Abstract
In this paper, we introduce the concepts of graph-preserving multi-valued mapping and a new type of multi-valued weak G-contraction on a metric space endowed with a directed graph G. We prove some coincidence point theorems for this type of multi-valued mapping and a surjective mapping under some conditions. Several examples for these new concepts and some examples satisfying all conditions of our main results are also given. Our main results extend and generalize many coincidence point and fixed point theorems in partially ordered metric spaces.
MSC:47H04, 47H10.
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1 Introduction
Fixed point theory of multi-valued mappings plays an important role in science and applied science. It has applications in control theory, convex optimization, differential inclusions and economics.
For a metric space , we let and be the set of all nonempty closed bounded subsets of X and the set of all nonempty compact subsets of X, respectively. A point is a fixed point a multi-valued mapping if . The first well-known theorem for multi-valued contraction mappings was given by Nadler in 1969 [1].
Theorem 1.1 Let be a complete metric space and let T be a mapping from X into . Assume that there exists such that
Then there exists such that .
Nadler’s fixed point theorem for multi-valued contractive mappings has been extended in many directions (see [2–6]). Reich [7] proved the following fixed point theorem for multi-valued φ contraction mappings.
Theorem 1.2 Let be a complete metric space and let T be a mapping from X into . Assume that there exists a function such that
and
Then there exists such that .
The multi-valued mapping T considered by Reich [7] in Theorem 1.2 has compact values, that is, Tx is a nonempty compact subset of X for all . In 1988, Mizoguchi and Takahashi [8] relaxed the compactness assumption on T to closed and bounded subsets of X. They proved the following theorem which is a generalization of Nadler’s theorem.
Theorem 1.3 Let be a complete metric space and let . Assume that there exists a function such that
and
Then there exists such that .
In 2007, Berinde and Berinde [3] extended Theorem 1.1 to the class of multi-valued weak contractions.
Definition 1.4 ([3])
Let be a metric space and be a multi-valued mapping. T is said to be a multi-valued weak contraction or a multi-valued -weak contraction if there exist two constants and such that
They proved in [3], Theorem 3 that in a complete metric space every multi-valued -weak contraction has a fixed point. In the same paper, they also introduced a class of multi-valued mappings which is more general than that of weak contractions.
Definition 1.5 ([3])
Let be a metric space and a multi-valued mapping. T is said to be a generalized multi-valued -weak contraction if there exist and a function satisfying , for every , such that
They also showed that in a complete metric space, every generalized multi-valued -weak contraction has a fixed point (see [[3], Theorem 4]).
For the last ten years, many results concerning the existence of fixed points of both single-valued and multi-valued mappings in metric spaces endowed with a partial ordering have been established. The first result in this direction was given by Ran and Reurings [9] and they also presented its applications to linear and nonlinear matrix equations. After that many authors extended those results and studied fixed point theorems in partially ordered metric spaces (see [9–13]).
In 2008, Jachymski [14] introduced the concept of G-contraction and proved some fixed point results of G-contractions in a complete metric space endowed with a graph.
Definition 1.6 ([14])
Let be a metric space and let be a directed graph such that and contains all loops, i.e., .
We say that a mapping is a G-contraction if f preserves edges of G, i.e.,
and there exists such that
He showed in [14] that under some certain properties on , a G-contraction has a fixed point if and only if is nonempty. The mapping satisfying condition (1.1) is also called a graph-preserving mapping.
Recently, Beg and Butt [5] introduced the concept of G-contraction for a multi-valued mapping and proved some fixed point results of this kind of mappings.
Definition 1.7 ([5])
Let be a multi-valued mapping. The mapping T is said to be a G-contraction if there exists such that
and if and are such that
then .
They also showed that if is a complete metric space and a triple has Property A [14], then a G-contraction mapping has a fixed point if and only if is nonempty.
Recently, in 2013, Dinevari and Frigon [6] introduced a new concept of G-contraction which is weaker than that of Beg and Butt [5].
