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GPrešić operators on metric spaces endowed with a graph and fixed point theorems
Fixed Point Theory and Applications volume 2014, Article number: 127 (2014)
Abstract
The purpose of this paper is to introduce the Prešić type contraction in metric spaces endowed with a graph and to prove some fixed point results for the GPrešić operators in such spaces. The results proved here generalize, extend, and unify several comparable results in the existing literature. Some examples are included which illustrate the results proved herein.
MSC:47H10, 54H25.
1 Introduction and preliminaries
The famous Banach contraction mapping principle is one of the most simple and useful tools in the theory of fixed points. There are several generalizations of this famous principle. In 1965, Prešić [1, 2] gave a generalization of Banach contraction principle in product spaces which ensures the convergence of a particular sequence. Prešić proved the following theorem.
Theorem 1.1 Let $(X,d)$ be a complete metric space, k a positive integer and $T:{X}^{k}\to X$ be a mapping satisfying the following contractive type condition:
for every ${x}_{1},{x}_{2},\dots ,{x}_{k+1}\in X$, where ${q}_{1},{q}_{2},\dots ,{q}_{k}$ are nonnegative constants such that ${q}_{1}+{q}_{2}+\cdots +{q}_{k}<1$. Then there exists a unique point $x\in X$ such that $T(x,x,\dots ,x)=x$. Moreover, if ${x}_{1},{x}_{2},\dots ,{x}_{k}$ are arbitrary points in X and for $n\in \mathbb{N}$, ${x}_{n+k}=T({x}_{n},{x}_{n+1},\dots ,{x}_{n+k1})$, then the sequence $\{{x}_{n}\}$ is convergent and ${lim}_{n\to \mathrm{\infty}}{x}_{n}=T({lim}_{n\to \mathrm{\infty}}{x}_{n},{lim}_{n\to \mathrm{\infty}}{x}_{n},\dots ,{lim}_{n\to \mathrm{\infty}}{x}_{n})$.
A mapping T satisfying (1.1) is called a Prešić operator. Note that the above theorem in the case $k=1$ reduces to the wellknown Banach contraction mapping principle. Therefore Theorem 1.1 is a generalization of the Banach fixed point theorem.
Prešić type operators have applications in solving nonlinear difference equations and in the study of convergence of sequences; for example, see [1–4]. In the recent years, several authors generalized and extend the result of Prešić in different directions. For more on the generalizations of Prešić type operators the reader is referred to [5–19].
The existence of fixed point in metric spaces endowed with a partial order was investigated by Ran and Reurings [20] and then by Nieto and RodríguezLopez [21, 22]. For the related results see, for instance, [23–27] and references therein. In [7] Malhotra et al. (see also [15, 19]) considered the Prešić type mappings in partially ordered sets and proved the ordered version of theorem of Prešić. This result is further generalized by Shukla et al. [16].
Kirk et al. [28] introduced the notion of cyclic operators and generalized the Banach contraction principle by proving the fixed point results for cyclic operators. Since then, many authors have made their contribution in this area; see, for example, [29–33] and the references cited therein. Very recently, Shukla and Abbas [18] generalized both the notions of cyclic operators and of Prešić operators by introducing the notion of cyclicPrešić operators.
On the other hands, Jachymski [34] initiate the study of fixed point theorems in metric spaces endowed with graphs. Jachymski generalized the Banach contraction principle and unified the results of Ran and Reurings [20], Nieto and RodríguezLopez [21, 22] and Edelstein [35]. For other related results, see, for instance, [23, 36–41].
In this paper, we generalize the result of Prešić in the metric spaces endowed with a graph. The notion of GPrešić operators is introduced and fixed point results for such operators are proved. The results of this paper generalize and unify the results of Prešić [1, 2], Luong and Thuan [19] and Shukla and Abbas [18] for the spaces endowed with a graph, also these results generalize the results of Ran and Reurings [20], Nieto and RodríguezLopez [21, 22] and Kirk et al. [28] in product spaces. Some examples are provided which illustrate the results proved herein.
First we recall some definitions and results which will be needed in the sequel.
