Fixed Point Theory and Applications

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Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces

Fixed Point Theory and Applications20122012:95

https://doi.org/10.1186/1687-1812-2012-95

©  Gordji et al; licensee Springer. 2012

Received: 26 September 2011

Accepted: 12 June 2012

Published: 12 June 2012

Abstract

In this paper, we introduce the concept of a mixed weakly monotone pair of mappings and prove some coupled common fixed point theorems for a contractive-type mappings with the mixed weakly monotone property in partially ordered metric spaces. Our results are generalizations of the main results of Bhaskar and Lakshmikantham and Kadelburg et al.

Mathematics Subject Classification 2000: 54H25.

Keywords

common fixed point mixed weakly monotone mappings partially ordered metric space.

1. Introduction

In 1922, Banach gave a theorem, which is well-known as Banach's Fixed Point Theorem (or Banach's Contractive Principle) to establish the existence of solutions for nonlinear operator equations and integral equations. Since then, because of their simplicity and usefulness, it has become a very popular tools in solving the existence problems in many branches of mathematical analysis. Since then, many authors have extended, improved and generalized Banach's theorem in several ways [1–11].

Recently, the existence of coupled fixed points for some kinds of contractive-type mappings in partially ordered metric spaces, (ordered) cone metric spaces, fuzzy metric spaces and other spaces with applications has been investigated by some authors, for example, Bhaskar and Lakshmikantham [5], Cho et al. [12–14], Dhage et al. [15], Gordji et al. [16, 17], Kadelburg et al. [18], Nieto and Lopez [10], Ran and Rarings [11], Sintunavarat et al. [19, 20], Yang et al. [21] and others.

Especially, in [5], Bhaskar and Lakshmikantham introduced the notions of a mixed monotone mapping and a coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings and discussed the existence and uniqueness of solution for periodic boundary value problems.

Definition 1.1. [5] Let (X, ≤) be a partially ordered set and f: X × X → X be a mapping. We say that f has the mixed monotone property on X if, for any x, y ∈ X,
x 1 , x 2 ∈ X , x 1 ≤ x 2 ⇒ f ( x 1 , y ) ≤ f ( x 2 , y )
and
y 1 , y 2 ∈ X , y 1 ≤ y 2 ⇒ f ( x , y 1 ) ≥ f ( x , y 2 ) .

Definition 1.2. [5] An element (x, y) ∈ X × X is called a coupled fixed point of a mapping F: X × X → X if x = F (x, y) and y = F (y, x).

Theorem 1.3. [5] Let (X, ≤, d) be a partially ordered complete metric space. Let f: X × X → X be a mapping having the mixed monotone property on X. Assume that there exists k ∈ [0, 1) with
d ( f ( x , y ) , f ( u , v ) ) ≤ k 2 ( d ( x , u ) + d ( y , v ) )
for all x, y, u, v ∈ X with x ≤ u and y ≥ v. Also, suppose that either
  1. (1)

    f is continuous or

     
  2. (2)

    X has the following properties:

     
  3. (a)

    if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;

     
  4. (b)

    if {y n } is a decreasing sequence y n → y, then y n ≥ y for all n ≥ 1.

     

If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0) and y0 ≥ f(y0, x0), then f has a coupled fixed point in X.

Very recently, Kadelburg et al. [18] proved the following theorem on cone metric spaces.

Theorem 1.4. [18] Let (X, ≤, d) be an ordered cone metric space. Let (f, g) be a weakly increasing pair of self-mappings on X with respect to ≤. Suppose that the following conditions hold:
  1. (1)
    there exist p, q, r, s, t ≥ 0 satisfying p + q + r + s + t < 1 and q = r or s = t such that
    d ( f x , g y ) ≤ p d ( x , y ) + q d ( x , f x ) + s d ( x , g y ) + t d ( y , f x )
     
for all comparable x, y ∈ X;
  1. (2)

    f or g is continuous or, if a nondecreasing {x n } converges to a point x ∈ X, then x n ≤ x for all n ≥ 1.

     

Then f and g have a common fixed point in X.

Note that a pair (f, g) of self-mappings on partially ordered set (X, ≤) is said to be weakly increasing if fx ≤ gfx and gx ≤ fgx for all x ∈ X.

Now, we introduce the following concept of the mixed weakly increasing property of mappings.

