Existence Theorems for Solvability of a Functional Equation Arising in Dynamic Programming

Abstract

The paper deals with the existence, uniqueness, and iterative approximations of solutions for the functional equations arising in dynamic programming of multistage decision making processes in Banach spaces BC(S) and B(S) and complete metric space BB(S), respectively. Our main results extend, improve, and generalize the results due to several authors. Some examples are also given to demonstrate the advantage of our results over existing one in the literature.

1. Introduction

In this paper, we introduce and study the existence and uniqueness of solutions for the following functional equation arising in dynamic programming of multistage decision processes: where “opt” denotes the “sup” or “inf,” and stand for the state and decision vectors, respectively, represents the transformation of the processes, and denotes the optimal return function with initial state .

It is clear that (1) includes many functional equations and system of functional equations as special case, respectively. Bellman [1] was the first to investigate the existence and uniqueness of solutions for the following functional equation: in a complete metric space .

Bhakta and Mitra [2] obtained the existence and uniqueness of solutions for the functional equations in a Banach space and in , respectively. Bhakta and Choudhury [3] established the existence of solutions for the functional equations (2) in .

In 2003, Liu and Ume [4] pointed out that the form of the functional equations of dynamic programming is as follows: In 2004, Liu et al. [5] obtained an existence, uniqueness, and iterative approximation of solutions for the functional equation In 2006, Liu et al. [6] provided the sufficient conditions which ensure the existence and uniqueness and iterative approximation of solution for the functional equation In 2007, Liu and Kang [7] studied the following functional equation: and gave an existence and uniqueness result of solution for the functional equation.

In 2011, Jiang et al. [8] investigated the following functional equation: and gave some existence and uniqueness results and iterative approximations of solutions for the functional equation in .

In Section 2, we recall some basic concepts, notations, and lemmas. In Section 3, we utilize the fixed point theorem due to Boyd and Wong [9] to establish the existence, uniqueness, and iterative approximation of solution for the functional equation (1) in Banach spaces and complete metric spaces. Also, we construct some nontrivial examples to explain our results. The results presented here generalize, improve, and extend the results of Bellman [1], Bhakta and Mitra [2], Bhakta and Choudhury [3], Liu and Ume [4], Liu et al. [5], Liu et al. [6], Liu and Kang [7], Jiang et al. [8], Pathak and Deepmala [10], and Liu [11]. The existence problems of solutions of various functional equations arising in dynamic programming are both theoretical and practical interest. Fixed point theorems are applied in various fields (see [12–16]). Here, we use a fixed point theorem to show the solvability of the functional equation arising in dynamic programming.

2. Preliminaries

Throughout this paper, we assume that , , and . For any , denotes the largest integer not exceeding and and are real Banach spaces. Let be the state space and let be decision space. Define  ,  ,   ,   ,   ,   ,   . Here, and are Banach spaces with . For any positive integer and , , let where . Then, is a countable family of pseudometrices on . A sequence in is said to converge to a point if for any , as . It is clear that is a complete metric space. A metric space is said to metrically convex if for each , there is a for which . Clearly, any Banach space is metrically convex.

Theorem 1 (see [9]). Suppose that is a complete metrically convex metric space and that satisfies where satisfies for , where and denotes the closure of . Then, has a fixed point and , for each .

Lemma 2 (see [8]). . Then,

Lemma 3 (see [8]). (i) Assume that is a mapping such that is bounded for some . Then,
(ii) Assume that is a mapping such that and is bounded for some . Then,

3. Main Results

In this section, we discuss existence and uniqueness of solutions in and .

Theorem 4. Let be compact. Let and for and satisfy the following conditions: (), and are bounded for .()for each , , , as uniformly for and .(), for some and . Then, the functional equation (1) possesses a unique solution and converges to for each , where is defined by

Proof. Let and and ; it follows from , and compactness of that there exists a constant , , and such that Thus, is bounded.
In light of , (15), (17), and Lemmas 2 and 3, we deduce that for all with which implies that is continuous at . Thus, is a self-mapping on .
Given , and . Suppose that . Then, such that Using (20), we arrive at which implies that where . Letting , we get Similarly, we conclude that (23) holds for .
Theorem 1 ensures that has a unique fixed point and converges to for each . It is obvious that is also a unique solution of the functional equation (1) in . This completes the proof.

Dropping the compactness of and in the proof of Theorem 4, we get the following result.

Theorem 5. Let and for and satisfy conditions and . Then, the functional equation (1) possesses a unique solution and converges to for each , where is defined by (15).

Remark 6. If , for and , then Theorems 4 and 5 reduce to the results of Jiang et al. [8]. The example below shows that Theorems 4 and 5 extend substantially the results in [8].

