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.

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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 Haussdorf 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 metrizability.

Suppose, that $X$ is compact, convex and metrizable in locally convex Haussdorf topological vector space. Then any l.s.c. map $F$ 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 metrizability.