Definition 1.5. Let (X, ≤) be a partially ordered set and f, g: X × X → X be mappings. We say that a pair (f, g) has the mixed weakly monotone property on X if, for any x, y ∈ X,
x ≤ f ( x , y ) , y ≥ f ( y , x ) ⇒ f ( x , y ) ≤ g ( f ( x , y ) , f ( y , x ) ) , f ( y , x ) ≥ g ( f ( y , x ) , f ( x , y ) )
and
x ≤ g ( x , y ) , y ≥ g ( y , x ) ⇒ g ( x , y ) ≤ f ( g ( x , y ) , g ( y , x ) ) , g ( y , x ) ≥ f ( g ( y , x ) , g ( x , y ) ) .
Example 1.6. Consider an ordered cone metric space (ℝ, ≤, d), where ≤ represents the usual order relation and d is a usual metric on ℝ and let f, g: ℝ × ℝ → ℝ be two functions defined by
f ( x , y ) = x - 2 y , g ( x , y ) = x - y .

Then a pair (f, g) has the mixed weakly monotone property.

Example 1.7. Consider an ordered cone metric space (ℝ, ≤, d), where ≤ represents the usual order relation and d is a usual metric on ℝ and let f, g: ℝ × ℝ → ℝ be two functions defined by
f ( x , y ) = x - y + 1 , g ( x , y ) = 2 x - 3 y .
Then both mappings f and g have the mixed monotone property, but a pair (f, g) has not the mixed weakly monotone property. To see this, for any 9 8 , 7 8 ∈ ℝ 2 , we have
9 8 ≤ f 9 8 , 7 8 , 7 8 ≥ f 7 8 , 9 8 ,
but
f 9 8 , 7 8 ≰ g f 9 8 , 7 8 , f 7 8 , 9 8 , f 7 8 , 9 8 ≥ g f 7 8 , 9 8 , f 9 8 , 7 8 .

The purpose of this paper is to present some coupled common fixed point theorems for a pair of mappings with the mixed weakly monotone property in a partially ordered metric space. Our results generalize the main results of Bhaskar and Lakshmikantham [5], Kadelburg et al. [18] and others.

2. Coupled common fixed point theorems

Let (X, ≤, d) be a partially ordered complete metric space. Now, we consider the product space X × X with following partial order: for all (x, y), (u, v) ∈ X × X,
( x , y ) ≤ ( u , v ) ⇔ x ≤ u , y ≥ v .
Also, let (X × X, D) be a metric space with the following metric:
D ( ( x , y ) , ( u , v ) ) : = d ( x , u ) + d ( y , v )

for all (x, y), (u, v) ∈ X × X.

Theorem 2.1. Let (X, ≤, d) be a partially ordered complete metric space. Let f, g: X × X → X be the mappings such that a pair (f, g) has the mixed weakly monotone property on X. Suppose that there exist p, q, r, s ≥ 0 with p + q + r + 2s < 1 such that
d ( f ( x , y ) , g ( u , v ) ) ≤ p 2 D ( ( x , y ) , ( u , v ) ) + q 2 D ( ( x , y ) , ( f ( x , y ) , f ( y , x ) ) ) + r 2 D ( ( u , v ) , ( g ( u , v ) , g ( v , u ) ) ) + s 2 D ( ( x , y ) , ( g ( u , v ) , g ( v , u ) ) ) + s 2 D ( ( u , v ) , ( f ( x , y ) , f ( y , x ) ) )
(2.1)

for all x, y, u, v ∈ X with x ≤ u and y ≥ v. Let x0, y0 ∈ X be such that x0 ≤ f(x0, y0), y0 ≥ f(y0, x0) or x0 ≤ g(x0, y0), y0 ≥ g(y0, x0). If f or g is continuous, then f and g have a coupled common fixed point in X.

