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Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph

Fixed Point Theory and Applications20132013:20

https://doi.org/10.1186/1687-1812-2013-20

  • Received: 12 October 2012
  • Accepted: 14 January 2013
  • Published:

Abstract

In this paper, we initiate a study of fixed point results in the setup of partial metric spaces endowed with a graph. The concept of a power graphic contraction pair of two mappings is introduced. Common fixed point results for such maps without appealing to any form of commutativity conditions defined on a partial metric space endowed with a directed graph are obtained. These results unify, generalize and complement various known comparable results from the current literature.

MSC:47H10, 54H25, 54E50.

Keywords

  • partial metric space
  • common fixed point
  • directed graph
  • power graphic contraction pair

1 Introduction and preliminaries

Consistent with Jachymski [1], let X be a nonempty set and d be a metric on X. A set { ( x , x ) : x X } is called a diagonal of X × X and is denoted by Δ. Let G be a directed graph such that the set V ( G ) of its vertices coincides with X and E ( G ) is the set of the edges of the graph with Δ E ( G ) . Also assume that the graph G has no parallel edges. One can identify a graph G with the pair ( V ( G ) , E ( G ) ) . Throughout this paper, the letters , R + , ω and will denote the set of real numbers, the set of nonnegative real numbers, the set of nonnegative integers and the set of positive integers, respectively.

Definition 1.1 [1]

A mapping f : X X is called a Banach G-contraction or simply G-contraction if

(a1) for each x , y X with ( x , y ) E ( G ) , we have ( f ( x ) , f ( y ) ) E ( G ) ,

(a2) there exists α ( 0 , 1 ) such that for all x , y X with ( x , y ) E ( G ) implies that d ( f ( x ) , f ( y ) ) α d ( x , y ) .

Let X f : = { x X : ( x , f ( x ) ) E ( G )  or  ( f ( x ) , x ) E ( G ) } .

Recall that if f : X X , then a set { x X : x = f ( x ) } of all fixed points of f is denoted by F ( f ) . A self-mapping f on X is said to be
  1. (1)

    a Picard operator if F ( f ) = { x } and f n ( x ) x as n for all x X ;

     
  2. (2)

    a weakly Picard operator if F ( f ) and for each x X , we have f n ( x ) x F ( f ) as n ;

     
  3. (3)
    orbitally continuous if for all x , a X , we have
    lim k f n k ( x ) = a implies lim i f ( f n k ( x ) ) = f ( a ) .
     

The following definition is due to Chifu and Petrusel [2].

Definition 1.2 An operator f : X X is called a Banach G-graphic contraction if

(b1) for each x , y X with ( x , y ) E ( G ) , we have ( f ( x ) , f ( y ) ) E ( G ) ,

(b2) there exists α [ 0 , 1 ) such that
d ( f ( x ) , f 2 ( x ) ) α d ( x , f ( x ) ) for all  x X f .

If x and y are vertices of G, then a path in G from x to y of length k N is a finite sequence { x n } , n { 0 , 1 , 2 , , k } of vertices such that x 0 = x , x k = y and ( x i 1 , x i ) E ( G ) for i { 1 , 2 , , k } .

Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if G ˜ is connected, where G ˜ denotes the undirected graph obtained from G by ignoring the direction of edges. Denote by G 1 the graph obtained from G by reversing the direction of edges. Thus,
E ( G 1 ) = { ( x , y ) X × X : ( y , x ) E ( G ) } .
Since it is more convenient to treat G ˜ as a directed graph for which the set of its edges is symmetric, under this convention, we have that
E ( G ˜ ) = E ( G ) E ( G 1 ) .
If G is such that E ( G ) is symmetric, then for x V ( G ) , the symbol [ x ] G denotes the equivalence class of the relation R defined on V ( G ) by the rule:
y R z  if there is a path in  G  from  y  to  z .

A graph G is said to satisfy the property (A) (see also [2]) if for any sequence { x n } in V ( G ) with x n x as n and ( x n , x n + 1 ) E ( G ) for n N implies that ( x n , x ) E ( G ) .

Jachymski [1] obtained the following fixed point result for a mapping satisfying the Banach G-contraction condition in metric spaces endowed with a graph.

Theorem 1.3 [1]

Let ( X , d ) be a complete metric space and G be a directed graph and let the triple ( X , d , G ) have a property (A). Let f : X X be a G-contraction. Then the following statements hold:
  1. 1.

    F f if and only if X f ;

     
  2. 2.

    if X f and G is weakly connected, then f is a Picard operator;

     
  3. 3.

    for any x X f we have that f | [ x ] G ˜ is a Picard operator;

     
  4. 4.

    if f E ( G ) , then f is a weakly Picard operator.

