Skip to content

Advertisement

  • Research
  • Open Access

Ω-Distance and coupled fixed point in G-metric spaces

Fixed Point Theory and Applications20132013:208

https://doi.org/10.1186/1687-1812-2013-208

  • Received: 29 April 2013
  • Accepted: 19 July 2013
  • Published:

Abstract

Saadati et al. (Math. Comput. Model. 52:797-801, 2010) introduced the concept of Ω-distance in generalized metric spaces and studied some nice fixed point theorems. Very recently, Jleli and Samet (Fixed Point Theory Appl. 2012:210, 2012) showed that some of the fixed point theorems in G-metric spaces can be obtained from quasi-metric space. In this paper, we utilize the concept of Ω-distance in the sense of Saadati et al. to establish some common coupled fixed point results. Also, we introduce an example to support the useability of our results. Note that the method of Jleli and Samet cannot be used in our results.

MSC:47H10, 54H25.

Keywords

  • coupled fixed point
  • Ω-distance

1 Introduction

In 2006, Mustafa and Sims [1] introduced a generalization of metric spaces, the G-metric spaces, which assigns to each triple of elements a non-negative real number. Very recently, Jleli and Samet [2] showed that some of the fixed point theorems in G-metric spaces can be obtained from quasi-metric spaces. For some works in G-metric spaces, see [336]. In 2010, Saadati et al. [26] introduced the concept of Ω-distance and studied some nice fixed point theorems (also, see [13]). Meanwhile, Bhaskar and Lakshmikantam [37] introduced the concept of coupled fixed point and proved several fixed point theorems. Lakshmikantam and Ćirić [38] generalized the concept of coupled fixed point to the the concept of coupled coincidence point of two mappings [39]. After that, many authors established coupled fixed point results (please, see [1944]). In the present paper, we utilize the concept of Ω-distance to establish some coupled fixed point results. Also, we introduce an example to support the useability of our study.

2 Preliminaries

Definition 2.1 ([1])

Let X be an nonempty set. The mapping G : X × X × X X is called G-metric if the following axioms are fulfilled:
  1. (1)

    G ( x , y , z ) = 0 if x = y = z (the coincidence);

     
  2. (2)

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

     
  3. (3)

    G ( x , x , z ) G ( x , y , z ) for each triple ( x , y , z ) from X × X × X with z y ;

     
  4. (4)

    G ( x , y , z ) = G ( p { x , y , z } ) for each permutation of { x , y , z } (the symmetry);

     
  5. (5)

    G ( x , y , z ) G ( x , a , a ) + G ( a , y , z ) for each x, y, z and a in X (the rectangle inequality).

     

Definition 2.2 ([1])

Consider X a G-metric space and ( x n ) a sequence in G.
  1. (1)

    ( x n ) is called G-Cauchy sequence if for each ϵ > 0 there is a positive integer n 0 so that for all m , n , l n 0 , G ( x n , x m , x l ) < ϵ .

     
  2. (2)

    ( x n ) is said to be G-convergent to x X if for each ϵ > 0 there is a positive integer n 0 such that G ( x m , x n , x ) < ϵ for each m , n n 0 .

     

Definition 2.3 ([26])

Consider ( X , G ) a G-metric space and Ω : X × X × X [ 0 , + ) . The mapping Ω is called an Ω-distance on X if it satisfies the three conditions in the following:
  1. (1)

    Ω ( x , y , z ) Ω ( x , a , a ) + Ω ( a , y , z ) for all x, y, z, a from X.

     
  2. (2)

    For each x, y from X, Ω ( x , y , ) , Ω ( x , , y ) : X [ 0 , + ) are lower semi-continuous.

     
  3. (3)

    for each ϵ > 0 there is δ > 0 , so that Ω ( x , a , a ) δ and Ω ( a , y , z ) δ imply G ( x , y , z ) ϵ .

     

The following lemma [13, 26] is going to be very helpful in computing the limits of several sequences.

Lemma 2.1 Let X be a metric space, endowed with metric G, and let Ω be an Ω-distance on X. ( x n ) , ( y n ) are sequences in X, ( α n ) and ( β n ) are sequences in [ 0 , + ) with lim n + α n = lim n + β n = 0 . If x, y, z and a X , then
  1. (1)

    If Ω ( y , x n , x n ) α n and Ω ( x n , y , z ) β n for n N , then G ( y , y , z ) < ϵ , and, by consequence, y = z .

     
  2. (2)

    Inequalities Ω ( y n , x n , x n ) α n and Ω ( x n , y m , z ) β n for m > n imply G ( y n , y m , z ) 0 , hence y n z .

     
  3. (3)

    If Ω ( x n , x m , x l ) α n for l , m , n N with n m l , then ( x n ) is a G-Cauchy sequence.

     
  4. (4)

    If Ω ( x n , a , a ) α n , n N , then ( x n ) is a G-Cauchy sequence.

     

Definition 2.4 ([37])

Consider X a nonempty set. A pair ( x , y ) X × X is called coupled fixed point of mapping F : X × X X if
F ( x , y ) = x , F ( y , x ) = y .

Definition 2.5 ([38])

Let X be a nonempty set. The element ( x , y ) X × X is a coupled coincidence point of mappings F : X × X X and g : X X if
F ( x , y ) = g x , F ( y , x ) = g y .

