There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:

Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff topological vector space. Then any l.s.c. map $F:X\rightarrow X$ which is closed and convex valued has a fixed point, i.e. $x\in F(x)$.

The question is, what happens if we drop the assumption that $X$ is metrizable. Xian Wu in his paper "A new fixed point theorem and its application" left it as an open problem, after giving the proof using a metrizability.