Some applications of the Kakutani fixed point theorem (Q1088189)

The famous Ky Fan theorem, leading to many non-trivial applications in nonlinear functional analysis, is based on the theorem on closed coverings of a simplex. The authors prove a likewise theorem using open coverings. The main theorem is as follows. Let C be a nonempty closed convex subset in a Hausdorff topological vector space E and \(F: C\to 2^ C\) be a map such that F(x) is closed for each \(x\in C\) and \(F^{-1}(y)\) is convex for each \(y\in C\), and \(C\subset \cup F(x_ i)\) for some finite set \(\{x_ 1,...,x_ n\}\). Then there exists w such that \(w\in F(w)\).