Open Access

Coupled fixed point theorems for α-ψ-contractive type mappings in partially ordered metric spaces

  • Mohammad Mursaleen1,
  • Syed Abdul Mohiuddine2Email author and
  • Ravi P Agarwal2, 3
Fixed Point Theory and Applications20122012:228

DOI: 10.1186/1687-1812-2012-228

Received: 17 May 2012

Accepted: 29 November 2012

Published: 18 December 2012

Abstract

The object of this paper is to determine some coupled fixed point theorems for nonlinear contractive mappings in the framework of a metric space endowed with partial order. We also prove the uniqueness of a coupled fixed point for such mappings in this setup.

MSC:47H10, 54H25, 34B15.

Keywords

coupled fixed point contractive mapping partially ordered set metric space

1 Introduction

Fixed point theory is a very useful tool in solving a variety of problems in control theory, economic theory, nonlinear analysis and global analysis. The Banach contraction principle [1] is the most famous, simplest and one of the most versatile elementary results in fixed point theory. A huge amount of literature is witnessed on applications, generalizations and extensions of this principle carried out by several authors in different directions, e.g., by weakening the hypothesis, using different setups, considering different mappings.

Many authors obtained important fixed point theorems, e.g., Abbas et al. [2], Agarwal et al. [3, 4], Bhaskar and Lakshmikantham [5], Choudhury and Kundu [6], Choudhury and Maity [7], Ćirić et al. [8], Luong and Thuan [9], Nieto and López [10, 11], Ran and Reurings [12] and Samet [13] presented some new results for contractions in partially ordered metric spaces. In [14], Ilić and Rakočević determined some common fixed point theorems by considering the maps on cone metric spaces. Recently, Haghi et al. [15] have shown that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations. For more detail on fixed point theory and related concepts, we refer to [1634] and the references therein.

In [5], Bhaskar and Lakshmikantham introduced the notions of mixed monotone property and coupled fixed point for the contractive mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq1_HTML.gif, where X is a partially ordered metric space, and proved some coupled fixed point theorems for a mixed monotone operator. As an application of the coupled fixed point theorems, they determined the existence and uniqueness of the solution of a periodic boundary value problem. Recently, Lakshmikantham and Ćirić [35] have proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Most recently, Samet et al. [36] have defined α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces.

The aim of this paper is to determine some coupled fixed point theorems for generalized contractive mappings in the framework of partially ordered metric spaces.

2 Definitions and preliminary results

We start with the definition of a mixed monotone property and a coupled fixed point and state the related results.

Definition 2.1 ([5])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq2_HTML.gif be a partially ordered set and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif be a mapping. Then a map F is said to have the mixed monotone property if F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq4_HTML.gif is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif,
x 1 , x 2 X , x 1 x 2 implies F ( x 1 , y ) F ( x 2 , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equa_HTML.gif
and
y 1 , y 2 X , y 1 y 2 implies F ( x , y 1 ) F ( x , y 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equb_HTML.gif

Definition 2.2 ([5])

An element ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq6_HTML.gif is said to be a coupled fixed point of the mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif if
F ( x , y ) = x and F ( y , x ) = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equc_HTML.gif

Theorem 2.3 ([5])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq2_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%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif is a complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_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%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq8_HTML.gif with
d ( F ( x , y ) , F ( u , v ) ) k 2 [ d ( x , u ) + d ( y , v ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equd_HTML.gif
for all x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq9_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq10_HTML.gif. If there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq11_HTML.gif such that
x 0 F ( x 0 , y 0 ) and y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Eque_HTML.gif

then there exist x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif such that F ( x , y ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq12_HTML.gif and F ( y , x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq13_HTML.gif.

Theorem 2.4 ([5])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq2_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%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif is a complete metric space. Assume that X has the following property:
  1. (i)

    if a non-decreasing sequence ( x n ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq14_HTML.gif, then x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq15_HTML.gif for all n;

     
  2. (ii)

    if a non-increasing sequence ( y n ) y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq16_HTML.gif, then y y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq17_HTML.gif for all n.

     
Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif be a 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%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq8_HTML.gif with
d ( F ( x , y ) , F ( u , v ) ) k 2 [ d ( x , u ) + d ( y , v ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equf_HTML.gif
for all x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq9_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq10_HTML.gif. If there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq11_HTML.gif such that
x 0 F ( x 0 , y 0 ) and y 0 F ( y 0 , x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equg_HTML.gif

then there exist x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif such that F ( x , y ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq12_HTML.gif and F ( y , x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq13_HTML.gif.

