Open Access

Some results on a tripled fixed point for nonlinear contractions in partially ordered G-metric spaces

Fixed Point Theory and Applications20122012:179

DOI: 10.1186/1687-1812-2012-179

Received: 13 April 2012

Accepted: 2 October 2012

Published: 17 October 2012

Abstract

Berinde and Borcut (Nonlinear Anal. 74(15):4889-4897, 2011) have quite recently defined the notion of a triple fixed point and proved some interesting results related to this concept in a partially ordered metric space. In this work we prove some triple fixed point theorem for a mixed monotone mapping satisfying nonlinear contractions in the framework of a generalized metric space endowed with partial order while the idea of a generalized metric space introduced by Mustafa and Sims (J. Nonlinear Convex Anal. 7:289-297, 2006). Further we prove the uniqueness of a coupled fixed point for such a mapping in this setting.

Keywords

tripled fixed point partially ordered set mixed monotone mapping generalized metric space

1 Introduction and preliminaries

The Banach contraction principle [1] is the most famous, simplest and one of the most versatile elementary results in fixed point theory. Fixed point theory is a very useful tool in solving a variety of problems in control theory, economic theory, nonlinear analysis and global analysis. 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 etc.

In 2006, Bhaskar and Lakshmikantham [2] initiated the study of a coupled fixed point and proved some coupled fixed point theorems for a mixed monotone operator in a partially ordered metric space. As an application of the coupled fixed point theorems, they obtained the existence and uniqueness of the solution of a periodic boundary value problem. In recent past, Lakshmikantham and Ćirić [3] determined some coupled coincidence and coupled common fixed point theorems for nonlinear contractions in partially ordered complete metric spaces. Most recently, the concept of a triple fixed point has been studied in partially ordered complete metric spaces for nonlinear contractions by Berinde and Borcut [4], who obtained the existence and uniqueness theorems for contractive type mappings in this setup which was later on studied by many authors. A large list of references can be found, for example, in the papers [523].

The concept of a generalized metric space was introduced and studied by Mustafa and Sims [24] and was later used to determine coupled fixed point theorems and related results by a number of authors [2532]. We shall assume throughout this paper that the symbols and will denote the set of real and natural numbers respectively. Now, we recall some definitions, notations and preliminary results which we will use throughout the paper.

Given a nonempty set X, a mapping G : X × X × X R is called a generalized metric (for short, G-metric) on X and ( X , G ) a generalized metric space or simply a G-metric space if the following conditions are satisfied:
  1. (i)

    G ( x , y , z ) = 0 if x = y = z ,

     
  2. (ii)

    G ( x , x , y ) > 0 for all x , y X and x y ,

     
  3. (iii)

    G ( x , x , y ) G ( x , y , z ) for all x , y , z X and y z ,

     
  4. (iv)

    G ( x , y , z ) = G ( x , z , y ) = G ( y , z , x ) = (symmetry in all three variables),

     
  5. (v)

    G ( x , y , z ) G ( x , a , a ) + G ( a , y , z ) for all x , y , z , a X (rectangle inequality).

     

Example 1.1 ([24])

Let ( R , d ) be a usual metric space. Define a function G s by
G s ( x , y , z ) = d ( x , y ) + d ( y , z ) + d ( x , z ) ,

for all x , y , z R . Then ( R , G s ) is a G-metric space.

The concepts of convergence and Cauchy sequences and continuous functions in a G-metric space are studied in [24].

Let ( X , G ) be a G-metric space. Then a sequence ( x n ) is said to be convergent in ( X , G ) or simply G-convergent to x X if for every ϵ > 0 there exists N N such that G ( x n , x m , x ) < ϵ for all n , m N .

Let ( X , G ) be a G-metric space. Then ( x n ) is said to be Cauchy in ( X , G ) or simply G-Cauchy if for every ϵ > 0 there exists N N such that G ( x n , x m , x k ) < ϵ for all n , m , k N . A G-metric space ( X , G ) is said to be complete if every G-Cauchy sequence is G-convergent.

Let ( X , G ) be a G-metric space and f : X X be a mapping. Then f is said to be G-continuous at a point x X if and only if it is G-sequentially continuous at x; that is, whenever ( x n ) is G-convergent to x, we have ( f ( x n ) ) is G-convergent to f ( x ) .

Proposition 1.2 ([24])

Let ( X , G ) be a G-metric space and ( x n ) be a sequence in X. Then, for all x X , the following statements are equivalent:
  1. (i)

    ( x n ) is G-convergent to x.

