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

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