Applications of KKM-map principle (Q1611448)

The following is the main result (Theorem 3) extending a result of \textit{J. B. Prolla} [Numer. Funct. Anal. Optimization 5, 449-455 (1983; Zbl 0513.41015)]: Let \(X\) be a nonempty, convex, weakly compact subset of a normed linear space \(E\) and let \(g\) be a strongly continuous, almost affine and onto map on \(X\). Then for each strongly continuous map \(f\) from \(X\) into \(E\), there exists a \(y_0\) in \(X\) such that \(d(g(y_0, f(y_0))= d(f(y_0),X)\). Some special cases are also discussed.