Solvability Of Solutions And Iterative Approximations For A Functional Equation Arising In Dynamic Programming

This paper introduces and studies the following functional equation arising in dynamic programming of multistage decision processes where λ,μ∈[0,1] are constants and such that λ+μ≤1and m∈N,opt denotes the sup or inf, x,y stand for the state and decision vectors, respectively, ui,vi,wi,pi,qi,ri:S×Dâ†'R are nappi ngs, ai,bi,ci:S×Dâ†'S represent the transformations of the processes for i∈{1,2,…,m}, f(x) denotes the optinal return function with initial state x.This paper discusses the existence, uni queness and iterative approxi nations of bounded solutions and continuous bounded solutions in Banach spaces BC(S) and B(S), respectively, for the above functional equation by utilizing the Banachfixed point theoremand Mann iterative scherrss, and studies properties of solutions, which is bounded in each bounded subset of S, in the complete metric spaces BB(S) by using the fixed point theoremin [20] and the iterative algorithms for the functional equation. The text presents four theorens, and in each proof of theorern one or two mappings are constructed, and then the nappi ngs which are used in the process of proof are proved to satisfy the conditions of the theorens. FinalIy, according to the correspondi ng fixed point theorens and iterative schemes, the conditions to ensure the existence of solutions of the equation are obtained.The results presented in this paper extend essentially the known results in this field. In the end, four nontrivial examples are constructed as applications of our results.