Random versions of Kakutani-Ky Fan's fixed point theorems

Abstract

The following results are proved: (1) X be either a locally convex Lusin space 1 1 A topological space X is called a Lusin space if it is separated and if there exist a Polish space P and a bijective continuous mapping of P onto X (N. Bourbaki, “Topologie générale,” Chapitre IX, 1974). or a locally convex metrizable (not necessarily separable) space, let Γ be a weakly upper semicontinuous random multimapping defined on a convex compact subspace of X taking convex weakly compact values and satisfying the Browder-Halpern's “inward” condition. Then Γ has a fixed point. (2) In an arbitrary metric space, a continuous random multimapping Γ (with stochastic complete domain) has fixed points, whenever the corresponding deterministic fixed point theorem for Γ holds.