Selecting families
Let \(X\) be a topological space and \(\{Y_i\mid i\in I\}\) a family of spaces. For a given family of multimaps \(\{A_i\mid i\in I\}\), a family of continuous maps \(\{f_i\mid i\in I\}\) is said to be a selecting family provided for every \(x\in X\) there exists \(i\in I\) such that \(f_i(x)\in A_i(x)\). The main result is an analogue of the Michael-Browder theorem stating the existence of a selecting family for a family \(\{A_i\mid i\in I\}\) with convex values \(A_i(x)\), open fibres \(A_i^{-1}(y)\) and such that for every \(x\in X\) there exists \(i\in I\) with \(A_i(x)\neq \emptyset\). As applications the authors obtain versions of the Ky Fan-Browder fixed point theorem for systems of multimaps and minimax inequalities of Ky Fan and von Neumann-Sion for systems of functions.
scientific article; zbMATH DE number 856951