Open Access

Multidimensional fixed points for generalized ψ-quasi-contractions in quasi-metric-like spaces

Journal of Inequalities and Applications20142014:27

https://doi.org/10.1186/1029-242X-2014-27

Received: 7 October 2013

Accepted: 17 December 2013

Published: 24 January 2014

Abstract

In this paper, we introduce the concept of a quasi-metric-like space, and by defining the w-compatibility of two mappings, we obtain multidimensional coincidence point and multidimensional fixed point theorems for generalized ψ-quasi-contractions in quasi-metric-like spaces. Our results extend the fixed point theorems in Vetro and Radenović (Appl. Math. Comput. 219:1594-1600, 2012) and references therein.

MSC:47H10, 54H25.

Keywords

quasi-metric-like space w-compatibility coincidence point fixed point

1 Introduction and preliminaries

In 1987, Guo and Lakshmikantham [1] initiated the study of the coupled fixed point. In 2010, Samet and Vetro [2] presented the concept of a fixed point of N-order as an extension of the coupled fixed point.

Definition 1.1 ([2])

Let X be a non-empty set and let F : X N X ( N 2 ) be a given mapping. An element ( x 1 , x 2 , , x N ) X N is called a fixed point of N-order of the mapping F if
F ( x 1 , x 2 , , x N 1 , x N ) = x 1 , F ( x 2 , x 3 , , x N , x 1 ) = x 2 , F ( x N , x 1 , x 2 , , x N 1 ) = x N .

Subsequently, a number of papers occurred on tripled fixed point and quadruple fixed point theory (see, e.g., [310]). Berzig and Samet [11] discussed the existence of the fixed point of N-order for m-mixed monotone mappings in complete ordered metric spaces. Very recently, Roldán et al. [12] extended the notion of the fixed point of N-order to the Φ-fixed point and obtained some Φ-fixed point theorems for a mixed monotone mapping in partially ordered complete metric spaces. Afterward, many results on multidimensional fixed points have been established (see, e.g., [1318]).

Matthews [19] introduced the notion of a partial metric space where the self-distance does not need to be zero. By generalizing the partial metric, Hitzler and Seda [20] presented the concept of a dislocated metric which was redefined as a metric-like by Amini-Harandi [21]. The existence of fixed points in dislocated metric (metric-like) spaces has been discussed by many authors (see, e.g., [2230]).

Definition 1.2 ([20, 21])

A mapping σ : X × X [ 0 , + ) , where X is a nonempty set, is said to be a dislocated metric (metric-like) on X if, for any x , y , z X , the following three conditions hold true:

(σ 1) σ ( x , y ) = σ ( y , x ) = 0 x = y ;

(σ 2) σ ( x , y ) = σ ( y , x ) ;

(σ 3) σ ( x , z ) σ ( x , y ) + σ ( y , z ) .

The pair ( X , σ ) is then called a dislocated metric (metric-like) space.

Karapınar et al. [31] introduced the notion of quasi-partial metric spaces and studied some fixed point theorems on quasi-partial metric spaces.

Definition 1.3 ([31])

A quasi-partial metric on a nonempty set X is a function q : X × X R + which satisfies:

(QPM1) If 0 q ( x , x ) = q ( x , y ) = q ( y , y ) , then x = y ,

(QPM2) q ( x , x ) q ( x , y ) ,

(QPM3) q ( x , x ) q ( y , x ) , and

(QPM4) q ( x , z ) + q ( y , y ) q ( x , y ) + q ( y , z ) ,

for all x , y , z X . The pair ( X , q ) is called a quasi-partial metric space.

In this paper, similar to the notation of Amini-Harandi [21], we define a quasi-metric-like space generalizing the metric-like space and the quasi-partial metric space. Furthermore, we discuss the existence and uniqueness of a multidimensional fixed point for a generalized g-ψ-quasi-contractive mapping in quasi-metric-like spaces using the new w-compatibility of two mappings.

2 A quasi-metric-like space

Definition 2.1 A mapping ρ : X × X [ 0 , + ) , where X is a nonempty set, is said to be a quasi-metric-like on X if, for any x , y , z X , the following conditions hold:

(ρ 1) ρ ( x , y ) = 0 x = y ;

(ρ 2) ρ ( x , z ) ρ ( x , y ) + ρ ( y , z ) .

The pair ( X , ρ ) is called a quasi-metric-like space.

Definition 2.2 Let ( X , ρ ) be a quasi-metric-like space. Then
  1. (1)
    A sequence { x n } converges to a point x X if and only if
    lim n + ρ ( x n , x ) = lim n + ρ ( x , x n ) = ρ ( x , x ) .
     
In this case, x is called a ρ-limit of { x n } .
  1. (2)

    A sequence { x n } is called a Cauchy sequence in ( X , ρ ) if lim m , n + ρ ( x m , x n ) and lim m , n + ρ ( x n , x m ) exist and are finite.