Let $(X,d)$ be a metric space. Let Δ denote the diagonal of the Cartesian product $X\times X$. Consider a directed graph G such that the set $V(G)$ of its vertices coincides with X, and the set $E(G)$ of its edges contains all loops, that is, $E(G)\supseteq \mathrm{\Delta}$. We assume that G has no parallel edges, so we can identify G with the pair $(V(G),E(G))$. Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.
By ${G}^{1}$ we denote the conversion of a graph G, that is, the graph obtained from G by reversing the direction of edges. Thus we have
The letter $\tilde{G}$ denotes the undirected graph obtained from G by ignoring the direction of edges. Actually, it will be more convenient for us to treat $\tilde{G}$ as a directed graph for which the set of its edges is symmetric. Under this convention,
If x and y are vertices in a graph G, then a path in G from x to y of length l is a sequence ${\{{x}_{i}\}}_{i=0}^{l}$ of $l+1$ vertices such that ${x}_{0}=x$, ${x}_{l}=y$ and $({x}_{i1},{x}_{i})\in E(G)$ for $i=1,\dots ,l$. A graph G is called connected if there is a path between any two vertices of G. G is weakly connected if $\tilde{G}$ is connected.
Throughout this paper we assume that X is a nonempty set, G is a directed graph such that $V(G)=X$ and $E(G)\supseteq \mathrm{\Delta}$.
Now we can state our main results.
2 Main results
First we define some notions which will be useful in the sequel.
Definition 2.1 Let $(X,d)$ be a metric space endowed with a graph G, k a positive integer and $T:{X}^{k}\to X$ be a mapping. A point $x\in X$ is called a fixed point of T if $T(x,x,\dots ,x)=x$. The set of all fixed points of T is denoted by $Fix(T)$. Let ${x}_{1},{x}_{2},\dots ,{x}_{k}\in X$ be arbitrary points in X. Then the sequence $\{{x}_{n}\}$ defined by ${x}_{n+k}=T({x}_{n},{x}_{n+1},\dots ,{x}_{n+k1})$ for all $n\in \mathbb{N}$ is called a PrešićPicard sequence (in short PPsequence) with initial values ${x}_{1},{x}_{2},\dots ,{x}_{k}$. The mapping T is called a PrešićPicard operator (in short PPoperator) on X, if T has a unique fixed point $u\in X$ and every PPsequence in X converges to u. The mapping T is called a weakly PrešićPicard operator (in short WPPoperator) on X, if for every PPsequence $\{{x}_{n}\}$ in X, ${lim}_{n\to \mathrm{\infty}}{x}_{n}$ exists (it may depend on the initial values ${x}_{1},{x}_{2},\dots ,{x}_{k}$) and is a fixed point of T. Let ${P}_{T}^{k}$ denotes the set of all paths ${\{{x}_{i}\}}_{i=1}^{k}$ of k vertices such that $({x}_{k},T({x}_{1},{x}_{2},\dots ,{x}_{k}))\in E(G)$, that is,
The space $(X,d)$ is said to have property (P) if:
Now we define a GPrešić operator on a metric space endowed with a graph.
Definition 2.2 Let $(X,d)$ be a metric space endowed with a graph G and k be a positive integer. Let $T:{X}^{k}\to X$ be a mapping satisfying the following conditions:
(GP1) if ${\{{x}_{i}\}}_{i=1}^{k+1}$ be any path in G then $(T({x}_{1},\dots ,{x}_{k}),T({x}_{2},\dots ,{x}_{k+1}))\in E(G)$;
(GP2) for every path ${\{{x}_{i}\}}_{i=1}^{k+1}$ in G we have
where ${q}_{i}$’s are nonnegative reals such that ${\sum}_{i=1}^{k}{q}_{i}<1$.
Then T is called a GPrešić operator.
Example 2.3 Any constant function $T:{X}^{k}\to X$ is a GPrešić operator since $E(G)$ contains all loops.
Example 2.4 Let T be any Prešić operator, that is, T satisfies (1.1); then T is a ${G}_{0}$Prešić operator where the graph ${G}_{0}$ is defined by $E({G}_{0})=X\times X$.
Remark 2.5 In the case $k=1$, GPrešić operator reduces into a Gcontraction (see Jachymski [34]). Proposition 2.1 of Jachymski [34] shows that if T is a Gcontraction then it is both ${G}^{1}$contraction and $\tilde{G}$contraction. But in the case $k>1$ a GPrešić operator need not be a ${G}^{1}$Prešić operator or a $\tilde{G}$Prešić operator, as shown in the following example.