     

Gwodzdz-Lukawska and Jachymski [3] developed the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed with a directed graph. Bojor [4] obtained a fixed point of a φ-contraction in metric spaces endowed with a graph (see also [5]). For more results in this direction, we refer to [2, 6, 7].

On the other hand, Mathews [8] introduced the concept of a partial metric to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle more suitable in this context. For examples, related definitions and work carried out in this direction, we refer to [919] and the references mentioned therein. Abbas et al. [20] proved some common fixed points in partially ordered metric spaces (see also [21]). Gu and He [22] proved some common fixed point results for self-maps with twice power type Φ-contractive condition. Recently, Gu and Zhang [23] obtained some common fixed point theorems for six self-mappings with twice power type contraction condition.

Throughout this paper, we assume that a nonempty set X = V ( G ) is equipped with a partial metric p, a directed graph G has no parallel edge and G is a weighted graph in the sense that each vertex x is assigned the weight p ( x , x ) and each edge ( x , y ) is assigned the weight p ( x , y ) . As p is a partial metric on X, the weight assigned to each vertex x need not be zero and whenever a zero weight is assigned to some edge ( x , y ) , it reduces to a loop ( x , x ) .

Also, the subset W ( G ) of V ( G ) is said to be complete if for every x , y W ( G ) , we have ( x , y ) E ( G ) .

Definition 1.4 Self-mappings f and g on X are said to form a power graphic contraction pair if
  1. (a)

    for every vertex v in G, ( v , f v ) and ( v , g v ) E ( G ) ,

     
  2. (b)
    there exists ϕ : R + R + an upper semi-continuous and nondecreasing function with ϕ ( t ) < t for each t > 0 such that
    p δ ( f x , g y ) ϕ ( p α ( x , y ) p β ( x , f x ) p γ ( y , g y ) )
    (1.1)
     

for all ( x , y ) E ( G ) holds, where α , β , γ 0 with δ = α + β + γ ( 0 , ) .

If we take f = g , then the mapping f is called a power graphic contraction.

The aim of this paper is to investigate the existence of common fixed points of a power graphic contraction pair in the framework of complete partial metric spaces endowed with a graph. Our results extend and strengthen various known results [8, 12, 13, 24].

2 Common fixed point results

We start with the following result.

Theorem 2.1 Let ( X , p ) be a complete partial metric space endowed with a directed graph G. If f , g : X X form a power graphic contraction pair, then the following hold:
  1. (i)

    F ( f ) or F ( g ) if and only if F ( f ) F ( g ) .

     
  2. (ii)

    If u F ( f ) F ( g ) , then the weight assigned to the vertex u is 0.

     
  3. (iii)

    F ( f ) F ( g ) provided that G satisfies the property (A).

     
  4. (iv)

    F ( f ) F ( g ) is complete if and only if F ( f ) F ( g ) is a singleton.

     
Proof To prove (i), let u F ( f ) . By the given assumption, ( u , g u ) E ( G ) . Assume that we assign a non-zero weight to the edge ( u , g u ) . As ( u , u ) E ( G ) and f and g form a power graphic contraction, we have
p δ ( u , g u ) = p δ ( f u , g u ) ϕ ( p α ( u , u ) p β ( u , f u ) p γ ( u , g u ) ) = ϕ ( p α + β ( u , u ) p γ ( u , g u ) ) ϕ ( p α + β ( u , g u ) p γ ( u , g u ) ) = ϕ ( p δ ( u , g u ) ) < p δ ( u , g u ) ,

a contradiction. Hence, the weight assigned to the edge ( u , g u ) is zero and so u = g u . Therefore, u F ( f ) F ( g ) . Similarly, if u F ( g ) , then we have u F ( f ) . The converse is straightforward.

Now, let u F ( f ) F ( g ) . Assume that the weight assigned to the vertex u is not zero, then from (1.1), we have
p δ ( u , u ) = p δ ( f u , g u ) ϕ ( p α ( u , u ) p β ( u , f u ) p γ ( u , g u ) ) = ϕ ( p α + β + γ ( u , u ) ) = ϕ ( p δ ( u , u ) ) < p δ ( u , u ) ,

a contradiction. Hence, (ii) is proved.

To prove (iii), we will first show that there exists a sequence { x n } in X with f x 2 n = x 2 n + 1 and g x 2 n + 1 = x 2 n + 2 for all n N with ( x n , x n + 1 ) E ( G ) , and lim n p ( x n , x n + 1 ) = 0 .