     
  2. (3)
    The quasi-metric-like-like space ( X , ρ ) is called complete if, for every Cauchy sequence { x n } in X, there is some x X such that
    lim n + ρ ( x n , x ) = lim n + ρ ( x , x n ) = ρ ( x , x ) = lim m , n + ρ ( x m , x n ) = lim m , n + ρ ( x n , x m ) .
     

Every quasi-partial metric space is a quasi-metric-like space. Below we give an example of a quasi-metric-like space.

Example 2.3 Let X = { 0 , 1 } , and let
ρ ( x , y ) = { 2 , if  x = y = 0 ; 1 , if  x = 0 , y = 1 ; 3 2 , if  x = 1 , y = 0 ; 0 , if  x = y = 1 .

Then ( X , ρ ) is a quasi-metric-like space, but ρ ( 0 , 0 ) ρ ( 1 , 0 ) , so ( X , ρ ) is not a quasi-partial metric space.

Remark 2.4 Every metric-like space is a quasi-metric-like space. Because the limit of a convergent sequence in metric-like space is not necessarily unique [25], the ρ-limit of a convergent sequence in quasi-metric-like spaces is not necessarily unique.

3 Main results

In this section, we establish the coincidence point and fixed point of r-order theorems, and an illustrative example is employed to show the validity of our results.

Definition 3.1 Let X be a nonempty set, and let g : X X and let F : X r X ( r 2 ) be two given mappings. An element ( x 1 , x 2 , , x r ) X r is called a coincidence point of r-order of F : X r X and g : X X if
g ( x 1 ) = F ( x 1 , x 2 , , x r 1 , x r ) , g ( x 2 ) = F ( x 2 , x 3 , , x r , x 1 ) , g ( x r ) = F ( x r , x 1 , x 2 , , x r 1 ) .

If g is the identity mapping on X, then ( x 1 , x 2 , , x r ) X r is a fixed point of r-order of the mapping F.

Throughout this paper, we denote all of the coincidence points of r-order of F : X r X and g : X X by C ( F , g , r ) .

By Ψ we denote the set of real functions ψ : [ 0 , + ) [ 0 , + ) which have the following properties:
  1. (i)

    ψ is nondecreasing;

     
  2. (ii)

    ψ ( 0 ) = 0 ;

     
  3. (iii)

    lim t + ( t ψ ( t ) ) = + ;

     
  4. (iv)

    lim s t + ψ ( s ) < t for all t > 0 .

     

From (iv) and ψ ( t ) lim s t + ψ ( s ) < t , we deduce that ψ ( t ) < t for all t > 0 [32].

Vetro and Radenović [32] introduced the concept of a g-ψ-quasi-contraction. We present the following definition as a generalization of the g-ψ-quasi-contraction.

Definition 3.2 Let ( X , ρ ) be a quasi-metric-like space, g : X X and let F : X r X ( r 2 ). F is called a generalized g-ψ-quasi-contraction if there exists ψ : [ 0 , + ) [ 0 , + ) such that
ρ ( F ( x 1 , x 2 , , x r ) , F ( y 1 , y 2 , , y r ) ) ψ ( M ( x 1 , x 2 , , x r ; y 1 , y 2 , , y r ) ) ,
(1)
where
M ( x 1 , x 2 , , x r ; y 1 , y 2 , , y r ) = max { ρ ( g x 1 , g y 1 ) , ρ ( g x 2 , g y 2 ) , , ρ ( g x r , g y r ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , ρ ( g x 2 , F ( x 2 , x 3 , , x r , x 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g y 1 , F ( y 1 , y 2 , , y r ) ) , ρ ( g y 2 , F ( y 2 , y 3 , , y r , y 1 ) ) , , ρ ( g y r , F ( y r , y 1 , , y r 1 ) ) , ρ ( g x 1 , F ( y 1 , y 2 , , y r ) ) , ρ ( g x 2 , F ( y 2 , y 3 , , y r , y 1 ) ) , , ρ ( g x r , F ( y r , y 1 , , y r 1 ) ) , ρ ( g y 1 , F ( x 1 , x 2 , , x r ) ) , ρ ( g y 2 , F ( x 2 , x 3 , , x r , x 1 ) ) , , ρ ( g y r , F ( x r , x 1 , , x r 1 ) ) } ,
(2)

for any ( x 1 , x 2 , , x r ) , ( y 1 , y 2 , , y r ) X r .

If g is the identity mapping, then F is a generalized ψ-quasi-contraction.

Definition 3.3 Let X be a nonempty set. The mappings g : X X and F : X r X ( r 2 ) are called w-compatible if
F ( g ( x 1 ) , g ( x 2 ) , , g ( x r ) ) = g ( F ( x 1 , x 2 , , x r ) ) ,

whenever ( x 1 , x 2 , , x r ) C ( F , g , r ) .

