On \(\epsilon\)-optimal continuous selectors and their application in discounted dynamic programming (Q1078075)

Let X, Y be topological spaces, \(\phi: X\to 2^ Y\) a multifunction, and u a real-valued function defined on the graph of \(\phi\). We give sufficient conditions for the existence of a continuous selector f for \(\phi\) such that \(u(x,f(x))\geq \sup \{u(x,y):\) \(y\in \phi (x)\}- \epsilon\), \(x\in X\), where \(\epsilon >0\). This result is applied to the stochastic discounted dynamic programming problem. We establish the existence of an \(\epsilon\)-optimal continuous stationary policy.