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Common fixed point results of Ciric-Suzuki-type inequality for multivalued maps in compact metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 7 (2014)
The aim of this paper is to introduce a new type of multivalued operators similar to those of Ciric-Suzuki type. A common fixed point theorem for multivalued maps on metric spaces satisfying Ciric-Suzuki-type inequality is proved. Applications to certain functional equations arising in dynamic programming are also discussed.
MSC:54E50, 54H25, 45G05.
1 Introduction and preliminaries
In 1937, Von Neumann  initiated fixed point theory for multivalued mappings in the study of game theory. Indeed, the fixed point theorems for multivalued mappings are quite useful in control theory and have been frequently used in solving many problems of economics and game theory. Successively, Nadler  initiated the development of the geometric fixed point theory for multivalued mappings. He used the concept of the Hausdorff metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case.
Consistent with Nadler (, p.620), and will denote a metric space and the collection of all nonempty closed subsets of X, respectively. For and ,
The hyperspace is called the generalized Hausdorff metric space induced by the metric d on X.
Theorem 1.1 
Define a nonincreasing function φ from into by
Let be a complete metric space and . Assume that there exists such that for every ,
Then there exists such that .
Theorem 1.2 
Let be a compact metric space, and let f be a mapping on X. Assume that
for . Then f has a unique fixed point.
In 2011, Haghi et al.  gave a very useful lemma which we need in our work.
Lemma 1.3 
Let X be a nonempty set and be a function. Then there exists a subset such that and is one-to-one.
In this manuscript, the well-known results of Suzuki , Edelstein  and Doric and Lazovic  have been merged to complement a multitude of related results from the literature. Moreover, we use Lemma 1.3 to obtain a common fixed point theorem for multivalued maps on a metric space. Finally, as an application, the existence of common solutions of certain functional equations arising in dynamic programming is proved.
2 Fixed point results
Theorem 2.1 Let be a compact metric space, and let . Assume that there exists such that
for all . Then T has a fixed point.
Proof Let be a sequence in X. We have that is a closed set, so we can find a sequence in X such that and . Now, put . X is compact, without loss of generality, we may assume that the sequences and converge to v and w in X, respectively. Let
We want to show that . Suppose not, that is, . Thus, we can choose such that
for with . Thus for . This implies
Since , . Thus
Moreover, . So
Let n attend to ∞, then we have
but . Thus, . From (2.1), we have
which is a contradiction to the definition of β. Hence, .
We want to prove that T has a fixed point. Assume, on the contrary, that for any . We have and thus , then .
First, we want to show that
We have and , that is, there exist such that
so we have, for any ,
Thus , which implies , where
but , thus
We have . Then (2.3) becomes
By taking , we obtain
If , then
and if , then . Thus, for any ,
Second, we want to prove that for every ,
If , it is trivial. Let . Then, for every , there exists a sequence such that . For all , by using (2.2) we have
If , then . When , we have
Hence, we find that .
If , then . Thus,
Since , therefore . Now, for every , we have
which implies (2.4).
Finally, from (2.4) for any , we get
We have , thus . If , then we obtain
As , we obtain that . Since Tv is closed, thus we reach the conclusion . This contradicts the assumption that T has no fixed point. Hence, T has a fixed point. □
Example 2.2 Let be endowed with the usual metric and be defined by
with . Then T satisfies the assumptions in Theorem 2.1.
Proof If , it is trivial. For we have the following:
and or ( and )
Hence, the assumptions of Theorem 2.1 are satisfied and is a fixed point. □
3 Common fixed point results
Theorem 3.1 Let be a metric space, Y be a nonempty set, and such that and is a compact subspace of X. Assume that for there exists such that
and implies . Then T and g have a coincidence point, that is, there exists such that . If , then g, T have a common fixed point provided that at v.
Proof By Lemma 1.3, there exists such that and is one-to-one. We define a map by .
For , whenever , i.e., . Also, we have . Thus, G is a mapping. Now, for we have
By Theorem 2.1, G has a fixed point in , that is, there exists such that . Further, if , we have . Thus, . Now, we want to show that . Suppose , then we have
Since , and , we obtain
We have for each . Since is a closed set, then
but . Thus, . Since , we have
which is a contradiction. Therefore, . □
Corollary 3.2 Let be a metric space, Y be a nonempty set, such that and is a compact subspace of X. Assume that for there exists such that
and implies . Then f and g have a unique coincidence point. If , then f, g have a unique common fixed point provided that f and g commute at v.
Proof The proof of this corollary follows from Theorem 3.1 by taking . We need to prove that v is a unique coincidence point of f and g. Suppose, to the contrary, that there exists such that and . Then the inequalities and are satisfied. Thus, we have , where
Therefore, , that is, , this is impossible due to . Hence, v is the unique coincidence point. Moreover, if , we have but . Thus, , i.e., gv is a coincidence point. Because of the uniqueness of the coincidence point, we obtain that . Then . □
4 An application
Many authors have studied the existence and uniqueness of solutions of functional equations and system of functional equations in dynamic programming by using diverse fixed point theorems (cf. [14–16]).
Throughout this section, we suppose that U and V are Banach spaces, is the state space, is the decision space and ℝ is the field of real numbers. Let denote the set of all the bounded real-valued functions on W. For an arbitrary , define . Then is a Banach space.
The return functions of the continuous decision process are defined by the functional equations
where x and y represent the state and decision vectors, respectively, represents the transformation of the process. Moreover, and are bounded functions.
In this article, we prove the existence and uniqueness of the common solution of functional equations (4.1) and (4.2) arising in dynamic programming, using Corollary 3.2.
Let the maps f and g be defined by
Suppose that the following conditions hold.
(Q1) For any , there exists such that
(Q2) There exists such that
Theorem 4.1 Suppose that conditions (Q1) and (Q2) are satisfied and is a closed and bounded subspace of . Assume that there exists such that for every , and , the inequality
then functional equations (4.1) and (4.2) have a unique common bounded solution in W.
Proof Note that, due to q, , G and F are bounded, f and g are self-maps of . is a complete metric space, where d is the metric defined by the supremum norm on . Since is a closed and bounded subspace of , then is complete and bounded. That is, is compact. Conditions (Q1) and (Q2) imply that and f and g commute at their coincidence points.
Let ε be an arbitrary positive real number, and . For , choose such that
where and .
Furthermore, from the definition of f, we have
If we suppose that the following inequality holds
From (4.3), (4.6) and (4.7), we obtain that
Similarly, (4.4), (4.5) and (4.7) imply
Thus, from (4.8) and (4.9) we have
Since ε is arbitrary, therefore for any , we have
Therefore, by Corollary 3.2, f and g have a unique common fixed point, and hence functional equations (4.1) and (4.2) have a unique bounded common solution. □
Remark 4.2 Fukhar-ud-din et al.  have established fixed point results on a non-compact domain in uniformly convex metric spaces for a single-valued map satisfying a contractive condition closely related to the Suzuki condition employed in Theorem 1.2. It will be interesting to extend the results of this paper in this general setup (cf. Open problem 1 on p.4759 in ).
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The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-DIP-2012-31.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final version.
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Shaddad, F., Noorani, M.S.M. & Alsulami, S.M. Common fixed point results of Ciric-Suzuki-type inequality for multivalued maps in compact metric spaces. J Inequal Appl 2014, 7 (2014). https://doi.org/10.1186/1029-242X-2014-7
- metric space
- common fixed point
- Urysohn integral equations