Theorem 3.4 Let ( X , ρ ) be a quasi-metric-like space, g : X X and F : X r X ( r 2 ). Suppose that F is a generalized g-ψ-quasi-contraction with ψ Ψ . If F ( X r ) g ( X ) and g ( X ) is a complete subspace of X, then C ( F , g , r ) is nonempty.

Proof Let ( x 1 0 , x 2 0 , , x r 0 ) X r . Since F ( X r ) g ( X ) , we can construct a sequence { ( x 1 n , x 2 n , , x r n ) } such that
g ( x i n ) = F ( x i n 1 , x i + 1 n 1 , , x r n 1 , x 1 n 1 , , x i 1 n 1 ) for  i = 1 , 2 , , r .
Define
O n ( x 1 0 , x 2 0 , , x r 0 ) = { g x 1 0 , g x 2 0 , , g x r 0 , g x 1 1 , g x 2 1 , , g x r 1 , , g x 1 n , g x 2 n , , g x r n } , O ( x 1 0 , x 2 0 , , x r 0 ) = { g x 1 0 , g x 2 0 , , g x r 0 , g x 1 1 , g x 2 1 , , g x r 1 , , g x 1 n , g x 2 n , , g x r n , } , δ n ( x 1 0 , x 2 0 , , x r 0 ) = diam ( O n ( x 1 0 , x 2 0 , , x r 0 ) ) = sup { ρ ( x , y ) : x , y O n ( x 1 0 , x 2 0 , , x r 0 ) } .

If there exists n 0 N such that δ n 0 ( x 1 0 , x 2 0 , , x r 0 ) = 0 , then for any 0 k n 0 1 , ( x 1 k , x 2 k , , x r k ) C ( F , g , r ) .

We suppose that δ n ( x 1 0 , x 2 0 , , x r 0 ) > 0 , for all n N .

Step 1. We shall prove that for each n N ,
δ n ( x 1 0 , x 2 0 , , x r 0 ) = max { sup 1 i , l r , 0 s n ρ ( g x i 0 , g x l s ) , sup 1 i , l r , 0 s n ρ ( g x l s , g x i 0 ) } .
(3)
In fact, for any 1 i , l r , 1 j , s n , we have
ρ ( g x i j , g x l s ) = ρ ( F ( x i j 1 , x i + 1 j 1 , , x r j 1 , x 1 j 1 , , x i 1 j 1 ) , F ( x l s 1 , x l + 1 s 1 , , x r s 1 , x 1 s 1 , , x l 1 s 1 ) ) ψ ( M ( x i j 1 , , x r j 1 , x 1 j 1 , , x i 1 j 1 ; x l s 1 , , x r s 1 , x 1 s 1 , , x l 1 s 1 ) ) ,
(4)
where
M ( x i j 1 , , x r j 1 , x 1 j 1 , , x i 1 j 1 ; x l s 1 , , x r s 1 , x 1 s 1 , , x l 1 s 1 ) = max { ρ ( g x i j 1 , g x l s 1 ) , ρ ( g x i + 1 j 1 , g x l + 1 s 1 ) , , ρ ( g x i 1 j 1 , g x l 1 s 1 ) , ρ ( g x i j 1 , F ( x i j 1 , x i + 1 j 1 , , x r j 1 , x 1 j 1 , , x i 1 j 1 ) ) , ρ ( g x l s 1 , F ( x l s 1 , x l + 1 s 1 , , x r s 1 , x 1 s 1 , , x l 1 s 1 ) ) , ρ ( g x i + 1 j 1 , F ( x i + 1 j 1 , x i + 2 j 1 , , x r j 1 , x 1 j 1 , , x i j 1 ) ) , ρ ( g x l + 1 s 1 , F ( x l + 1 s 1 , x l + 2 s 1 , , x r s 1 , x 1 s 1 , , x l s 1 ) ) , , ρ ( g x i 1 j 1 , F ( x i 1 j 1 , x i j 1 , , x r j 1 , x 1 j 1 , , x i 2 j 1 ) ) , ρ ( g x l 1 s 1 , F ( x l 1 s 1 , x l s 1 , , x r s 1 , x 1 s 1 , , x l 2 s 1 ) ) , ρ ( g x l s 1 , F ( x i j 1 , x i + 1 j 1 , , x r j 1 , x 1 j 1 , , x i 1 j 1 ) ) , ρ ( g x i j 1 , F ( x l s 1 , x l + 1 s 1 , , x r s 1 , x 1 s 1 , , x l 1 s 1 ) ) , ρ ( g x l + 1 s 1 , F ( x i + 1 j 1 , x i + 2 j 1 , , x r j 1 , x 1 j 1 , , x i j 1 ) ) , ρ ( g x i + 1 j 1 , F ( x l + 1 s 1 , x l + 2 s 1 , , x r s 1 , x 1 s 1 , , x l s 1 ) ) , , ρ ( g x l 1 s 1 , F ( x i 1 j 1 , x i j 1 , , x r j 1 , x 1 j 1 , , x i 2 j 1 ) ) , ρ ( g x i 1 j 1 , F ( x l 1 s 1 , x l s 1 , , x r s 1 , x 1 s 1 , , x l 2 s 1 ) ) } = max { ρ ( g x i j 1 , g x l s 1 ) , ρ ( g x i + 1 j 1 , g x l + 1 s 1 ) , , ρ ( g x i 1 j 1 , g x l 1 s 1 ) , ρ ( g x i j 1 , g x i j ) , ρ ( g x l s 1 , g x l s ) , ρ ( g x i + 1 j 1 , g x i + 1 j ) , ρ ( g x l + 1 s 1 , g x l + 1 s ) , , ρ ( g x i 1 j 1 , g x i 1 j ) , ρ ( g x l 1 s 1 , g x l 1 s ) , ρ ( g x l s 1 , g x i j ) , ρ ( g x i j 1 , g x l s ) , ρ ( g x l + 1 s 1 , g x i + 1 j ) , ρ ( g x i + 1 j 1 , g x l + 1 s ) , , ρ ( g x l 1 s 1 , g x i 1 j ) , ρ ( g x i 1 j 1 , g x l 1 s ) } .
So, for 1 i , l r , 1 j , s n , we have
ρ ( g x i j , g x l s ) ψ ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ) < δ n ( x 1 0 , x 2 0 , , x r 0 ) .
(5)

