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Fixed point of setvalued graph contractive mappings
Journal of Inequalities and Applications volume 2013, Article number: 252 (2013)
Abstract
Let (X,d) be a metric space and let F, H be two setvalued 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 setvalued 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 setvalued mappings with a general contractive condition. The aim of this paper is to study the existence of common fixed points for setvalued 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
where
Mapping D is said to be a Hausdorff metric induced by d.
Definition 1.1 Let F:X\to X be a setvalued 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 setvalued mapping F if x\in Fx.
Definition 1.2 A metric space (X,d) is called a εchainable 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 εchainable 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 setvalued 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 setvalued mappings.
Theorem 1.6 [2]
Let (X,d) be a εchainable complete metric space for some \epsilon >0 and let F:X\to C(X) be a setvalued 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].
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
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}_{i1},{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
and if u\in Fx and v\in Hy are 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.
Let ⪯ be a partial order in X. Define the graph G:={G}_{1} by
and G:={G}_{2} 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}_{i1} 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 setvalued 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
Using Lemma 1.13, we have the existence of {x}_{2}\in H{x}_{1} such that
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
and Lemma 1.13 gives the existence of {x}_{3}\in F{x}_{2} such that
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
Next we show that ({x}_{n}) is a Cauchy sequence in X. Let m>n. Then
because \kappa \in (0,1), 1{\kappa}^{mn}<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.
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 13 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 13. 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 εchainable 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
The εchainability 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 14 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 singlevalued, then we improve the Banach contraction theorem.

2.
If F is a singlevalued mapping, then Theorem 2.5(2, 5) with the graph {G}_{1} improves [[19], Theorem 2.2].

3.
If F is a singlevalued mapping, then Theorem 2.5(2, 5) with the graph {G}_{2} improves [[20], Theorem 2.1].

4.
If F=H is a singlevalued mapping, then Theorem 2.1 and Theorem 2.5 partially generalize [[22], Theorem 3.2].

5.
If we take F=H as singlevalued 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].
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IB gave the idea. ARB wrote the initial draft. IB and ARB finalized the manuscript. All authors read and approved the final manuscript. Correspondence was mainly done by IB.
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Beg, I., Butt, A.R. Fixed point of setvalued graph contractive mappings. J Inequal Appl 2013, 252 (2013). https://doi.org/10.1186/1029242X2013252
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DOI: https://doi.org/10.1186/1029242X2013252
Keywords
 fixed point
 directed graph
 metric space
 setvalued mapping