Approximate selection theorems and their applications (Q1364829)

Let \(X\) be a paracompact space, \(Y\) a locally convex linear topological space, and \(F:X\to Y\) a multifunction with convex values. \(F\) is called sub-lsc if for each \(x\in X\) and each neighbourhood of zero in \(Y\) there is a \(z\in F(x)\) and a neighbourhood \(U(x)\) of \(x\) in \(X\) such that for each \(y\in U(x)\), \(z\in F(y)+V\). It is proved that \(F\) is sub-lsc if and only if for each neighbourhood \(V\) of zero in \(Y\) there is a continuous \(V\)-approximate selection \(f:X\to Y\), i.e. \(f(x)\in F(x)+V\). Moreover, the paper contains two theorems concerning \(V\)-approximate selections avoiding an usc multifunction \(T\) topologically separated from \(F\), a remark on Michael's selection theorem, a fixed point theorem for lsc multifunctions with compact range and the existence of equilibrium for generalized games.