Example 7. Let , , ; then, Theorem 5 ensures that the following functional equation: possesses a unique solution in . However, the corresponding results in [8] are not applicable for the functional equation (24) because We point out that the functional equation (24) possesses also a unique solution in .

In our next results, we discuss properties of solutions in .

Theorem 8. Let and for and satisfy the following conditions: (), and are bounded on for and , , for ,there exists a constant such that . Then, the functional equation (1) possesses a unique solution and converges to for each , where is defined by (15).

Proof. It follows from and that for each and , there exists and such that By virtue of , (26), we know that Thus, is bounded; that is, is self-mapping on .
Let , and . Suppose that . Then, there exists satisfying In terms of the above, and Lemma 2, we get which implies that In a similar way, we can show that (30) holds for . It follows that where for . As in (31), we get that It follows from Theorem 2.2 in [3] that has a unique fixed point and converges to for each . Obviously, is also a unique solution of the functional equation (1). This completes the proof.

Remark 9. (1) If , , and , then Theorem 8 reduces to Theorem 3.4 of Bhakta and Choudhury [3] and a result of Bellman [1].
(2) If , then Theorem 8 reduces to a result of Liu et al. [5].
(3) If we replace by and , then Theorem 8 reduces to Theorem 4.1 of Liu et al. [6], which, in turn, generalizes the results in [1, 3].
(4) Theorem 3.3 of Jiang et al. [8] is a particular case of Theorem 8.
(5) Theorem 4.1 of Pathak and Deepmala [10] is reduced to Theorem 8, whenever .

Theorem 10. Let and for , and let be in satisfying the following conditions: . . . Then, the functional equation (1) possesses a solution that satisfies the following conditions: the sequence is defined by  , for all ;  , for all converges to . for any , and , for all .() is unique with respect to condition .

Proof. Since is in , it is easy to verify that First of all we assert that the mapping defined by (15) is nonexpansive on on account of (33) and ; we obtain that which implies that a constant with By virtue of , , (15), (35), and Lemmas 2 and 3, we deduce that Thus, is self-mapping on .
As in the proof of Theorem 8, by we immediately conclude that for and , which yields that for . That is, is nonexpansive.
Now we assert that for each , Now by we see that That is, (39) is true for . Suppose that (39) holds for some . From ()–() we know that Hence, (39) holds for .
Next we claim that is a cauchy sequence in . Given and . Let , . Suppose that . Then, we select such that In view of (42), , and Lemma 2, we infer that which means that for some and .
Similarly, we conclude that the above inequality (44) holds for . Proceeding in this way, we select and for such that It follows from , (33), (39), (44), and (45) that which implies that As in the above inequality, we deduce that which yields that is a cauchy sequence in since , for each . Suppose that converges to some . Notice that is nonexpansive. It follows that That is, . Thus, the functional equation (1) possess a solution .
Now we show that holds. Let , , and , for . Put . Then, there exists a positive integer satisfying By (39), , and (50), we infer that, for , which means that .
Finally, we show that holds. Suppose that the functional equation (1) possesses another solution , which satisfies condition . Let and . If , then there exists such that Using Lemma 2, , and (52), we get which implies that for some and . Similarly, we conclude that (54) holds for . Proceeding in this way, we select and for satisfying Combining (54) and (55), we obtain that Letting in the above inequalities, by we get that As in the above inequality, we know that . This completes the proof.

Remark 11. (1) Theorem 10 extends, improves, and unifies the results in Liu et al. [5, 6].
(2) In case and , then Theorem 10 reduces to Theorem 3.4 of Jiang et al. [8], which, in turn, generalizes and unifies the results of Bellman [1], Bhakta and Choudhury [3], Liu [11], and Liu and Ume [4].
(3) If we replace , , , , , , and , for , , where is a constant in and is a positive constant, then Theorem 10 reduces to Theorem of Liu and Ume [4], which, in turn, generalizes Theorem in [3], Theorem in [2], and a result in [1].
(4) Dropping condition and replacing , , , , then Theorem 10 reduces to Theorem 3.4 of Liu and Kang [7], which, in turn, generalizes the results in [1, 3, 7].
The example below demonstrates that Theorem 10 generalizes, extends, and unifies the results in [1–8, 11].

Example 12. Let , , , , for all . Consider the following functional equation: It follows from Theorem 10 that the functional equation (58) possesses a solution . However, the corresponding results in [1–8, 11] are not valid for the functional equation (58) because does not hold for , where is a positive constant; also

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the editor and the anonymous referees for their valuable comments and suggestions improving the exposition of the paper. The research of the author is supported by the Department of Science and Technology, Government of India, under INSPIRE Fellowship Program no. DST/INSPIRE Fellowship/2013/.