     
  2. (ii)

    G ( x n , x n , x ) 0 as n .

     
  3. (iii)

    G ( x n , x , x ) 0 as n .

     
  4. (iv)

    G ( x n , x m , x ) 0 as n , m .

     

Proposition 1.3 ([24])

Let ( X , G ) be a G-metric space and ( x n ) be a sequence in X. Then the following statements are equivalent:
  1. (i)

    ( x n ) is G-Cauchy.

     
  2. (ii)

    For every ϵ > 0 there exists N N such that G ( x n , x m , x m ) < ϵ for all n , m N .

     

Lemma 1.4 ([24])

If ( X , G ) is a G-metric space, then G ( x , y , y ) 2 G ( y , x , x ) for all x , y X .

Let ( X , G ) be a G-metric space and F : X × X × X X be a mapping. Then a map F is said to be continuous [28] in ( X , G ) if for every G-convergent sequence x n x , y n y and z n z , ( F ( x n , y n , z n ) ) is G-convergent to F ( x , y , z ) .

Bhaskar and Lakshmikantham [2] defined and studied the concepts of a mixed monotone property and a coupled fixed point in a partially ordered metric space. Quite recently, the notions of the mixed monotone property for the mapping F : X × X × X X and a tripled fixed point were introduced by Berinde and Borcut [4] as follows.

Let ( X , ) be a partially ordered set and F : X × X × X X be a mapping. Then a map F is said to have the mixed monotone property if F ( x , y , z ) is monotone non-decreasing in x and z, and is monotone non-increasing in y; that is, for any x , y , z X ,
x 1 , x 2 X , x 1 x 2 implies F ( x 1 , y , z ) F ( x 2 , y , z ) , y 1 , y 2 X , y 1 y 2 implies F ( x , y 1 , z ) F ( x , y 2 , z )
and
z 1 , z 2 X , z 1 z 2 implies F ( x , y , z 1 ) F ( x , y , z 2 ) .
An element ( x , y , z ) X × X × X is said to be a tripled fixed point of the mapping F : X × X × X X if
F ( x , y , z ) = x , F ( y , x , y ) = y and F ( z , y , x ) = z .

The main results of Berinde and Borcut are as follows.

Theorem 1.5 ([4])

Let ( X , ) be a partially ordered set and suppose there is a metric d on X such that ( X , d ) is a complete metric space. Let F : X × X × X X be a mapping having the mixed monotone property on X. Assume that there exist constants j , k , l [ 0 , 1 ) with j + k + l < 1 for which
d ( F ( x , y , z ) ) + d ( F ( u , v , w ) ) j d ( x , u ) + k d ( y , v ) + l d ( z , w )
for all x u , y v , z w . Assume that either
  1. (a)

    F is continuous or

     
  2. (b)

    X has the following property:

     
  3. (i)

    if a non-decreasing sequence ( x n ) is G-convergent to x, then x n x for all n,

     
  4. (ii)

    if a non-increasing sequence ( y n ) is G-convergent to y, then y n y for all n.

     

If there exist x 0 , y 0 , z 0 X such that x 0 F ( x 0 , y 0 , z 0 ) , y 0 F ( y 0 , x 0 , y 0 ) and z 0 F ( z 0 , y 0 , x 0 ) , then there exist x , y , z X such that F ( x , y , z ) = x , F ( y , x , y ) = y and F ( z , y , x ) = z .

Motivated by [4], we determine in this paper some triple fixed point theorems for nonlinear contractions in the framework of partially ordered generalized metric spaces and obtain uniqueness theorems for contractive type mappings in this setting.

2 Main results

In this section, we establish some tripled fixed point results by considering maps on generalized metric spaces endowed with partial order. Before proceeding further, first, we define the following function which will be used in our results.

Let ( x n ) , ( y n ) and ( z n ) be any three sequences of nonnegative real numbers. Denote with Θ the set of all functions θ : [ 0 , ) 3 [ 0 , 1 ) which, satisfying θ ( x n , y n , z n ) 1 , implies x n , y n , z n 0 . An example of such a function is as follows:
θ ( x , y , z ) = { ln ( 1 + k 1 x + k 2 y + k 3 z ) k 1 x + k 2 y + k 3 z ; at least one of  x , y , z > 0  and  k 1 , k 2 , k 3 ( 0 , 1 ) , [ 0 , 1 ) ; x = 0 = y = z .

