HTML.gif

then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif, that is, F has a coupled fixed point.

Proof First, we define a subset M X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq54_HTML.gif by
M = { ( a , b , c , d ) X 4 : a c , b d } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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
d ( F ( x , y ) , F ( u , v ) ) φ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ35_HTML.gif
(3.29)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq12_HTML.gif with ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq138_HTML.gif. Since x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq15_HTML.gif such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equab_HTML.gif
we get
( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , x 0 , y 0 ) M . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equac_HTML.gif
For the assumption (b), for any two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif such that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif is a non-decreasing sequence in X with x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq139_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif is a non-increasing sequence in X with y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq140_HTML.gif, we have
x 1 x 2 x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equad_HTML.gif
and
y 1 y 2 y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equae_HTML.gif

for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif. Therefore, we have ( x , y , x n , y n ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x , y ) , ( z , t ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq141_HTML.gif, there exists ( u , v ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq114_HTML.gif such that x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif, y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif and z u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq142_HTML.gif, t v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq143_HTML.gif. Then F has a unique coupled fixed point.

Proof First, we define a subset M X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq54_HTML.gif by
M = { ( a , b , c , d ) X 4 : a c , b d } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x , y ) , ( z , t ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq113_HTML.gif, there exists ( u , v ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq114_HTML.gif such that x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif, y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif and z u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq142_HTML.gif, t v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq143_HTML.gif, we can conclude that ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq115_HTML.gif and ( z , t , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif with
d ( F ( x , y ) , F ( u , v ) ) k ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ36_HTML.gif
(3.30)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq12_HTML.gif for which x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif. If there exists x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq15_HTML.gif such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equag_HTML.gif

then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif.

Proof Taking φ ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq112_HTML.gif for some k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif in Corollary 3.6(a), we can get the conclusion. □

Corollary 3.9 (Bhaskar and Lakshmikantham [21])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq9_HTML.gif be a partially ordered set and suppose that there is a metric d on X such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space. Suppose that X has the following property:
  1. (i)

    if { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq18_HTML.gif is a non-decreasing sequence with { x n } x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq19_HTML.gif, then x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq20_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif,

     
  2. (ii)

    if { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq22_HTML.gif is a non-increasing sequence with { y n } y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq23_HTML.gif, then y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq24_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq66_HTML.gif.

     
Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq10_HTML.gif be a continuous mapping having the mixed monotone property on X. Assume that there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq11_HTML.gif with
d ( F ( x , y ) , F ( u , v ) ) k ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ37_HTML.gif
(3.31)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq12_HTML.gif for which x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq13_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq14_HTML.gif. If there exists x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq15_HTML.gif such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equah_HTML.gif

then there exists x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq7_HTML.gif such that x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq16_HTML.gif and y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq17_HTML.gif.

Proof Taking φ ( t ) = k t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq112_HTML.gif for some k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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:
x ( t ) = 0 T f ( t , x ( s ) , y ( s ) ) d s , t [ 0 , T ] ; y ( t ) = 0 T f ( t , y ( s ) , x ( s ) ) d s , t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ38_HTML.gif
(4.1)

where T is a real number such that T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq144_HTML.gif and f : [ 0 , T ] × R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq145_HTML.gif.

Let X = C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq146_HTML.gif denote the space of R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq147_HTML.gif-valued continuous functions on the interval [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq148_HTML.gif. We endowed X with the metric d : X × X R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq149_HTML.gif defined by
d ( x , y ) = sup t [ 0 , T ] | x ( t ) y ( t ) | , x , y X . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equai_HTML.gif

It is clear that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq4_HTML.gif is a complete metric space.

Now, we consider the following assumptions:

Definition 4.1 An element α , β C ( [ 0 , T ] , R ) × C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq150_HTML.gif is called a coupled lower and upper solution of the integral equation (4.1) if α ( t ) β ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq151_HTML.gif and
α ( t ) 0 T f ( t , α ( s ) , β ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equaj_HTML.gif
and
β ( t ) 0 T f ( t , β ( s ) , α ( s ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equak_HTML.gif

for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq152_HTML.gif.

(1) f : [ 0 , T ] × R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq153_HTML.gif is continuous;

(2) for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq154_HTML.gif and for all x , y , u , v R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq155_HTML.gif for which x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq156_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq157_HTML.gif, we have
0 f ( t , x , y ) f ( t , u , v ) 1 T φ ( x u + v y 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equal_HTML.gif

where φ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq55_HTML.gif is continuous, non-decreasing and satisfies 0 = φ ( 0 ) < φ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq56_HTML.gif and lim r t + φ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq57_HTML.gif for each t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq158_HTML.gif and ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq159_HTML.gif hold. Then the integral equations (4.1) have the unique solution ( x ˜ , y ˜ ) C ( [ 0 , T ] , R ) × C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq160_HTML.gif if there exists a coupled lower and upper solution for (4.1).

Proof Define the mapping F : C ( [ 0 , T ] , R ) × C ( [ 0 , T ] , R ) C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq161_HTML.gif by
F ( x , y ) ( t ) = 0 T f ( t , x ( s ) , y ( s ) ) d s , x , y C ( [ 0 , T ] , R ) , t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equam_HTML.gif

Let M = { ( x , y , u , v ) X 4 : x ( t ) u ( t )  and  y ( t ) v ( t )  for all  t [ 0 , T ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq162_HTML.gif. It is obvious that M is an F-invariant subset of X 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x ˜ , y ˜ ) C ( [ 0 , T ] , R ) × C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq160_HTML.gif.