3 Main results

Theorem 3.1 Let ( X , G ) be a G-metric space and Ω an Ω-distance on X such that X is Ω-bounded. g : X X and F : X × X X are mappings. Suppose there exists k [ 0 , 1 ) such that for each x, y, z, x , y and z in X
Ω ( F ( x , y ) , F ( x , y ) , F ( z , z ) ) + Ω ( F ( y , x ) , F ( y , x ) , F ( z , z ) ) k max { Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) , Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) , Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) , Ω ( F ( x , y ) , g x , g z ) + Ω ( F ( y , x ) , g y , g z ) , Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) , Ω ( F ( x , y ) , F ( x , y ) , g z ) + Ω ( F ( y , x ) , F ( y , x ) , g z ) } .
Consider also that the following conditions hold true:
  1. (1)

    F ( X × X ) g X ;

     
  2. (2)

    gX is a complete subspace of X with respect to the topology, induced by G;

     
  3. (3)
    If F ( u , v ) g u or F ( v , u ) g v , then
    inf { Ω ( g x , F ( x , y ) , g u ) + Ω ( g y , F ( y , x ) , g v ) + Ω ( g x , g u , F ( x , y ) ) + Ω ( g y , g v , F ( y , x ) ) } > 0 .
     

Then, F and g have a unique coupled coincidence point ( u , v ) . Moreover, F ( u , v ) = g u = g v = F ( v , u ) .

Proof Consider x 0 X and y 0 X . Because F ( X × X ) g X , there exist x 1 and y 1 in X such that g x 1 = F ( x 0 , y 0 ) and y 1 = F ( y 0 , x 0 ) . By continuing the process, we obtain two sequences, ( x n ) and ( y n ) , with the properties
g x n + 1 = F ( x n , y n ) , g y n + 1 = F ( y n , x n )
Using the contraction condition, we obtain
Ω ( g x n , g x n + 1 , g x n + s ) + Ω ( g y n , g y n + 1 , g y n + s ) = Ω ( F ( x n 1 , y n 1 ) , F ( x n , y n ) , F ( x n + s 1 , y n + s 1 ) ) + Ω ( F ( y n 1 , x n 1 ) , F ( y n , x n ) , F ( y n + s 1 , x n + s 1 ) ) k max { Ω ( g x n 1 , g x n , g x n + s 1 ) + Ω ( g y n 1 , g y n , g y n + s 1 ) , Ω ( g x n , g x n 1 , g x n + s 1 ) + Ω ( g y n , g y n 1 , g y n + s 1 ) , Ω ( g x n 1 , g x n + 1 , g x n + s 1 ) + Ω ( g y n 1 , g y n + 1 , g y n + s 1 ) , Ω ( g x n , g x n , g x n + s 1 ) + Ω ( g y n , g y n , g y n + s 1 ) , Ω ( g x n , g x n + 1 , g x n + s 1 ) + Ω ( g y n , g y n + 1 , g y n + s 1 ) } .
By applying the contraction inequality repeatedly, we get that
Ω ( g x n , g x n + 1 , g x n + s ) + Ω ( g y n , g y n + 1 , g y n + s ) k n 1 max ( i , j , t ) A { Ω ( g x i , g x j , g x t ) + Ω ( g y i , g y j , g y t ) } ,
(1)

where A = { ( i , j , t ) | 1 i n , 1 j n + 1 , s + 1 t n + s 1 } .

Since X is Ω-bounded, there is M > 0 such that Ω ( x , y , z ) < M for each triple ( x , y , z ) X × X × X . Hence, relation (1) becomes
Ω ( g x n , g x n + 1 , g x n + s ) + Ω ( g y n , g y n + 1 , g y n + s ) 2 k n 1 M .
Consider now l > m > n > 0 , l , m , n N . The following relations hold true:
Ω ( g x n , g x m , g x l ) Ω ( g x n , g x n + 1 , g x n + 1 ) + Ω ( g x n + 1 , g x m , g x l ) Ω ( g x n , g x n + 1 , g x n + 1 ) + Ω ( g x n + 1 , g x n + 2 , g x n + 2 ) + + Ω ( g x m 1 , g x m , g x l ) ,
(2)
and, also
Ω ( g y n , g y m , g y l ) Ω ( g y n , g y n + 1 , g y n + 1 ) + Ω ( g y n + 1 , g y m , g y l ) Ω ( g y n , g y n + 1 , g y n + 1 ) + Ω ( g y n + 1 , g y n + 2 , g y n + 2 ) + + Ω ( g y m 1 , g y m , g y l ) .
(3)
Making the sum of relations (2) and (3), and using inequality (1), it follows that
Ω ( g x n , g x m , g x l ) + Ω ( g y n , g y m , g y l ) 2 M ( k n 1 + k n + + k m 2 ) 2 M k n 1 1 1 k .

Lemma 2.1, part (3), implies that ( g x n ) and ( g y n ) are G-Cauchy sequences. Since gX is a complete G-subspace of X, there are gu and gv in gX such that g x n g u and g y n g v .

