Fixed point theorem for convex, closed multivalued mapping - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:26:19Z http://mathoverflow.net/feeds/question/61309 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61309/fixed-point-theorem-for-convex-closed-multivalued-mapping Fixed point theorem for convex, closed multivalued mapping Maciej S. 2011-04-11T16:31:02Z 2011-05-12T07:29:48Z <p>There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:</p> <p>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. $x\in F(x)$.</p> <p>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.</p>