Fixed point theorem for multifunctions in topological vector spaces

The authors prove the following generalizations of the Kakutani fixed point theorem: Let S be a nonempty subset of a Hausdorff topological vector space X and F be a closed convex valued map such that either the graph of F is closed and the set clF(S) is locally convex or the graph is weakly closed. Then F has a fixed point if and only if there is a compact convex set K such that F(x) intersects the set K, for any x \in S. They also give a positive answer to a conjecture concerning strongly inward nonexpansive compact valued-maps defined on a subset of a uniformly convex Banach space.