Recent Contributions to Fixed Point Theory and Its ApplicationsView this Special Issue
A Suzuki Type Coupled Fixed Point Theorem for Generalized Multivalued Mapping
We obtain a new Suzuki type coupled fixed point theorem for a multivalued mapping from into , satisfying a generalized contraction condition in a complete metric space. Our result unifies and generalizes various known comparable results in the literature. We also give an application to certain functional equations arising in dynamic programming.
1. Introduction and Preliminaries
Theorem 1 (see ). Let be a complete metric space. Let be a self-map and let be defined by
Let and . Suppose that for all . Then, has a unique fixed point , and holds for every .
Let be a metric space. We denote by the family of all nonempty, closed bounded subsets of . Let H be a Hausdorff metric; that is, for , where .
Nadler  proved multivalued extension of the Banach contraction principle as follows.
Let be a complete metric space and let be a mapping from into . Assume that there exists such that for all . Then, there exists such that . Many fixed point theorems have been proved by various authors as a generalization of Nadler’s theorem [6–9]. One of the general fixed point theorems for a generalized multivalued mapping appears in .
Theorem 2 (see ). Let be a complete metric space and let be a mapping from into . Assume that there exists a function from into defined by such that for all . Then, there exists such that .
Bhaskar and Lakshmikantham  introduced the concept of coupled fixed point for a mapping from to and established some coupled fixed point theorems in partially ordered sets. As an application, they studied the existence and uniqueness of solution for a periodic boundary value problem associated with a first order ordinary differential equation.
Definition 3. Let be a metric space and . An element is called a coupled fixed point of if and .
The aim of this paper is to obtain coupled fixed point for a multivalued mapping which satisfies the generalized contraction condition in complete metric spaces. Our results unify, extend, and generalize various known comparable results in the literature.
2. Main Results
Now, we shall prove our main result.
Firstly, we define a nonincreasing function from into by
Theorem 4. Let be a complete metric space and let be a mapping from into . Assume that there exists such that implies for all . Then, there exist such that and .
Proof. Let be a real number such that , and such that and . Since and , then
and, as ,
Thus, from assumption (10), we have
Adding (14) and (16), we have
If , then we have as ; a contradiction. Therefore,
Again, if , , we get and if , we get
Using triangle inequality, we obtain This implies
Hence, there exist with and such that
Thus, we construct such sequences and in such that , , and Then, we have
Hence, we conclude that in both cases the sequences and are Cauchy sequences. Since is complete, there are some points and in such that and .
Now, we shall show that for all and . Since and , there exists such that for all . Therefore
from (29) we have
Then, from (10) we have
By adding, (32) and (33), we get
Letting tends to , we obtain
Hence, we have:
Now, we shall prove that and .
Case 1. First, we consider the case . Suppose, on contrary, that and . Let and be such that . Since and imply and , from (10), we have
On the other hand, since , from (10), we get
By adding, (38) and (39), we have
And, from (40), we have
Therefore, we obtain
This is a contradiction. As a result, we have and .
Case 2. Now, we consider the case . We shall first prove that for all . If and , then the previous equation (44) obviously holds. Hence, let us assume and . Then, for every , there exist sequences and such that
Then for all , we have
If , then we get
Letting , we have
Thus and, from (10), we obtain (44). If , then
Letting , we have and, thus from condition (56), we obtain (44).
Finally, from (44), we obtain
Hence, as , we obtain
This implies that and .
Corollary 5. Let be a complete metric space and let be a mapping from into . Assume that there exists such that implies for all , where the function is defined as in Theorem 4. Then, there exist such that and .
Corollary 6. Let be a complete metric space and let be a mapping from into . Assume that there exists such that implies for all , where the function is defined as in Theorem 4. Then, there exist such that and .
3. An Application
The existence and uniqueness of solutions of functional equations and system of functional equations arising in dynamic programming have been studied by using various fixed point theorems. In this paper, we shall prove the existence and uniqueness of a solution for a class of functional equations using Corollary 6.
In this section, we assume that and are Banach spaces. , , and is a field of real numbers. Let denote the set of all the real valued functions on . It is known that endowed with the metric is a complete metric space. According to Bellman and Lee, the basic form of the functional equation of dynamic programming is given as where and represent the state and decision vectors, respectively, represents the transformation of the process, and represents the optimal return function with initial state . In this section, we study the existence and uniqueness of a solution of the following functional equation: where and are bounded functions. In this section, we study the existence and uniqueness of a solution of (59) in the following new form. Let a function be defined as in Theorem 4 and let the mapping be defined by
Theorem 7. Suppose that there exists such that for every , , and , the inequality implies where Then, the functional equation (59) has a unique bounded solution in .
Proof. Note that is a map from onto and that is a complete metric space, where is the metric defined in (57). Let be an arbitrary positive real number, and .For arbitrary and so that From the definition of mapping , we have If inequality (61) holds, then from (64) and (67), we get Similarly, (65) and (66) implies that Hence, from (68) and (69), we have Since inequality (70) is true for any and arbitrary , then (61) implies Therefore, all the conditions of Corollary 6 are met for the mapping , and hence the functional equation (59) has a unique bounded solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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