On Solvability And Iterative Approximations Of A Functional Equation

This paper is concerned with a new functional equation arising in dynamic programming of multistage decision process, where λ∈[0,1] is a constant, x and y stand for the state and decision vectors, respectively, a,b and c denote for the transformations of the processes, and f{x) is the optimal return function with initial state x.The first chapter states the importance of the study for the new functional equation. When λ takes special values, the above equation brings many cases of the study in [1-20], and the paper introduces the development process of functional equations simply.The second chapter gives some related definitions and important lemmas.The third chapter using the Banach fixed point theorem and iterative algorithms, ensure the existence, uniqueness and iterative approximations of solutions for the functional equation in the Banach spaces BC(S) and B(S) are provided. Next, utilizing the Liu-Ume-Kang fixed point theorem, we prove the existence, uniqueness and iterative approximations of solutions for the functional equation in the complete metric space BB(S). The last, we use the nonexpansive method to discuss the the existence, uniqueness and the other properties of solutions for the functional equation in the complete metric space BB(S). In this chapter, four Theorems are included, the results presented in this paper extend and unify some known results in [5,6,9,,11-13,15-17,19].The fourth chapter, four nontrivial examples are given to illustrate the results present in this paper, at the same time, four examples explain that the results obtained in this paper are more general than the others.