Hence, equation (3) is true.

Step 2. Now, we claim that for each n N , lim n + δ n ( x 1 0 , x 2 0 , , x r 0 ) < + . For this, we distinguish three cases.

Since the sequence { δ n ( x 1 0 , x 2 0 , , x r 0 ) } is nondecreasing, there exists lim n + δ n ( x 1 0 , x 2 0 , , x r 0 ) .

Case 1. If, for all n N , δ n ( x 1 0 , x 2 0 , , x r 0 ) = diam { g x 1 0 , g x 2 0 , , g x r 0 } , then the claim holds.

Case 2. Suppose that there exist n 0 N , 1 i 0 , l 0 r , and 1 s 0 n 0 such that
δ n 0 ( x 1 0 , x 2 0 , , x r 0 ) = ρ ( g x i 0 0 , g x l 0 s 0 ) ,
then, for any n n 0 , there exist 1 i , l r , and 1 s n such that
δ n ( x 1 0 , x 2 0 , , x r 0 ) ρ ( g x i 0 , g x l s ) .
By equation (5), we obtain
δ n ( x 1 0 , x 2 0 , , x r 0 ) ρ ( g x i 0 , g x l 1 ) + ρ ( g x l 1 , g x l s ) ρ ( g x i 0 , g x l 1 ) + ψ ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ) ,
which implies that
δ n ( x 1 0 , x 2 0 , , x r 0 ) ψ ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ) ρ ( g x i 0 , g x l 1 ) .
(6)
Suppose that lim n + δ n ( x 1 0 , x 2 0 , , x r 0 ) = + , from the property (iii) of ψ, we have
lim n + ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ψ ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ) ) = + .
Nevertheless, by equation (6), we get
lim n + ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ψ ( δ n ( x 1 0 , x 2 0 , , x r 0 ) ) ) ρ ( g x i 0 , g x l 1 ) ,

which is a contradiction. Thus, lim n + δ n ( x 1 0 , x 2 0 , , x r 0 ) < + .

Case 3. If there exist n 1 N , 1 i 1 , l 1 r , and 1 s 1 n 1 such that
δ n 1 ( x 1 0 , x 2 0 , , x r 0 ) = ρ ( g x l 1 s 1 , g x i 1 0 ) ,

the proof is similar to Case 2.

Step 3. Next, we prove that, for every 1 i r , { g x i n } is a Cauchy sequence in ( X , ρ ) .

Let
O ( g x 1 p , g x 2 p , , g x r p ) = { g x 1 p , g x 2 p , , g x r p , g x 1 p + 1 , g x 2 p + 1 , , g x r p + 1 , } ,
and let
δ ( x 1 p , x 2 p , , x r p ) = diam ( O ( g x 1 p , g x 2 p , , g x r p ) ) , p = 0 , 1 , 2 , .
Then,
δ ( x 1 p , x 2 p , , x r p ) δ ( x 1 0 , x 2 0 , , x r 0 ) = lim n + δ n ( x 1 0 , x 2 0 , , x r 0 ) < + , p = 0 , 1 , 2 , .
Since
0 δ ( x 1 p + 1 , x 2 p + 1 , , x r p + 1 ) δ ( x 1 p , x 2 p , , x r p ) , p = 0 , 1 , 2 , ,
there exists δ 0 such that
lim p + δ ( x 1 p , x 2 p , , x r p ) = δ .
If δ > 0 , using the monotonicity of { δ ( x 1 p , x 2 p , , x r p ) } and the property (iv) of ψ, we conclude that
lim p + ψ ( δ ( x 1 p , x 2 p , , x r p ) ) = lim δ ( x 1 p , x 2 p , , x r p ) δ + ψ ( δ ( x 1 p , x 2 p , , x r p ) ) < δ .
(7)
However, by equation (4), we have, for any p 0 ,
δ ( x 1 p + 1 , x 2 p + 1 , , x r p + 1 ) ψ ( δ ( x 1 p , x 2 p , , x r p ) ) ,
which implies that
δ = lim p + δ ( x 1 p + 1 , x 2 p + 1 , , x r p + 1 ) lim p + ψ ( δ ( x 1 p , x 2 p , , x r p ) ) ,

