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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

• 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 $\left\{\left(x,x\right):x\in X\right\}$ is called a diagonal of $X×X$ and is denoted by Δ. Let G be a directed graph such that the set $V\left(G\right)$ of its vertices coincides with X and $E\left(G\right)$ is the set of the edges of the graph with $\mathrm{\Delta }\subseteq E\left(G\right)$. Also assume that the graph G has no parallel edges. One can identify a graph G with the pair $\left(V\left(G\right),E\left(G\right)\right)$. Throughout this paper, the letters , ${\mathbb{R}}^{+}$, 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 $f:X\to X$ is called a Banach G-contraction or simply G-contraction if

(a1) for each $x,y\in X$ with $\left(x,y\right)\in E\left(G\right)$, we have $\left(f\left(x\right),f\left(y\right)\right)\in E\left(G\right)$,

(a2) there exists $\alpha \in \left(0,1\right)$ such that for all $x,y\in X$ with $\left(x,y\right)\in E\left(G\right)$ implies that $d\left(f\left(x\right),f\left(y\right)\right)\le \alpha d\left(x,y\right)$.

Let .

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

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

2. (2)

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

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

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

(b1) for each $x,y\in X$ with $\left(x,y\right)\in E\left(G\right)$, we have $\left(f\left(x\right),f\left(y\right)\right)\in E\left(G\right)$,

(b2) there exists $\alpha \in \left[0,1\right)$ 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 $\left\{{x}_{n}\right\}$, $n\in \left\{0,1,2,\dots ,k\right\}$ of vertices such that ${x}_{0}=x$, ${x}_{k}=y$ and $\left({x}_{i-1},{x}_{i}\right)\in E\left(G\right)$ for $i\in \left\{1,2,\dots ,k\right\}$.

Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if $\stackrel{˜}{G}$ is connected, where $\stackrel{˜}{G}$ 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,
$E\left({G}^{-1}\right)=\left\{\left(x,y\right)\in X×X:\left(y,x\right)\in E\left(G\right)\right\}.$
Since it is more convenient to treat $\stackrel{˜}{G}$ as a directed graph for which the set of its edges is symmetric, under this convention, we have that
$E\left(\stackrel{˜}{G}\right)=E\left(G\right)\cup E\left({G}^{-1}\right).$
If G is such that $E\left(G\right)$ is symmetric, then for $x\in V\left(G\right)$, the symbol ${\left[x\right]}_{G}$ denotes the equivalence class of the relation R defined on $V\left(G\right)$ by the rule:

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

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

${F}_{f}\ne \mathrm{\varnothing }$ if and only if ${X}_{f}\ne \mathrm{\varnothing }$;

2. 2.

if ${X}_{f}\ne \mathrm{\varnothing }$ and G is weakly connected, then f is a Picard operator;

3. 3.

for any $x\in {X}_{f}$ we have that $f{|}_{{\left[x\right]}_{\stackrel{˜}{G}}}$ is a Picard operator;

4. 4.

if $f\subseteq E\left(G\right)$, 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 $X=V\left(G\right)$ 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 $p\left(x,x\right)$ and each edge $\left(x,y\right)$ is assigned the weight $p\left(x,y\right)$. 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 $\left(x,y\right)$, it reduces to a loop $\left(x,x\right)$.

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

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, $\left(v,fv\right)$ and $\left(v,gv\right)\in E\left(G\right)$,

2. (b)
there exists $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ an upper semi-continuous and nondecreasing function with $\varphi \left(t\right) for each $t>0$ such that
${p}^{\delta }\left(fx,gy\right)\le \varphi \left({p}^{\alpha }\left(x,y\right){p}^{\beta }\left(x,fx\right){p}^{\gamma }\left(y,gy\right)\right)$
(1.1)

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

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

Theorem 2.1 Let $\left(X,p\right)$ 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:
1. (i)

$F\left(f\right)\ne \mathrm{\varnothing }$ or $F\left(g\right)\ne \mathrm{\varnothing }$ if and only if $F\left(f\right)\cap F\left(g\right)\ne \mathrm{\varnothing }$.

2. (ii)

If $u\in F\left(f\right)\cap F\left(g\right)$, then the weight assigned to the vertex u is 0.

3. (iii)

$F\left(f\right)\cap F\left(g\right)\ne \mathrm{\varnothing }$ provided that G satisfies the property (A).

