Let (\Omega, \Sigma) be a measurable space, with \sum a sigma-algebra of subsets of \Omega, 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: \Omega \times E \rightarrow KC(X) satisfying an inwardness condition and itself being a 1-\chi-contractive nonexpansive mapping has a random fixed point. We also prove that a multivalued nonexpansive, non-self operator T:\Omega\times E\rightarrow KC(X) with a uniformly convex X satisfying an inwardness condition has a random fixed point.
KUMAM, POOM and PLUBTIENG, SOMYOT (2006) "Some Random Fixed Point Theorems for Non-Self Nonexpansive Random Operators," Turkish Journal of Mathematics: Vol. 30: No. 4, Article 2. Available at: https://journals.tubitak.gov.tr/math/vol30/iss4/2