Let ϵ > 0 . From the lower semi-continuity of Ω, we get
Ω ( g x n , g x m , g u ) lim inf p + Ω ( g x n , g x m , g x p ) ϵ , m n ,
(4)
Ω ( g y n , g y m , g v ) lim inf p + Ω ( g y n , g y m , g y p ) ϵ , m n ,
(5)
Ω ( g x n , g u , g x l ) lim inf p + Ω ( g x n , g x p , g x l ) ϵ , l n ,
(6)
Ω ( g y n , g v , g y l ) lim inf p + Ω ( g y n , g y p , g y l ) ϵ , l n .
(7)
Suppose that F ( u , v ) g u or F ( v , u ) g v . Applying hypotheses (3) of the theorem, and using inequalities (4)-(7), we obtain
0 < inf { Ω ( g x n , F ( x n , y n ) , g u ) + Ω ( g y n , F ( y n , x n ) , g v ) + Ω ( g x n , g u , F ( x n , y n ) ) + Ω ( g y n , g v , F ( y n , x n ) ) } 4 ϵ

for each ϵ > 0 , which is a contradiction.

Therefore, F ( u , v ) = g u and F ( v , u ) = g v .

Using the contraction condition from the hypotheses, we get
Ω ( F ( u , v ) , g x n + 1 , g x n + 1 ) + Ω ( F ( v , u ) , g y n + 1 , g y n + 1 ) = Ω ( F ( u , v ) , F ( x n , y n ) , F ( x n , y n ) ) + Ω ( F ( v , u ) , F ( y n , x n ) , F ( y n , x n ) ) k max { Ω ( g u , g x n , g x n ) + Ω ( g v , g y n , g y n ) , Ω ( g x n , g u , g x n ) + Ω ( g y n , g v , g y n ) , Ω ( g u , g x n + 1 , g x n ) + Ω ( g v , g y n + 1 , g y n ) Ω ( g x n , g u , g x n ) + Ω ( g y n , g v , g y n ) } .
We apply repeatedly the contraction inequality, and we obtain
Ω ( F ( u , v ) , g x n + 1 , g x n + 1 ) + Ω ( F ( v , u ) , g y n + 1 , g y n + 1 ) k n max { Ω ( g u , g x i , g x 1 ) + Ω ( g v , g y i , g y 1 ) , Ω ( g x j , g u , g x 1 ) + Ω ( g y j , g v , g y 1 ) | 1 i n , 1 j n + 1 } .
Since X is Ω-bounded, it follows that
Ω ( F ( u , v ) , g x n + 1 , g x n + 1 ) + Ω ( F ( v , u ) , g y n + 1 , g y n + 1 ) 2 M k n .
(8)
In a similar manner, it can be proved that
Ω ( g x n + 1 , F ( u , v ) , F ( v , u ) ) + Ω ( g y n + 1 , F ( v , u ) , F ( u , v ) ) 2 M k n .
(9)

Taking into account (8), (9) and the first statement of Lemma 2.1, we get g u = g v .

We will prove now the uniqueness of the coupled coincidence point of F and g.

Suppose ( u , v ) and ( u , v ) are coupled coincidence points of F and g. Using the contraction condition, we obtain
Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) k ( Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) ) ,

hence Ω ( g u , g u , g u ) = Ω ( g v , g v , g v ) = 0 .

On the other hand,
Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) k max { Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) , Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) }
(10)
and
Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) k max { Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) , Ω ( g u , g u , g u ) + Ω ( g v , g v , g v ) } .
(11)

Relations (10) and (11) imply that Ω ( g u , g u , g u ) = Ω ( g u , g u , g u ) = 0 and also Ω ( g v , g v , g v ) = Ω ( g v , g v , g v ) = 0 . Lemma 2.1 imposes that g u = g u and g v = g v , and the uniqueness is proved. □

If we take g = I d X in Theorem 3.1, we easily get the following.

Corollary 3.1 Let ( X , G ) be a complete G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. Suppose F : X × X X is a mapping for which there exists k [ 0 , 1 ) such that for each x, y, z, x , y and z in X
Ω ( F ( x , y ) , F ( x , y ) , F ( z , z ) ) + Ω ( F ( y , x ) , F ( y , x ) , F ( z , z ) ) k max { Ω ( x , x , z ) + Ω ( y , y , z ) , Ω ( x , x , z ) + Ω ( y , y , z ) , Ω ( x , F ( x , y ) , z ) + Ω ( y , F ( y , x ) , z ) , Ω ( F ( x , y ) , x , z ) + Ω ( F ( y , x ) , y , z ) , Ω ( x , F ( x , y ) , z ) + Ω ( y , F ( y , x ) , z ) , Ω ( F ( x , y ) , F ( x , y ) , z ) + Ω ( F ( y , x ) , F ( y , x ) , z ) } .
Consider also that if F ( u , v ) u or F ( v , u ) v , then
inf { Ω ( x , F ( x , y ) , u ) + Ω ( y , F ( y , x ) , v ) + Ω ( F ( x , y ) , u , x ) + Ω ( F ( y , x ) , v , y ) } > 0 .

Then, F has a unique coupled fixed point ( u , v ) . Moreover, F ( u , v ) = u = v = F ( v , u ) .