4. (iv)

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

Proof To prove (i), let $u\in F\left(f\right)$. By the given assumption, $\left(u,gu\right)\in E\left(G\right)$. Assume that we assign a non-zero weight to the edge $\left(u,gu\right)$. As $\left(u,u\right)\in E\left(G\right)$ and f and g form a power graphic contraction, we have
$\begin{array}{rcl}{p}^{\delta }\left(u,gu\right)& =& {p}^{\delta }\left(fu,gu\right)\\ \le & \varphi \left({p}^{\alpha }\left(u,u\right){p}^{\beta }\left(u,fu\right){p}^{\gamma }\left(u,gu\right)\right)\\ =& \varphi \left({p}^{\alpha +\beta }\left(u,u\right){p}^{\gamma }\left(u,gu\right)\right)\\ \le & \varphi \left({p}^{\alpha +\beta }\left(u,gu\right){p}^{\gamma }\left(u,gu\right)\right)\\ =& \varphi \left({p}^{\delta }\left(u,gu\right)\right)\\ <& {p}^{\delta }\left(u,gu\right),\end{array}$

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

Now, let $u\in F\left(f\right)\cap F\left(g\right)$. Assume that the weight assigned to the vertex u is not zero, then from (1.1), we have
$\begin{array}{rcl}{p}^{\delta }\left(u,u\right)& =& {p}^{\delta }\left(fu,gu\right)\\ \le & \varphi \left({p}^{\alpha }\left(u,u\right){p}^{\beta }\left(u,fu\right){p}^{\gamma }\left(u,gu\right)\right)\\ =& \varphi \left({p}^{\alpha +\beta +\gamma }\left(u,u\right)\right)\\ =& \varphi \left({p}^{\delta }\left(u,u\right)\right)\\ <& {p}^{\delta }\left(u,u\right),\end{array}$

a contradiction. Hence, (ii) is proved.

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

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 $\left({x}_{0},f{x}_{0}\right)\in E\left(G\right)$, so $\left({x}_{0},{x}_{1}\right)\in E\left(G\right)$. Also, $\left({x}_{1},g{x}_{1}\right)\in E\left(G\right)$ gives $\left({x}_{1},{x}_{2}\right)\in E\left(G\right)$. Continuing this way, we define a sequence $\left\{{x}_{n}\right\}$ in X such that $\left({x}_{n},{x}_{n+1}\right)\in E\left(G\right)$ with $f{x}_{2n}={x}_{2n+1}$ and $g{x}_{2n+1}={x}_{2n+2}$ for $n\in \mathbb{N}$.