3 Main results

In this section, we establish some coupled fixed point results by considering maps on metric spaces endowed with partial order.

Denote by Ψ the family of non-decreasing functions ψ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq18_HTML.gif such that n = 1 ψ n ( t ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq19_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif, where ψ n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq21_HTML.gif is the n th iterate of ψ satisfying (i) ψ 1 ( { 0 } ) = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq22_HTML.gif, (ii) ψ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq23_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif and (iii) lim r t + ψ ( r ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq24_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif.

Lemma 3.1 If ψ : [ 0 , ] [ 0 , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq25_HTML.gif is non-decreasing and right continuous, then ψ n ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq26_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq27_HTML.gif for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq28_HTML.gif if and only if ψ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq23_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif.

Definition 3.2 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif be a partially ordered metric space and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif be a mapping. Then a map F is said to be ( α , ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq29_HTML.gif-contractive if there exist two functions α : X 2 × X 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq30_HTML.gif and ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq31_HTML.gif such that
α ( ( x , y ) , ( u , v ) ) d ( F ( x , y ) , F ( u , v ) ) ψ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equh_HTML.gif

for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq32_HTML.gif with x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq9_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq10_HTML.gif.

Definition 3.3 Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif and α : X 2 × X 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq30_HTML.gif be two mappings. Then F is said to be ( α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq33_HTML.gif-admissible if
α ( ( x , y ) , ( u , v ) ) 1 α ( ( F ( x , y ) , F ( y , x ) ) , ( F ( u , v ) , F ( v , u ) ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equi_HTML.gif

for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq32_HTML.gif.

Theorem 3.4 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq2_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%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif is a complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif be a mapping having the mixed monotone property of X. Suppose that there exist ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq31_HTML.gif and α : X 2 × X 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq30_HTML.gif such that for x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq32_HTML.gif, the following holds:
α ( ( x , y ) , ( u , v ) ) d ( F ( x , y ) , F ( u , v ) ) ψ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ1_HTML.gif
(3.1)
for all x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq9_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq10_HTML.gif. Suppose also that
  1. (i)

    F is ( α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq33_HTML.gif-admissible,

     
  2. (ii)
    there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq11_HTML.gif such that
    α ( ( x 0 , y 0 ) , ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) ) ) 1 and α ( ( y 0 , x 0 ) , ( F ( y 0 , x 0 ) , F ( x 0 , y 0 ) ) ) 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equj_HTML.gif
     
  3. (iii)

    F is continuous.

