# Fixed point of set-valued graph contractive mappings

- Ismat Beg
^{1}Email author and - Asma Rashid Butt
^{2}

**2013**:252

https://doi.org/10.1186/1029-242X-2013-252

© Beg and Butt; licensee Springer 2013

**Received: **26 November 2012

**Accepted: **30 April 2013

**Published: **17 May 2013

## Abstract

Let $(X,d)$ be a metric space and let *F*, *H* be two set-valued mappings on *X*. We obtained sufficient conditions for the existence of a common fixed point of the mappings *F*, *H* in the metric space *X* endowed with a graph *G* such that the set of vertices of *G*, $V(G)=X$ and the set of edges of *G*, $E(G)\subseteq X\times X$.

**MSC:**47H10, 47H04, 47H07, 54C60, 54H25.

## Keywords

## 1 Introduction and preliminaries

Edelstein [1] generalized classical Banach’s contraction mapping principle and Nadler [2] proved Banach’s fixed point theorem for set-valued mappings. Recently several extensions of Nadler’s theorem in different directions were obtained; see [3–15]. Beg and Azam [5] extended Edelstein’s theorem by considering a pair of set-valued mappings with a general contractive condition. The aim of this paper is to study the existence of common fixed points for set-valued graph contractive mappings in metric spaces endowed with a graph *G*. Our results improve/generalize [1, 2, 16] and several other known results in the literature.

*X*. For $A,B\in CB(X)$, let

Mapping *D* is said to be a *Hausdorff metric* induced by *d*.

**Definition 1.1** Let $F:X\to X$ be a set-valued mapping, *i.e.*, $X\ni x\mapsto Fx$ is a subset of *X*. A point $x\in X$ is said to be a *fixed point* of the set-valued mapping *F* if $x\in Fx$.

**Definition 1.2**A metric space $(X,d)$ is called a

*ε*-chainable metric space for some $\epsilon >0$ if given $x,y\in X$, there is $n\in N$ and a sequence ${({x}_{i})}_{i=0}^{n}$ such that

Let $FixF:=\{x\in X:x\in Fx\}$ denote the set of fixed points of the mapping *F*.

**Definition 1.3** Let $(X,d)$ be a metric space, $\epsilon >0$, $0\le \kappa <1$ and $x,y\in X$. A mapping $f:X\to X$ is called $(\epsilon ,\kappa )$ uniformly locally contractive if $0<d(x,y)<\epsilon \Rightarrow d(fx,fy)<\kappa d(x,y)$.

The following significant generalization of Banach’s contraction principle [[17], Theorem 2.1 ] was obtained by Edelstein [1].

**Theorem 1.4** [1]

*Let* $(X,d)$ *be a* *ε*-*chainable complete metric space*. *If* $f:X\to X$ *is a* $(\epsilon ,\kappa )$ *uniformly locally contractive mapping*, *then* *f* *has a unique fixed point*.

Afterwards, in 1969, Nadler [2] proved a set-valued extension of Banach’s theorem and obtained the following result.

**Theorem 1.5** [2]

*Let*$(X,d)$

*be a complete metric space and*$F:X\to CB(X)$.

*If there exists*$\kappa \in (0,1)$

*such that*

*then* *F* *has a fixed point in* *X*.

Nadler [2] also extended Edelstein’s theorem for set-valued mappings.

**Theorem 1.6** [2]

*Let*$(X,d)$

*be a*

*ε*-

*chainable complete metric space for some*$\epsilon >0$

*and let*$F:X\to C(X)$

*be a set*-

*valued mapping such that*

*Fx*

*is a nonempty compact subset of*

*X*.

*If*

*F*

*satisfies the following condition*:

*then* *F* *has a fixed point*.

Consider a directed graph *G* such that the set of its vertices coincides with *X* (*i.e.*, $V(G):=X$) and the set of its edges $E(G):=\{(x,y):(x,y)\in X\times X,x\ne y\}$. We assume that *G* has no parallel edges and weighted graph by assigning to each edge the distance between the vertices; for details about definitions in graph theory, see [18].