Example 2.6 Let $X=\{1,2,3,4,5\}$ and $(X,d)$ be the metric space with usual metric d and G be the graph defined by
For $k=2$, define a mapping $T:{X}^{2}\to X$ by $T(3,5)=2$, $T(5,3)=3$, $T(2,2)=T(3,3)=1$, $T(5,5)=2$, and $T(x,y)=min\{x,y\}$ for all other values of x and y. Then T is a GPrešić operator with ${q}_{1}=0$, ${q}_{2}=1/2$.
On the other hand, T is not a ${G}^{1}$Prešić operator. Indeed,
and so ${\{{x}_{i}\}}_{i=1}^{3}$, where ${x}_{1}=5$, ${x}_{2}=3$, ${x}_{3}=1$ is a path in ${G}^{1}$. Now $d(T(5,3),T(3,1))=d(3,1)=2$ and $d(5,3)=2$, $d(3,1)=2$, so there are no nonnegative constants ${q}_{1}$, ${q}_{2}$ such that ${q}_{1}+{q}_{2}<1$ and $d(T(5,3),T(3,1))\le {q}_{1}d(5,3)+{q}_{2}d(3,1)$. Because $E(\tilde{G})\supseteq E({G}^{1})$ we have ${\{{x}_{i}\}}_{i=1}^{3}$, where ${x}_{1}=5$, ${x}_{2}=3$, ${x}_{3}=1$ is also a path in $\tilde{G}$ and so T is not a $\tilde{G}$Prešić operator.
Remark 2.7 Let $(X,d)$ be a metric space endowed with a graph G, k a positive integer and $T:{X}^{k}\to X$ be a GPrešić operator. If $(x,y)\in E(\tilde{G})$ then $d(T(x,x,\dots ,x),T(y,y,\dots ,y))<d(x,y)$.
Proof Suppose $(x,y)\in E(\tilde{G})=E(G)\cup E({G}^{1})$. If $(x,y)\in E(G)$ then since $E(G)\supseteq \mathrm{\Delta}$ so by (GP2) we have
Similarly, if $(x,y)\in E({G}^{1})$ we obtain the same result. □
Theorem 2.8 Let $(X,d)$ be a metric space endowed with a graph G and k be a positive integer. Suppose $T:{X}^{k}\to X$ be a GPrešić operator and ${P}_{T}^{k}\ne \mathrm{\varnothing}$. Then for every path ${\{{x}_{i}\}}_{i=1}^{k}$ in ${P}_{T}^{k}$, the PPsequence with initial values ${x}_{1},{x}_{2},\dots ,{x}_{k}$ is a Cauchy sequence.
Proof Let ${\{{x}_{i}\}}_{i=1}^{k}$ be a path in ${P}_{T}^{k}$, then by definition we have
Let $\{{x}_{n}\}$ be the PPsequence with initial values ${x}_{1},{x}_{2},\dots ,{x}_{k}$, that is,
Then by (2.1), (2.2) we have $({x}_{k},{x}_{k+1})\in E(G)$. Therefore by (2.1), (2.2) and (GP1) we have $({x}_{k+1},{x}_{k+2})\in E(G)$ and ${\{{x}_{i}\}}_{i=1}^{k+2}$ is a path of $k+2$ vertices in G. In a similar way, we find that, for any fix $n>1$, ${\{{x}_{i}\}}_{i=1}^{n}$ is a path of n vertices in G.
For notational convenience, let ${d}_{n}=d({x}_{n},{x}_{n+1})$ for all $n\in \mathbb{N}$ and
We shall show that
By the definition of μ, it is clear that (2.3) is true for $n=1,2,\dots ,k$. Now let the following k inequalities:
be the induction hypothesis.
Since ${\{{x}_{i}\}}_{i=1}^{n}$ is a path for all $n\in \mathbb{N}$ we obtain from (GP2)
and the inductive proof of (2.3) is complete. Now we shall show that the sequence $\{{x}_{n}\}$ is a Cauchy sequence. If $m,n\in \mathbb{N}$ with $m>n$ then by (2.3) we have
Since $\theta ={[{\sum}_{i=1}^{k}{q}_{i}]}^{1/k}<1$, it follows from the above inequality that ${lim}_{n,m\to \mathrm{\infty}}d({x}_{n},{x}_{m})=0$, that is, the sequence $\{{x}_{n}\}$ is a Cauchy sequence. □
Note that the above theorem cannot be considered as an existence theorem for GPrešić operator even when the space $(X,d)$ is complete. The following example verifies this fact.
