Abstract
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.
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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 be a complete metric space and let be a class of all nonempty closed and bounded subsets of X. For , let
where
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 .
The following significant generalization of Banach’s contraction principle [[17], Theorem 2.1 ] was obtained by Edelstein [1].
Theorem 1.4 [1]
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 [2] proved a set-valued extension of Banach’s theorem and obtained the following result.
Theorem 1.5 [2]
Let be a complete metric space and . If there exists 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 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 [18].
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
then .
Definition 1.12 A partial order is a binary relation ⪯ over a set X which satisfies the following conditions:
-
1.
(reflexivity);
-
2.
if and , then (antisymmetry);
-
3.
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
and by
The class of -contractive mappings was considered in [19] and that of -contractive mappings in [20].
The weak connectivity of or means, given , there is a sequence such that , and for all , and 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 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.
-
1.
For any , F, have a common fixed point.
-
2.
If and G is weakly connected, then F, H have a common fixed point in X.
-
3.
If , then F, have a common fixed point.
-
4.
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 .
-
2.
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.
-
3.
It follows easily from 1 and 2.
-
4.
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.
-
1.
For any , has a fixed point.
-
2.
If and G is weakly connected, then F has a fixed point in X.
-
3.
If , then has a fixed point.
-
4.
If , then F has a fixed point.
-
5.
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. □
Remark 2.6
-
1.
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.
-
2.
If F is a single-valued mapping, then Theorem 2.5(2, 5) with the graph improves [[19], Theorem 2.2].
-
3.
If F is a single-valued mapping, then Theorem 2.5(2, 5) with the graph improves [[20], Theorem 2.1].
-
4.
If is a single-valued mapping, then Theorem 2.1 and Theorem 2.5 partially generalize [[22], Theorem 3.2].
-
5.
If we take as single-valued mappings in Corollary 2.4, then we have [[1], Theorem 5.2].
-
6.
If we take , 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 set-valued graph contractive mappings. J Inequal Appl 2013, 252 (2013). https://doi.org/10.1186/1029-242X-2013-252
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DOI: https://doi.org/10.1186/1029-242X-2013-252