Now, we are ready to prove our main results.

Theorem 2.1 Let ( X , ) be a partially ordered set and G be a G-metric on X such that ( X , G ) is a complete G-metric space. Suppose that F : X × X × X X is a continuous mapping having the mixed monotone property. Assume that there exists θ Θ such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ1_HTML.gif
(2.1)

for all x , y , z , s , t , u , p , q , r X with x s p and y t q and z u r , where either s p or t q or u r . If there exist x 0 , y 0 , z 0 X such that x 0 F ( x 0 , y 0 , z 0 ) , y 0 F ( y 0 , x 0 , y 0 ) and z 0 F ( z 0 , y 0 , x 0 ) , then F has a tripled fixed point; that is, there exist x , y , z X such that F ( x , y , z ) = x , F ( y , x , y ) = y and F ( z , y , x ) = z .

Proof Let x 0 , y 0 , z 0 X be such that x 0 F ( x 0 , y 0 , z 0 ) , y 0 F ( y 0 , x 0 , y 0 ) and z 0 F ( z 0 , y 0 , x 0 ) . We can choose x 1 , y 1 , z 1 X such that x 1 = F ( x 0 , y 0 , z 0 ) , y 1 = F ( y 0 , x 0 , y 0 ) and z 1 = F ( z 0 , y 0 , x 0 ) . Write
x n + 1 = F ( x n , y n , z n ) , y n + 1 = F ( y n , x n , y n ) , and z n + 1 = F ( z n , y n , x n )
(2.2)
for all n 1 . Due to the mixed monotone property of F, we can find x 2 x 1 x 0 , y 2 y 1 y 0 and z 2 z 1 z 0 . By straightforward calculation, we obtain
x 0 x 1 x 2 x n + 1 , y 0 y 1 y 2 y n + 1 , z 0 z 1 z 2 z n + 1 .
Assume that there exists a nonnegative integer n such that
G ( x n + 1 , x n + 1 , x n ) + G ( y n + 1 , y n + 1 , y n ) + G ( z n + 1 , z n + 1 , z n ) = 0 .
It follows that
G ( x n + 1 , x n + 1 , x n ) = 0 = G ( y n + 1 , y n + 1 , y n ) = G ( z n + 1 , z n + 1 , z n ) .
From the definition of G-metric space, we have x n + 1 = x n , y n + 1 = y n and z n + 1 = z n . It follows from (2.2) that ( x n , y n , z n ) is a triple fixed point of F. Now, we suppose that for all nonnegative integer n
G ( x n + 1 , x n + 1 , x n ) + G ( y n + 1 , y n + 1 , y n ) + G ( z n + 1 , z n + 1 , z n ) 0 .
Using (2.1) and (2.2), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ3_HTML.gif
(2.3)
which implies
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ4_HTML.gif
(2.4)
For all n N , write
γ n = G ( x n + 1 , x n + 1 , x n ) + G ( y n + 1 , y n + 1 , y n ) + G ( z n + 1 , z n + 1 , z n ) ,
then a sequence ( γ n ) is monotone decreasing. Therefore, there exists some γ 0 such that
lim n γ n = lim n [ G ( x n + 1 , x n + 1 , x n ) + G ( y n + 1 , y n + 1 , y n ) + G ( z n + 1 , z n + 1 , z n ) ] = γ .
We shall claim that γ = 0 . On the contrary, suppose that γ > 0 , we have from (2.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ5_HTML.gif
(2.5)
Letting n , we get
θ ( G ( x n , x n , x n 1 ) + G ( y n , y n , y n 1 ) + G ( z n , z n , z n 1 ) ) 1 .
Using the property of the function θ, we have
G ( x n , x n , x n 1 ) , G ( y n , y n , y n 1 ) , G ( z n , z n , z n 1 ) 0 as  n .
So, we have
G ( x n , x n , x n 1 ) + G ( y n , y n , y n 1 ) + G ( z n , z n , z n 1 ) 0 as  n ,
which is a contradiction in virtue of (2.5). Thus, γ = 0 . From (2.4) we have
G ( x n + 1 , x n + 1 , x n ) + G ( y n + 1 , y n + 1 , y n ) + G ( z n + 1 , z n + 1 , z n ) 0 .
(2.6)
Now, we have to show that ( x n ) , ( y n ) and ( z n ) are Cauchy sequences in the G-metric space ( X , G ) . On the contrary, suppose that at least one of ( x n ) , ( y n ) or ( z n ) is not a Cauchy sequence in ( X , G ) . Then there exists ϵ > 0 for which we can find subsequences ( x k ( j ) ) , ( x l ( j ) ) of ( x n ) ; ( y k ( j ) ) , ( y l ( j ) ) of ( y n ) and ( z k ( j ) ) , ( z l ( j ) ) of the sequence ( z n ) with k ( j ) > l ( j ) j for all j N such that
α j = G ( x k ( j ) , x k ( j ) , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z l ( j ) ) ϵ .
(2.7)
We may also assume
G ( x k ( j ) 1 , x k ( j ) 1 , x l ( j ) ) + G ( y k ( j ) 1 , y k ( j ) 1 , y l ( j ) ) + G ( z k ( j ) 1 , z k ( j ) 1 , z l ( j ) ) < ϵ ,