Proof. Suppose that x0 ≤ f(x0, y0) and y0 ≥ f (y0, x0) and let
f ( x 0 , y 0 ) = x 1 , f ( y 0 , x 0 ) = y 1 .
From the mixed weakly monotone property of the pair (f, g), we have
x 1 = f ( x 0 , y 0 ) ≤ g ( f ( x 0 , y 0 ) , f ( y 0 , x 0 ) ) = g ( x 1 , y 1 )
and
y 1 = f ( y 0 , x 0 ) ≥ g ( f ( y 0 , x 0 ) , f ( x 0 , y 0 ) ) = g ( y 1 , x 1 ) .
Let
g ( x 1 , y 1 ) = x 2 , g ( y 1 , x 1 ) = y 2 .
Then we have
g ( x 1 , y 1 ) ≤ f ( g ( x 1 , y 1 ) , g ( y 1 , x 1 ) ) = f ( x 2 , y 2 )
and
g ( y 1 , x 1 ) ≥ f ( g ( y 1 , x 1 ) , g ( x 1 , y 1 ) ) = f ( y 2 , x 2 ) .
Continuously, let
x 2 n + 1 = f ( x 2 n , y 2 n ) , y 2 n + 1 = f ( y 2 n , x 2 n )
and
x 2 n + 2 = g ( x 2 n + 1 , y 2 n + 1 ) , y 2 n + 2 = g ( y 2 n + 1 , x 2 n + 1 )
for all n ≥ 1. Then we can easily verify that
x 0 ≤ x 1 ≤ x 2 ≤ ⋯ ≤ x n ≤ x n + 1 ≤ ⋯
and
y 0 ≥ y 1 ≥ y 2 ≥ ⋯ ≥ y n ≥ y n + 1 ≥ ⋯ .
Similarly, from the condition x0 ≤ g(x0, y0) and y0 ≥ g(y0, x0), one can show that the sequences {x n } and {y n } are increasing and decreasing, respectively. Thus, applying (2.1), we obtain
d ( x 2 n + 1 , x 2 n + 2 ) = d ( f ( x 2 n , y 2 n ) , g ( x 2 n + 1 , y 2 n + 1 ) ) ≤ p 2 D ( ( x 2 n , y 2 n ) , ( x 2 n + 1 , y 2 n + 1 ) ) + q 2 D ( ( x 2 n , y 2 n ) , ( f ( x 2 n , y 2 n ) , f ( y 2 n , x 2 n ) ) ) + r 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( g ( x 2 n + 1 , y 2 n + 1 ) , g ( y 2 n + 1 , x 2 n + 1 ) ) ) + s 2 D ( ( x 2 n , y 2 n ) , ( g ( x 2 n + 1 , y 2 n + 1 ) , g ( y 2 n + 1 , x 2 n + 1 ) ) ) + s 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( f ( x 2 n , y 2 n ) , f ( y 2 n , x 2 n ) ) ) = p 2 D ( ( x 2 n , y 2 n ) , ( x 2 n + 1 , y 2 n + 1 ) ) + q 2 D ( ( x 2 n , y 2 n ) , ( x 2 n + 1 , y 2 n + 1 ) ) + r 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + s 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 1 , y 2 n + 1 ) ) + s 2 D ( ( x 2 n , y 2 n ) , ( x 2 n + 2 , y 2 n + 2 ) ) ≤ p + q 2 D ( ( x 2 n , y 2 n ) , ( x 2 n + 1 , y 2 n + 1 ) ) + r 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + s 2 [ D ( ( x 2 n , y 2 n ) , ( x 2 n + 1 , y 2 n + 1 ) ) + D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) ] = p + q + s 2 D ( ( x 2 n , y 2 n ) , ( x 2 n + 1 , y 2 n + 1 ) ) + r + s 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) .
Hence it follows that
d ( x 2 n + 1 , x 2 n + 2 ) ≤ p + q + s 2 ( d ( x 2 n , x 2 n + 1 ) + d ( y 2 n , y 2 n + 1 ) ) + r + s 2 ( d ( x 2 n + 1 , x 2 n + 2 ) + d ( y 2 n + 1 , y 2 n + 2 ) )
(2.2)
for all n ≥ 1. Similarly, we have
d ( y 2 n + 1 , y 2 n + 2 ) ≤ p + q + s 2 ( d ( y 2 n , y 2 n + 1 ) + d ( x 2 n , x 2 n + 1 ) ) + r + s 2 ( d ( y 2 n + 1 , y 2 n + 2 ) + d ( x 2 n + 1 , x 2 n + 2 ) )
(2.3)
for all n ≥ 1. Thus it follows from (2.2) and (2.3) that
d ( x 2 n + 1 , x 2 n + 2 ) + d ( y 2 n + 1 , y 2 n + 2 ) ≤ p + q + s 1 - ( r + s ) ( d ( x 2 n , x 2 n + 1 ) + d ( y 2 n , y 2 n + 1 ) )
(2.4)
for all n ≥ 1. Moreover, if we apply (2.1), then we have
d ( x 2 n + 2 , x 2 n + 3 ) = d ( g ( x 2 n + 1 , y 2 n + 1 ) , f ( x 2 n + 2 , y 2 n + 2 ) ) ≤ p 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + q 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( g ( x 2 n + 1 , y 2 n + 1 ) , g ( y 2 n + 1 , x 2 n + 1 ) ) ) + r 2 D ( ( x 2 n + 2 , y 2 n + 2 ) , ( f ( x 2 n + 2 , y 2 n + 2 ) , f ( y 2 n + 2 , x 2 n + 2 ) ) ) + s 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( f ( x 2 n + 2 , y 2 n + 2 ) , f ( y 2 n + 2 , x 2 n + 2 ) ) ) + s 2 D ( ( x 2 n + 2 , y 2 n + 2 ) , ( g ( x 2 n + 1 , y 2 n + 1 ) , g ( y 2 n + 1 , x 2 n + 1 ) ) ) = p 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + q 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + r 2 D ( ( x 2 n + 2 , y 2 n + 2 ) , ( x 2 n + 3 , y 2 n + 3 ) ) + s 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 3 , y 2 n + 3 ) ) + s 2 D ( ( x 2 n + 2 , y 2 n + 2 ) , ( x 2 n + 2 , y 2 n + 2 ) ) ≤ p + q 2 D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + r 2 D ( ( x 2 n + 2 , y 2 n + 2 ) , ( x 2 n + 3 , y 2 n + 3 ) ) + s 2 [ D ( ( x 2 n + 1 , y 2 n + 1 ) , ( x 2 n + 2 , y 2 n + 2 ) ) + D ( ( x 2 n + 2 , y 2 n + 2 ) , ( x 2 n + 3 , y 2 n + 3 ) ) ] .
Hence it follows that
d ( x 2 n + 2 , x 2 n + 3 ) ≤ p + q + s 2 ( d ( x 2 n + 1 , x 2 n + 2 ) + d ( y 2 n + 1 , y 2 n + 2 ) ) + r + s 2 ( d ( x 2 n + 2 , x 2 n + 2 ) + d ( y 2 n + 3 , y 2 n + 3 ) )
(2.5)
for all n ≥ 1. Similarly, we have
d ( y 2 n + 2 , y 2 n + 3 ) ≤ p + q + s 2 ( d ( y 2 n + 1 , y 2 n + 2 ) + d ( x 2 n + 1 , x 2 n + 2 ) ) + r + s 2 ( d ( y 2 n + 2 , y 2 n + 2 ) + d ( x 2 n + 3 , x 2 n + 3 ) ) .
(2.6)
Thus, using (2.5) and (2.6), we have
d ( x 2 n + 2 , x 2 n + 3 ) + d ( y 2 n + 2 , y 2 n + 3 ) ≤ p + q + s 1 - ( r + s ) ( d ( x 2 n + 1 , x 2 n + 2 ) + d ( y 2 n + 1 , y 2 n + 2 ) )
(2.7)
for all n ≥ 1. Also, it follows from (2.4) and (2.7) that
d ( x 2 n + 2 , x 2 n + 3 ) + d ( y 2 n + 2 , y 2 n + 3 ) ≤ p + q + s 1 - ( r + s ) 2 ( d ( x 2 n , x 2 n + 1 ) + d ( y 2 n , y 2 n + 1 ) )
(2.8)
for all n ≥ 1. Let A = p + q + s 1 - ( r + s ) . Then 0 ≤ A < 1 and
d ( x 2 n + 1 , x 2 n + 2 ) + d ( y 2 n + 1 , y 2 n + 2 ) ≤ A ( d ( x 2 n , x 2 n + 1 ) + d ( y 2 n , y 2 n + 1 ) ) ≤ A 3 ( d ( x 2 n - 2 , x 2 n - 1 ) + d ( y 2 n - 2 , y 2 n - 1 ) ) ≤ A 5 ( d ( x 2 n - 4 , x 2 n - 3 ) + d ( y 2 n - 4 , y 2 n - 3 ) ) ≤ ⋯ ≤ A 2 n + 1 ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) )
and
d ( x 2 n + 2 , x 2 n + 3 ) + d ( y 2 n + 2 , y 2 n + 3 ) ≤ A 2 ( d ( x 2 n , x 2 n + 1 ) + d ( y 2 n , y 2 n + 1 ) ) ≤ A 4 ( d ( x 2 n - 2 , x 2 n - 1 ) + d ( y 2 n - 2 , y 2 n - 1 ) ) ≤ A 6 ( d ( x 2 n - 4 , x 2 n - 3 ) + d ( y 2 n - 4 , y 2 n - 3 ) ) ≤ ⋯ ≤ A 2 n + 2 ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) )
for all n ≥ 1. Now, for all m, n ≥ 1 with n ≤ m, we have
d ( x 2 n + 1 , x 2 m + 1 ) + d ( y 2 n + 1 , y 2 m + 1 ) ≤ ( d ( x 2 n + 1 , x 2 n + 2 ) + d ( y 2 n + 1 , y 2 n + 2 ) ) + ( d ( x 2 n + 2 , x 2 n + 3 ) + d ( y 2 n + 2 , y 2 n + 3 ) ) + ⋯ + ( d ( x 2 m , x 2 m + 1 ) + d ( y 2 m , y 2 m + 1 ) ) ≤ ( A 2 n + 1 + A 2 n + 2 + ⋯ + A 2 m ) ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) ≤ A 2 n + 1 1 - A ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) .
Similarly, we have
d ( x 2 n , x 2 m + 1 ) + d ( y 2 n , y 2 m + 1 ) ≤ ( A 2 n + A 2 n + 1 + A 2 n + 2 + ⋯ + A 2 m ) ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) ≤ A 2 n 1 - A ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) , d ( x 2 n , x 2 m ) + d ( y 2 n , y 2 m ) ≤ ( A 2 n + A 2 n + 1 + A 2 n + 2 + ⋯ + A 2 m - 1 ) ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) ≤ A 2 n 1 - A ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) )
and
d ( x 2 n + 1 , x 2 m ) + d ( y 2 n + 1 , y 2 m ) ≤ ( A 2 n + 1 + A 2 n + 1 + A 2 n + 2 + ⋯ + A 2 m - 1 ) ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) ≤ A 2 n + 1 1 - A ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) ) .
Hence, for all m, n ≥ 1 with n ≤ m, it follows that
d ( x n , x m ) + d ( y n , y m ) ≤ A 2 n 1 - A ( d ( x 0 , x 1 ) + d ( y 0 , y 1 ) )
and so, since 0 ≤ A < 1, we can conclude that
d ( x n , x m ) + d ( y n , y m ) → 0

