Fixed point of set-valued graph contractive mappings

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.

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.

Let $(X,d)$ be a complete metric space and let $CB(X)$ be a class of all nonempty closed and bounded subsets of X. For $A,B\in CB(X)$, let

$$D(A,B):=max\{\underset{b\in B}{sup}d(b,A),\underset{a\in A}{sup}d(a,B)\},$$

where

$$d(a,B):=\underset{b\in B}{inf}d(a,b).$$

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

$$x_0=x,x_n=y\text{and}d(x_{i-1},x_i)<\epsilon \text{for }i=1,\dots ,n.$$

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

$$D(Fx,Fy)\le \kappa d(x,y)\mathit{\text{for all}}x,y\in X,$$

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:

$$x,y\in X\mathit{\text{and}}0<d(x,y)<\epsilon \Rightarrow D(Fx,Fy)<\kappa d(x,y),$$

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].

We can identify 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

$$E(\tilde{G}):=E(G)\cup E\left(G^{-1}\right).$$

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

$$(x\ne y),(x,y)\in E(G)\Rightarrow D(Fx,Hy)<\kappa d(x,y),$$

and if $u\in Fx$ and $v\in Hy$ are such that

$$d(u,v)<d(x,y),$$

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. $x⪯x$ (reflexivity);

• 2. if $x⪯y$ and $y⪯x$, then $x=y$ (antisymmetry);

• 3. if $x⪯y$ and $y⪯z$, then $x⪯z$ (transitivity);

for all x, y and z in X.

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

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

Let ⪯ be a partial order in X. Define the graph $G:=G_1$ by

$$E(G_1):=\{(x,y)\in X\times X:x⪯y,x\ne y\},$$

and $G:=G_2$ by

$$E(G_2):=\{(x,y)\in X\times X:x⪯y\vee y⪯x,x\ne y\}.$$

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 $D(A,B)<ϵ$, then for each $a\in A$ there exists an element $b\in B$ such that $d(a,b)<ϵ$.

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

$$D(F{x}_0,H{x}_1)<\kappa d(x_0,x_1).$$

Using Lemma 1.13, we have the existence of $x_2\in H{x}_1$ such that

$$d(x_1,x_2)<\kappa d(x_0,x_1).$$

(1)

Again, because 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

$$D(F{x}_2,H{x}_1)<\kappa d(x_1,x_2)<{\kappa }^{2}d(x_0,x_1),$$

and Lemma 1.13 gives the existence of $x_3\in F{x}_2$ such that

$$d(x_2,x_3)<{\kappa }^{2}d(x_0,x_1).$$

(2)

Continuing in this way, we have $x_{2n+1}\in F{x}_{2n}$ and $x_{2n+2}\in H{x}_{2n+1}$, $n=0,1,2,\dots$ . Also, $(x_n,x_{n+1})\in E(G)$ such that

$$d(x_n,x_{n+1})<{\kappa }^{n}d(x_0,x_1).$$

(3)

Next we show that $(x_n)$ is a Cauchy sequence in X. Let $m>n$. Then

$$\begin{array}{rcl}d(x_n,x_m)& \le & d(x_n,x_{n+1})+d(x_{n+1},x_{n+2})+d(x_{n+2},x_{n+3})+\cdots +d(x_{m-1},x_m)\\ <& [{\kappa }^n+{\kappa }^{n+1}+{\kappa }^{n+2}+\cdots +{\kappa }^{m-1}]d(x_0,x_1)\\ =& {\kappa }^n[1+\kappa +{\kappa }^2+\cdots +{\kappa }^{m-n-1}]d(x_0,x_1)\\ =& {\kappa }^n\left[\frac{1-{\kappa }^{m-n}}{1-\kappa }\right]d(x_0,x_1)\end{array}$$

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

$$D(F{x}_n,Hx)<\kappa d(x_n,x).$$

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 $(x,x_n)\in E(G)$,

$$D(Fx,H{x}_n)<\kappa d(x,x_n).$$

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

Next as $(x_n,x_{n+1})\in E(G)$, also $(x_n,x)\in E(G)$ for $n\in N$. We infer that $(x_0,x_1,\dots ,x_n,x)$ is a path in 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

$$0<d(x,y)<\epsilon \Rightarrow D(Fx,Hx)<\kappa d(x,y).$$

Then F and H have a common fixed point.

Proof Consider the graph G as $V(G):=X$ and

$$E(G):=\{(x,y)\in X\times X:0<d(x,y)<\epsilon \}.$$

(4)

The ε-chainability of $(X,d)$ means G is connected. If $(x,y)\in E(G)$, then

$$D(Fx,Hy)<\kappa d(x,y)<\kappa \epsilon <\epsilon$$

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].