We may assume that the weight assigned to each edge $\left({x}_{2n},{x}_{2n+1}\right)$ is non-zero for all $n\in \mathbb{N}$. If not, then ${x}_{2k}={x}_{2k+1}$ for some k, so $f{x}_{2k}={x}_{2k+1}={x}_{2k}$, and thus ${x}_{2k}\in F\left(f\right)$. Hence, ${x}_{2k}\in F\left(f\right)\cap F\left(g\right)$ by (i). Now, since $\left({x}_{2n},{x}_{2n+1}\right)\in E\left(G\right)$, so from (1.1), we have
$\begin{array}{rcl}{p}^{\delta }\left({x}_{2n+1},{x}_{2n+2}\right)& =& {p}^{\delta }\left(f{x}_{2n},g{x}_{2n+1}\right)\\ \le & \varphi \left({p}^{\alpha }\left({x}_{2n},{x}_{2n+1}\right){p}^{\beta }\left({x}_{2n},f{x}_{2n}\right){p}^{\gamma }\left({x}_{2n+1},g{x}_{2n+1}\right)\right)\\ =& \varphi \left({p}^{\alpha }\left({x}_{2n},{x}_{2n+1}\right){p}^{\beta }\left({x}_{2n},{x}_{2n+1}\right){p}^{\gamma }\left({x}_{2n+1},{x}_{2n+2}\right)\right)\\ =& \varphi \left({p}^{\alpha +\beta }\left({x}_{2n},{x}_{2n+1}\right){p}^{\gamma }\left({x}_{2n+1},{x}_{2n+2}\right)\right)\\ <& {p}^{\alpha +\beta }\left({x}_{2n},{x}_{2n+1}\right){p}^{\gamma }\left({x}_{2n+1},{x}_{2n+2}\right),\end{array}$
which implies that
${p}^{\alpha +\beta }\left({x}_{2n+1},{x}_{2n+2}\right)<{p}^{\alpha +\beta }\left({x}_{2n},{x}_{2n+1}\right),$
a contradiction if $\alpha +\beta =0$. So, take $\alpha +\beta >0$, and we have
$p\left({x}_{2n+1},{x}_{2n+2}\right)
for all $n\in \mathbb{N}$. Again from (1.1), we have
$\begin{array}{rcl}{p}^{\delta }\left({x}_{2n+2},{x}_{2n+3}\right)& =& {p}^{\delta }\left(g{x}_{2n+1},f{x}_{2n+2}\right)\\ =& {p}^{\delta }\left(f{x}_{2n+2},g{x}_{2n+1}\right)\\ \le & \varphi \left({p}^{\alpha }\left({x}_{2n+2},{x}_{2n+1}\right){p}^{\beta }\left({x}_{2n+2},f{x}_{2n+2}\right){p}^{\gamma }\left({x}_{2n+1},g{x}_{2n+1}\right)\right)\\ =& \varphi \left({p}^{\alpha }\left({x}_{2n+1},{x}_{2n+2}\right){p}^{\beta }\left({x}_{2n+2},{x}_{2n+3}\right){p}^{\gamma }\left({x}_{2n+1},{x}_{2n+2}\right)\right)\\ =& \varphi \left({p}^{\alpha +\gamma }\left({x}_{2n+1},{x}_{2n+2}\right){p}^{\beta }\left({x}_{2n+2},{x}_{2n+3}\right)\right)\\ <& {p}^{\alpha +\gamma }\left({x}_{2n+1},{x}_{2n+2}\right){p}^{\beta }\left({x}_{2n+2},{x}_{2n+3}\right),\end{array}$
which implies that
${p}^{\alpha +\gamma }\left({x}_{2n+2},{x}_{2n+3}\right)<{p}^{\alpha +\gamma }\left({x}_{2n+1},{x}_{2n+2}\right).$
We arrive at a contradiction in case $\alpha +\gamma =0$. Therefore, we must take $\alpha +\gamma >0$; consequently, we have
$p\left({x}_{2n+2},{x}_{2n+3}\right)
for all $n\in \mathbb{N}$. Hence,
${p}^{\delta }\left({x}_{n},{x}_{n+1}\right)\le \varphi \left({p}^{\delta }\left({x}_{n-1},{x}_{n}\right)\right)<{p}^{\delta }\left({x}_{n-1},{x}_{n}\right)$
(2.1)
for all $n\in \mathbb{N}$. Therefore, the decreasing sequence of positive real numbers $\left\{{p}^{\delta }\left({x}_{n},{x}_{n+1}\right)\right\}$ converges to some $c\ge 0$. If we assume that $c>0$, then from (2.1) we deduce that
$0
a contradiction. So, $c=0$, that is, ${lim}_{n\to \mathrm{\infty }}{p}^{\delta }\left({x}_{n},{x}_{n+1}\right)=0$ and so we have ${lim}_{n\to \mathrm{\infty }}p\left({x}_{n},{x}_{n+1}\right)=0$. Also,
${p}^{\delta }\left({x}_{n},{x}_{n+1}\right)\le \varphi \left({p}^{\delta }\left({x}_{n-1},{x}_{n}\right)\right)\le \cdots \le {\varphi }^{n}\left({p}^{\delta }\left({x}_{0},{x}_{1}\right)\right).$
(2.2)
Now, for $m,n\in \mathbb{N}$ with $m>n$,
$\begin{array}{rcl}{p}^{\delta }\left({x}_{n},{x}_{m}\right)& \le & {p}^{\delta }\left({x}_{n},{x}_{n+1}\right)+{p}^{\delta }\left({x}_{n+1},{x}_{n+2}\right)+\cdots +{p}^{\delta }\left({x}_{m-1},{x}_{m}\right)\\ -{p}^{\delta }\left({x}_{n+1},{x}_{n+1}\right)-{p}^{\delta }\left({x}_{n+2},{x}_{n+2}\right)-\cdots -{p}^{\delta }\left({x}_{m-1},{x}_{m-1}\right)\\ \le & {\varphi }^{n}\left({p}^{\delta }\left({x}_{0},{x}_{1}\right)\right)+{\varphi }^{n+1}\left({p}^{\delta }\left({x}_{0},{x}_{1}\right)\right)+\cdots +{\varphi }^{m-1}\left({p}^{\delta }\left({x}_{0},{x}_{1}\right)\right)\end{array}$
implies that ${p}^{\delta }\left({x}_{n},{x}_{m}\right)$ converges to 0 as $n,m\to \mathrm{\infty }$. That is, ${lim}_{n,m\to \mathrm{\infty }}p\left({x}_{n},{x}_{m}\right)=0$. Since $\left(X,p\right)$ is complete, following similar arguments to those given in Theorem 2.1 of [9], there exists a $u\in X$ such that ${lim}_{n,m\to \mathrm{\infty }}p\left({x}_{n},{x}_{m}\right)={lim}_{n\to \mathrm{\infty }}p\left({x}_{n},u\right)=p\left(u,u\right)=0$. By the given hypothesis, $\left({x}_{2n},u\right)\in E\left(G\right)$ for all $n\in \mathbb{N}$. We claim that the weight assigned to the edge $\left(u,gu\right)$ is zero. If not, then as f and g form a power graphic contraction, so we have
$\begin{array}{rcl}{p}^{\delta }\left({x}_{2n+1},u\right)& =& {p}^{\delta }\left(f{x}_{2n},gu\right)\\ \le & \varphi \left({p}^{\alpha }\left({x}_{2n},u\right){p}^{\beta }\left({x}_{2n},f{x}_{2n}\right){p}^{\gamma }\left(u,gu\right)\right)\\ =& \varphi \left({p}^{\alpha }\left({x}_{2n},u\right){p}^{\beta }\left({x}_{2n},{x}_{2n+1}\right){p}^{\gamma }\left(u,gu\right)\right).\end{array}$
(2.3)
We deduce, by taking upper limit as $n\to \mathrm{\infty }$ in (2.3), that
$\begin{array}{rcl}{p}^{\delta }\left(u,gu\right)& \le & \underset{n\to \mathrm{\infty }}{lim sup}\varphi \left({p}^{\alpha }\left({x}_{2n},u\right){p}^{\beta }\left({x}_{2n},{x}_{2n+1}\right){p}^{\gamma }\left(u,gu\right)\right)\\ \le & \varphi \left({p}^{\alpha }\left(u,u\right){p}^{\beta }\left(u,u\right){p}^{\gamma }\left(u,gu\right)\right)\\ \le & \varphi \left({p}^{\alpha +\beta +\gamma }\left(u,gu\right)\right)\\ <& {p}^{\delta }\left(u,gu\right),\end{array}$