as n → ∞, which implies that d(x n , x m ) → 0 and d(y n , y m ) → 0 as m, n → ∞. Therefore, the sequences {x n } and {y n } are Cauchy sequences in X. Since (X, d) is a complete metric space, then there exist x, y ∈ X such that x n → x and y n → y as n → ∞.

Suppose that f is a continuous. Then we have
x = lim k → ∞ x 2 k + 1 = lim k → ∞ f ( x 2 k , y 2 k ) = f ( lim k → ∞ x 2 k , lim k → ∞ y 2 k ) = f ( x , y )
and
y = lim k → ∞ y 2 k + 1 = lim k → ∞ f ( y 2 k , x 2 k ) = f ( lim k → ∞ y 2 k , lim k → ∞ x 2 k ) = f ( y , x ) .
Taking x = u and y = v in (2.1), we have
d ( f ( x , y ) , g ( x , y ) ) + d ( f ( y , x ) , g ( y , x ) ) ≤ p 2 D ( ( x , y ) , ( x , y ) ) + q 2 D ( ( x , y ) , f ( x , y ) , f ( y , x ) ) + r 2 D ( ( x , y ) , g ( x , y ) , g ( y , x ) ) + s 2 D ( ( x , y ) , g ( x , y ) , g ( y , x ) ) + s 2 D ( ( x , y ) , f ( x , y ) , f ( y , x ) ) p 2 D ( ( y , x ) , ( y , x ) ) + q 2 D ( ( y , x ) , f ( y , x ) , f ( x , y ) ) + r 2 D ( ( y , x ) , g ( y , x ) , g ( x , y ) ) + s 2 D ( ( y , x ) , g ( y , x ) , g ( x , y ) ) + s 2 D ( ( y , x ) , f ( y , x ) , f ( x , y ) ) .
Hence we have
d ( x , g ( x , y ) ) + d ( y , g ( y , x ) ) ≤ ( r + s ) ( d ( x , g ( x , y ) ) + d ( y , g ( y , x ) ) )
and so, since r + s < 1, we can get that
d ( x , g ( x , y ) ) = 0 , d ( y , g ( y , x ) ) = 0 .

Hence (x, y) is a coupled common fixed point of f and g.

Similarly, we can prove that (x, y) is a coupled common fixed point of f and g when g is a continuous mapping. This completes the proof.   □

Theorem 2.2. Let (X, ≤, d) be a partially ordered complete metric space. Assume that X has the following property:
  1. (1)

    if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;

     
  2. (2)

    if {y n } is a decreasing sequence with y n → y, then y n ≥ y for all n ≥ 1.

     
Let f, g: X × X → X be the mappings such that a pair (f, g) has the mixed weakly monotone property on X. Also, suppose that there exist p, q, r, s ≥ 0 with p + q + r + 2s < 1 such that
d ( f ( x , y ) , g ( u , v ) ) ≤ p 2 D ( ( x , y ) , ( u , v ) ) + q 2 D ( ( x , y ) , ( f ( x , y ) , f ( y , x ) ) ) + r 2 D ( ( u , v ) , ( g ( u , v ) , g ( v , u ) ) ) + s 2 D ( ( x , y ) , ( g ( u , v ) , g ( v , u ) ) ) + s 2 D ( ( u , v ) , ( f ( x , y ) , f ( y , x ) ) )

for all x, y, u, v ∈ X with x ≤ u and y ≥ v. If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0), y0 ≥ f(y0, x0) or x0 ≤ g(x0, y0), y0 ≥ g(y0, x0), then f and g have a coupled common fixed point in X.