     
If there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq11_HTML.gif such that x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq34_HTML.gif and y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq35_HTML.gif, then F has a coupled fixed point; that is, there exist x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif such that
F ( x , y ) = x and F ( y , x ) = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equk_HTML.gif
Proof Let x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq11_HTML.gif be such that α ( ( x 0 , y 0 ) , ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq36_HTML.gif and α ( ( y 0 , x 0 ) , ( F ( y 0 , x 0 ) , F ( x 0 , y 0 ) ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq37_HTML.gif and x 0 F ( x 0 , y 0 ) = x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq38_HTML.gif (say) and y 0 F ( y 0 , x 0 ) = y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq39_HTML.gif (say). Let x 2 , y 2 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq40_HTML.gif be such that F ( x 1 , y 1 ) = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq41_HTML.gif and F ( y 1 , x 1 ) = y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq42_HTML.gif. Continuing this process, we can construct two sequences ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq43_HTML.gif and ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq44_HTML.gif in X as follows:
x n + 1 = F ( x n , y n ) and y n + 1 = F ( y n , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equl_HTML.gif
for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq45_HTML.gif. We will show that
x n x n + 1 and y n y n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ2_HTML.gif
(3.2)
for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq45_HTML.gif. We will use the mathematical induction. Let n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq46_HTML.gif. Since x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq34_HTML.gif and y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq35_HTML.gif and as x 1 = F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq47_HTML.gif and y 1 = F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq48_HTML.gif, we have x 0 x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq49_HTML.gif and y 0 y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq50_HTML.gif. Thus, (3.2) hold for n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq46_HTML.gif. Now suppose that (3.2) hold for some fixed n, n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq45_HTML.gif. Then, since x n x n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq51_HTML.gif and y n y n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq52_HTML.gif and by the mixed monotone property of F, we have
x n + 2 = F ( x n + 1 , y n + 1 ) F ( x n , y n + 1 ) F ( x n , y n ) = x n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equm_HTML.gif
and
y n + 2 = F ( y n + 1 , x n + 1 ) F ( y n , x n + 1 ) F ( y n , x n ) = y n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equn_HTML.gif
From above, we conclude that
x n + 1 x n + 2 and y n + 1 y n + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equo_HTML.gif
Thus, by the mathematical induction, we conclude that (3.2) hold for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq45_HTML.gif. If for some n we have ( x n + 1 , y n + 1 ) = ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq53_HTML.gif, then F ( x n , y n ) = x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq54_HTML.gif and F ( y n , x n ) = y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq55_HTML.gif; that is, F has a coupled fixed point. Now, we assumed that ( x n + 1 , y n + 1 ) ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq56_HTML.gif for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq45_HTML.gif. Since F is ( α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq33_HTML.gif-admissible, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equp_HTML.gif
Thus, by the mathematical induction, we have
α ( ( x n , y n ) , ( x n + 1 , y n + 1 ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ3_HTML.gif
(3.3)
and similarly,
α ( ( y n , x n ) , ( y n + 1 , x n + 1 ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ4_HTML.gif
(3.4)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq57_HTML.gif. Using (3.1) and (3.3), we obtain
d ( x n , x n + 1 ) = d ( F ( x n 1 , y n 1 ) , F ( x n , y n ) ) α ( ( x n 1 , y n 1 ) , ( x n , y n ) ) d ( F ( x n 1 , y n 1 ) , F ( x n , y n ) ) ψ ( d ( x n 1 , x n ) + d ( y n 1 , y n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ5_HTML.gif
(3.5)
Similarly, we have
d ( y n , y n + 1 ) = d ( F ( y n 1 , y n 1 ) , F ( y n , y n ) ) α ( ( y n 1 , x n 1 ) , ( y n , x n ) ) d ( F ( y n 1 , x n 1 ) , F ( y n , x n ) ) ψ ( d ( y n 1 , y n ) + d ( x n 1 , x n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ6_HTML.gif
(3.6)
Adding (3.5) and (3.6), we get
d ( x n , x n + 1 ) + d ( y n , y n + 1 ) 2 ψ ( d ( x n 1 , x n ) + d ( y n 1 , y n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equq_HTML.gif
Repeating the above process, we get
d ( x n , x n + 1 ) + d ( y n , y n + 1 ) 2 ψ n ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equr_HTML.gif
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq57_HTML.gif. For ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq58_HTML.gif there exists n ( ϵ ) N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq59_HTML.gif such that
n n ( ϵ ) ψ n ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) 2 ) < ϵ / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equs_HTML.gif
Let n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq60_HTML.gif be such that m > n > n ( ϵ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq61_HTML.gif. Then, by using the triangle inequality, we have
d ( x n , x m ) + d ( y n , y m ) 2 k = n m 1 d ( x k , x k + 1 ) + d ( y k , y k + 1 ) 2 k = n m 1 ψ k ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) 2 ) n n ( ϵ ) ψ n ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) 2 ) < ϵ / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equt_HTML.gif
This implies that d ( x n , x m ) + d ( y n , y m ) < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq62_HTML.gif. Since
d ( x n , x m ) d ( x n , x m ) + d ( y n , y m ) < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equu_HTML.gif
and
d ( y n , y m ) d ( x n , x m ) + d ( y n , y m ) < ϵ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equv_HTML.gif
and hence ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq43_HTML.gif and ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq44_HTML.gif are Cauchy sequences in ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif. Since ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif is a complete metric space and hence ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq43_HTML.gif and ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq44_HTML.gif are convergent in ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif. Then there exist x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif such that
lim n x n = x and lim n y n = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equw_HTML.gif
Since F is continuous and x n + 1 = F ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq63_HTML.gif and y n + 1 = F ( y n , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq64_HTML.gif, taking limit n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq27_HTML.gif, we get
x = lim n x n = lim n F ( x n 1 , y n 1 ) = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equx_HTML.gif
and
y = lim n y n = lim n F ( y n 1 , x n 1 ) = F ( y , x ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equy_HTML.gif

that is, F ( x , y ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq12_HTML.gif and F ( y , x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq13_HTML.gif and hence F has a coupled fixed point. □

In the next theorem, we omit the continuity hypothesis of F.

