Some Random Fixed Point Theorems for Non-Self Nonexpansive Random Operators

Let (W, S) be a measurable space, with \sum a sigma-algebra of subsets of W, and let E be a nonempty bounded closed convex and separable subset of a Banach space X, whose characteristic of noncompact convexity is less than 1. We prove that a multivalued nonexpansive, non-self operator T: W \times E \rightarrow KC(X) satisfying an inwardness condition and itself being a 1-c-contractive nonexpansive mapping has a random fixed point. We also prove that a multivalued nonexpansive, non-self operator T:W\times E\rightarrow KC(X) with a uniformly convex X satisfying an inwardness condition has a random fixed point.