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

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Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph Mujahid Abbas1 and Talat Nazir2* *

Correspondence: [email protected] Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, 22060, Pakistan Full list of author information is available at the end of the article 2

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 [], 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 . 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  ⊆ 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, 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 . [] A mapping f : X → X is called a Banach G-contraction or simply Gcontraction if (a ) for each x, y ∈ X with (x, y) ∈ E(G), we have (f (x), f (y)) ∈ E(G), (a ) there exists α ∈ (, ) such that for all x, y ∈ X with (x, y) ∈ E(G) implies that d(f (x), f (y)) ≤ αd(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 () a Picard operator if F(f ) = {x∗ } and f n (x) → x∗ as n → ∞ for all x ∈ X; () a weakly Picard operator if F(f ) = ∅ and for each x ∈ X, we have f n (x) → x∗ ∈ F(f ) as n → ∞; © 2013 Abbas and Nazir; licensee Springer. 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.

Abbas and Nazir Fixed Point Theory and Applications 2013, 2013:20 http://www.fixedpointtheoryandapplications.com/content/2013/1/20

() orbitally continuous if for all x, a ∈ X, we have lim f nk (x) = a

k→∞

implies

  lim f f nk (x) = f (a).

i→∞

The following definition is due to Chifu and Petrusel []. Definition . An operator f : X → X is called a Banach G-graphic contraction if (b ) for each x, y ∈ X with (x, y) ∈ E(G), we have (f (x), f (y)) ∈ E(G