Corollary 3.2 Let ( X , G ) be a G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. g : X X and F : X × X X are mappings. Suppose that there exists k 1 , k 2 , k 3 , k 4 , k 5 , k 6 [ 0 , 1 ) with k 1 + k 2 + k 3 + k 4 + k 5 + k 6 < 1 such that for each x, y, z, x , y and z in X
Ω ( F ( x , y ) , F ( x , y ) , F ( z , z ) ) + Ω ( F ( y , x ) , F ( y , x ) , F ( z , z ) ) k 1 ( Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) ) + k 2 ( Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) ) + k 3 ( Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) ) + k 4 ( Ω ( F ( x , y ) , g x , g z ) + Ω ( F ( y , x ) , g y , g z ) ) + k 5 ( Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) ) + k 6 ( Ω ( F ( x , y ) , F ( x , y ) , g z ) + Ω ( F ( y , x ) , F ( y , x ) , g z ) ) .
Consider also that the following conditions hold true:
  1. (1)

    F ( X × X ) g X ;

     
  2. (2)

    gX is a complete subspace of X with respect to the topology induced by G;

     
  3. (3)
    If F ( u , v ) g u or F ( v , u ) g v , then
    inf { Ω ( g x , F ( x , y ) , g u ) + Ω ( g y , F ( y , x ) , g v ) + Ω ( g x , g u , F ( x , y ) ) + Ω ( g y , g v , F ( y , x ) ) } > 0 .
     

Then, F and g have a unique coupled coincidence point ( u , v ) . Moreover, F ( u , v ) = g u = g v = F ( v , u ) .

Proof Follows from Theorem 3.1 by noting that
k 1 Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) + k 2 Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) + k 3 Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) + k 4 Ω ( F ( x , y ) , g x , g z ) + Ω ( F ( y , x ) , g y , g z ) + k 5 Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) , + k 6 Ω ( F ( x , y ) , F ( x , y ) , g z ) + Ω ( F ( y , x ) , F ( y , x ) , g z ) k max { Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) , Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) , Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) , Ω ( F ( x , y ) , g x , g z ) + Ω ( F ( y , x ) , g y , g z ) , Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) , Ω ( F ( x , y ) , F ( x , y ) , g z ) + Ω ( F ( y , x ) , F ( y , x ) , g z ) } .

 □

If we take g = I d X in Corollary 3.2, we easily get the following.

Corollary 3.3 Let ( X , G ) be a complete G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. Suppose F : X × X X is a mapping, for which there exists k 1 , k 2 , k 3 , k 4 , k 5 , k 6 [ 0 , 1 ) with k 1 + k 2 + k 3 + k 4 + k 5 + k 6 < 1 such that for each x, y, z, x , y and z in X
Ω ( F ( x , y ) , F ( x , y ) , F ( z , z ) ) + Ω ( F ( y , x ) , F ( y , x ) , F ( z , z ) ) k 1 ( Ω ( x , x , z ) + Ω ( y , y , z ) ) + k 2 ( Ω ( x , x , z ) + Ω ( y , y , z ) ) + k 3 ( Ω ( x , F ( x , y ) , z ) + Ω ( y , F ( y , x ) , z ) ) + k 4 ( Ω ( F ( x , y ) , x , z ) + Ω ( F ( y , x ) , y , z ) ) + k 5 ( Ω ( x , F ( x , y ) , z ) + Ω ( y , F ( y , x ) , z ) ) + k 6 ( Ω ( F ( x , y ) , F ( x , y ) , z ) + Ω ( F ( y , x ) , F ( y , x ) , z ) ) .
Consider also that if F ( u , v ) u or F ( v , u ) v , then
inf { Ω ( x , F ( x , y ) , u ) + Ω ( y , F ( y , x ) , v ) + Ω ( F ( x , y ) , u , x ) + Ω ( F ( y , x ) , v , y ) } > 0 .

Then, F has a unique coupled fixed point ( u , v ) . Moreover, F ( u , v ) = u = v = F ( v , u ) .

By modifying the contraction condition, we get the following theorem.

Theorem 3.2 Let ( X , G ) be a G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. g : X X and F : X × X X are mappings. Suppose that there exist k 1 , k 2 [ 0 , 1 ) with k 1 + k 2 < 1 such that for each x, y, z, x , y and z in X
Ω ( F ( x , y ) , g x , F ( z , z ) ) + Ω ( F ( y , x ) , g y , F ( z , z ) ) k 1 max { Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) , Ω ( g x , g x , g z ) + Ω ( g y , g y , g z ) , Ω ( F ( x , y ) , g x , g z ) + Ω ( F ( y , x ) , g y , g z ) , Ω ( g x , F ( x , y ) , g z ) + Ω ( g y , F ( y , x ) , g z ) } + k 2 ( Ω ( F ( x , y ) , F ( x , y ) , F ( z , z ) ) + Ω ( F ( y , x ) , F ( y , x ) , F ( z , z ) ) ) ,

and the conditions (1)-(3) from Theorem  3.1 hold.

Then, F and g have a unique coupled coincidence point ( u , v ) . Moreover, F ( u , v ) = g u = g v = F ( v , u ) .

