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

- Mujahid Abbas
^{1}and - Talat Nazir
^{2}Email author

**2013**:20

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

© Abbas and Nazir; licensee Springer 2013

**Received: **12 October 2012

**Accepted: **14 January 2013

**Published: **30 January 2013

## 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

## 1 Introduction and preliminaries

Consistent with Jachymski [1], let *X* be a nonempty set and *d* be a metric on *X*. A set $\{(x,x):x\in X\}$ is called a diagonal of $X\times X$ 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 $\mathrm{\Delta}\subseteq E(G)$. 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, ${\mathbb{R}}^{+}$, *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 $f:X\to X$ is called a Banach *G*-contraction or simply *G*-contraction if

(a_{1}) for each $x,y\in X$ with $(x,y)\in E(G)$, we have $(f(x),f(y))\in E(G)$,

(a_{2}) there exists $\alpha \in (0,1)$ such that for all $x,y\in X$ with $(x,y)\in E(G)$ implies that $d(f(x),f(y))\le \alpha d(x,y)$.

Let ${X}^{f}:=\{x\in X:(x,f(x))\in E(G)\text{or}(f(x),x)\in E(G)\}$.

*f*is denoted by $F(f)$. A self-mapping

*f*on

*X*is said to be

- (1)
a Picard operator if $F(f)=\{{x}^{\ast}\}$ and ${f}^{n}(x)\to {x}^{\ast}$ as $n\to \mathrm{\infty}$ for all $x\in X$;

- (2)
a weakly Picard operator if $F(f)\ne \mathrm{\varnothing}$ and for each $x\in X$, we have ${f}^{n}(x)\to {x}^{\ast}\in F(f)$ as $n\to \mathrm{\infty}$;

- (3)orbitally continuous if for all $x,a\in X$, we have$\underset{k\to \mathrm{\infty}}{lim}{f}^{{n}_{k}}(x)=a\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}\underset{i\to \mathrm{\infty}}{lim}f({f}^{{n}_{k}}(x))=f(a).$

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

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

(b_{1}) for each $x,y\in X$ with $(x,y)\in E(G)$, we have $(f(x),f(y))\in E(G)$,

_{2}) there exists $\alpha \in [0,1)$ such that

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

*G*is connected if there is a path between any two vertices and it is weakly connected if $\tilde{G}$ is connected, where $\tilde{G}$ 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,

*G*is such that $E(G)$ is symmetric, then for $x\in V(G)$, the symbol ${[x]}_{G}$ denotes the equivalence class of the relation

*R*defined on $V(G)$ by the rule:

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

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*$f:X\to X$

*be a*

*G*-

*contraction*.

*Then the following statements hold*:

- 1.
${F}_{f}\ne \mathrm{\varnothing}$

*if and only if*${X}_{f}\ne \mathrm{\varnothing}$; - 2.
*if*${X}_{f}\ne \mathrm{\varnothing}$*and**G**is weakly connected*,*then**f**is a Picard operator*; - 3.
*for any*$x\in {X}_{f}$*we have that*$f{|}_{{[x]}_{\tilde{G}}}$*is a Picard operator*; - 4.
*if*$f\subseteq E(G)$,*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 [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 $X=V(G)$ 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 $x,y\in W(G)$, we have $(x,y)\in E(G)$.

**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*, $(v,fv)$ and $(v,gv)\in E(G)$, - (b)there exists $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ an upper semi-continuous and nondecreasing function with $\varphi (t)<t$ for each $t>0$ such that${p}^{\delta}(fx,gy)\le \varphi ({p}^{\alpha}(x,y){p}^{\beta}(x,fx){p}^{\gamma}(y,gy))$(1.1)

for all $(x,y)\in E(G)$ holds, where $\alpha ,\beta ,\gamma \ge 0$ with $\delta =\alpha +\beta +\gamma \in (0,\mathrm{\infty})$.