Proof. Following the proof of Theorem 2.1, we only have to show that
f ( x , y ) = g ( x , y ) = x , f ( y , x ) = g ( y , x ) = y .
It is clear that
D ( ( x , y ) , ( f ( x , y ) , f ( y , x ) ) ) ≤ D ( ( x , y ) , ( x 2 k + 2 , y 2 k + 2 ) ) + D ( ( x 2 k + 2 , y 2 k + 2 ) , ( f ( x , y ) , f ( y , x ) ) ) = D ( ( x , y ) , ( x 2 k + 2 , y 2 k + 2 ) ) + D ( ( g ( x 2 k + 1 , y 2 k + 1 ) , g ( y 2 k + 1 , x 2 k + 1 ) ) , ( f ( x , y ) , f ( y , x ) ) ) = D ( ( x , y ) , ( x 2 k + 2 , y 2 k + 2 ) ) + d ( g ( x 2 k + 1 , y 2 k + 1 ) , f ( x , y ) ) + d ( f ( y , x ) , g ( y 2 k + 1 , x 2 k + 1 ) ) ≤ D ( ( x , y ) , ( x 2 k + 2 , y 2 k + 2 ) ) + p 2 D ( ( x 2 k + 1 , y 2 k + 1 ) , ( x , y ) ) + q 2 D ( ( x 2 k + 1 , y 2 k + 1 ) , ( g ( x 2 k + 1 , y 2 k + 1 ) , g ( y 2 k + 1 , x 2 k + 1 ) ) ) + r 2 D ( ( x , y ) , ( f ( x , y ) , f ( y , x ) ) ) + s 2 D ( ( x 2 k + 1 , y 2 k + 1 ) , ( f ( x , y ) , f ( y , x ) ) ) + s 2 D ( ( x , y ) , ( g ( x 2 k + 1 , y 2 k + 1 ) , g ( y 2 k + 1 , x 2 k + 1 ) ) ) + p 2 D ( ( y , x ) , ( y 2 k + 1 , x 2 k + 1 ) ) + q 2 D ( ( y , x ) , ( f ( y , x ) , f ( x , y ) ) ) + r 2 D ( ( y 2 k + 1 , x 2 k + 1 ) , ( g ( y 2 k + 1 , x 2 k + 1 ) , g ( x 2 k + 1 , y 2 k + 1 ) ) ) + s 2 D ( ( y , x ) , ( g ( y 2 k + 1 , x 2 k + 1 ) , g ( x 2 k + 1 , y 2 k + 1 ) ) ) + s 2 D ( ( y 2 k + 1 , x 2 k + 1 ) , ( f ( y , x ) , f ( x , y ) ) )
and so
d ( x , f ( x , y ) ) + d ( y , f ( y , x ) ) ≤ d ( x , x 2 k + 2 ) + d ( y , y 2 k + 2 ) + p ( d ( x 2 k + 1 , x ) + d ( y 2 k + 1 , y ) ) + q 2 ( d ( x 2 k + 1 , x 2 k + 2 ) + d ( y 2 k + 1 , y 2 k + 2 ) + d ( x , f ( x , y ) ) + d ( y , f ( y , x ) ) ) + r 2 ( d ( x , f ( x , y ) ) + d ( y , f ( y , x ) ) + d ( y 2 k + 1 , y 2 k + 2 ) + d ( x 2 k + 1 , x 2 k + 2 ) ) + s ( d ( x 2 k + 2 , x ) + d ( y 2 k + 2 , y ) + d ( x 2 k + 1 , f ( x , y ) ) + d ( y 2 k + 1 , f ( y , x ) ) ) .
(2.9)
Letting k → ∞ in (2.9), we obtain
d ( x , f ( x , y ) ) + d ( y , f ( y , x ) ) ≤ q + r + 2 s 2 [ d ( x , f ( x , y ) ) + d ( y , f ( y , x ) ) ] .
Since q + r + 2 s 2 < 1 , we have
d ( x , f ( x , y ) ) + d ( y , f ( y , x ) ) = 0

and so f(x, y) = x and f(y, x) = y. Similarly, we can show that g(x, y) = x and g(y, x) = y. Therefore, (x, y) is a coupled common fixed point of f and g. This completes the proof.   □

Now, we give an example to illustrate Theorem 2.1 as follows:

Example 2.3. Consider (ℝ, ≤, d), where ≤ represents the usual order relation and d is a usual metric on ℝ and let f, g: ℝ × ℝ → ℝ be two functions defined by
f ( x , y ) = 6 x - 3 y + 33 36 , g ( x , y ) = 8 x - 4 y + 44 48 .
Then a pair (f, g) has the mixed weakly monotone property and
d ( f ( x , y ) , g ( u , v ) ) = | f ( x , y ) - g ( u , v ) | = 6 x - 3 y + 33 36 - 8 x - 4 y + 44 48 ≤ 1 6 | x - u | + 1 12 | y - v | ≤ 1 6 ( | x - u | + | y - v | ) .

By putting p = 1 3 and q = r = s = 0 in (2.1), we see that (1, 1) is a unique coupled common fixed point of f and g.

Corollary 2.4. In Theorems 2.1 and 2.2, if X is a total ordered set, then a coupled common fixed point of f and g is unique and x = y.

