Fixed point of set-valued graph contractive mappings
© Beg and Butt; licensee Springer 2013
Received: 26 November 2012
Accepted: 30 April 2013
Published: 17 May 2013
Let 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, and the set of edges of G, .
MSC:47H10, 47H04, 47H07, 54C60, 54H25.
1 Introduction and preliminaries
Edelstein  generalized classical Banach’s contraction mapping principle and Nadler  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  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.
Mapping D is said to be a Hausdorff metric induced by d.
Definition 1.1 Let be a set-valued mapping, i.e., is a subset of X. A point is said to be a fixed point of the set-valued mapping F if .
Let denote the set of fixed points of the mapping F.
Definition 1.3 Let be a metric space, , and . A mapping is called uniformly locally contractive if .
Theorem 1.4 
Let be a ε-chainable complete metric space. If is a uniformly locally contractive mapping, then f has a unique fixed point.
Afterwards, in 1969, Nadler  proved a set-valued extension of Banach’s theorem and obtained the following result.
Theorem 1.5 
then F has a fixed point in X.
Nadler  also extended Edelstein’s theorem for set-valued mappings.
Theorem 1.6 
then F has a fixed point.
Consider a directed graph G such that the set of its vertices coincides with X (i.e., ) and the set of its edges . 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 .
Definition 1.7 A subgraph of a graph G is a graph H such that and and for any edge , .
Definition 1.8 Let x and y be vertices in a graph G. A path in G from x to y of length n () is a sequence of vertices such that , and for .
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 is connected.
Assume that G is such that is symmetric, and x is a vertex in G, then the subgraph 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 defined on by the rule R ( if there is a path from u to v) is such that .
Property A: For any sequence in X, if and for , then .
if and , then (antisymmetry);
if and , then (transitivity);
for all x, y and z in X.
A set with a partial order ⪯ is called a partially ordered set.
Let be a partially ordered set and . Elements x and y are said to be comparable elements of X if either or .
The weak connectivity of or means, given , there is a sequence such that , and for all , and are comparable.
Lemma 1.13 If with , then for each there exists an element such that .
Lemma 1.14 Let be a sequence in and for . If and , then .
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 is a complete metric space and G is a directed graph such that is symmetric.
For any , F, have a common fixed point.
If and G is weakly connected, then F, H have a common fixed point in X.
If , then F, have a common fixed point.
If , then F, H have a common fixed point.
because , .
Therefore as implies that is a Cauchy sequence and hence converges to some point (say) x in the complete metric space X.
Now we have to show that .
Since and , therefore by Lemma 1.14, .
Now, by following the same arguments as above, .
Since , so there exists , and since G is weakly connected, therefore , and by 1, mappings F and H have a common fixed point in X.
It follows easily from 1 and 2.
implies that all are such that there exists some with so and by 2 and 3. F, H have a fixed point. □
Remark 2.2 Replace by in conditions 1-3 of Theorem 2.1, then the conclusion remains true. That is, if , then we have , which follows easily from 1-3. Similarly, in condition 4, we can replace by .
Corollary 2.3 is a direct consequence of Theorem 2.1(1).
Corollary 2.3 Let be a complete metric space and let the triple have the property A. If G is weakly connected, then graph contractive mappings such that for some have a common fixed point.
Then F and H have a common fixed point.
and by using Lemma 1.13, for each , we have the existence of such that , which implies . Hence F and H are graph contractive mappings. Also, has property A. Indeed, if and for , then for sufficiently large n, therefore . So, by Theorem 2.1(2), F and H have a common fixed point. □
For any , has a fixed point.
If and G is weakly connected, then F has a fixed point in X.
If , then has a fixed point.
If , then F has a fixed point.
If , then .
Proof Statements 1-4 can be proved by taking in Theorem 2.1 and 5 obtained from Remark 2.2.
Note that the assumption that is symmetric is not needed in our Theorem 2.5. □
If we assume G is such that , 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.
If F is a single-valued mapping, then Theorem 2.5(2, 5) with the graph improves [, Theorem 2.2].
If F is a single-valued mapping, then Theorem 2.5(2, 5) with the graph improves [, Theorem 2.1].
If is a single-valued mapping, then Theorem 2.1 and Theorem 2.5 partially generalize [, Theorem 3.2].
If we take as single-valued mappings in Corollary 2.4, then we have [, Theorem 5.2].
If we take , then Corollary 2.4 becomes Theorem 1.5 due to .
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