Let x 0 be an arbitrary point of X. If f x 0 = x 0 , then the proof is finished, so we assume that f x 0 x 0 . As ( x 0 , f x 0 ) E ( G ) , so ( x 0 , x 1 ) E ( G ) . Also, ( x 1 , g x 1 ) E ( G ) gives ( x 1 , x 2 ) E ( G ) . Continuing this way, we define a sequence { x n } in X such that ( x n , x n + 1 ) E ( G ) with f x 2 n = x 2 n + 1 and g x 2 n + 1 = x 2 n + 2 for n N .

We may assume that the weight assigned to each edge ( x 2 n , x 2 n + 1 ) is non-zero for all n N . If not, then x 2 k = x 2 k + 1 for some k, so f x 2 k = x 2 k + 1 = x 2 k , and thus x 2 k F ( f ) . Hence, x 2 k F ( f ) F ( g ) by (i). Now, since ( x 2 n , x 2 n + 1 ) E ( G ) , so from (1.1), we have
p δ ( x 2 n + 1 , x 2 n + 2 ) = p δ ( f x 2 n , g x 2 n + 1 ) ϕ ( p α ( x 2 n , x 2 n + 1 ) p β ( x 2 n , f x 2 n ) p γ ( x 2 n + 1 , g x 2 n + 1 ) ) = ϕ ( p α ( x 2 n , x 2 n + 1 ) p β ( x 2 n , x 2 n + 1 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ) = ϕ ( p α + β ( x 2 n , x 2 n + 1 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ) < p α + β ( x 2 n , x 2 n + 1 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ,
which implies that
p α + β ( x 2 n + 1 , x 2 n + 2 ) < p α + β ( x 2 n , x 2 n + 1 ) ,
a contradiction if α + β = 0 . So, take α + β > 0 , and we have
p ( x 2 n + 1 , x 2 n + 2 ) < p ( x 2 n , x 2 n + 1 )
for all n N . Again from (1.1), we have
p δ ( x 2 n + 2 , x 2 n + 3 ) = p δ ( g x 2 n + 1 , f x 2 n + 2 ) = p δ ( f x 2 n + 2 , g x 2 n + 1 ) ϕ ( p α ( x 2 n + 2 , x 2 n + 1 ) p β ( x 2 n + 2 , f x 2 n + 2 ) p γ ( x 2 n + 1 , g x 2 n + 1 ) ) = ϕ ( p α ( x 2 n + 1 , x 2 n + 2 ) p β ( x 2 n + 2 , x 2 n + 3 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ) = ϕ ( p α + γ ( x 2 n + 1 , x 2 n + 2 ) p β ( x 2 n + 2 , x 2 n + 3 ) ) < p α + γ ( x 2 n + 1 , x 2 n + 2 ) p β ( x 2 n + 2 , x 2 n + 3 ) ,
which implies that
p α + γ ( x 2 n + 2 , x 2 n + 3 ) < p α + γ ( x 2 n + 1 , x 2 n + 2 ) .
We arrive at a contradiction in case α + γ = 0 . Therefore, we must take α + γ > 0 ; consequently, we have
p ( x 2 n + 2 , x 2 n + 3 ) < p ( x 2 n + 1 , x 2 n + 2 )
for all n N . Hence,
p δ ( x n , x n + 1 ) ϕ ( p δ ( x n 1 , x n ) ) < p δ ( x n 1 , x n )
(2.1)
for all n N . Therefore, the decreasing sequence of positive real numbers { p δ ( x n , x n + 1 ) } converges to some c 0 . If we assume that c > 0 , then from (2.1) we deduce that
0 < c lim sup n ϕ ( p δ ( x n 1 , x n ) ) ϕ ( c ) < c ,
a contradiction. So, c = 0 , that is, lim n p δ ( x n , x n + 1 ) = 0 and so we have lim n p ( x n , x n + 1 ) = 0 . Also,
p δ ( x n , x n + 1 ) ϕ ( p δ ( x n 1 , x n ) ) ϕ n ( p δ ( x 0 , x 1 ) ) .
(2.2)
Now, for m , n N with m > n ,
p δ ( x n , x m ) p δ ( x n , x n + 1 ) + p δ ( x n + 1 , x n + 2 ) + + p δ ( x m 1 , x m ) p δ ( x n + 1 , x n + 1 ) p δ ( x n + 2 , x n + 2 ) p δ ( x m 1 , x m 1 ) ϕ n ( p δ ( x 0 , x 1 ) ) + ϕ n + 1 ( p δ ( x 0 , x 1 ) ) + + ϕ m 1 ( p δ ( x 0 , x 1 ) )
implies that p δ ( x n , x m ) converges to 0 as n , m . That is, lim n , m p ( x n , x m ) = 0 . Since ( X , p ) is complete, following similar arguments to those given in Theorem 2.1 of [9], there exists a u X such that lim n , m p ( x n , x m ) = lim n p ( x n , u ) = p ( u , u ) = 0 . By the given hypothesis, ( x 2 n , u ) E ( G ) for all n N . We claim that the weight assigned to the edge ( u , g u ) is zero. If not, then as f and g form a power graphic contraction, so we have
p δ ( x 2 n + 1 , u ) = p δ ( f x 2 n , g u ) ϕ ( p α ( x 2 n , u ) p β ( x 2 n , f x 2 n ) p γ ( u , g u ) ) = ϕ ( p α ( x 2 n , u ) p β ( x 2 n , x 2 n + 1 ) p γ ( u , g u ) ) .
(2.3)
We deduce, by taking upper limit as n in (2.3), that
p δ ( u , g u ) lim sup n ϕ ( p α ( x 2 n , u ) p β ( x 2 n , x 2 n + 1 ) p γ ( u , g u ) ) ϕ ( p α ( u , u ) p β ( u , u ) p γ ( u , g u ) ) ϕ ( p α + β + γ ( u , g u ) ) < p δ ( u , g u ) ,