Proof Let x 0 and y 0 be elements of X. Since F ( X × X ) g X , there exist x 1 and y 1 in X such that g x 1 = F ( x 0 , y 0 ) and y 1 = F ( y 0 , x 0 ) . Repeating this procedure, we obtain two sequences, ( x n ) and ( y n ) , with the properties
g x n + 1 = F ( x n , y n ) , g y n + 1 = F ( y n , x n ) .
The contraction condition implies that
Ω ( g x n , g x n + 1 , g x n + s ) + Ω ( g y n , g y n + 1 , g y n + s ) = Ω ( F ( x n 1 , y n 1 ) , g x n + 1 , F ( x n + s 1 , y n + s 1 ) ) + Ω ( F ( y n 1 , x n 1 ) , g y n + 1 , F ( y n + s 1 , x n + s 1 ) ) k 1 max { Ω ( g x n 1 , g x n + 1 , g x n + s 1 ) + Ω ( g y n 1 , g y n + 1 , g y n + s 1 ) , Ω ( g x n + 1 , g x n 1 , g x n + s 1 ) + Ω ( g y n + 1 , g y n 1 , g y n + s 1 ) , Ω ( g x n , g x n + 1 , g x n + s 1 ) + Ω ( g y n , g y n + 1 , g y n + s 1 ) , Ω ( g x n + 1 , g x n , g x n + s 1 ) + Ω ( g y n + 1 , g y n , g y n + s 1 ) } + k 2 ( Ω ( g x n , g x n + 1 , g x n + s ) + Ω ( g y n , g y n + 1 , g y n + s ) ) ,
which leads us to
Ω ( g x n , g x n + 1 , g x n + s ) + Ω ( g y n , g y n + 1 , g y n + s ) k max { Ω ( g x n 1 , g x n + 1 , g x n + s 1 ) + Ω ( g y n 1 , g y n + 1 , g y n + s 1 ) , Ω ( g x n + 1 , g x n 1 , g x n + s 1 ) + Ω ( g y n + 1 , g y n 1 , g y n + s 1 ) , Ω ( g x n , g x n + 1 , g x n + s 1 ) + Ω ( g y n , g y n + 1 , g y n + s 1 ) , Ω ( g x n + 1 , g x n , g x n + s 1 ) + Ω ( g y n + 1 , g y n , g y n + s 1 ) } ,

where k = k 1 1 k 2 < 1 .

Following the same steps, as we did in Theorem 3.1, the conclusion is straightforward. □

Theorem 3.2 leads us to a coupled fixed point property, by considering g = I d X .

Corollary 3.4 Let ( X , G ) be a complete G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. Suppose that F : X × X X is a mapping, for which there exist k 1 , k 2 [ 0 , 1 ) with k 1 + k 2 < 1 such that for each x, y, z, x , y and z in X
Ω ( F ( x , y ) , x , F ( z , z ) ) + Ω ( F ( y , x ) , y , F ( z , z ) ) k 1 max { Ω ( x , x , z ) + Ω ( y , y , z ) , Ω ( x , x , z ) + Ω ( y , y , z ) , Ω ( F ( x , y ) , x , z ) + Ω ( F ( y , x ) , y , z ) , Ω ( x , F ( x , y ) , z ) + Ω ( y , F ( y , x ) , z ) } + k 2 ( Ω ( F ( x , y ) , F ( x , y ) , F ( z , z ) ) + Ω ( F ( y , x ) , F ( y , x ) , F ( z , z ) ) ) ,
and if F ( u , v ) u or F ( v , u ) v , then
inf { Ω ( x , F ( x , y ) , u ) + Ω ( y , F ( y , x ) , v ) + Ω ( x , u , F ( x , y ) ) + Ω ( y , v , F ( y , x ) ) } > 0 .

Then, F has a coupled fixed point ( u , v ) . Moreover, F ( u , v ) = u = v = F ( v , u ) .

Theorem 3.3 Let ( X , G ) be a G-metric space, and let Ω be an Ω-distance on X. Consider F : X × X X , g : X X and ϕ : g X R + such that
Ω ( g x , F ( x , y ) , F ( z , z ) ) + Ω ( g y , F ( y , x ) , F ( z , z ) ) ϕ ( g x ) + ϕ ( g y ) + ϕ ( g z ) + ϕ ( g z ) ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) ) ϕ ( F ( z , z ) ) ϕ ( F ( z , z ) )
for all x , y , z , z X . Suppose that the following conditions are fulfilled:
  1. (1)

    F ( X × X ) g X .

     
  2. (2)

    gX is a complete subspace of X with respect to the topology, induced by G.

     
  3. (3)

    There exists k > 0 such that Ω ( x , x , y ) k Ω ( x , y , y ) holds for all x , y X .

     
  4. (4)
    If F ( u , v ) g u or F ( v , u ) g v , then
    inf { Ω ( g x , F ( x , y ) , g u ) + Ω ( g y , F ( y , x ) , g v ) + Ω ( g x , g u , F ( x , y ) ) + Ω ( g y , g v , F ( y , x ) ) } > 0 .
     

Then F and g have a coupled coincidence point ( u , v ) .

Proof Consider ( x 0 , y 0 ) a pair from X × X . As F ( X × X ) g X , there exist ( x 1 , y 1 ) X × X so that g x 1 = F ( x 0 , y 0 ) , g y 1 = F ( y 0 , x 0 ) .

