Solvability and algorithms for functional equations originating from dynamic programming (Q536885)

The purpose of the paper is to study the functional equation \[ f(x)=\text{opt}_{y\in D}\text{opt}\{p(x,y),q(x,y)f(a(x,y)),r(x,y)f(b(x,y)),s(x,y)f(c(x,y))\}, \] where the operator ``\(\text{opt}\)'' may stand for any of \(\sup,\inf,\max,\min\). This equation arises in dynamic programming of multistage decision processes. Using the Banach fixed point theorem as a main tool, existence and uniqueness theorems and iterative approximations of the solutions are obtained. Eight nontrivial examples are also discussed.