Example 2.9 Let $X=[0,1]$, then $(X,d)$ is a complete metric space, where d is usual metric on X. Let the graph G be defined by
For $k=2$, define a mapping $T:{X}^{2}\to X$ by
Then it is easy to see that T is a GPrešić operator with ${q}_{1}={q}_{2}=1/3$. For any pair ${x}_{1},{x}_{2}\in (0,1]$ with ${x}_{2}\le {x}_{1}$, $\frac{{x}_{1}}{3}\le {x}_{2}$ we have $({x}_{1},{x}_{2}),({x}_{2},T({x}_{1},{x}_{2}))\in E(G)$, that is, ${\{{x}_{i}\}}_{i=1}^{2}\in {P}_{T}^{k}$ so ${P}_{T}^{k}\ne \mathrm{\varnothing}$. Thus, all the conditions of Theorem 2.8 are satisfied and $(X,d)$ is a complete metric space, but T has no fixed point.
The following is an existence theorem for a GPrešić operator and provides a sufficient condition for a GPrešić operator to be a WPPoperator.
Theorem 2.10 Let $(X,d)$ be a complete metric space endowed with a graph G and k be a positive integer. Suppose $T:{X}^{k}\to X$ be a GPrešić operator and ${P}_{T}^{k}\ne \mathrm{\varnothing}$. Then for every path ${\{{x}_{i}\}}_{i=1}^{k}$ in ${P}_{T}^{k}$, the PPsequence with initial values ${x}_{1},{x}_{2},\dots ,{x}_{k}$ is a Cauchy sequence. In addition, if $(X,d)$ has the property (P) then $T{}_{{P}_{T}^{k}}$ is a WPPoperator.
Proof From Theorem 2.8 it follows that for every path ${\{{x}_{i}\}}_{i=1}^{k}$ in ${P}_{T}^{k}$, the PPsequence with initial values ${x}_{1},{x}_{2},\dots ,{x}_{k}$ is a Cauchy sequence. Also, by following the arguments similar to those in Theorem 2.8 we have ${\{{x}_{i}\}}_{i=1}^{n}$ is a path in G for all $n\in \mathbb{N}$. Now, by completeness of X there exists $u\in X$ such that
We shall show that u is a fixed point of T. By the property (P) there exists a subsequence $\{{x}_{{n}_{p}}\}$ such that $({x}_{{n}_{p}},u)\in E(G)$ for all $p\in \mathbb{N}$. Therefore for any $p\in \mathbb{N}$ with ${n}_{p}>k$ by (GP2) we obtain
Since ${lim}_{n\to \mathrm{\infty}}{x}_{n}=u$, letting $p\to \mathrm{\infty}$ in the above inequality we obtain $d(u,T(u,u,\dots ,u))=0$, that is, $T(u,u,\dots ,u)=u$. Thus u is a fixed point of T. □
In the above theorem the fixed point of the mapping T may not be unique; moreover, T may not be a PPoperator and may have infinitely many fixed points as illustrated in the following example.
Example 2.11 Let $X=\mathbb{N}={\bigcup}_{k\in {\mathbb{N}}_{0}}{\mathcal{N}}_{k}$, where ${\mathbb{N}}_{0}=\mathbb{N}\cup \{0\}$ and ${\mathcal{N}}_{k}=\{{2}^{k}(2n1):n\in \mathbb{N}\}$ for all $k\in {\mathbb{N}}_{0}$. Let G be defined by
Then $(X,d)$ is a complete metric space where d is the usual metric on X. For $k=2$, define a mapping $T:{X}^{2}\to X$ by
Now it is easy to see that T is a GPrešić operator with ${q}_{1}={q}_{2}=1/4$. For all pairs ${x}_{1},{x}_{2}\in {\mathcal{N}}_{2k}$, $k\in \mathbb{N}$ we have $({x}_{1},{x}_{2}),({x}_{2},T({x}_{1},{x}_{2}))\in E(G)$, that is, ${\{{x}_{i}\}}_{i=1}^{2}\in {P}_{T}^{k}$ so ${P}_{T}^{k}\ne \mathrm{\varnothing}$. Also the property (P) is satisfied trivially in this case. Thus, all the conditions of Theorem 2.10 are satisfied and T has infinitely many fixed points, precisely $Fix(T)=\{{2}^{2k}:k\in {\mathbb{N}}_{0}\}$. Therefore T is not a PPoperator but WPPoperator on ${P}_{T}^{k}$.