a contradiction. Hence, u = g u and u F ( f ) F ( g ) by (i).

Finally, to prove (iv), suppose the set F ( f ) F ( g ) is complete. We are to show that F ( f ) F ( g ) is a singleton. Assume on the contrary that there exist u and v such that u , v F ( f ) F ( g ) but u v . As ( u , v ) E ( G ) and f and g form a power graphic contraction, so
0 < p δ ( u , v ) = p δ ( f u , f v ) ϕ ( p α ( u , v ) p β ( u , f u ) p γ ( v , g v ) ) = ϕ ( p α ( u , v ) p β ( u , u ) p γ ( v , v ) ) ϕ ( p δ ( u , v ) ) ,

a contradiction. Hence, u = v . Conversely, if F ( f ) F ( g ) is a singleton, then it follows that F ( f ) F ( g ) is complete. □

Corollary 2.2 Let ( X , p ) be a complete partial metric space endowed with a directed graph G. If we replace (1.1) by
p δ ( f s x , g t y ) ϕ ( p α ( x , y ) p β ( x , f s x ) p γ ( y , g t y ) ) ,
(2.4)

where α , β , γ 0 with δ = α + β + γ ( 0 , ) and s , t N , then the conclusions obtained in Theorem 2.1 remain true.

Proof It follows from Theorem 2.1, that F ( f s ) F ( g t ) is a singleton provided that F ( f s ) F ( g t ) is complete. Let F ( f s ) F ( g t ) = { w } , then we have f ( w ) = f ( f s ( w ) ) = f s + 1 ( w ) = f s ( f ( w ) ) , and g ( w ) = g ( g t ( w ) ) = g t + 1 ( w ) = g t ( g ( w ) ) implies that fw and gw are also in F ( f s ) F ( g t ) . Since F ( f s ) F ( g t ) is a singleton, we deduce that w = f w = g w . Hence, F ( f ) F ( g ) is a singleton. □

The following remark shows that different choices of α, β and γ give a variety of power graphic contraction pairs of two mappings.

Remarks 2.3 Let ( X , p ) be a complete partial metric space endowed with a directed graph G.

(R1) We may replace (1.1) with the following:
p 3 ( f x , g y ) ϕ ( p ( x , y ) p ( x , f x ) p ( y , g y ) )
(2.5)

to obtain conclusions of Theorem 2.1. Indeed, taking α = β = γ = 1 in Theorem 2.1, one obtains (2.5).

(R2) If we replace (1.1) by one of the following condition:
(2.6)
(2.7)
(2.8)
then the conclusions obtained in Theorem 2.1 remain true. Note that
  1. (i)

    if we take α = β = 1 and γ = 0 in (1.1), then we obtain (2.6),

     
  2. (ii)

    take α = γ = 1 , β = 0 in (1.1) to obtain (2.7),

     
  3. (iii)

    use β = γ = 1 , α = 0 in (1.1) and obtain (2.8).

     
(R3) Also, if we replace (1.1) by one of the following conditions:
(2.9)
(2.10)
(2.11)
then the conclusions obtained in Theorem 2.1 remain true. Note that
  1. (iv)

    take α = 1 and β = γ = 0 in (1.1) to obtain (2.9),

     
  2. (v)

    to obtain (2.10), take β = 1 , α = γ = 0 in (1.1),

     
  3. (vi)

    if one takes γ = 1 , α = β = 0 in (1.1), then we obtain (2.11).

     

Remark 2.4 If we take f = g in a power graphic contraction pair, then we obtain fixed point results for a power graphic contraction.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Lahore University of Management Sciences, Lahore, 54792, Pakistan
(2)
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, 22060, Pakistan

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© Abbas and Nazir; 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.

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