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Fixed point of set-valued graph contractive mappings
Journal of Inequalities and Applications volume 2013, Article number: 252 (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.
Let be a complete metric space and let be a class of all nonempty closed and bounded subsets of X. For , let
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 .
Definition 1.2 A metric space is called a ε-chainable metric space for some if given , there is and a sequence such that
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 
Let be a complete metric space and . If there exists such that
then F has a fixed point in X.
Nadler  also extended Edelstein’s theorem for set-valued mappings.
Theorem 1.6 
Let be a ε-chainable complete metric space for some and let 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., ) 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 .
We can identify G as . denotes the conversion of a graph G, the graph obtained from G by reversing the direction of its edges. denotes the undirected graph obtained from G by ignoring the direction of edges of G. We consider 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 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 .
Definition 1.11 Let be a metric space and . The mappings F, H are said to be graph contractive if there exists such that
and if and are such that
Definition 1.12 A partial order is a binary relation ⪯ over a set X which satisfies the following conditions:
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 .
Let ⪯ be a partial order in X. Define the graph by
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.
Theorem 2.1 Let be graph contractive mappings and let the triple have the property A. Set . Then the following statements hold.
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.
Proof 1. Let , then there exists such that . Since F, H are graph contractive mappings, we have
Using Lemma 1.13, we have the existence of such that
Again, because F, H are graph contractive , also , since is symmetric, we have
and Lemma 1.13 gives the existence of such that
Continuing in this way, we have and , . Also, such that
Next we show that is a Cauchy sequence in X. Let . Then
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 .
For n even: By property A, we have . Therefore, by using graph contractivity, we have
Since and , therefore by Lemma 1.14, .
For n odd: As ,
Now, by following the same arguments as above, .
Next as , also for . We infer that is a path in G and so .
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.
Corollary 2.4 Let be a ε-chainable complete metric space for some . Let be such that there exists with
Then F and H have a common fixed point.
Proof Consider the graph G as and
The ε-chainability of means G is connected. If , then
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. □
Theorem 2.5 Let be a graph contractive mapping and let the triple have the property A. Set . Then the following statements hold.
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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The authors declare that they have no competing interests.
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 set-valued graph contractive mappings. J Inequal Appl 2013, 252 (2013). https://doi.org/10.1186/1029-242X-2013-252
- fixed point
- directed graph
- metric space
- set-valued mapping