We continue the process, and we obtain two sequences ( x n ) , ( y n ) from X, having the properties that
g x n + 1 = F ( x n , y n ) , g y n + 1 = F ( y n , x n ) .
Using the contraction condition, we get
Ω ( g x n , g x n + 1 , g x n + 1 ) + Ω ( g y n , g y n + 1 , g y n + 1 ) = Ω ( g x n , F ( x n , y n ) , F ( x n , y n ) ) + Ω ( g y n , F ( y n , x n ) , F ( y n , x n ) 2 Ω ( g x n ) + 2 Ω ( g y n ) 2 Ω ( g x n + 1 ) 2 Ω ( g y n + 1 ) .
(12)
For m > n , the first part of the definition Ω-distance and (12) yields
Ω ( g x n , g x m , g x m ) + Ω ( g y n , g y m , g y m ) k = n m 1 [ Ω ( g x k , g x k + 1 , g x k + 1 ) + Ω ( g y k , g y k + 1 , g y k + 1 ) ] .
(13)
Let
S n = k = 0 n [ Ω ( g x k , g x k + 1 , g x k + 1 ) + Ω ( g y k , g y k + 1 , g y k + 1 ) ] .
According to (12),
S n 2 ϕ ( g x 0 ) + 2 ϕ ( g y 0 ) 2 ϕ ( g x n + 1 ) 2 ϕ ( g y n + 1 ) 2 ϕ ( g x 0 ) + 2 ϕ ( g y 0 ) .
Thus, ( S n ) is an increasing bounded sequence, so
lim n + s n = n = 0 [ Ω ( g x k , g x k + 1 , g x k + 1 ) + Ω ( g y k , g y k + 1 , g y k + 1 ) ]

exists.

Now, we shall show that ( g x n ) and ( g y n ) are G-Cauchy sequences in gX. Consider ϵ > 0 . By part (3) of the definition of an Ω-distance, we choose δ > 0 such that if Ω ( x , a , a ) < δ and Ω ( x , y , z ) < δ , then G ( x , y , z ) < ϵ . Let η = min { δ , δ k } .

Using the fact that
n = 0 + [ Ω ( g x k , g x k + 1 , g x k + 1 ) + Ω ( g y k , g y k + 1 , g y k + 1 ) ] < +
and letting n + in (13), we choose n 0 N such that
Ω ( g x n , g x m , g x m ) + Ω ( g y n , g y m , g y m ) < η δ

for all m > n n 0 .

Thus,
Ω ( g x n , g x m , g x m ) < δ
and
Ω ( g y n , g y m , g y m ) < δ
for all m > n n 0 . Also we have
Ω ( g x m , g x m , g x l ) + Ω ( g y m , g y m , g y l ) k [ Ω ( g x m , g x m , g x l ) + Ω ( g y m , g y m , g y l ) ] < k η δ
for all l > m n 0 . Thus,
Ω ( g x m , g x m , g x l ) < δ
and
Ω ( g y m , g y m , g y l ) < δ
for all m > n n 0 . Thus, by part (3) of the definition of Ω-distance, we have
G ( g x n , g x m , g x l ) < ϵ
and
G ( g y n , g y m , g y l ) < ϵ

for l > m > n n 0 .

Therefore, ( g x n ) and ( g y n ) are G-Cauchy sequences. As gX is G-complete, it follows that there are u, v X so that lim n + g x n = g u and lim n + g v n = g v .

Since Ω is lower semi-continuous in its second and third variable, we obtain, for ϵ > 0
Ω ( g x n , g x m , g u ) lim inf p + Ω ( g x n , g x m , g x p ) ϵ , m n ,
(14)
Ω ( g y n , g y m , g v ) lim inf p + Ω ( g y n , g y m , g y p ) ϵ , m n ,
(15)
Ω ( g x n , g u , g x l ) lim inf p + Ω ( g x n , g x p , g x l ) ϵ , l n ,
(16)
Ω ( g y n , g v , g y l ) lim inf p + Ω ( g y n , g y p , g y l ) ϵ , l n .
(17)
We make the sum of inequalities (14), (15), (16) and (17). It follows that
0 < inf { Ω ( g x n , F ( x n , y n ) , g u ) + Ω ( g y n , F ( y n , x n ) , g v ) + Ω ( g x n , g u , F ( x n , y n ) ) + Ω ( g y n , g v , F ( y n , x n ) ) } 4 ϵ ,

for each ϵ > 0 , which contradicts the hypothesis.

Hence, F ( u , v ) = g u and F ( v , u ) = g v , that is, ( u , v ) is a coupled coincidence point of F and g. □

By considering g = I d X , we get the following corrolary.

Corollary 3.5 Let ( X , G ) be a complete G-metric space, and let Ω be an Ω-distance on X. Consider F : X × X X and ϕ : X R + such that
Ω ( x , F ( x , y ) , F ( z , z ) ) + Ω ( y , F ( y , x ) , F ( z , z ) ) ϕ ( x ) + ϕ ( y ) + ϕ ( z ) + ϕ ( z ) ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) ) ϕ ( F ( z , z ) ) ϕ ( F ( z , z ) )
for all x , y , z , z X . Suppose that the following conditions are fulfilled:
  1. (1)

    There exists k > 0 such that Ω ( x , x , y ) k Ω ( x , y , y ) holds for all x , y X .