In the next theorem a condition for the uniqueness of a fixed point of a GPrešić operator is provided.
Theorem 2.12 Let $(X,d)$ be a complete metric space endowed with a graph G, k be a positive integer and $T:{X}^{k}\to X$ be a GPrešić operator such that all the conditions of Theorem 2.10 are satisfied, then $T{}_{{P}_{T}^{k}}$ is a WPPoperator. In addition, if the subgraph ${G}_{F}^{T}$ is weakly connected, where $V({G}_{F}^{T})=Fix(T)$ and $E({G}_{F}^{T})\subseteq E(G)$, then $T{}_{{P}_{T}^{k}}$ is a PPoperator.
Proof The existence of a fixed point follows from Theorem 2.8. Suppose ${G}_{F}^{T}$ is weakly connected and $u,v\in Fix(T)$ with $u\ne v$. Since ${G}_{F}^{T}$ is weakly connected, there exists a path ${\{{x}_{i}\}}_{i=0}^{l}$ of $l+1$ vertices with ${x}_{0}=u$, ${x}_{l}=v$ and $({x}_{i},{x}_{i+1})\in E({\tilde{G}}_{F}^{T})$ for $0\le i\le l1$.
Note that T is also ${G}_{F}^{T}$Prešić operator and so by Remark 2.7 we have $u=v$. Thus, the fixed point of T is unique and $T{}_{{P}_{T}^{k}}$ is a PPoperator. □
Remark 2.13 In Example 2.11 the fixed point of the operator is not unique. Note that $Fix(T)=\{{2}^{2k}:k\in {\mathbb{N}}_{0}\}$ is not weakly connected. Indeed, $({2}^{{k}_{1}},{2}^{{k}_{2}})\notin E(G)$ for all ${k}_{1},{k}_{2}\in {\mathbb{N}}_{0}$ with ${k}_{1}\ne {k}_{2}$. Therefore, Example 2.11 shows that when considering the uniqueness, the additional condition ‘${G}_{F}^{T}$ is weakly connected’ in Theorem 2.12 cannot be relaxed.
Now we derive some results as a consequences of the above results. For this first we state some definitions about the Prešić type mappings which can be found in [7, 18].
Definition 2.14 [7]
Let a nonempty set X is equipped with a partial order ‘⊑’ such that $(X,d)$ is a metric space, then $(X,\u2291,d)$ is called an ordered metric space. A sequence $\{{x}_{n}\}$ in X is said to be nondecreasing with respect to ‘⊑’ if ${x}_{1}\u2291{x}_{2}\u2291\cdots \u2291{x}_{n}\u2291\cdots $ . Let k be a positive integer and $T:{X}^{k}\to X$ be a mapping, then T is said to be nondecreasing with respect to ‘⊑’ if for any finite nondecreasing sequence ${\{{x}_{i}\}}_{i=1}^{k+1}$ we have $T({x}_{1},{x}_{2},\dots ,{x}_{k})\u2291T({x}_{2},{x}_{3},\dots ,{x}_{k+1})$. T is said to be an ordered Prešić type contraction if:
(OP1) T is nondecreasing with respect to ‘⊑’;
(OP2) there exist nonnegative real numbers ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{k}$ such that ${\sum}_{i=1}^{k}{\alpha}_{i}<1$ and
for all ${x}_{1},{x}_{2},\dots ,{x}_{k+1}\in X$ with ${x}_{1}\u2291{x}_{2}\u2291\cdots \u2291{x}_{k+1}$.
