- Open Access
Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph
© Abbas and Nazir; licensee Springer 2013
Received: 12 October 2012
Accepted: 14 January 2013
Published: 30 January 2013
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.
1 Introduction and preliminaries
Consistent with Jachymski , let X be a nonempty set and d be a metric on X. A set is called a diagonal of and is denoted by Δ. Let G be a directed graph such that the set of its vertices coincides with X and is the set of the edges of the graph with . Also assume that the graph G has no parallel edges. One can identify a graph G with the pair . Throughout this paper, the letters ℝ, , ω and ℕ 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 
A mapping is called a Banach G-contraction or simply G-contraction if
(a1) for each with , we have ,
(a2) there exists such that for all with implies that .
a Picard operator if and as for all ;
a weakly Picard operator if and for each , we have as ;
- (3)orbitally continuous if for all , we have
The following definition is due to Chifu and Petrusel .
Definition 1.2 An operator is called a Banach G-graphic contraction if
(b1) for each with , we have ,
If x and y are vertices of G, then a path in G from x to y of length is a finite sequence , of vertices such that , and for .
A graph G is said to satisfy the property (A) (see also ) if for any sequence in with as and for implies that .
Jachymski  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 
if and only if ;
if and G is weakly connected, then f is a Picard operator;
for any we have that is a Picard operator;
if , then f is a weakly Picard operator.
Gwodzdz-Lukawska and Jachymski  developed the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed with a directed graph. Bojor  obtained a fixed point of a φ-contraction in metric spaces endowed with a graph (see also ). For more results in this direction, we refer to [2, 6, 7].
On the other hand, Mathews  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 [9–19] and the references mentioned therein. Abbas et al.  proved some common fixed points in partially ordered metric spaces (see also ). Gu and He  proved some common fixed point results for self-maps with twice power type Φ-contractive condition. Recently, Gu and Zhang  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 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 and each edge is assigned the weight . 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 , it reduces to a loop .
Also, the subset of is said to be complete if for every , we have .
for every vertex v in G, and ,
- (b)there exists an upper semi-continuous and nondecreasing function with for each such that(1.1)
for all holds, where with .
If we take , 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.
or if and only if .
If , then the weight assigned to the vertex u is 0.
provided that G satisfies the property (A).
is complete if and only if is a singleton.
a contradiction. Hence, the weight assigned to the edge is zero and so . Therefore, . Similarly, if , then we have . The converse is straightforward.
a contradiction. Hence, (ii) is proved.
To prove (iii), we will first show that there exists a sequence in X with and for all with , and .
Let be an arbitrary point of X. If , then the proof is finished, so we assume that . As , so . Also, gives . Continuing this way, we define a sequence in X such that with and for .
a contradiction. Hence, and by (i).
a contradiction. Hence, . Conversely, if is a singleton, then it follows that is complete. □
where with and , then the conclusions obtained in Theorem 2.1 remain true.
Proof It follows from Theorem 2.1, that is a singleton provided that is complete. Let , then we have , and implies that fw and gw are also in . Since is a singleton, we deduce that . Hence, 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 be a complete partial metric space endowed with a directed graph G.
to obtain conclusions of Theorem 2.1. Indeed, taking in Theorem 2.1, one obtains (2.5).
if we take and in (1.1), then we obtain (2.6),
take , in (1.1) to obtain (2.7),
use , in (1.1) and obtain (2.8).
take and in (1.1) to obtain (2.9),
to obtain (2.10), take , in (1.1),
if one takes , in (1.1), then we obtain (2.11).
Remark 2.4 If we take in a power graphic contraction pair, then we obtain fixed point results for a power graphic contraction.
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