     
  2. (2)
    If F ( u , v ) u or F ( v , u ) v , then
    inf { Ω ( x , F ( x , y ) , u ) + Ω ( y , F ( y , x ) , v ) + Ω ( x , u , F ( x , y ) ) + Ω ( y , v , F ( y , x ) ) } > 0 .
     

Then F has coupled fixed point ( u , v ) .

Now, we introduce the following example to support the useability of our result.

Example 3.1 Let X = [ 0 , 1 ] . Define
G : X × X × X R + , G ( x , y , z ) = | x y | + | x z | + | y z |
and
Ω : X × X × X R + , Ω ( x , y , z ) = | x y | + | x z | .
Also define
F : X × X X , F ( x , y ) = 1 2 x ; g : X X , g x = x ; ϕ : X R + , ϕ ( x ) = 4 x .
Then,
  1. (1)

    ( X , G ) is a complete G-metric space.

     
  2. (2)

    Ω is Ω-distance.

     
  3. (3)

    Ω ( x , x , y ) 2 Ω ( x , y , y ) for all x , y X .

     
  4. (4)

    F ( X × X ) g X .

     
  5. (5)
    For x , y , z , z X we have
    Ω ( x , F ( x , y ) , F ( z , z ) ) + Ω ( y , F ( y , x ) , F ( z , z ) ) ϕ ( x ) + ϕ ( y ) + ϕ ( z ) + ϕ ( z ) ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) ) ϕ ( F ( z , z ) ) ϕ ( F ( z , z ) ) .
     
  6. (6)
    If F ( u , v ) u or F ( v , u ) v , then
    inf { Ω ( x , F ( x , y ) , u ) + Ω ( y , F ( y , x ) , v ) + Ω ( x , u , F ( x , y ) ) + Ω ( y , v , F ( y , x ) ) } > 0 .
     
Proof The proof of (1), (2), (3) and (4) is clear. To prove (5) given x , y , z , z X .
Ω ( x , F ( x , y ) , F ( z , z ) ) + Ω ( y , F ( y , x ) , F ( z , z ) ) = Ω ( x , 1 2 x , 1 2 z ) + Ω ( y , 1 2 y , 1 2 z ) = 1 2 x + | x 1 2 z | + 1 2 y + | y 1 2 z | 3 2 x + 1 2 z + 3 2 y + 1 2 z 2 x + 2 y + 2 z + 2 z = ϕ ( x ) + ϕ ( y ) + ϕ ( z ) + ϕ ( z ) ϕ ( F ( x , y ) ) ϕ ( F ( y , x ) ) ϕ ( F ( z , z ) ) ϕ ( F ( z , z ) ) .
To prove (6), let F ( u , v ) u or F ( v , u ) v . Then u 0 or v 0 . Thus,
inf { Ω ( x , F ( x , y ) , u ) + Ω ( y , F ( y , x ) , v ) + Ω ( x , u , F ( x , y ) ) + Ω ( y , v , F ( y , x ) ) : x , y X } = inf { Ω ( x , 1 2 x , u ) + Ω ( y , 1 2 y , v ) + Ω ( x , u , 1 2 x ) + Ω ( y , v , 1 2 y ) : x , y X } = inf { x + 2 | x u | + y + 2 | y v | : x , y X } = inf { x + 2 | x u | : x X } + inf { y + 2 | y v | : y X } u + v > 0 .

So, F and g satisfy all the hypotheses of Corollary 3.5. Hence the mappings F and g have a coupled coincidence point, Here ( 0 , 0 ) is the coupled coincidence point of F and g. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Hashemite University, Zarqa, Jordan
(2)
Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independenţei, Bucharest, 060042, Romania

