A general stochastic fixed-point theorem for continuous random operators on stochastic domains

Abstract

We prove that under certain conditions a continuous random operator with stochastic domain has a random fixed point provided that each realization has a (deterministic) fixed point. As a by-product we obtain a selection theorem for the interior of certain convex-valued measurable correspondences. We apply our results to obtain stochastic Krasnoselski- and Rothe-type theorems and existence results for random differential and integral equations.