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

Finally, to prove (iv), suppose the set $F\left(f\right)\cap F\left(g\right)$ is complete. We are to show that $F\left(f\right)\cap F\left(g\right)$ is a singleton. Assume on the contrary that there exist u and v such that $u,v\in F\left(f\right)\cap F\left(g\right)$ but $u\ne v$. As $\left(u,v\right)\in E\left(G\right)$ and f and g form a power graphic contraction, so
$\begin{array}{rcl}0& <& {p}^{\delta }\left(u,v\right)={p}^{\delta }\left(fu,fv\right)\\ \le & \varphi \left({p}^{\alpha }\left(u,v\right){p}^{\beta }\left(u,fu\right){p}^{\gamma }\left(v,gv\right)\right)\\ =& \varphi \left({p}^{\alpha }\left(u,v\right){p}^{\beta }\left(u,u\right){p}^{\gamma }\left(v,v\right)\right)\\ \le & \varphi \left({p}^{\delta }\left(u,v\right)\right),\end{array}$

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

Corollary 2.2 Let $\left(X,p\right)$ be a complete partial metric space endowed with a directed graph G. If we replace (1.1) by
${p}^{\delta }\left({f}^{s}x,{g}^{t}y\right)\le \varphi \left({p}^{\alpha }\left(x,y\right){p}^{\beta }\left(x,{f}^{s}x\right){p}^{\gamma }\left(y,{g}^{t}y\right)\right),$
(2.4)

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

Proof It follows from Theorem 2.1, that $F\left({f}^{s}\right)\cap F\left({g}^{t}\right)$ is a singleton provided that $F\left({f}^{s}\right)\cap F\left({g}^{t}\right)$ is complete. Let $F\left({f}^{s}\right)\cap F\left({g}^{t}\right)=\left\{w\right\}$, then we have $f\left(w\right)=f\left({f}^{s}\left(w\right)\right)={f}^{s+1}\left(w\right)={f}^{s}\left(f\left(w\right)\right)$, and $g\left(w\right)=g\left({g}^{t}\left(w\right)\right)={g}^{t+1}\left(w\right)={g}^{t}\left(g\left(w\right)\right)$ implies that fw and gw are also in $F\left({f}^{s}\right)\cap F\left({g}^{t}\right)$. Since $F\left({f}^{s}\right)\cap F\left({g}^{t}\right)$ is a singleton, we deduce that $w=fw=gw$. Hence, $F\left(f\right)\cap F\left(g\right)$ 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 $\left(X,p\right)$ be a complete partial metric space endowed with a directed graph G.

(R1) We may replace (1.1) with the following:
${p}^{3}\left(fx,gy\right)\le \varphi \left(p\left(x,y\right)p\left(x,fx\right)p\left(y,gy\right)\right)$
(2.5)

to obtain conclusions of Theorem 2.1. Indeed, taking $\alpha =\beta =\gamma =1$ 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 $\alpha =\beta =1$ and $\gamma =0$ in (1.1), then we obtain (2.6),

2. (ii)

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

3. (iii)

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

2. (v)

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

3. (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.

## 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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