Random fixed point theorems for multivalued nonexpansive non-self-random operators

Let (Ω,Σ) be a measurable space, with Σ a sigma-algebra of subset of Ω, and let C be a nonempty bounded closed convex separable subset of a Banach space X, whose characteristic of noncompact convexity is less than 1, KC(X) the family of all compact convex subsets of X. We prove that a multivalued nonexpansive non-self-random operator T:Ω×C→KC(X), 1-χ-contractive mapping, satisfying an inwardness condition has a random fixed point.