References

  1. Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.MathSciNetMATHGoogle Scholar
  2. Jleli M, Samet B: Remarks on G -metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 210Google Scholar
  3. Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Appl. Math. Comput. 2010, 217: 195–202. 10.1016/j.amc.2010.05.042MathSciNetView ArticleMATHGoogle Scholar
  4. Abbas M, Nazir T, Radenović S: Common fixed point of generalized weakly contractive maps in partially ordered G -metric spaces. Appl. Math. Comput. 2012, 218: 9383–9395. 10.1016/j.amc.2012.03.022MathSciNetView ArticleMATHGoogle Scholar
  5. Aydi H: A fixed point result involving a generalized weakly contractive condition in G -metric spaces. Bull. Math. Anal. Appl. 2011, 3(4):180–188.MathSciNetMATHGoogle Scholar
  6. Aydi H: A common fixed point of integral type contraction in generalized metric spaces. J. Adv. Math. Stud. 2012, 5(1):111–117.MathSciNetMATHGoogle Scholar
  7. 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 ArticleMathSciNetMATHGoogle Scholar
  8. Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput. Math. Appl. 2011, 62(11):4222–4229. 10.1016/j.camwa.2011.10.007MathSciNetView ArticleMATHGoogle Scholar
  9. Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for ( ψ , ϕ ) -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012, 63(1):298–309. 10.1016/j.camwa.2011.11.022MathSciNetView ArticleMATHGoogle Scholar
  10. Chandok, S, Mustafa, Z, Postolache, M: Coupled common fixed point theorems for mixed g-monotone mappings in partially ordered G-metric spaces. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. (in press)Google Scholar
  11. Gajić L, Crvenković ZL: A fixed point result for mappings with contractive iterate at a point in G -metric spaces. Filomat 2011, 25: 53–58.MathSciNetMATHGoogle Scholar
  12. Gajić L, Crvenković ZL: On mappings with contractive iterate at a point in generalized metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 458086Google Scholar
  13. Gholizadeh L, Saadati R, Shatanawi W, Vaezpour SM: Contractive mapping in generalized, ordered metric spaces with application in integral equations. Math. Probl. Eng. 2011., 2011: Article ID 380784Google Scholar
  14. Kaewcharoen A, Kaewkhao A: Common fixed points for single-valued and multi-valued mappings in G -metric spaces. Int. J. Math. Anal. 2011, 5: 1775–1790.MathSciNetMATHGoogle Scholar
  15. 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
  16. Mustafa Z, Sims B: Fixed point theorems for contractive mapping in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175Google Scholar
  17. Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011, 48: 304–319.MathSciNetMATHGoogle Scholar
  18. Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed points results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028Google Scholar
  19. Nashine HK, Shatanawi W: Coupled common fixed point theorems for pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. 2011, 62(4):1984–1993. 10.1016/j.camwa.2011.06.042MathSciNetView ArticleMATHGoogle Scholar
  20. Ciric L, Hussain N, Akbar F, Ume JS: Common fixed point for Banach operator pairs from the set of best approximations. Bull. Belg. Math. Soc. Simon Stevin 2009, 16: 319–336.MathSciNetMATHGoogle Scholar
  21. Ciric L, Hussain N, Cakic N: Common fixed points for Ciric type f -weak contraction with applications. Publ. Math. (Debr.) 2010, 76(1–2):31–49.MathSciNetMATHGoogle Scholar
  22. Ciric L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060MathSciNetView ArticleMATHGoogle Scholar
  23. Rhoades BE, Abbas M: Necessary and sufficient condition for common fixed point theorems. J. Adv. Math. Stud. 2009, 2(2):97–102.MathSciNetMATHGoogle Scholar
  24. Popa V, Patriciu AM: A general fixed point theorem for mappings satisfying an ϕ -implicit relation in complete G -metric spaces. G.U. Journal of Science 2012, 25(2):403–408.Google Scholar
  25. Radenović S, Pantelić S, Salimi P, Vujaković J: A note on some tripled coincidence point results in G -metric spaces. Int. J. Math. Sci. Eng. Appl. 2012, 6(6):23–38.MathSciNetGoogle Scholar
  26. Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered G -metric spaces. Math. Comput. Model. 2010, 52: 797–801. 10.1016/j.mcm.2010.05.009MathSciNetView ArticleMATHGoogle Scholar
  27. 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 ArticleMATHGoogle Scholar
  28. Fadail ZM, Ahmad AGB, Radenović S: Common fixed point and fixed point results under c -distance in cone metric spaces. Appl. Math. Sci. Lett. 2013, 1(2):47–52.Google Scholar
  29. Radenović S: Remarks on some recent coupled coincidence point results in symmetric G -metric spaces. J. Oper. 2013., 2013: Article ID 290525Google Scholar
  30. Kadelburg Z, Nashine HK, Radenović S: Common coupled fixed point results in partially ordered G -metric spaces. Bull. Math. Anal. Appl. 2012, 4(2):51–63.MathSciNetMATHGoogle Scholar
  31. Radenović S: Remarks on some coupled coincidence point results in partially ordered metric spaces. Arab J. Math. Sci. 2013. 10.1016/j.ajmsc.2013.02.003Google Scholar
  32. Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650Google Scholar
  33. Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat. 2011, 40(3):441–447.MathSciNetMATHGoogle Scholar
  34. Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870Google Scholar
  35. Shatanawi W, Chauhan S, Postolache M, Abbas M, Radenović S: Common fixed points for contractive mappings of integral type in G -metric spaces. J. Adv. Math. Stud. 2013, 6(1):53–72.MathSciNetMATHGoogle Scholar
  36. 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
  37. Bhaskar TG, Laksmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017MathSciNetView ArticleMATHGoogle Scholar
  38. Laksmikantham 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 ArticleMATHGoogle Scholar
  39. Choudhury BS, Metiya N, Postolache M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 152Google Scholar
  40. Aydi H, Karapınar 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
  41. Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012, 218(12):6727–6732. 10.1016/j.amc.2011.12.038MathSciNetView ArticleMATHGoogle Scholar
  42. Aydi H, Abbas M, Postolache M: Coupled coincidence points for hybrid pair of mappings via mixed monotone property. J. Adv. Math. Stud. 2012, 5(1):118–126.MathSciNetMATHGoogle Scholar
  43. Shatanawi W: Some common coupled fixed point results in cone metric spaces. Int. J. Math. Anal. 2010, 4: 2381–2388.MathSciNetMATHGoogle Scholar
  44. Shatanawi W: Partially ordered cone metric spaces and coupled fixed point results. Comput. Math. Appl. 2010, 60: 2508–2515. 10.1016/j.camwa.2010.08.074MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Shatanawi and Pitea; licensee Springer 2013

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.

Advertisement