Proof. If (x*, y*) ∈ X × X is another coupled common fixed point of f and g, then, by the use of (2.1), we have
d ( x , x * ) + d ( y , y * ) = d ( f ( x , y ) , g ( x * , y * ) ) + d ( f ( y , x ) , g ( y * , x * ) ) ≤ p 2 D ( ( x , y ) , ( x * , y * ) ) + q 2 D ( ( x , y ) , ( f ( x , y ) , f ( y , x ) ) ) + r 2 D ( ( x * , y * ) , ( g ( x * , y * ) , g ( y * , x * ) ) ) + s 2 D ( ( x , y ) , ( g ( x * , y * ) , g ( y * , x * ) ) ) + s 2 D ( ( x * , y * ) , ( f ( x , y ) , f ( y , x ) ) ) + p 2 D ( ( y , x ) , ( y * , x * ) ) + q 2 D ( ( y , x ) , ( f ( y , x ) , f ( x , y ) ) ) + r 2 D ( ( y * , x * ) , ( g ( y * , x * ) , g ( x * , y * ) ) ) + s 2 D ( ( y , x ) , ( g ( y * , x * ) , g ( x * , y * ) ) ) + s 2 D ( ( y * , x * ) , ( f ( y , x ) , f ( x , y ) ) ) = p ( d ( x , x * ) ) + d ( y , y * ) + 2 s ( d ( x , x * ) ) + d ( y , y * )
and hence
d ( x , x * ) + d ( y , y * ) = ( p + 2 s ) ( d ( x , x * ) + d ( y , y * ) ) .

Since q + 2s < 1, we have d(x, x*) + d(y, y*) = 0, which implies that x = x* and y = y*.

On the other hand, we have
d ( x , y ) = d ( f ( x , y ) , g ( y , x ) ) ≤ p 2 D ( ( x , y ) , ( y , x ) ) + s D ( ( x , y ) , ( y , x ) ) = ( p + 2 s ) d ( x , y ) .

Since p + 2s < 1, we have d(x, y) = 0 and x = y. This completes the proof.   □

Let f: X × X → X be a mapping. Now, we denote
f n + 1 ( x , y ) = f ( f n ( x , y ) , f n ( y , x ) )

for all x, y ∈ X and n ≥ 1.

Remark 2.5. Let (X, ≤, d) be a partially ordered complete metric space. Let f: X × X → X be a mapping with the mixed monotone property on X. Then, for each n ≥ 1, a pair (f n , f n ) has the mixed weakly monotone property on X. In fact, let x ≤ f n (x, y) and y ≤ f n (y, x). Then it follows from the mixed monotone property of f that
f ( x , y ) ≤ f ( f n ( x , y ) , y ) ≤ f ( f n ( x , y ) , f n ( y , x ) ) = f n + 1 ( x , y ) , f ( y , x ) ≥ f ( f n ( y , x ) , x ) ≥ f ( f n ( y , x ) , f n ( x , y ) ) = f n + 1 ( y , x )
and
f 2 ( x , y ) = f ( f ( x , y ) , f ( y , x ) ) ≤ f ( f n + 1 ( x , y ) , f n + 1 ( y , x ) ) = f n + 2 ( x , y ) , f 2 ( y , x ) = f ( f ( y , x ) , f ( x , y ) ) ≥ f ( f n + 1 ( y , x ) , f n + 1 ( x , y ) ) = f n + 2 ( y , x ) .
Continuously, we have
f n ( x , y ) ≤ f n + n ( x , y ) , f n ( y , x ) ≥ f n + n ( y , x ) .
Hence we have
f n ( x , y ) ≤ f n ( f n ( x , y ) , f n ( y , x ) ) , f n ( y , x ) ≥ f n ( f n ( y , x ) , f n ( x , y ) ) ,

which implies that the pair (f n , f n ) has the mixed weakly monotone property on X.

Corollary 2.6. Let (X, ≤, d) be a partially ordered complete metric space. Let f: X × X → X be a mapping with the mixed monotone property on X. Assume that there exist p, q, r, s ≥ 0 with p + q + r + 2s < 1 such that
d ( f ( x , y ) , f ( u , v ) ) ≤ p 2 D ( ( x , y ) , ( u , v ) ) + q 2 D ( ( x , y ) , ( f ( x , y ) , f ( y , x ) ) ) + r 2 D ( ( u , v ) , ( f ( u , v ) , f ( v , u ) ) ) + s 2 D ( ( x , y ) , ( f ( u , v ) , f ( v , u ) ) ) + s 2 D ( ( u , v ) , ( f ( x , y ) , f ( y , x ) ) )
for all x, y, u, v ∈ X with x ≤ u and y ≥ v. Moreover, suppose that either
  1. (1)

    f is continuous or

     
  2. (2)

    X has the following properties:

     
  3. (a)

    if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;

     
  4. (b)

    if {y n } is a decreasing sequence with y n → y, then y n ≥ y for all n ≥ 1.