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*$f,g:X\to X$

*form a power graphic contraction pair*,

*then the following hold*:

- (i)
$F(f)\ne \mathrm{\varnothing}$

*or*$F(g)\ne \mathrm{\varnothing}$*if and only if*$F(f)\cap F(g)\ne \mathrm{\varnothing}$. - (ii)
*If*$u\in F(f)\cap F(g)$,*then the weight assigned to the vertex**u**is*0. - (iii)
$F(f)\cap F(g)\ne \mathrm{\varnothing}$

*provided that**G**satisfies the property*(A). - (iv)
$F(f)\cap F(g)$

*is complete if and only if*$F(f)\cap F(g)$*is a singleton*.

*Proof*To prove (i), let $u\in F(f)$. By the given assumption, $(u,gu)\in E(G)$. Assume that we assign a non-zero weight to the edge $(u,gu)$. As $(u,u)\in E(G)$ and

*f*and

*g*form a power graphic contraction, we have

a contradiction. Hence, the weight assigned to the edge $(u,gu)$ is zero and so $u=gu$. Therefore, $u\in F(f)\cap F(g)\ne \mathrm{\varnothing}$. Similarly, if $u\in F(g)$, then we have $u\in F(f)$. The converse is straightforward.

*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 $\{{x}_{n}\}$ in *X* with $f{x}_{2n}={x}_{2n+1}$ and $g{x}_{2n+1}={x}_{2n+2}$ for all $n\in \mathbb{N}$ with $({x}_{n},{x}_{n+1})\in E(G)$, and ${lim}_{n\to \mathrm{\infty}}p({x}_{n},{x}_{n+1})=0$.

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

*k*, so $f{x}_{2k}={x}_{2k+1}={x}_{2k}$, and thus ${x}_{2k}\in F(f)$. Hence, ${x}_{2k}\in F(f)\cap F(g)$ by (i). Now, since $({x}_{2n},{x}_{2n+1})\in E(G)$, so from (1.1), we have

*f*and

*g*form a power graphic contraction, so we have

a contradiction. Hence, $u=gu$ and $u\in F(f)\cap F(g)$ by (i).

*u*and

*v*such that $u,v\in F(f)\cap F(g)$ but $u\ne v$. As $(u,v)\in E(G)$ and

*f*and

*g*form a power graphic contraction, so

a contradiction. Hence, $u=v$. Conversely, if $F(f)\cap F(g)$ is a singleton, then it follows that $F(f)\cap F(g)$ 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*

*where* $\alpha ,\beta ,\gamma \ge 0$ *with* $\delta =\alpha +\beta +\gamma \in (0,\mathrm{\infty})$ *and* $s,t\in \mathbb{N}$, *then the conclusions obtained in Theorem* 2.1 *remain true*.

*Proof* It follows from Theorem 2.1, that $F({f}^{s})\cap F({g}^{t})$ is a singleton provided that $F({f}^{s})\cap F({g}^{t})$ is complete. Let $F({f}^{s})\cap F({g}^{t})=\{w\}$, 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 $F({f}^{s})\cap F({g}^{t})$. Since $F({f}^{s})\cap F({g}^{t})$ is a singleton, we deduce that $w=fw=gw$. Hence, $F(f)\cap F(g)$ 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*.

to obtain conclusions of Theorem 2.1. Indeed, taking $\alpha =\beta =\gamma =1$ in Theorem 2.1, one obtains (2.5).

- (i)
if we take $\alpha =\beta =1$ and $\gamma =0$ in (1.1), then we obtain (2.6),

- (ii)
take $\alpha =\gamma =1$, $\beta =0$ in (1.1) to obtain (2.7),

- (iii)
use $\beta =\gamma =1$, $\alpha =0$ in (1.1) and obtain (2.8).

- (iv)
take $\alpha =1$ and $\beta =\gamma =0$ in (1.1) to obtain (2.9),

- (v)
to obtain (2.10), take $\beta =1$, $\alpha =\gamma =0$ in (1.1),

- (vi)
if one takes $\gamma =1$, $\alpha =\beta =0$ 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

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