A random fixed point theorem for a multivalued contraction mapping

Abstract

Some results on measurability of multivalued mappings are given. Then using them, the following random fixed point theorem is proved: Theorem. Let X be a Polish space, (T, A) a measurable space. Let F: T × X → CB(X) be a mapping such that for each x ϵ X, F(·, x) is measurable and for each t ϵ T, F(t, ·) is k (t)-contraction, where k: T → [0,1) is measurable. Then there exists a measurable mapping u: T → X such that for every t ϵ T, u(t) ϵ F(t, u(t)). © 1977 Pacific Journal of Mathematics. All rights reserved.