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

The following results are proved: (1) X be either a locally convex Lusin space¹ 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.