which contradicts equation (7). Therefore, lim p + δ ( x 1 p , x 2 p , , x r p ) = δ = 0 , that is, for every 1 i r , { g x i n } is a Cauchy sequence in ( X , ρ ) .

Step 4. Finally, we prove that C ( F , g , r ) is nonempty.

Since g ( X ) is a complete subspace of X, there exist u i = g x i , i = 1 , 2 , , r , such that
lim n + ρ ( g x i , g x i n ) = lim n + ρ ( g x i n , g x i ) = lim m , n + ρ ( g x i n , g x i m ) = lim m , n + ρ ( g x i m , g x i n ) = ρ ( g x i , g x i ) = ρ ( u i , u i ) = 0 .
(8)
For 1 i r , n N , from
ρ ( g x i n + 1 , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) ρ ( g x i n + 1 , g x i ) + ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) )
and
ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) ρ ( g x i , g i n + 1 ) ρ ( g x i n + 1 , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) )
we get
lim n + ρ ( g x i n + 1 , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) = ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) .
(9)
For any 1 i r , n N , we have
ρ ( g x i n + 1 , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) = ρ ( F ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ) , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) ψ ( M ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ; x i , x i + 1 , , x r , x 1 , , x i 1 ) ) ,
(10)
where
M ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ; x i , x i + 1 , , x r , x 1 , , x i 1 ) = max { ρ ( g x i n , g x i ) , ρ ( g x i + 1 n , g x i + 1 ) , , ρ ( g x i 1 n , g x i 1 ) , ρ ( g x i n , F ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ) ) , ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , ρ ( g x i + 1 n , F ( x i + 1 n , x i + 2 n , , x r n , x 1 n , , x i n ) ) , ρ ( g x i + 1 , F ( x i + 1 , x i + 2 , , x r , x 1 , , x i ) ) , , ρ ( g x r n , F ( x r n , x 1 n , , x r 1 n ) ) , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 n , F ( x 1 n , x 2 n , , x r n ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 n , F ( x i 1 n , x i n , , x r n , x 1 n , , x i 2 n ) ) , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) , ρ ( g x i , F ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ) ) , ρ ( g x i n , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , ρ ( g x i + 1 , F ( x i + 1 n , x i + 2 n , , x r n , x 1 n , , x i n ) ) , ρ ( g x i + 1 n , F ( x i + 1 , x i + 2 , , x r , x 1 , , x i ) ) , , ρ ( g x r , F ( x r n , x 1 n , , x r 1 n ) ) , ρ ( g x r n , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 n , x 2 n , , x r n ) ) , ρ ( g x 1 n , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 n , x i n , , x r n , x 1 n , , x i 2 n ) ) , ρ ( g x i 1 n , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } .
By equations (8) and (9), for any ε > 0 , there exists n 0 N , and, for every n > n 0 and 1 i r , we have
max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } M ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ; x i , x i + 1 , , x r , x 1 , , x i 1 ) max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } + ε .
(11)
Thus, for each 1 i r ,
lim n + M ( x i n , x i + 1 n , , x r n , x 1 n , , x i 1 n ; x i , x i + 1 , , x r , x 1 , , x i 1 ) = max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } .
(12)
If
max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } > 0 ,
using equations (9), (10), (11), and (12) and the property (iv) of ψ, we obtain, for every 1 i r ,
ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) = lim n + ρ ( g x i n + 1 , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) lim n + ψ ( M ( x i n , , x r n , x 1 n , , x i 1 n ; x i , , x r , x 1 , , x i 1 ) ) < max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } .
(13)
By the arbitrariness of i in equation (13), we deduce that
max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } < max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } ,
which is a contradiction. Therefore,
max { ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) , , ρ ( g x r , F ( x r , x 1 , , x r 1 ) ) , ρ ( g x 1 , F ( x 1 , x 2 , , x r ) ) , , ρ ( g x i 1 , F ( x i 1 , x i , , x r , x 1 , , x i 2 ) ) } = 0 ,

which implies that ρ ( g x i , F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) = 0 , i = 1 , 2 , , r .

