Selecting families and their applications
Extending the result of \textit{P.\,Deguire} and \textit{M.\,Lassonde} [Topol.\ Methods Nonlinear Anal.\ 5, No.\,2, 261--269 (1995; Zbl 0877.54017)], the authors present the following theorem. Let \(X\) be a paracompact space and \(\{Y_i\}_{i \in I}\) a family of convex subsets of Hausdorff topological vector spaces. If \(\mathcal T=\{T_i:X \multimap Y_i\}_{i \in I}\) is a family of multimaps with convex values satisfying the condition \[ X = \bigcup_{i \in I} \bigcup_{y_i \in Y_i} \operatorname{int} T^-_i(y_i), \] then \(\mathcal T\) has a selecting family, i.e., there exists a family of continuous maps \(\{f_i: X \to Y_i\}_{i \in I}\) such that for each \(x \in X\), there exists \(i \in I\) such that \(f_i(x) \in T_i(x)\). As application, some fixed component theorems of Browder-Fan type, an intersection result, a maximal element theorem, and a minimax inequality are considered.
scientific article; zbMATH DE number 5293421
scientific article