Now, let ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq115_HTML.gif. Using ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq159_HTML.gif, for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq154_HTML.gif, we have
| F ( x , y ) ( t ) F ( u , v ) ( t ) | = 0 T [ f ( t , x ( s ) , y ( s ) ) f ( t , u ( s ) , v ( s ) ) ] d s 1 T 0 T φ ( x ( s ) u ( s ) + v ( s ) y ( s ) 2 ) d s 1 T 0 T φ ( sup z [ 0 , T ] | x ( z ) u ( z ) | + sup z [ 0 , T ] | y ( z ) v ( z ) | 2 ) d s = φ ( sup z [ 0 , T ] | x ( z ) u ( z ) | + sup z [ 0 , T ] | y ( z ) v ( z ) | 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equan_HTML.gif
which implies that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equ39_HTML.gif
(4.2)
Therefore, we get
d ( F ( x , y ) , F ( u , v ) ) φ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_Equao_HTML.gif

for all ( x , y , u , v ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x 0 , y 0 ) C ( [ 0 , T ] , R ) × C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq163_HTML.gif such that ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , x 0 , y 0 ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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 ( x ˜ , y ˜ ) C ( [ 0 , T ] , R ) × C ( [ 0 , T ] , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_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

References

  1. Border KC: Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, New York; 1985.View Article
  2. Cataldo, A, Lee, EA, Liu, X, Matsikoudis, ED, Zheng, H: A constructive Fixed point theorem and the feedback semantics of timed systems. Technical Report UCB/EECS-2006–4, EECS Dept., University of California, Berkeley (2006)
  3. Guo Y: A generalization of Banach’s contraction principle for some non-obviously contractive operators in a cone metric space. Turk. J. Math. 2012, 36: 297–304.
  4. Hyvärinen A: Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 1999, 10(3):626–634. 10.1109/72.761722View Article
  5. Noumsi A, Derrien S, Quinton P: Acceleration of a content based image retrieval application on the RDISK cluster. IEEE International Parallel and Distributed Processing Symposium 2006.
  6. Yantir A, Gulsan Topal S: Positive solutions of nonlinear m-point BVP on time scales. Int. J. Differ. Equ. 2008, 3(1):179–194. 0973–6069
  7. Badii M: Existence of periodic solutions for the thermistor problem with the Joule-Thomson effect. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2008, 54: 1–10. 10.1007/s11565-008-0041-5MathSciNetView Article
  8. Arvanitakis AD: A proof of the generalized Banach contraction conjecture. Proc. Am. Math. Soc. 2003, 131: 3647–3656. 10.1090/S0002-9939-03-06937-5MathSciNetView Article
  9. Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9MathSciNetView Article
  10. Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 93
  11. Sintunavarat W, Kumam P:Weak condition for generalized multi-valued ( f , α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq165_HTML.gif-weak contraction mappings. Appl. Math. Lett. 2011, 24: 460–465. 10.1016/j.aml.2010.10.042MathSciNetView Article
  12. Sintunavarat W, Kumam P: Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011., 2011: Article ID 3
  13. Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25: 52–57. 10.1016/j.aml.2011.05.047MathSciNetView Article
  14. Sintunavarat W, Kumam P:Common fixed point theorems for generalized JH https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq166_HTML.gifv-operator classes and invariant approximations. J. Inequal. Appl. 2011., 2011: Article ID 67
  15. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView Article
  16. Cho YJ, Saadati R, Wang S: Common fixed point theorems on generalized distance in order cone metric spaces. Comput. Math. Appl. 2011, 61: 1254–1260. 10.1016/j.camwa.2011.01.004MathSciNetView Article
  17. Graily E, Vaezpour SM, Saadati R, Cho YJ: Generalization of fixed point theorems in ordered metric spaces concerning generalized distance. Fixed Point Theory Appl. 2011., 2011: Article ID 30
  18. Nieto JJ, Lopez RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205–2212. 10.1007/s10114-005-0769-0MathSciNetView Article
  19. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040MathSciNetView Article
  20. Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl. 1987, 11: 623–632. 10.1016/0362-546X(87)90077-0MathSciNetView Article
  21. Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017MathSciNetView Article
  22. Abbas M, Sintunavarat W, Kumam P: Coupled fixed point in partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
  23. Cho YJ, He G, Huang NJ: The existence results of coupled quasi-solutions for a class of operator equations. Bull. Korean Math. Soc. 2010, 47: 455–465.MathSciNetView Article
  24. Cho YJ, Shah MH, Hussain N: Coupled fixed points of weakly F -contractive mappings in topological spaces. Appl. Math. Lett. 2011, 24: 1185–1190. 10.1016/j.aml.2011.02.004MathSciNetView Article
  25. Cho YJ, Rhoades BE, Saadati R, Samet B, Shantawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8
  26. Gordji ME, Cho YJ, Baghani H: Coupled fixed point theorems for contractions in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2011, 54: 1897–1906. 10.1016/j.mcm.2011.04.014View Article
  27. Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
  28. Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for weak contraction mapping under F -invariant set. Abstr. Appl. Anal. 2012., 2012: Article ID 324874
  29. Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93
  30. Sintunavarat W, Cho YJ, Kumam P: Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128
  31. Sintunavarat W, Petruşel A, Kumam P:Common coupled fixed point theorems for w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-170/MediaObjects/13663_2012_Article_267_IEq167_HTML.gif-compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo 2012. doi:10.1007/s12215–012–0096–0
  32. Samet B, Vetro C: Coupled fixed point F -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1: 46–56.MathSciNetView Article

Copyright

© Sintunavarat et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. 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.