*G*as $(V(G),E(G))$. ${G}^{-1}$ denotes the conversion of a graph

*G*, the graph obtained from

*G*by reversing the direction of its edges. $\tilde{G}$ denotes the undirected graph obtained from

*G*by ignoring the direction of edges of

*G*. We consider $\tilde{G}$ as a directed graph for which the set if its edges is symmetric, thus we have

**Definition 1.7** A *subgraph* of a graph *G* is a graph *H* such that $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$ and for any edge $(x,y)\in E(H)$, $x,y\in V(H)$.

**Definition 1.8** Let *x* and *y* be vertices in a graph *G*. A *path* in *G* from *x* to *y* of length *n* ($n\in N\cup \{0\}$) is a sequence ${({x}_{i})}_{i=0}^{n}$ of $n+1$ vertices such that ${x}_{0}=x$, ${x}_{n}=y$ and $({x}_{i-1},{x}_{i})\in E(G)$ for $i=1,2,\dots ,n$.

**Definition 1.9** The number of edges in *G* constituting the path is called the *length of the path*.

**Definition 1.10** A graph *G* is *connected* if there is a path between any two vertices of *G*.

If a graph *G* is not connected, then it is called *disconnected*. Moreover, *G* is weakly connected if $\tilde{G}$ is connected.

Assume that *G* is such that $E(G)$ is symmetric, and *x* is a vertex in *G*, then the subgraph ${G}_{x}$ consisting of all edges and vertices, which are contained in some path in *G* beginning at *x*, is called the component of *G* containing *x*. In this case the equivalence class ${[x]}_{G}$ defined on $V(G)$ by the rule *R* ($uRv$ if there is a path from *u* to *v*) is such that $V({G}_{x})={[x]}_{G}$.

*Property* A: For any sequence ${({x}_{n})}_{n\in N}$ in *X*, if ${x}_{n}\to x$ and $({x}_{n},{x}_{n+1})\in E(G)$ for $n\in N$, then $({x}_{n},x)\in E(G)$.

**Definition 1.11**Let $(X,d)$ be a metric space and $F,H:X\to CB(X)$. The mappings

*F*,

*H*are said to be graph contractive if there exists $\kappa \in (0,1)$ such that

then $(u,v)\in E(G)$.

**Definition 1.12**A

*partial order*is a binary relation ⪯ over a set

*X*which satisfies the following conditions:

- 1.
$x\u2aafx$ (reflexivity);

- 2.
if $x\u2aafy$ and $y\u2aafx$, then $x=y$ (antisymmetry);

- 3.
if $x\u2aafy$ and $y\u2aafz$, then $x\u2aafz$ (transitivity);

for all *x*, *y* and *z* in *X*.

A set with a partial order ⪯ is called a *partially ordered set*.

Let $(X,\u2aaf)$ be a partially ordered set and $x,y\in X$. Elements *x* and *y* are said to be *comparable elements* of *X* if either $x\u2aafy$ or $y\u2aafx$.

*X*. Define the graph $G:={G}_{1}$ by

The class of ${G}_{1}$-contractive mappings was considered in [19] and that of ${G}_{2}$-contractive mappings in [20].

The weak connectivity of ${G}_{1}$ or ${G}_{2}$ means, given $x,y\in X$, there is a sequence ${({x}_{i})}_{i=0}^{n}$ such that ${x}_{0}=x$, ${x}_{n}=y$ and for all $i=1,\dots ,n$, ${x}_{i-1}$ and ${x}_{i}$ are comparable.

We shall make use of the following lemmas due to Nadler [2], Assad and Kirk [21] in the proof of our results in next section.

**Lemma 1.13** *If* $A,B\in CB(X)$ *with* $D(A,B)<\u03f5$, *then for each* $a\in A$ *there exists an element* $b\in B$ *such that* $d(a,b)<\u03f5$.

