Selections of multifunctions of two variables
We establish a very general selection theorem for multifunctions of two variables. From it, of which we cannot give here the extremely complex statement, a long series of consequences is then derived. We obtain, for instance, the following continuous selection theorem, here stated in a simplified version. Theorem. Let T, X be two paracompact topological spaces, Y a Banach space and F a multifunction from \(T\times X\) into Y, with non-empty, closed and convex values. Assume that, for each fixed \(x\in X\), the set F(T,x) is bounded, the multifunction F(.,x) is lower semicontinuous and, for each of its continuous selections g, one has \[ \lim_{z\to x}\sup_{t\in T}d(g(t),F(t,z))=0, \] where \(d(g(t),F(t,z))=\inf \{\| g(t)-v\|:\) \(v\in F(t,z)\}\). Under such hypotheses, F admits a continuous selection on \(T\times X\). This theorem is not a direct consequence of the classical Michael selection theorem, since, in general, the product of two paracompact spaces fails to be paracompact.
existence theorem of differential inclusions in Banach spaces
scientific article; zbMATH DE number 3879694