(2.8)
by choosing k ( j ) to be the smallest number exceeding l ( j ) for which (2.7) holds. From (2.7) and (2.8), and using the rectangle inequality, we obtain
ϵ α j = G ( x k ( j ) , x k ( j ) , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z l ( j ) ) G ( x k ( j ) , x k ( j ) , x k ( j ) 1 ) + G ( x k ( j ) 1 , x k ( j ) 1 , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y k ( j ) 1 ) + G ( y k ( j ) 1 , y k ( j ) 1 , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z k ( j ) 1 ) + G ( z k ( j ) 1 , z k ( j ) 1 , z l ( j ) ) < G ( x k ( j ) , x k ( j ) , x k ( j ) 1 ) + G ( y k ( j ) , y k ( j ) , y k ( j ) 1 ) + G ( z k ( j ) , z k ( j ) , z k ( j ) 1 ) + ϵ .
Letting j in the above inequality and using (2.6), we get
α j = G ( x k ( j ) , x k ( j ) , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z l ( j ) ) ϵ .
(2.9)
Again, by using the rectangle inequality, we obtain
α j = G ( x k ( j ) , x k ( j ) , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z l ( j ) ) G ( x k ( j ) , x k ( j ) , x k ( j ) + 1 ) + G ( x k ( j ) + 1 , x k ( j ) + 1 , x l ( j ) + 1 ) + G ( x l ( j ) + 1 , x l ( j ) + 1 , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y k ( j ) + 1 ) + G ( y k ( j ) + 1 , y k ( j ) + 1 , y l ( j ) + 1 ) + G ( y l ( j ) + 1 , y l ( j ) + 1 , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z k ( j ) + 1 ) + G ( z k ( j ) + 1 , z k ( j ) + 1 , z l ( j ) + 1 ) + G ( z l ( j ) + 1 , z l ( j ) + 1 , z l ( j ) ) = γ l ( j ) + G ( x k ( j ) , x k ( j ) , x k ( j ) + 1 ) + G ( x k ( j ) + 1 , x k ( j ) + 1 , x l ( j ) + 1 ) + G ( y k ( j ) , y k ( j ) , y k ( j ) + 1 ) + G ( y k ( j ) + 1 , y k ( j ) + 1 , y l ( j ) + 1 ) + G ( z k ( j ) , z k ( j ) , z k ( j ) + 1 ) + G ( z k ( j ) + 1 , z k ( j ) + 1 , z l ( j ) + 1 ) .
By using Lemma 1.4, the above inequality becomes
α j γ l ( j ) + 2 G ( x k ( j ) + 1 , x k ( j ) + 1 , x k ( j ) ) + 2 G ( y k ( j ) + 1 , y k ( j ) + 1 , y k ( j ) ) + 2 G ( z k ( j ) + 1 , z k ( j ) + 1 , z k ( j ) ) + G ( x k ( j ) + 1 , x k ( j ) + 1 , x l ( j ) + 1 ) + G ( y k ( j ) + 1 , y k ( j ) + 1 , y l ( j ) + 1 ) + G ( z k ( j ) + 1 , z k ( j ) + 1 , z l ( j ) + 1 ) .
This implies
α j γ l ( j ) + 2 γ k ( j ) + G ( x k ( j ) + 1 , x k ( j ) + 1 , x l ( j ) + 1 ) + G ( y k ( j ) + 1 , y k ( j ) + 1 , y l ( j ) + 1 ) + G ( z k ( j ) + 1 , z k ( j ) + 1 , z l ( j ) + 1 ) .
(2.10)
Using (2.1) and (2.2), (2.10) becomes
α j γ l ( j ) + 2 γ k ( j ) + G ( F ( x k ( j ) , y k ( j ) , z k ( j ) ) , F ( x k ( j ) , y k ( j ) , z k ( j ) ) , F ( x l ( j ) , y l ( j ) , z l ( j ) ) ) + G ( F ( y k ( j ) , x k ( j ) , y k ( j ) ) , F ( y k ( j ) , x k ( j ) , y k ( j ) ) , F ( y l ( j ) , x l ( j ) , y l ( j ) ) ) + G ( F ( z k ( j ) , y k ( j ) , x k ( j ) ) , F ( z k ( j ) , y k ( j ) , x k ( j ) ) , F ( z l ( j ) , y l ( j ) , x l ( j ) ) ) θ ( G ( x k ( j ) , x k ( j ) , x l ( j ) ) , G ( y k ( j ) , y k ( j ) , y l ( j ) ) , G ( z k ( j ) , z k ( j ) , z l ( j ) ) ) ( G ( x k ( j ) , x k ( j ) , x l ( j ) ) + G ( y k ( j ) , y k ( j ) , y l ( j ) ) + G ( z k ( j ) , z k ( j ) , z l ( j ) ) ) + γ l ( j ) + 2 γ k ( j ) = θ ( G ( x k ( j ) , x k ( j ) , x l ( j ) ) , G ( y k ( j ) , y k ( j ) , y l ( j ) ) , G ( z k ( j ) , z k ( j ) , z l ( j ) ) ) α j + γ l ( j ) + 2 γ k ( j ) .
It follows that
α j γ l ( j ) 2 γ k ( j ) α j θ ( G ( x k ( j ) , x k ( j ) , x l ( j ) ) , G ( y k ( j ) , y k ( j ) , y l ( j ) ) , G ( z k ( j ) , z k ( j ) , z l ( j ) ) ) < 1 .
Taking the limit as j , we obtain
θ ( G ( x k ( j ) , x k ( j ) , x l ( j ) ) , G ( y k ( j ) , y k ( j ) , y l ( j ) ) , G ( z k ( j ) , z k ( j ) , z l ( j ) ) ) 1 .
Using the property θ ( x n , y n ) 1 implies x n , y n 0 , we get
G ( x k ( j ) , x k ( j ) , x l ( j ) ) , G ( y k ( j ) , y k ( j ) , y l ( j ) ) , G ( z k ( j ) , z k ( j ) , z l ( j ) ) 0 .