**Lemma 1.14** *Let* $\{{A}_{n}\}$ *be a sequence in* $CB(X)$ *and* ${lim}_{n\to \mathrm{\infty}}D({A}_{n},A)=0$ *for* $A\in CB(X)$. *If* ${x}_{n}\in {A}_{n}$ *and* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},x)=0$, *then* $x\in A$.

## 2 Common fixed point

We begin with the following theorem that gives the existence of a common fixed point (not necessarily unique) in metric spaces endowed with a graph for the set-valued mappings. Further, we assume that $(X,d)$ is a complete metric space and *G* is a directed graph such that $E(G)$ is symmetric.

**Theorem 2.1**

*Let*$F,H:X\to CB(X)$

*be graph contractive mappings and let the triple*$(X,d,G)$

*have the property*A.

*Set*${X}_{F}:=\{x\in X:(x,u)\in E(G)\mathit{\text{for some}}u\in Fx\}$.

*Then the following statements hold*.

- 1.
*For any*$x\in {X}_{F}$,*F*, $H{|}_{{[x]}_{G}}$*have a common fixed point*. - 2.
*If*${X}_{F}\ne \mathrm{\varnothing}$*and**G**is weakly connected*,*then**F*,*H**have a common fixed point in**X*. - 3.
*If*${X}^{\mathrm{\prime}}:=\bigcup \{{[x]}_{G}:x\in {X}_{F}\}$,*then**F*, $H{|}_{{X}^{\mathrm{\prime}}}$*have a common fixed point*. - 4.
*If*$F\subseteq E(G)$,*then**F*,*H**have a common fixed point*.

*Proof*1. Let ${x}_{0}\in {X}_{F}$, then there exists ${x}_{1}\in F{x}_{0}$ such that $({x}_{0},{x}_{1})\in E(G)$. Since

*F*,

*H*are graph contractive mappings, we have

*F*,

*H*are graph contractive $({x}_{1},{x}_{2})\in E(G)$, also $({x}_{2},{x}_{1})\in E(G)$, since $E(G)$ is symmetric, we have

*X*. Let $m>n$. Then

because $\kappa \in (0,1)$, $1-{\kappa}^{m-n}<1$.

Therefore $d({x}_{n},{x}_{m})\to 0$ as $n\to \mathrm{\infty}$ implies that $({x}_{n})$ is a Cauchy sequence and hence converges to some point (say) *x* in the complete metric space *X*.

Now we have to show that $x\in Fx\cap Hx$.

*For*

*n*

*even*: By property A, we have $({x}_{n},x)\in E(G)$. Therefore, by using graph contractivity, we have

Since ${x}_{n+1}\in F{x}_{n}$ and ${x}_{n}\to x$, therefore by Lemma 1.14, $x\in Hx$.

*For*

*n*

*odd*: As $(x,{x}_{n})\in E(G)$,

Now, by following the same arguments as above, $x\in Fx$.

*G*and so $x\in {[{x}_{0}]}_{G}$.

- 2.
Since ${X}_{F}\ne \mathrm{\varnothing}$, so there exists ${x}_{0}\in {X}_{F}$, and since

*G*is weakly connected, therefore ${[{x}_{0}]}_{G}=X$, and by 1, mappings*F*and*H*have a common fixed point in*X*. - 3.
It follows easily from 1 and 2.

- 4.
$F\subseteq E(G)$ implies that all $x\in X$ are such that there exists some $u\in Fx$ with $(x,u)\in E(G)$ so ${X}_{F}=X$ and by 2 and 3.

*F*,*H*have a fixed point. □

**Remark 2.2** Replace ${X}_{F}$ by ${X}_{H}:=\{x\in X:(x,u)\in E(G)\text{for some}u\in Hx\}$ in conditions 1-3 of Theorem 2.1, then the conclusion remains true. That is, if ${X}_{F}\cup {X}_{H}\ne \mathrm{\varnothing}$, then we have $FixF\cap FixH\ne \mathrm{\varnothing}$, which follows easily from 1-3. Similarly, in condition 4, we can replace $F\subseteq E(G)$ by $H\subseteq E(G)$.

