Abstract
Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: C → KC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × C → KC(C) has a random fixed point.
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Kumam, P., Plubtieng, S. The characteristic of noncompact convexity and random fixed point theorem for set-valued operators. Czech Math J 57, 269–279 (2007). https://doi.org/10.1007/s10587-007-0060-x
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DOI: https://doi.org/10.1007/s10587-007-0060-x