Definition 1.8 ([6])
Let be a map with nonempty values. We say that T is a G-contraction (in the sense of Dinevari and Frigon) if there exists such that
for all and all , there exists such that
They showed that under some properties on a metric space which is weaker than Property A, a multi-valued G-contraction with closed values has a fixed point (see [6], Theorem 2.10 and Corollary 2.11). We note that the concept of G-contraction for multi-valued mappings does not concern the concept of graph-preserving as seen for single-valued mappings. Motivated by this observation and those previous works, we are interested in introducing the concept of graph-preserving for multi-valued mappings and study their fixed point theorem in a complete metric space endowed with a graph.
2 Preliminaries
Let be a metric space and be the set of all nonempty closed bounded subsets of X. For and , define
Denote by H the Pompeiu-Hausdorff metric induced by d, see [4], that is,
The following two lemmas, which can be found in [1] or [8], are useful for our main results.
Lemma 2.1 ([1])
Let be a metric space. If and , then, for each , there exists such that
Lemma 2.2 ([8])
Let be a metric space in , be a sequence in X such that . Let be a function satisfying for every . Suppose that is a non-increasing sequence such that
where and . Then is a Cauchy sequence in X.
Let be a directed graph where is a set of vertices of the graph and be a set of its edges. Assume that G has no parallel edges. If x and y are vertices in G, then a path in G from x to y of length is a sequence of vertices such that , , for . A graph G is connected if there is a path between any two vertices of G.
A partial order is a binary relation ≤ over the set X which satisfies the followings conditions:
-
1.
(reflexivity);
-
2.
If and , then (antisymmetry);
-
3.
If and , then (transitivity)
for all . A set with a partial order ≤ is called a partially ordered set. We write if and .
Definition 2.3 Let be a partially ordered set. For each ,
Definition 2.4 Let be a metric space endowed with a partial order ≤. Let be surjective and , T is said to be g-increasing if for any ,
In the case , the identity map, the mapping T is called an increasing mapping.
Example 2.5 Let have the usual relation ≤ and and be defined by
and for . It is easy to see that T is g-increasing.
Definition 2.6 Let X be a nonempty set and be a graph such that , and let . Then T is said to be graph-preserving if
Example 2.7 Let , where = ∪ ∪ ∪ ∪ . Define by
We will show that T is a graph-preserving mapping. Let .
If or or or , where , then and .
If , , then , and , , , . And we see that , and . Hence T is graph-preserving.
Example 2.8 Let , where and . Define by
It is easy to see that T is graph-preserving but not G-contraction in the sense of Dinevari and Frigon [6] since for all and for any .
Definition 2.9 Let X be a nonempty set and be a graph such that , and . Then T is said to be g-graph-preserving if for any such that
Example 2.10 Let and = ∪ ∪ ∪ ∪ . Let be defined as in Example 2.7 and let be defined by
We will show that T is g-graph-preserving. Let .
If for , then and , and , , , .
If or or , then and .
If , then and and , , and .
If , then and and . Hence T is g-graph-preserving.
3 Main results
We start with defining a new type of multi-valued mappings.
Definition 3.1 Let be a metric space, be a directed graph such that , and . T is said to be a multi-valued weak G-contraction with respect to g or -G-contraction if there exists a function satisfying for every and with
for all such that .
Remark 3.2 If , where and , , then a -G-contraction is a generalized multi-valued -weak contraction.
Property A ([14])
For any sequence in X, if and for , then there is a subsequence with for .
Theorem 3.3 Let be a complete metric space and be a directed graph such that , and let be a surjective mapping. If is a multi-valued mapping satisfying the following properties:
-
(1)
T is a g-graph-preserving mapping;
-
(2)
there exists such that for some ;
-
(3)
X has Property A;
-
(4)
T is a -G-contraction;
then there exists such that .
Proof Since g is surjective, there exists such that . By (2) we obtain . We can choose such that
By Lemma 2.1, there exists such that
Since , , and T is a g-graph-preserving mapping, we have . Moreover, by (3.1) and (3.2), we get
Next, we can choose such that
By Lemma 2.1, there exists such that
By the above two inequalities and , we get
By induction, we obtain a sequence in X and a sequence of positive integers with the property that for each , , and
and
Therefore for any , i.e., is a non-increasing sequence. Thus it follows from Lemma 2.2 that is a Cauchy sequence in X. Since X is complete, there exists such that . By assumption (3), we have a subsequence such that for any . Thus we get
Since converges to as , it follows that . Since Tu is closed, we conclude that . □
Corollary 3.4 Let be a metric space endowed with a partial order ≤, be surjective and be a multivalued mapping. Suppose that
-
(1)
T is g-increasing;
-
(2)
there exist and such that ;
-
(3)
for each sequence such that for all and converges to , for some , then for all ;
-
(4)
there exists satisfying for every and such that
for any with ;
-
(5)
the metric d is complete.