Therefore, α j 0 , which is a contradiction and hence ( x n ) , ( y n ) and ( z n ) are Cauchy sequences in the G-metric space ( X , G ) . Since ( X , G ) is a complete G-metric space, hence ( x n ) , ( y n ) and ( z n ) are G-convergent. Then there exist x , y , z X such that ( x n ) , ( y n ) and ( z n ) are G-convergent to x, y and z respectively. Since F is continuous. Letting n in (2.2), we get x = F ( x , y , z ) , y = F ( y , x , y ) and z = F ( z , y , x ) . Thus, we conclude that F has a tripled fixed point. □

Theorem 2.2 Let ( X , ) be a partially ordered set and G be a G-metric on X such that ( X , G ) is a complete G-metric space. Suppose that there exist θ Θ and a mapping F : X × X × X X having the mixed monotone property such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ11_HTML.gif
(2.11)
for all x , y , z , s , t , u , p , q , r X with x s p , y t q and z u r where either s p or t q or u r . Assume that X has the following property:
  1. (i)

    if a non-decreasing sequence ( x n ) is G-convergent to x ( ( z n ) is G-convergent to z), then x n x ( z n x respectively) for all n,

     
  2. (ii)

    if a non-increasing sequence ( y n ) is G-convergent to y, then y n y for all n.

     

If there exist x 0 , y 0 , z 0 X such that x 0 F ( x 0 , y 0 , z 0 ) , y 0 F ( y 0 , x 0 , y 0 ) and z 0 F ( z 0 , y 0 , x 0 ) then F has a triple fixed point.

