Common fixed point results of Ciric-Suzuki-type inequality for multivalued maps in compact metric spaces
© Shaddad et al.; licensee Springer. 2014
Received: 22 July 2013
Accepted: 4 December 2013
Published: 2 January 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.
The hyperspace is called the generalized Hausdorff metric space induced by the metric d on X.
Theorem 1.1 
Then there exists such that .
Theorem 1.2 
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
for all . Then T has a fixed point.
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 .
Hence, we find that .
which implies (2.4).
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. □
with . Then T satisfies the assumptions in Theorem 2.1.
- (2)and or ( and )
Hence, the assumptions of Theorem 2.1 are satisfied and is a fixed point. □
3 Common fixed point results
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 .
which is a contradiction. Therefore, . □
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.
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.
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.
Suppose that the following conditions hold.
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.
where and .
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 ).
The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-DIP-2012-31.
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