     

If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0) and y0 ≥ f(y0, x0), then f has a coupled fixed point in X.

Proof. Taking f = g in Theorems 2.1, 2.2 and using Remark 2.5, we can get the conclusion.   □

Corollary 2.7. Let (X, ≤, d) be a partially ordered complete metric space. Let f: X × X → X be a mapping with the mixed monotone property on X. Assume that there exists k ∈ [0, 1) with
d ( f ( x , y ) , f ( u , v ) ) ≤ k 2 ( d ( x , u ) + d ( y , v ) )
for all x, y, u, v ∈ X with x ≤ u and y ≥ v. Also, suppose that either
  1. (1)

    f is continuous or

     
  2. (2)

    X has the following properties:

     
  3. (a)

    if {x n } is an increasing sequence with x n → x, then x n ≤ x for all n ≥ 1;

     
  4. (b)

    if {y n } is a decreasing sequence with y n → y, then y n ≥ y for all n ≥ 1.

     

If there exist x0, y0 ∈ X such that x0 ≤ f(x0, y0) and y0 ≥ f(y0, x0), then f has a coupled fixed point in X.

Proof. Taking f = g, p = k and q = r = s = 0 in Theorems 2.1, 2.2 and using Remark 2.5, we can get the conclusion.   □

Declarations

Acknowledgements

The authors would like to thank anonymous referees for their valuable comments and suggestions. This research was also supported by 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, Semnan University
(2)
Department of Mathematics Education, The RINS Gyeongsang National University

References

  1. Agarwal RP, O'Regan D: Fixed point theory for admissible type maps with applications. Fixed Point Theory Appl 2009, 2009: 22. Article ID 439176Google Scholar
  2. Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput Math Appl 2010, 59: 1095–1100. 10.1016/j.camwa.2009.05.010MathSciNetView ArticleGoogle Scholar
  3. Amini-Harandi A, Farajzadeh AP, O'Regan D, Agarwal RP: Coincidence point, best approximation, and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces. Fixed Point Theory Appl 2008, 2008: 8. Article ID 543154Google Scholar
  4. Ariza-Ruiz , Jimnez-Melado A: A continuation method for weakly contractive mappings under the interior condition. Fixed Point Theory Appl 2009, 2009: 8. Article ID 809315View ArticleGoogle Scholar
  5. 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 ArticleGoogle Scholar
  6. Chang SS, Cho YJ, Huang NJ: Coupled fixed point theorems with applications. J Korean Math Soc 1996, 33: 575–585.MathSciNetGoogle Scholar
  7. 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 ArticleGoogle Scholar
  8. Guo D, Cho YJ, Zhu J: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, Inc, New York 2004.Google Scholar
  9. Huang NJ, Fang YP, Cho YJ: Fixed point and coupled fixed point theorems for multi-valued increasing operators in ordered metric spaces. Volume 3. Edited by: Cho YJ, Kim JK, Kang SM. Fixed Point Theory and Applications Nova Science Publishers, Inc, New York; 2002:91–98.Google Scholar
  10. Nieto JJ, Lopez RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleGoogle Scholar
  11. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc Am Math Soc 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleGoogle Scholar
  12. 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: 8. 10.1186/1687-1812-2012-8View ArticleGoogle Scholar
  13. Cho YJ, Saadati R, Wang S: 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 ArticleGoogle Scholar
  14. 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 ArticleGoogle Scholar
  15. Dhage BC, O'Regan D, Agarwal RP: Common fixed point theorems for a pair of countably condensing mappings in ordered banach spaces. J Appl Math Stoch Anal 2003, 16: 243–248. 10.1155/S1048953303000182MathSciNetView ArticleGoogle Scholar
  16. Gordji ME, Baghani H, Cho YJ: 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 ArticleGoogle Scholar
  17. Gordji ME, Ramezani M: A generalization of Mizoguchi and Takahashi's theorem for single-valued mappings in partially ordered metric spaces. Nonlinear Anal 2001, 74: 4544–4549.MathSciNetView ArticleGoogle Scholar
  18. Kadelburg Z, Pavlovic M, Radenovic S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput Math Appl 2010, 59: 3148–3159. 10.1016/j.camwa.2010.02.039MathSciNetView ArticleGoogle Scholar
  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 ArticleGoogle Scholar
  20. 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: 81. 10.1186/1687-1812-2011-81MathSciNetView ArticleGoogle Scholar
  21. Yang SK, Bae JS, Cho SH: Coincidence and common fixed and periodic point theorems in cone metric spaces. Comput Math Appl 2011, 61: 170–177. 10.1016/j.camwa.2010.10.031MathSciNetView ArticleGoogle Scholar

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