Proof Proceeding along the same lines as in Theorem 2.1, we obtain a non-decreasing sequence ( x n ) converges to x, a non-increasing sequence ( y n ) converges to y and a non-decreasing sequence ( z n ) converges to z for some x , y , z X . Since x n x , y n y and z n z for all n. If x n = x , y n = y and z n = z for some n 0 , then by construction, x n + 1 = x , y n + 1 = y and z n + 1 = z . Thus, ( x , y , z ) is a tripled fixed point of F. So, we assume either x n x or y n y or z n z for all n 0 . Then by using (2.11) and the rectangle inequality, we have
G ( F ( x , y , z ) , x , x ) + G ( F ( y , x , y ) , y , y ) + G ( F ( z , y , x ) , z , z ) G ( F ( x , y , z ) , F ( x n , y n , z n ) , F ( x n , y n , z n ) ) + G ( F ( x n , y n , z n ) , x , x ) + G ( F ( y , x , y ) , F ( y n , x n , y n ) , F ( y n , x n , y n ) ) + G ( F ( y n , x n , y n ) , y , y ) + G ( F ( z , y , x ) , F ( z n , y n , x n ) , F ( z n , y n , x n ) ) + G ( F ( z n , y n , x n ) , z , z ) = G ( F ( x n , y n , z n ) , F ( x n , y n , z n ) , F ( x , y , z ) ) + G ( x n + 1 , x , x ) + G ( F ( y n , x n , y n ) , F ( y n , x n , y n ) , F ( y , x , y ) ) + G ( y n + 1 , y , y ) + G ( F ( z n , y n , x n ) , F ( z n , y n , x n ) , F ( z , y , x ) ) + G ( z n + 1 , z , z ) θ ( G ( x n , x n , x ) , G ( y n , y n , y ) , G ( z n , z n , z ) ) ( G ( x n , x n , x ) + G ( y n , y n , y ) + G ( z n , z n , z ) ) + G ( x n + 1 , x , x ) + G ( y n + 1 , y , y ) + G ( z n + 1 , z , z ) < G ( x n , x n , x ) + G ( y n , y n , y ) + G ( z n , z n , z ) + G ( x n + 1 , x , x ) + G ( y n + 1 , y , y ) + G ( z n + 1 , z , z ) .
Letting n in the above equation, we get
G ( F ( x , y , z ) , x , x ) + G ( F ( y , x , z ) , y , y ) + G ( F ( z , y , x ) , z , z ) = 0 .

Thus, x = F ( x , y , z ) , y = F ( y , x , z ) and z = F ( z , y , x ) and hence ( x , y , z ) is a tripled fixed point of F. □

Corollary 2.3 Let ( X , ) be a partially ordered set and G be a G-metric on X such that ( X , G ) is a complete G-metric space. Suppose that F : X × X × X X is a mapping having the mixed monotone property and assume that there exists μ Θ such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ12_HTML.gif
(2.12)
for all x , y , z , s , t , u , p , q , r X with x s p , y t q and z u r , where either s p or t q or u r . Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    X has the following property:

     
  3. (i)

    if a non-decreasing sequence ( x n ) is G-convergent to x ( ( z n ) is G-convergent to z), then x n x ( z n x respectively) for all n,

     
  4. (ii)

    if a non-increasing sequence ( y n ) is G-convergent to y, then y n y for all n.

     

If there exist x 0 , y 0 , z 0 X such that x 0 F ( x 0 , y 0 , z 0 ) , y 0 F ( y 0 , x 0 , y 0 ) and z 0 F ( z 0 , y 0 , x 0 ) , then F has a triple fixed point.

Proof For all x , y , z , s , t , u , p , q , r X , write
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ13_HTML.gif
(2.13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ14_HTML.gif
(2.14)
Adding (2.12), (2.13) and (2.14), we get
G ( F ( x , y , z ) , F ( s , t , u ) , F ( p , q , r ) ) + G ( F ( y , x , z ) , F ( t , s , u ) , F ( q , p , r ) ) + G ( F ( z , y , x ) , F ( u , t , s ) , F ( r , q , p ) ) 1 3 [ μ ( G ( x , s , p ) , G ( y , t , q ) , G ( z , u , r ) ) + μ ( G ( y , t , q ) , G ( x , s , p ) , G ( z , u , r ) ) + μ ( G ( z , u , r ) , G ( y , t , q ) , G ( x , s , p ) ) ] ( G ( x , s , p ) + G ( y , t , q ) + G ( z , u , r ) ) = θ ( G ( x , s , p ) , G ( y , t , q ) , G ( z , u , r ) ) ( G ( x , s , p ) + G ( y , t , q ) + G ( z , u , r ) ) ,

where θ ( β 1 , β 2 , β 3 ) = 1 3 [ μ ( β 1 , β 2 , β 3 ) + μ ( β 2 , β 1 , β 3 ) + μ ( β 3 , β 2 , β 1 ) ] for all β 1 , β 2 , β 3 [ 0 , ) . It is easy to verify that θ Θ . Applying Theorems 2.1 and 2.2, we get the desired result. □