Thus,
g x i = F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) , i = 1 , 2 , , r ,

that is, ( x 1 , x 2 , , x r ) C ( F , g , r ) . □

Theorem 3.5 Let ( X , ρ ) be a quasi-metric-like space. Let g : X X and let F : X r X ( r 2 ) be mappings satisfying all the conditions of Theorem  3.4. If F and g are w-compatible, then F and g have a unique coincidence point of r-order, which is a fixed point of g and a fixed point of r-order of F. Moreover, the coincidence point of r-order is of the form ( u , u , , u ) for some u X .

Proof Suppose that there exist ( x 1 , x 2 , , x r ) , ( x 1 , x 2 , , x r ) C ( F , g , r ) , that is, for each 1 i r ,
g x i = F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ,
(14)
g x i = F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) .
(15)
First, we prove that, for any 1 i , j , k r ,
g x i = g x j = g x k .
(16)
By equations (1), (14), and (15), for 1 i r 1 , we have
ρ ( g x i , g x i + 1 ) = ρ ( F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) , F ( x i + 1 , x i + 2 , , x r , x 1 , , x i ) ) ψ ( M ( x i , x i + 1 , , x r , x 1 , , x i 1 ; x i + 1 , x i + 2 , , x r , x 1 , , x i ) ) ,
(17)
where
M ( x i , x i + 1 , , x r , x 1 , , x i 1 ; x i + 1 , x i + 2 , , x r , x 1 , , x i ) = max { ρ ( g x 1 , g x 2 ) , , ρ ( g x r 1 , g x r ) , ρ ( g x r , g x 1 ) , ρ ( g x 2 , g x 1 ) , ρ ( g x 3 , g x 2 ) , , ρ ( g x r , g x r 1 ) , ρ ( g x 1 , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) } .
(18)
Set
ζ = max { ρ ( g x 1 , g x 2 ) , , ρ ( g x r 1 , g x r ) , ρ ( g x r , g x 1 ) , ρ ( g x 2 , g x 1 ) , ρ ( g x 3 , g x 2 ) , , ρ ( g x r , g x r 1 ) , ρ ( g x 1 , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) } .
(19)
Similarly, we have
ρ ( g x r , g x 1 ) ψ ( ζ ) , ρ ( g x 1 , g r ) ψ ( ζ )
(20)
and
ρ ( g x i + 1 , g x i ) ψ ( ζ ) , i = 1 , 2 , , r 1 .
(21)
By equations (1), (14), (15), and the monotonicity of ψ, for 1 i r , we also have
ρ ( g x i , g x i ) ψ ( max { ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) } ) ψ ( ζ )
(22)
and
ρ ( g x i , g x i ) ψ ( max { ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) } ) ψ ( ζ ) .
(23)
From equations (17) to (23), we can conclude that
ζ ψ ( ζ ) ,
which is a contradiction, unless ζ = 0 . So
max { ρ ( g x 1 , g x 2 ) , , ρ ( g x r 1 , g x r ) , ρ ( g x r , g x 1 ) , ρ ( g x 2 , g x 1 ) , ρ ( g x 3 , g x 2 ) , , ρ ( g x r , g x r 1 ) , ρ ( g x 1 , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) } = 0 ,
that is,
g x i = g x i + 1 , i = 1 , 2 , , r 1 ,
(24)
g x r = g x 1 .
(25)
On the other hand, for any 1 i r , we obtain
ρ ( g x i , g x i ) ψ ( M ( x i , , x r , x 1 , , x i 1 ; x i , , x r , x 1 , , x i 1 ) ) = ψ ( max { ρ ( g x i , g x i ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x i 1 , g x i 1 ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x i , g x i ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x i 1 , g x i 1 ) } ) .
(26)
Set
λ = max { ρ ( g x i , g x i ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x i 1 , g x i 1 ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x i , g x i ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x i 1 , g x i 1 ) } .
(27)
Similarly, for any
ξ { ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x r , g x r ) ρ ( g x i , g x i ) , , ρ ( g x r , g x r ) , ρ ( g x 1 , g x 1 ) , , ρ ( g x i 1 , g x i 1 ) } ,
we have
ξ ψ ( λ ) .
(28)
By equations (26), (27), and (28), we get
λ ψ ( λ ) ,
which is a contradiction, unless λ = 0 . That is,
g x i = g x i , i = 1 , 2 , , r .
(29)

Therefore, equations (24), (25), and (29) imply that equation (16) is true.

Next, we prove that the coincidence point of r-order is unique.

In view of equation (16), let g x i = u , i = 1 , 2 , , r .

Using the w-compatibility of F and g, we conclude that
g u = g ( g x i ) = g ( F ( x i , x i + 1 , , x r , x 1 , , x i 1 ) ) = F ( g x i , g x i + 1 , , g x r , g x 1 , , g x i 1 ) = F ( u , u , , u ) .

