Continuous selections for a family of nonconvex-valued mappings with noncompact domain
The existence of a continuous selection for multifunctions of the form \[ H(x) = F(x) \cap \bigcap^N_{i=1} \bigl(G_i(x) + f_i(x) \cdot B_i \bigr) \] is exhibited. \(H\) acts between a separable metric space \(S\) and the space \(L_1(T,X)\) where \(T\) is a topological space endowed with positive Radon measure adn \(X\) a separable Banach space. \(F\) and \(G_i: S\to L_1(T,X)\) are lsc with closed, decomposable values, \(f_i\) are single-valued lsc functions and \(B_i: =\{x\in L_1(T,X) : P_i(x) <1\}\) where the \(P_i\) constitute a family of continuous seminorms with some special properties. The paper contains applications of such selection and numerous proofs that many particular seminorms are suitable.
scientific article; zbMATH DE number 917064
scientific article