Theorem 3.5 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq2_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%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif is a complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif be a mapping such that F has the mixed monotone property. Assume that there exist ψ Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq31_HTML.gif and a mapping α : X 2 × X 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq30_HTML.gif such that
α ( ( x , y ) , ( u , v ) ) d ( F ( x , y ) , F ( u , v ) ) ψ ( d ( x , u ) + d ( y , v ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equz_HTML.gif
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq32_HTML.gif with x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq9_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq10_HTML.gif. Suppose that
  1. (i)

    conditions (i) and (ii) of Theorem 3.4 hold,

     
  2. (ii)
    if ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq43_HTML.gif and ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq44_HTML.gif are sequences in X such that
    α ( ( x n , y n ) , ( x n + 1 , y n + 1 ) ) 1 and α ( ( y n , x n ) , ( y n + 1 , x n + 1 ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equaa_HTML.gif
     
for all n and lim n x n = x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq65_HTML.gif and lim n y n = y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq66_HTML.gif, then
α ( ( x n , y n ) , ( x , y ) ) 1 and α ( ( x n , y n ) , ( x , y ) ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equab_HTML.gif

If there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq11_HTML.gif such that x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq34_HTML.gif and y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq35_HTML.gif, then there exist x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif such that F ( x , y ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq12_HTML.gif and F ( y , x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq13_HTML.gif; that is, F has a coupled fixed point in X.

Proof Proceeding along the same lines as in the proof of Theorem 3.4, we know that ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq43_HTML.gif and ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq44_HTML.gif are Cauchy sequences in the complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq7_HTML.gif. Then there exist x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif such that
lim n x n = x and lim n y n = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ7_HTML.gif
(3.7)
On the other hand, from (3.3) and hypothesis (ii), we obtain
α ( ( x n , y n ) , ( x , y ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ8_HTML.gif
(3.8)
and similarly,
α ( ( y n , x n ) , ( y , x ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ9_HTML.gif
(3.9)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq57_HTML.gif. Using the triangle inequality, (3.8) and the property of ψ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq23_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif, we get
d ( F ( x , y ) , x ) d ( F ( x , y ) , F ( x n , y n ) ) + d ( x n + 1 , x ) α ( ( x n , y n ) , ( x , y ) ) d ( F ( x n , y n ) , F ( x , y ) ) + d ( x n + 1 , x ) ψ ( d ( x n , x ) + d ( y n , y ) 2 ) + d ( x n + 1 , x ) < d ( x n , x ) + d ( y n , y ) 2 + d ( x n + 1 , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equac_HTML.gif
Similarly, using (3.9), we obtain
d ( F ( y , x ) , y ) α ( ( y n , x n ) , ( y , x ) ) d ( F ( y n , x n ) , F ( y , x ) ) + d ( y n + 1 , y ) ψ ( d ( y n , y ) + d ( x n , x ) 2 ) + d ( y n + 1 , y ) < d ( y n , y ) + d ( x n , x ) 2 + d ( y n + 1 , y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equad_HTML.gif
Taking the limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq27_HTML.gif in the above two inequalities, we get
d ( F ( x , y ) , x ) = 0 and d ( F ( y , x ) , y ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equae_HTML.gif

Hence, F ( x , y ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq12_HTML.gif and F ( y , x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq13_HTML.gif. Thus, F has a coupled fixed point. □

In the following theorem, we will prove the uniqueness of the coupled fixed point. If ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq2_HTML.gif is a partially ordered set, then we endow the product X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq67_HTML.gif with the following partial order relation:
( x , y ) ( u , v ) x u , y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equaf_HTML.gif

for all ( x , y ) , ( u , v ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq68_HTML.gif.

Theorem 3.6 In addition to the hypothesis of Theorem 3.4, suppose that for every ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq69_HTML.gif, ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq70_HTML.gif in X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq67_HTML.gif, there exists ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq71_HTML.gif in X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq67_HTML.gif such that
α ( ( x , y ) , ( u , v ) ) 1 and α ( ( s , t ) , ( u , v ) ) 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equag_HTML.gif

and also assume that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq72_HTML.gif is comparable to ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq69_HTML.gif and ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq70_HTML.gif. Then F has a unique coupled fixed point.