So, u C ( F , g , r ) . By equation (16), we can deduce that g u = g x i , i = 1 , 2 , , r .

Thus,
u = g x i = g u = F ( u , u , , u ) .
(30)

Moreover, equations (16) and (30) imply that ( u , u , , u ) is the unique coincidence point of r-order of F and g, u is a fixed point of g, and ( u , u , , u ) is a fixed point of r-order of F. □

For each a ( 0 , 1 ) , setting ψ ( t ) = a t in Theorem 3.4 and Theorem 3.5, we obtain the following results.

Corollary 3.6 Let ( X , ρ ) be a quasi-metric-like space, let g : X X and let F : X r X ( r 2 ). Suppose there exists a ( 0 , 1 ) such that F is a generalized g-ψ-quasi-contraction with ψ ( t ) = a t . If F ( X r ) g ( X ) and g ( X ) is a complete subspace of X, then C ( F , g , r ) is nonempty.

Corollary 3.7 Let ( X , ρ ) be a quasi-metric-like space. Let g : X X and let F : X r X ( r 2 ) be mappings satisfying all the conditions of Corollary  3.6. If F and g are w-compatible, then F and g have a unique coincidence point of r-order, which is a fixed point of g and a fixed point of r-order of F. Moreover, the coincidence point of r-order is of the form ( u , u , , u ) for some u X .

Example 3.8 Let X = { 0 , 1 , 2 } , Define ρ : X × X [ 0 , + ) as follows:
ρ ( 0 , 0 ) = 0 , ρ ( 1 , 1 ) = 3 , ρ ( 2 , 2 ) = 1 2 , ρ ( 0 , 1 ) = 3 , ρ ( 0 , 2 ) = 3 2 , ρ ( 1 , 0 ) = 5 2 , ρ ( 2 , 0 ) = 3 , ρ ( 1 , 2 ) = 4 5 , ρ ( 2 , 1 ) = 4 .

Then ( X , ρ ) is a complete quasi-metric-like space.

Define g : X × X X by
g 0 = 1 , g 1 = 2 , g 2 = 0 ,
and F : X r X ( r 2 ) by
F ( x 1 , x 2 , , x r ) = { 0 , if  x 1 = x 2 = = x r ; min { x 1 , x 2 , , x r } , otherwise .

It is easy to prove that g and F satisfy all conditions of Theorem 3.4 by taking ψ ( t ) = 5 6 t , and the proof would be lengthy.

Here, ( 2 , 2 , , 2 ) C ( F , g , r ) .

Declarations

Acknowledgements

The authors thank the editor and the referees for their valuable comments and suggestions. The research was supported by the National Natural Science Foundation of China (11071108, 11361042) and the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147, 20114BAB201007), and it was supported partly by the Provincial Graduate Innovation Foundation of Jiangxi, China (YC2012-B004).

Authors’ Affiliations

(1)
Department of Mathematics, Nanchang University
(2)
Department of Mathematics, Jiangxi Agricultural University
(3)
Zastava Oružje AD

