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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 , 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 

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 .

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 ) 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  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 

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  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  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 $X=V\left(G\right)$ 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 , 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).

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:
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

## References 