Corollary 2.4 Let ( X , ) be a partially ordered set and G be a G-metric on X such that ( X , G ) is a complete G-metric space. Suppose that F : X × X × X X is a mapping having a mixed monotone property and assume that there exists k [ 0 , 1 ) such that
G ( F ( x , y , z ) , F ( s , t , u ) , F ( p , q , r ) ) k 3 ( G ( x , s , p ) + G ( y , t , q ) + G ( z , u , r ) )
for all x , y , z , s , t , u , p , q , r X with x s p , y t q and z u r , where either s p or t q or u r . Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    X has the following property:

     
  3. (i)

    if a non-decreasing sequence ( x n ) is G-convergent to x ( ( z n ) is G-convergent to z), then x n x ( z n x respectively) for all n,

     
  4. (ii)

    if a non-increasing sequence ( y n ) is G-convergent to y, then y n y for all n.

     

If there exist x 0 , y 0 , z 0 X such that x 0 F ( x 0 , y 0 , z 0 ) , y 0 F ( y 0 , x 0 , y 0 ) and z 0 F ( z 0 , y 0 , x 0 ) , then F has a triple fixed point.

Proof Taking μ ( β 1 , β 2 , β 3 ) = k in Theorems 2.1 and 2.2 for all β 1 , β 2 , β 3 [ 0 , ) and k [ 0 , 1 ) , we get the desired result. □

Remark 2.5 To assure the uniqueness of a coupled fixed point, we shall consider the following condition: If ( Y , ) is a partially ordered set, we endow the product Y × Y × Y with
( x , y , z ) ( u , v , w ) if and only if x u , y v , z w ,
(C1)

for all ( x , y , z ) , ( u , v , w ) Y × Y × Y .

Theorem 2.6 In addition to the hypothesis of Theorem  2.1, suppose that for all ( x , y , z ) , ( s , t , u ) X × X × X , there exists ( p , q , r ) X × X × X that is comparable with ( x , y , z ) and ( s , t , u ) . Then F has a unique triple fixed point.

Proof It follows from Theorem 2.1 that the set of coupled fixed points is nonempty. Suppose ( x , y , z ) and ( s , t , u ) are triple fixed points of the mapping F : X × X × X X ; that is, x = F ( x , y , z ) , y = F ( y , x , y ) , z = F ( z , y , x ) , s = F ( s , t , u ) , t = F ( t , s , t ) and u = F ( u , t , s ) . We shall now show that x = s , y = t and z = u . By assumption, there exists ( p , q , r ) in X × X × X that is comparable to ( x , y , z ) and ( s , t , u ) . Put p = p 0 , q = q 0 and r = r 0 , and choose p 1 , q 1 , r 1 X such that p 1 = F ( p 1 , q 1 , r 1 ) , q 1 = F ( q 1 , p 1 , q 1 ) and r 1 = F ( r 1 , q 1 , p 1 ) . Thus, we can define three sequences ( p n ) , ( q n ) and ( r n ) as
p n = F ( p n 1 , q n 1 , r n 1 ) , q n = F ( q n 1 , p n 1 , q n 1 ) and r n = F ( r n 1 , q n 1 , p n 1 ) .
Since ( p , q , r ) is comparable to ( x , y , z ) , we can assume that ( x , y , z ) ( p , q , r ) = ( p 0 , q 0 , r 0 ) . Then it is easy to show that ( p n , q n , r n ) and ( x , y , z ) are comparable; that is, ( x , y , z ) ( p n , q n , r n ) for all n. Thus, from (2.1) we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ16_HTML.gif
(2.15)
which implies
G ( p n , x , x ) + G ( q n , y , y ) + G ( r n , z , z ) < G ( p n 1 , x , x ) + G ( q n 1 , y , y ) + G ( r n 1 , z , z ) .
(2.16)
We see that the sequence ( G ( p n , x , x ) + G ( q n , y , y ) + G ( r n , z , z ) ) is decreasing, there exists some ξ 0 such that
G ( p n , x , x ) + G ( q n , y , y ) + G ( r n , z , z ) ξ as  n .
(2.17)
Now, we have to show that ξ = 0 . On the contrary, suppose that ξ > 0 . Following the same arguments as in the proof of Theorem 2.1, we obtain
θ ( G ( p n 1 , x , x ) , G ( q n 1 , y , y ) , G ( r n 1 , z , z ) ) 1 .
It follows that
G ( p n 1 , x , x ) , G ( q n 1 , y , y ) , G ( r n 1 , z , z ) 0 .
This implies that
G ( p n 1 , x , x ) + G ( q n 1 , y , y ) + G ( r n 1 , z , z ) 0 ,
which is not possible in virtue of (2.15). Hence, ξ = 0 . Therefore, (2.17) becomes
G ( p n , x , x ) + G ( q n , y , y ) + G ( r n , z , z ) 0 as  n .
(2.18)
Similarly, we can show that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ20_HTML.gif
(2.19)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ21_HTML.gif
(2.20)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-179/MediaObjects/13663_2012_Article_284_Equ22_HTML.gif
(2.21)