Corollary 2.3 is a direct consequence of Theorem 2.1(1).

**Corollary 2.3** *Let* $(X,d)$ *be a complete metric space and let the triple* $(X,d,G)$ *have the* property A. *If* *G* *is weakly connected*, *then graph contractive mappings* $F,H:X\to CB(X)$ *such that* $({x}_{0},{x}_{1})\in E(G)$ *for some* ${x}_{1}\in F{x}_{0}$ *have a common fixed point*.

**Corollary 2.4**

*Let*$(X,d)$

*be a*

*ε*-

*chainable complete metric space for some*$\epsilon >0$.

*Let*$F,H:X\to CB(X)$

*be such that there exists*$\kappa \in (0,1)$

*with*

*Then* *F* *and* *H* *have a common fixed point*.

*Proof*Consider the graph

*G*as $V(G):=X$ and

*ε*-chainability of $(X,d)$ means

*G*is connected. If $(x,y)\in E(G)$, then

and by using Lemma 1.13, for each $u\in Fx$, we have the existence of $v\in Hy$ such that $d(u,v)<\epsilon $, which implies $(u,v)\in E(G)$. Hence *F* and *H* are graph contractive mappings. Also, $(X,d,G)$ has *property* A. Indeed, if ${x}_{n}\to x$ and $d({x}_{n},{x}_{n+1})<\epsilon $ for $n\in N$, then $d({x}_{n},x)<\epsilon $ for sufficiently large n, therefore $({x}_{n},x)\in E(G)$. So, by Theorem 2.1(2), *F* and *H* have a common fixed point. □

**Theorem 2.5**

*Let*$F:X\to CB(X)$

*be a graph contractive mapping and let the triple*$(X,d,G)$

*have the property*A.

*Set*${X}_{F}:=\{x\in X:(x,u)\in E(G)\mathit{\text{for some}}u\in Fx\}$.

*Then the following statements hold*.

- 1.
*For any*$x\in {X}_{F}$, $F{|}_{{[x]}_{G}}$*has a fixed point*. - 2.
*If*${X}_{F}\ne \mathrm{\varnothing}$*and**G**is weakly connected*,*then**F**has a fixed point in**X*. - 3.
*If*${X}^{\mathrm{\prime}}:=\bigcup \{{[x]}_{G}:x\in {X}_{F}\}$,*then*$F{|}_{{X}^{\mathrm{\prime}}}$*has a fixed point*. - 4.
*If*$F\subseteq E(G)$,*then**F**has a fixed point*. - 5.
*If*${X}_{F}\ne \mathrm{\varnothing}$,*then*$FixF\ne \mathrm{\varnothing}$.

*Proof* Statements 1-4 can be proved by taking $F=H$ in Theorem 2.1 and 5 obtained from Remark 2.2.

Note that the assumption that $E(G)$ is symmetric is not needed in our Theorem 2.5. □

**Remark 2.6**

- 1.
If we assume

*G*is such that $E(G):=X\times X$, then clearly*G*is connected and our Theorem 2.5(2) improves Nadler’s theorem, and further if*F*is single-valued, then we improve the Banach contraction theorem. - 2.
If

*F*is a single-valued mapping, then Theorem 2.5(2, 5) with the graph ${G}_{1}$ improves [[19], Theorem 2.2]. - 3.
If

*F*is a single-valued mapping, then Theorem 2.5(2, 5) with the graph ${G}_{2}$ improves [[20], Theorem 2.1]. - 4.
If $F=H$ is a single-valued mapping, then Theorem 2.1 and Theorem 2.5 partially generalize [[22], Theorem 3.2].

- 5.
If we take $F=H$ as single-valued mappings in Corollary 2.4, then we have [[1], Theorem 5.2].

- 6.
If we take $F=H$, then Corollary 2.4 becomes Theorem 1.5 due to [2].

## Declarations

## Authors’ Affiliations

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