Proof From Theorem 3.4, the set of coupled fixed points is nonempty. Suppose ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq69_HTML.gif and ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq70_HTML.gif are coupled fixed points of the mappings F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif; that is, x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq73_HTML.gif, y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq74_HTML.gif and s = F ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq75_HTML.gif, t = F ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq76_HTML.gif. By assumption, there exists ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq72_HTML.gif in X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq67_HTML.gif such that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq72_HTML.gif is comparable to ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq69_HTML.gif and ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq70_HTML.gif. Put u = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq77_HTML.gif and v = v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq78_HTML.gif and choose u 1 , v 1 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq79_HTML.gif such that u 1 = F ( u 1 , v 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq80_HTML.gif and v 1 = F ( v 1 , u 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq81_HTML.gif. Thus, we can define two sequences ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq82_HTML.gif and v n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq83_HTML.gif as
u n + 1 = F ( u n , v n ) and v n + 1 = F ( v n , u n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equah_HTML.gif
Since ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq72_HTML.gif is comparable to ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq69_HTML.gif, it is easy to show that x u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq84_HTML.gif and y v 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq85_HTML.gif. Thus, x u n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq86_HTML.gif and y v n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq87_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq88_HTML.gif. Since for every ( x , y ) , ( s , t ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq89_HTML.gif, there exists ( u , v ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq90_HTML.gif such that
α ( ( x , y ) , ( u , v ) ) 1 and α ( ( s , t ) , ( u , v ) ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ10_HTML.gif
(3.10)
Since F is ( α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq33_HTML.gif-admissible, so from (3.10), we have
α ( ( x , y ) , ( u , v ) ) 1 α ( ( F ( x , y ) , F ( y , x ) ) , ( F ( u , v ) , F ( v , u ) ) ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equai_HTML.gif
Since u = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq77_HTML.gif and v = v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq78_HTML.gif, we get
α ( ( x , y ) , ( u , v ) ) 1 α ( ( F ( x , y ) , F ( y , x ) ) , ( F ( u 0 , v 0 ) , F ( v 0 , u 0 ) ) ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equaj_HTML.gif
Thus,
α ( ( x , y ) , ( u , v ) ) 1 α ( ( x , y ) , ( u 1 , v 1 ) ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equak_HTML.gif
Therefore, by the mathematical induction, we obtain
α ( ( x , y ) , ( u n , v n ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ11_HTML.gif
(3.11)
for all n and similarly, α ( ( y , x ) , ( v n , u n ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq91_HTML.gif. From (3.10) and (3.11), we get
d ( x , u n + 1 ) = d ( F ( x , y ) , F ( u n , v n ) ) α ( ( x , y ) , ( u n , v n ) ) d ( F ( x , y ) , F ( u n , v n ) ) ψ ( d ( x , u n ) + d ( y , v n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ12_HTML.gif
(3.12)
Similarly, we have
d ( y , v n + 1 ) = d ( F ( y , x ) , F ( v n , u n ) ) α ( ( y , x ) , ( v n , u n ) ) d ( F ( y , x ) , F ( v n , u n ) ) ψ ( d ( y , v n ) + d ( x , u n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ13_HTML.gif
(3.13)
Adding (3.12) and (3.13), we get
d ( x , u n + 1 ) + d ( y , v n + 1 ) 2 ψ ( d ( x , u n ) + d ( y , v n ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equal_HTML.gif
Thus,
d ( x , u n + 1 ) + d ( y , v n + 1 ) 2 ψ n ( d ( x , u 1 ) + d ( y , v 1 ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ14_HTML.gif
(3.14)
for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq88_HTML.gif. Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq27_HTML.gif in (3.14) and using Lemma 3.1, we get
lim n [ d ( x , u n + 1 ) + d ( y , v n + 1 ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equam_HTML.gif
This implies
lim n d ( x , u n + 1 ) = lim n d ( y , v n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ15_HTML.gif
(3.15)
Similarly, one can show that
lim n d ( s , u n + 1 ) = lim n d ( t , v n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equ16_HTML.gif
(3.16)

From (3.15) and (3.16), we conclude that x = s https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq92_HTML.gif and y = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq93_HTML.gif. Hence, F has a unique coupled fixed point. □

