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

This paper deals with solvability of the following functional equation arising in dynamic programming of multistage decision processes f(x) = opt y∈D { u(x, y)(p(x, y) + f(a(x, y))) + v(x, y) opt{q(x, y), f(b(x, y))} } , ∀x ∈ S. Using the Banach fixed point theorem and new iterative techniques, we obtain the existence and uniqueness of solutions for the above equation in the complete metric space BB(S) and the Banach spaces BC(S) and B(S), construct some iterative methods, prove their convergence and provide several error estimates between these iterative sequences generated by the iterative methods and the corresponding solutions, respectively. Four nontrivial examples illustrating applications of the results presented in this paper are provided.