Using (2.18)-(2.21), the rectangle inequality and taking the limit n , we obtain G ( s , x , x ) + G ( t , y , y ) + G ( u , z , z ) = 0 . Thus, we conclude that x = s , y = t and z = u . Hence, F has a unique triple fixed point. □

Similarly, we can prove the following statement:

Theorem 2.7 In addition to the hypothesis of Theorem  2.2, suppose that for all ( x , y , z ) , ( s , t , u ) X × X × X , there exists ( p , q , r ) X × X × X that is comparable with ( x , y , z ) and ( s , t , u ) . Then F has a unique triple fixed point.

Declarations

Acknowledgements

The authors have benefited from the reports of the anonymous referees, and they are thankful for their valuable comments on the first draft of this paper which improved the presentation and readability.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University

References

  1. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
  2. 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
  3. Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020MathSciNetView ArticleGoogle Scholar
  4. Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74(15):4889–4897. 10.1016/j.na.2011.03.032MathSciNetView ArticleGoogle Scholar
  5. Abbas M, Nazir T, Radenović S: Fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038MathSciNetView ArticleGoogle Scholar
  6. Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006MathSciNetView ArticleGoogle Scholar
  7. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151MathSciNetView ArticleGoogle Scholar
  8. Altun I, Erduran A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 508730Google Scholar
  9. Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157: 2778–2785. 10.1016/j.topol.2010.08.017MathSciNetView ArticleGoogle Scholar
  10. Aydi H, Karapinar E, Postolache M: Tripled coincidence point theorems for weak φ -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44Google Scholar
  11. Berzig M, Samet B: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 2012, 63: 1319–1334. 10.1016/j.camwa.2012.01.018MathSciNetView ArticleGoogle Scholar
  12. Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025MathSciNetView ArticleGoogle Scholar
  13. Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294Google Scholar
  14. Karapinar E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062MathSciNetView ArticleGoogle Scholar
  15. Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055MathSciNetView ArticleGoogle Scholar
  16. Nashine HK, Kadelburg Z, Radenović S:Coupled common fixed point theorems for w -compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 2012, 218: 5422–5432. 10.1016/j.amc.2011.11.029MathSciNetView ArticleGoogle Scholar
  17. Nashine HK, Shatanawi W: Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. 2011, 62: 1984–1993. 10.1016/j.camwa.2011.06.042MathSciNetView ArticleGoogle Scholar
  18. Nashine HK, Samet B:Fixed point results for mappings satisfying ( ψ , φ ) -weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 2201–2209. 10.1016/j.na.2010.11.024MathSciNetView ArticleGoogle Scholar
  19. Nieto JJ, Rodríguez-López R: 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
  20. Nieto JJ, Rodriguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleGoogle Scholar
  21. 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 ArticleGoogle Scholar
  22. Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026MathSciNetView ArticleGoogle Scholar
  23. 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
  24. Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.MathSciNetGoogle Scholar
  25. Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870Google Scholar
  26. Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31Google Scholar
  27. Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059View ArticleGoogle Scholar
  28. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036MathSciNetView ArticleGoogle Scholar
  29. Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered G -metric spaces. Math. Comput. Model. 2012, 55: 1601–1609. 10.1016/j.mcm.2011.10.058MathSciNetView ArticleGoogle Scholar
  30. Mohiuddine SA, Alotaibi A: On coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 897198Google Scholar
  31. Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175Google Scholar
  32. Tahat N, Aydi H, Karapinar E, Shatanawi W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48Google Scholar

Copyright

© Mohiuddine and Alotaibi; 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.