Generalized \(R\)-KKM theorems in topological space and their applications.
Let \(X\) be a nonempty set, \(Y\) be a topological space and \(T: X\to Y\) a multifunction with nonempty and compactly closed values (i.e. \(T(x)\cap K\) is closed for any compact \(K\subset Y\)). It is proved that \(\bigcap_{x\in X} T(x)\neq\emptyset\) if and only if \(T\) is generalized \(R\)-KKM and \(\bigcap_{x\in M} T(x)\) is compact for some finite \(M\subset X\). The approach is in the spirit of [\textit{R. Bieławski}, J. Math. Anal. Appl. 127, 155--171 (1987; Zbl 0638.52002)] and generalizes some recent results. A corresponding minimax and saddle-point theorem are deduced.
scientific article; zbMATH DE number 1992575