Then there exists such that .
Proof Define by and . Let such that . Then so . For any and , we have , i.e., . So T is graph-preserving. By assumption (2), there exist and such that , so . Hence (2) of Theorem 3.3 is satisfied. It is easy to see that (3) and (4) of Theorem 3.3 are also satisfied. Therefore Corollary 3.4 is obtained directly by Theorem 3.3. □
If we put for all in Corollary 3.4, we obtain the following result.
Corollary 3.5 Let be a metric space endowed with a partial order ≤ and be a multivalued mapping. Suppose that
-
(1)
T is increasing;
-
(2)
there exists such that ;
-
(3)
for each sequence such that for all and converges to x, for some , then for all ;
-
(4)
there exist and such that
for any with , where for every ;
-
(5)
the metric d is complete.
Then there exists such that .
Remark 3.6 Theorem 4 in [3] is directly obtained from Theorem 3.3 by setting , where , and for all .
Example 3.7 Let , , , = ∪ ∪ and be defined by
We note that
This means that T does not satisfy Nadler’s theorem. We will show that T is a weak contraction with and .
Let .
If ∈ ∈ , we have
If , we have
If , we have
If , we have
If , we have
Hence T is an -G-contraction. To show that T is graph-preserving, let . If or , then , , and we see that . If , then and . If or , then , and we see that . If , then and we see that . If , then and we see that . Hence T is graph-preserving. By the definition of T and G, we see that and , that is, condition (2) of Theorem 3.3 is satisfied. It is easy to see that X has Property A. Therefore all the conditions of Theorem 3.3 are satisfied, so T has a fixed point and we see that .
Next, we give an example of a map which lacks assumption (2) and has no fixed point.
Example 3.8 Let , , , and be defined by
and let be an identity map. The same as in Example 3.7, T is a graph-preserving mapping and T is a -G-contraction. Moreover, we can easily check that condition (2) of Theorem 3.3 does not hold and we note that T has no fixed point.
References
Nadler S: Multi-valued contraction mappings. Pac. J. Math. 1969, 20(2):475–488.
Alghamdi MA, Berinde V, Shahzad N: Fixed points of multivalued nonself almost contractions. J. Appl. Math. 2013., 2013: Article ID 621614
Berinde M, Berinde V: On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016
Berinde V, Pacurar M: The role of Pompeiu-Hausdorff metric in fixed point theory. Creative Math. Inform. 2013, 22(2):143–150.
Beg I, Butt AR: The contraction principle for set valued mappings on a metric space with graph. Comput. Math. Appl. 2010, 60: 1214–1219. 10.1016/j.camwa.2010.06.003
Dinevari T, Frigon M: Fixed point results for multivalued contractions on a metric space with a graph. J. Math. Anal. Appl. 2013, 405: 507–517. 10.1016/j.jmaa.2013.04.014
Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.
Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2003, 132(5):1435–1443.
Beg I, Butt AR: Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces. Math. Commun. 2010, 15: 65–75.
Beg I, Latif A: Common fixed point and coincidence point of generalized contractions in ordered metric spaces. Fixed Point Theory Appl. 2012. 10.1186/1687-1812-2012-229
Beg I, Butt AR: Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 2009, 71: 3699–3704. 10.1016/j.na.2009.02.027
Bhaskar TG, Laskhmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 2006, 65(7):1379–1393. 10.1016/j.na.2005.10.017
Jachymski J: The contraction principle for mappings on a metric with a graph. Proc. Am. Math. Soc. 2008, 1(136):1359–1373.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. The first author would like to thank Science Achievement Scholarship of Thailand (SAST). This paper was supported by Chiang Mai University, Chiang Mai, Thailand.
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Tiammee, J., Suantai, S. Coincidence point theorems for graph-preserving multi-valued mappings. Fixed Point Theory Appl 2014, 70 (2014). https://doi.org/10.1186/1687-1812-2014-70
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DOI: https://doi.org/10.1186/1687-1812-2014-70