Research Article | Open Access

# A Suzuki Type Coupled Fixed Point Theorem for Generalized Multivalued Mapping

**Academic Editor:**Mohamed Amine Khamsi

#### Abstract

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

In 2008, Suzuki [1] introduced a new type of mappings which generalize the well-known Banach contraction principle [2], and, further, Kikkawa and Suzuki [3] proved a Kannan [4] version of mappings.

Theorem 1 (see [3]). *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 [5] 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 [10].

Theorem 2 (see [11]). *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 [12] 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

Thus,

Since

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

This implies

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

And, therefore

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.

#### References

- T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 5, pp. 1861–1869, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,”
*Fundamenta Mathematicae*, vol. 3, pp. 133–181, 1922. View at: Google Scholar - M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 649749, 8 pages, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. Kannan, “Some results on fixed points. II,”
*The American Mathematical Monthly*, vol. 76, pp. 405–408, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. B. Nadler Jr., “Multi-valued contraction mappings,”
*Pacific Journal of Mathematics*, vol. 30, pp. 475–488, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 141, no. 1, pp. 177–188, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Z. Daffer and H. Kaneko, “Fixed points of generalized contractive multi-valued mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 192, no. 2, pp. 655–666, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. V. Semenov, “On fixed points of multivalued contractions,”
*Functional Analysis and Its Applications*, vol. 36, no. 2, pp. 159–161, 2002. View at: Publisher Site | Google Scholar | MathSciNet - L. Ćirić, “Multi-valued nonlinear contraction mappings,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 2716–2723, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. B. Ćirić, “Fixed points for generalized multi-valued contractions,”
*Matematički Vesnik*, vol. 9, no. 24, pp. 265–272, 1972. View at: Google Scholar | Zentralblatt MATH | MathSciNet - D. Doric and R. Lazović, “Some Suzuki-type fixed point theorems for generalized multivalued mappings and applications,”
*Fixed Point Theory and Applications*, vol. 2011, article 40, 2011. View at: Google Scholar | MathSciNet - T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 65, no. 7, pp. 1379–1393, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Pushpendra Semwal and Ramesh Chandra Dimri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.