- Research Article
- Open Access

# On Properties of Solutions for Two Functional Equations Arising in Dynamic Programming

- Zeqing Liu
^{1}, - JeongSheok Ume
^{2}Email author and - ShinMin Kang
^{3}

**2010**:905858

https://doi.org/10.1155/2010/905858

© Zeqing Liu et al. 2010

**Received:**12 July 2010**Accepted:**26 October 2010**Published:**31 October 2010

## Abstract

We introduce and study two new functional equations, which contain a lot of known functional equations as special cases, arising in dynamic programming of multistage decision processes. By applying a new fixed point theorem, we obtain the existence, uniqueness, iterative approximation, and error estimate of solutions for these functional equations. Under certain conditions, we also study properties of solutions for one of the functional equations. The results presented in this paper extend, improve, and unify the results according to Bellman, Bellman and Roosta, Bhakta and Choudhury, Bhakta and Mitra, Liu, Liu and Ume, and others. Two examples are given to demonstrate the advantage of our results over existing results in the literature.

## Keywords

- Positive Integer
- State Space
- Unique Solution
- Functional Equation
- Dynamic Programming

## 1. Introduction and Preliminaries

where and are real Banach spaces, is the state space, is the decision space, opt denotes the sup or inf, and stand for the state and decision vectors, respectively, , represent the transformations of the processes, and denotes the optimal return function with initial state . The rest of the paper is organized as follows. In Section 2, we state the definitions, notions, and a lemma and establish a new fixed point theorem, which will be used in the rest of the paper. The main results are presented in Section 3. By applying the new fixed point theorem, we establish the existence, uniqueness, iterative approximation, and error estimate of solutions for the functional equation (1.3) and (1.4). Under certain conditions, we also study other properties of solutions for the functional equations (1.4). The results present in this paper extend, improve, and unify the corresponding results according to Bellman [1], Bellman and Roosta [5], Bhakta and Choudhury [6], Bhakta and Mitra [7], Liu [8], Liu and Ume [11], and others. Two examples are given to demonstrate the advantage of our results over existing results in the literature.

## 2. A Fixed Point Theorem

then is a metric on . A sequence in is said to converge to a point if as for any and to be a Cauchy sequence if as for any .

Theorem 2.1.

where is some element in , then

(i) has a unique fixed point and for any ,

Proof.

which is a contradiction. Consequently, is a unique fixed point of .

This completes the proof.

Remark 2.2.

Theorem 2.1 extends Theorem 2.1 of Bhakta and Choudhury [6] and Theorem 1 of Boyd and Wong [12].

Lemma 2.3 (see [11]).

## 3. Properties of Solutions

where and , then is a countable family of pseudometrics on . It is clear that is a complete metric space.

Theorem 3.1.

Let and be mappings, and let be in , such that

(C2)for any and , there exists satisfying

Proof.

Notice that the functional equation (1.3) possesses a unique solution if and only if the mapping has a unique fixed point . Thus, Theorem 3.1 follows from Theorem 2.1. This completes the proof.

Remark 3.2.

The conditions of Theorem 3.1 are weaker than the conditions of Theorem 3.1 of Bhakta and Choudhury [6].

Theorem 3.3.

Let and be mappings for . Assume that the following conditions are satisfied:

(C5)there exists a constant such that

Proof.

where for . Thus, Theorem 3.3 follows from Theorem 3.1. This completes the proof.

Remark 3.4.

Theorem 2 of Bellman [1, page 121], the result of Bellman and Roosta [5, page 545], Theorem 3.3 of Bhakta and Choudhury [6], and Theorems 3.3 and 3.4 of Liu [8] are special cases of Theorem 3.3. The example below shows that Theorem 3.3 extends properly the results in [1, 5, 6, 8].

Example 3.5.

possesses a unique solution . However, the results in [1, 5, 6, 8] are not applicable.

Theorem 3.6.

Let and be mappings for , and, be in satisfying

then the functional equation (1.4) possesses a solution that satisfies the following conditions:

(C11) is unique with respect to condition (C10).

Proof.

which is a contradiction since .

Therefore, (3.32) holds for any .

which implies that . That is, is a solution of the functional equation (1.4).

by letting . Similarly, (3.49) also holds for . As , we know that . This completes the proof.

Remark 3.7.

Theorem 3.6 generalizes Theorem 1 of Bellman [1, page 119], Theorem 3.5 of Bhakta and Choudhury [6], Theorem 2.4 of Bhakta and Mitra [7], Theorem 3.5 of Liu [8] and Theorem 3.1 of Liu and Ume [11]. The following example reveals that Theorem 3.6 is indeed a generalization of the results in [1, 6–8, 11].

Example 3.8.

satisfies conditions (C6)–(C8). Consequently, Theorem 3.6 ensures that it has a solution that satisfies conditions (C9)–(C11). However, Theorem 1 of Bellman [1, page 119], Theorem 3.5 of Bhakta and Choudhury [6], Theorem 2.4 of Bhakta and Mitra [7], Theorem 3.5 of Liu [8], and Theorem 3.1 of Liu and Ume [11] are not applicable.

## Declarations

### Acknowledgment

This research is financially supported by Changwon National University in 2009-2010.

## Authors’ Affiliations

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