Definition 2.15 [18]
Let X be any nonempty set, k a positive integer, $T:{X}^{k}\to X$ an operator and ${A}_{1},{A}_{2},\dots ,{A}_{m}$ be subsets of X. Then $X={\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of X with respect to T if:

(1)
${A}_{i}$, $i=1,2,\dots ,m$ are nonempty sets;

(2)
$T({A}_{1}\times {A}_{2}\times \cdots \times {A}_{k})\subset {A}_{k+1}$, $T({A}_{2}\times {A}_{3}\times \cdots \times {A}_{k+1})\subset {A}_{k+2}$, … , $T({A}_{i}\times {A}_{i+1}\times \cdots \times {A}_{i+k1})\subset {A}_{i+k}$, … , where ${A}_{m+j}={A}_{j}$ for all $j\in \mathbb{N}$.
$T:{Y}^{k}\to Y$ is called a cyclicPrešić operator if the following conditions are met:
(CP1) $Y={\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of Y with respect to T;
(CP2) there exist nonnegative real numbers ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{k}$ such that ${\sum}_{i=1}^{k}{\alpha}_{i}<1$ and
for all ${x}_{1}\in {A}_{i},{x}_{2}\in {A}_{i+1},\dots ,{x}_{k+1}\in {A}_{i+k}$ ($i=1,2,\dots ,m$ where ${A}_{m+j}={A}_{j}$ for all $j\in \mathbb{N}$).
Now we give some consequences of our main results.
The following corollary is the fixed point result for the ordered Prešić type contractions (for details, see [7]).
Corollary 2.16 Let $(X,\u2291,d)$ be an ordered complete metric space. Let $T:{X}^{k}\to X$ be a mapping such that the following conditions hold:

(A)
T is an ordered Prešić type contraction;

(B)
there exist ${x}_{1},{x}_{2},\dots ,{x}_{k}\in X$ such that ${x}_{1}\u2291{x}_{2}\u2291\cdots \u2291{x}_{k}\u2291T({x}_{1},{x}_{2},\dots ,{x}_{k})$;

(C)
if a nondecreasing sequence $\{{x}_{n}\}$ converges to $x\in X$, then ${x}_{n}\u2291x$ for all $n\in \mathbb{N}$.
Then T has a fixed point $u\in X$. In addition, $Fix(T)$ is wellordered if and only if the fixed point of T is unique.
Proof Define a graph G by $V(G)=X$ and
Then (OP1) implies (GP1) and (OP2) implies (GP2) therefore T is a GPrešić operator. By condition (B) it follows that ${P}_{T}^{k}\ne \mathrm{\varnothing}$ and the path ${\{{x}_{i}\}}_{i=1}^{k}\in {P}_{T}^{k}$. Condition (C) insures that $(X,d)$ has the property (P) and wellorderedness of $Fix(T)$ implies that ${G}_{F}^{T}$ is weakly connected. By Theorem 2.12, T has a unique fixed point. □
Corollary 2.17 Let ${A}_{1},{A}_{2},\dots ,{A}_{m}$ be closed subsets of a complete metric space $(X,d)$, k a positive integer, and $Y={\bigcup}_{i=1}^{m}{A}_{i}$. Let $T:{Y}^{k}\to Y$ be a cyclicPrešić operator, then T has a fixed point $u\in {\bigcap}_{i=1}^{m}{A}_{i}$. In addition, fixed point of T is unique if and only if $Fix(T)\subset {\bigcap}_{i=1}^{m}{A}_{i}$.
Proof Define a graph G by $V(G)=X$ and
where ${A}_{m+j}={A}_{j}$ for all $j\in \mathbb{N}$. Note that Y is a complete subspace of X. Since $Y={\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of Y with respect to T, therefore (GP1) holds and (CP2) implies (GP2) and so T is a GPrešić operator. As ${A}_{i}$, $1\le i\le k$ are nonempty, therefore ${P}_{T}^{k}\ne \mathrm{\varnothing}$. Proposition 2.1 of [18] shows that the subspace $(Y,d)$ has the property (P). Finally, note that if $Fix(T)\subset {\bigcap}_{i=1}^{m}{A}_{i}$ then ${G}_{F}^{T}$ is weakly connected, therefore proof follows from Theorem 2.12. □
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support. The authors would like to thank the reviewers for their valuable suggestions.
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Keywords
 graph
 fixed point
 Prešić type mapping