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.
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1 Introduction and preliminaries
Consistent with Jachymski [1], 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 [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 .
Let .
Recall that if , then a set of all fixed points of f is denoted by . A self-mapping f on X is said to be
-
(1)
a Picard operator if and as for all ;
-
(2)
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 [2].
Definition 1.2 An operator is called a Banach G-graphic contraction if
(b1) for each with , we have ,
(b2) there exists such that
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 .
Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if is connected, where denotes the undirected graph obtained from G by ignoring the direction of edges. Denote by the graph obtained from G by reversing the direction of edges. Thus,
Since it is more convenient to treat as a directed graph for which the set of its edges is symmetric, under this convention, we have that
If G is such that is symmetric, then for , the symbol denotes the equivalence class of the relation R defined on by the rule:
A graph G is said to satisfy the property (A) (see also [2]) if for any sequence in with as and for implies that .
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 be a complete metric space and G be a directed graph and let the triple have a property (A). Let be a G-contraction. Then the following statements hold:
-
1.
if and only if ;
-
2.
if and G is weakly connected, then f is a Picard operator;
-
3.
for any we have that is a Picard operator;
-
4.
if , 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 [9–19] 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 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 .
Definition 1.4 Self-mappings f and g on X are said to form a power graphic contraction pair if
-
(a)
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.
Theorem 2.1 Let be a complete partial metric space endowed with a directed graph G. If form a power graphic contraction pair, then the following hold:
-
(i)
or if and only if .
-
(ii)
If , then the weight assigned to the vertex u is 0.
-
(iii)
provided that G satisfies the property (A).
-
(iv)
is complete if and only if is a singleton.
Proof To prove (i), let . By the given assumption, . Assume that we assign a non-zero weight to the edge . As and f and g form a power graphic contraction, we have
a contradiction. Hence, the weight assigned to the edge is zero and so . Therefore, . Similarly, if , then we have . The converse is straightforward.
Now, let . Assume that the weight assigned to the vertex u is not zero, then from (1.1), we have
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 .
We may assume that the weight assigned to each edge is non-zero for all . If not, then for some k, so , and thus . Hence, by (i). Now, since , so from (1.1), we have
which implies that
a contradiction if . So, take , and we have
for all . Again from (1.1), we have
which implies that
We arrive at a contradiction in case . Therefore, we must take ; consequently, we have
for all . Hence,
for all . Therefore, the decreasing sequence of positive real numbers converges to some . If we assume that , then from (2.1) we deduce that
a contradiction. So, , that is, and so we have . Also,
Now, for with ,
implies that converges to 0 as . That is, . Since is complete, following similar arguments to those given in Theorem 2.1 of [9], there exists a such that . By the given hypothesis, for all . We claim that the weight assigned to the edge is zero. If not, then as f and g form a power graphic contraction, so we have
We deduce, by taking upper limit as in (2.3), that
a contradiction. Hence, and by (i).
Finally, to prove (iv), suppose the set is complete. We are to show that is a singleton. Assume on the contrary that there exist u and v such that but . As and f and g form a power graphic contraction, so
a contradiction. Hence, . Conversely, if is a singleton, then it follows that is complete. □
Corollary 2.2 Let be a complete partial metric space endowed with a directed graph G. If we replace (1.1) by
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.
(R1) We may replace (1.1) with the following:
to obtain conclusions of Theorem 2.1. Indeed, taking in Theorem 2.1, one obtains (2.5).
(R2) If we replace (1.1) by one of the following condition:
then the conclusions obtained in Theorem 2.1 remain true. Note that
-
(i)
if we take and in (1.1), then we obtain (2.6),
-
(ii)
take , in (1.1) to obtain (2.7),
-
(iii)
use , in (1.1) and obtain (2.8).
(R3) Also, if we replace (1.1) by one of the following conditions:
then the conclusions obtained in Theorem 2.1 remain true. Note that
-
(iv)
take and in (1.1) to obtain (2.9),
-
(v)
to obtain (2.10), take , in (1.1),
-
(vi)
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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Abbas, M., Nazir, T. Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph. Fixed Point Theory Appl 2013, 20 (2013). https://doi.org/10.1186/1687-1812-2013-20
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DOI: https://doi.org/10.1186/1687-1812-2013-20