Example 3.7 (Linear case)

Let X = [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq94_HTML.gif and d : X × X R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq95_HTML.gif be a standard metric. Define a mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq3_HTML.gif by F ( x , y ) = 1 4 x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq96_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif. Consider a mapping α : X 2 × X 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq30_HTML.gif be such that
α ( ( x , y ) , ( u , v ) ) = { 1 if  x y , u v , 0 otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equan_HTML.gif
Since | x y u v | | x u | + | y v | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq97_HTML.gif holds for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq32_HTML.gif. Therefore, we have
d ( F ( x , y ) , F ( u , v ) ) = | x y 4 u v 4 | 1 4 ( | x u | + | y v | ) = 1 4 ( d ( x , u ) + d ( y , v ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equao_HTML.gif
It follows that
α ( ( x , y ) , ( u , v ) ) d ( F ( x , y ) , F ( u , v ) ) 1 4 ( d ( x , u ) + d ( y , v ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equap_HTML.gif

Thus (3.1) holds for ψ ( t ) = t / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq98_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif, and we also see that all the hypotheses of Theorem 3.4 are fulfilled. Then there exists a coupled fixed point of F. In this case, ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq99_HTML.gif is a coupled fixed point of F.

Example 3.8 (Nonlinear case)

Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq100_HTML.gif and d : X × X R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq101_HTML.gif be a standard metric. Define a mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq1_HTML.gif by F ( x , y ) = 1 4 ln ( 1 + | x | ) + 1 4 ln ( 1 + | y | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq102_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq5_HTML.gif. Consider a mapping α : X 2 × X 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq103_HTML.gif be such that
α ( ( x , y ) , ( u , v ) ) = { 1 if  x y , u v , 0 otherwise . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equaq_HTML.gif
Then we get
d ( F ( x , y ) , F ( u , v ) ) = | 1 4 ln ( 1 + | x | ) + 1 4 ln ( 1 + | y | ) 1 4 ln ( 1 + | u | ) 1 4 ln ( 1 + | v | ) | 1 4 | ln 1 + | x | 1 + | u | | + 1 4 | ln 1 + | y | 1 + | v | | 1 2 [ 1 2 ln ( 1 + | x u | ) + 1 2 ln ( 1 + | y v | ) ] 1 2 ln ( 2 + | x u | + | y v | 2 ) 1 2 ln ( 1 + | x u | + | y v | 2 ) = 1 2 ln ( 1 + d ( x , u ) + d ( y , v ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equar_HTML.gif
Thus,
α ( ( x , y ) , ( u , v ) ) d ( F ( x , y ) , F ( u , v ) ) 1 2 ln ( 1 + d ( x , y ) + d ( y , v ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_Equas_HTML.gif

Therefore (3.1) holds for ψ ( t ) = 1 2 ln ( 1 + t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq104_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq20_HTML.gif, and also the hypothesis of Theorem 3.4 is fulfilled. Then there exists a coupled fixed point of F. In this case, ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq99_HTML.gif is a coupled fixed point of F.

4 Concluding remark

The author of [33] recently established some coupled fixed point theorems in partially ordered metric spaces shortly by using some usual corresponding fixed point theorems on the metric space M = X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq105_HTML.gif. Note that if the right-hand side of the α-ψ-contractive type condition (3.1) is replaced by 1 2 ( d ( x , u ) + d ( y , v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq106_HTML.gif, then a very short proof similar to what followed in [33] can be provided for a coupled fixed point theorem of Theorem 3.4 type by making just use of the results in [36]. However, since the right-hand side of (3.1) is not of the form 1 2 ( d ( x , u ) + d ( y , v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq106_HTML.gif, specially for nonlinear functions ψ, then it is not possible to apply the method [33]. In this connection, notice that Example 3.7 works for both when the right-hand side is either 1 2 ( d ( x , u ) + d ( y , v ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-228/MediaObjects/13663_2012_Article_331_IEq106_HTML.gif or as in (3.1), but Example 3.8 works only for (3.1). Hence, our results are more interesting and different from the existing results of [33] and [36].

Declarations

Acknowledgements

The work of the second author was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. He acknowledges with thanks DSR technical and financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Aligarh Muslim University
(2)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(3)
Department of Mathematics, Texas A&M University-Kingsville

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© Mursaleen 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.