A random fixed point theorem for multivalued nonexpansive operators in uniformly convex Banach spaces

Abstract

Let (Ω, Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω, and let C be a nonempty, bounded, closed, convex, and separable subset of a uniformly convex Banach space X. It is shown that every multivalued nonexpansive random operator T: Ω × C → K(C) has a random fixed point, where K(C) is the family of all nonempty compact subsets of C endowed with the Hausdorff metric induced by the norm of X