References

  1. Guo DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 1987, 11: 623-632. 10.1016/0362-546X(87)90077-0MATHMathSciNetView ArticleGoogle Scholar
  2. Samet B, Vetro C: Coupled fixed point, f -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1: 46-56.MATHMathSciNetView ArticleGoogle Scholar
  3. Aydi H, Karapınar E, Shatanawi W: Tripled coincidence point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101 10.1186/1687-1812-2012-101Google Scholar
  4. Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 4889-4897. 10.1016/j.na.2011.03.032MATHMathSciNetView ArticleGoogle Scholar
  5. Borcut M, Berinde V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 5929-5936. 10.1016/j.amc.2011.11.049MATHMathSciNetView ArticleGoogle Scholar
  6. Borcut M: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 7339-7346. 10.1016/j.amc.2012.01.030MATHMathSciNetView ArticleGoogle Scholar
  7. Karapınar E, Luong NV: Quadruple fixed point theorems for nonlinear contractions. Comput. Math. Appl. 2012, 64: 1839-1848. 10.1016/j.camwa.2012.02.061MATHMathSciNetView ArticleGoogle Scholar
  8. Karapınar E, Shatanawi W, Mustafa Z: Quadruple fixed point theorems under nonlinear contractive conditions in partially ordered metric spaces. J. Appl. Math. 2012. 10.1155/2012/951912Google Scholar
  9. Mustafa Z: Mixed g -monotone property and quadruple fixed point theorems in partially ordered G -metric spaces using ( ϕ ψ ) contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 199 10.1186/1687-1812-2012-199Google Scholar
  10. Roldán A, Martínez-Moreno J, Roldán C: Tripled fixed point theorem in fuzzy metric spaces and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 29 10.1186/1687-1812-2013-29Google 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.2012.01.018MATHMathSciNetView ArticleGoogle Scholar
  12. Roldán A, Martínez-Moreno J, Roldán C: Multidimensional fixed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl. 2012, 396: 536-545. 10.1016/j.jmaa.2012.06.049MATHMathSciNetView ArticleGoogle Scholar
  13. Karapınar E, Roldán A: A note on ‘ n -tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’. J. Inequal. Appl. 2013., 2013: Article ID 567 10.1186/1029-242X-2013-567Google Scholar
  14. Karapınar E, Roldán A, Martínez-Moreno J, Roldán C: Meir-Keeler type multidimensional fixed point theorems in partially ordered metric spaces. Abstr. Appl. Anal. 2013. 10.1155/2013/406026Google Scholar
  15. Karapınar E, Roldán A, Roldán C, Martínez-Moreno J: A note on ‘ N -fixed point theorems for nonlinear contractions in partially ordered metric spaces’. Fixed Point Theory Appl. 2013., 2013: Article ID 310 10.1186/1687-1812-2013-310Google Scholar
  16. Roldán A, Karapınar E:Some multidimensional fixed point theorems on partially preordered G -metric spaces under ( ψ , φ ) -contractivity conditions. Fixed Point Theory Appl. 2013., 2013: Article ID 158 10.1186/1687-1812-2013-158Google Scholar
  17. Roldán A, Martínez-Moreno J, Roldán C, Cho YJ: Multidimensional coincidence point results for compatible mappings in partially ordered fuzzy metric spaces. Fuzzy Sets Syst. 2013. 10.1016/j.fss.2013.10.009Google Scholar
  18. Roldán A, Martínez-Moreno J, Roldán C, Karapınar E:Multidimensional fixed point theorems in partially ordered complete partial metric spaces under ( ψ , φ ) -contractivity conditions. Abstr. Appl. Anal. 2013. 10.1155/2013/634371Google Scholar
  19. Matthews SG: Partial metric topology. Ann. N. Y. Acad. Sci. 728. General Topology and Its Applications 1994, 183-197. Pro. 8th Summer Conf. Queen’s College, 1992Google Scholar
  20. Hitzler P, Seda AK: Dislocated topologies. J. Electr. Eng. 2000, 51: 3-7.MATHGoogle Scholar
  21. Amini-Harandi A: Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory Appl. 2012., 2012: Article ID 204 10.1186/1687-1812-2012-204Google Scholar
  22. Alghamdi MA, Hussain N, Salimiand P: Fixed point and coupled fixed point theorems on b -metric-like spaces. J. Inequal. Appl. 2013., 2013: Article ID 402 10.1186/1029-242X-2013-402Google Scholar
  23. Arshad M, Shoaib A, Beg I: Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space. Fixed Point Theory Appl. 2013., 2013: Article ID 115 10.1186/1687-1812-2013-115Google Scholar
  24. Hussain N, Roshan JR, Parvaneh V, Abbas M: Common fixed point results for weak contractive mappings in ordered b -dislocated metric spaces with applications. J. Inequal. Appl. 2013., 2013: Article ID 486 10.1186/1029-242X-2013-486Google Scholar
  25. Isik H, Türkoğlu D: Fixed point theorems for weakly contractive mappings in partially ordered metric-like spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 51 10.1186/1687-1812-2013-51Google Scholar
  26. Karapınar E, Salimi P: Dislocated metric space to metric spaces with some fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 222 10.1186/1687-1812-2013-222Google Scholar
  27. Kumari PS, Kumar VV, Sarma IR: Common fixed point theorems on weakly compatible maps on dislocated metric spaces. Math. Sci. 2012., 6: Article ID 71 10.1186/2251-7456-6-71Google Scholar
  28. Malhotra SK, Radenović S, Shukla S: Some fixed point results without monotone property in partially ordered metric-like spaces. J. Egypt. Math. Soc. 2013. 10.1016/j.joems.2013.06.010Google Scholar
  29. Sarma IR, Kumari PS: On dislocated metric spaces. Int. J. Math. Arch. 2012, 3: 72-77.Google Scholar
  30. Shukla S, Radenović S, Rajić VĆ: Some common fixed point theorems in 0- σ -complete metric-like spaces. Vietnam J. Math. 2013. 10.1007/s10013-013-0028-0Google Scholar
  31. Karapınar E, Erhan IM, Öztürk A: Fixed point theorems on quasi-partial metric spaces. Math. Comput. Model. 2013, 57: 2442-2448. 10.1016/j.mcm.2012.06.036MATHView ArticleGoogle Scholar
  32. Vetro F, Radenović S: Nonlinear ψ -quasi-contractions of Ćirić-type in partial metric spaces. Appl. Math. Comput. 2012, 219: 1594-1600. 10.1016/j.amc.2012.07.061MATHMathSciNetView ArticleGoogle Scholar

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© Zhu et al.; licensee Springer. 2014

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