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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 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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if he already gave the proof, how is it an open problem?? –  Suvrit Apr 11 '11 at 16:36
    
He proved the theorem as stated above, where X is metrizable. I have corrected this small mistake :-) –  Maciej S. Apr 11 '11 at 16:51
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Note that your statement is true with USC in place than LSC. Ky Fan's generalization (1951) of Kakutani fixed point theorem is stated for Hausdorff locally convex topological vector spaces, without any metrizability assumption. However, the assumption in Ky Fan's theorem is USC rather than LSC (see here: ncbi.nlm.nih.gov/pmc/articles/PMC1063516/?page=2 ) –  Pietro Majer Apr 11 '11 at 18:32
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I know that result. However, the difference is very deep. For example, for both l.s.c. and u.s.c. map it is possible to construct approximate continuous selection. Unfortunately, passing to the limit in our case is impossible due to the fact, that a l.s.c. map do not have the closed graph, in contrast to the u.s.c. case. Trying to prove by an approximation argument, we get only, that the diagonal meet the closure of the graph of given map, instead of intersecting the graph exactly. –  Maciej S. Apr 11 '11 at 19:05

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