A Random Fixed Point Theorem for Multivalued Nonexpansive Operators in Uniformly Convex Banach Spaces

Abstract

Let $(\Omega ,\Sigma )$ be a measurable space with $\Sigma$ a sigma-algebra of subsets of $\Omega$, 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 $T:\Omega \times C \to K(C)$ 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 $X$.