Multivalued maps and the existence of best approximants
The authors give the following generalization of \textit{Ky Fan}'s well- known result [Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112, 234-240 (1969; Zbl 0185.395)]: Let X be a Banach space and let M be an approximatively compact and convex subset of X. Let f: \(M\to X\) be a continuous map such that f(M) is relatively compact. Let g: \(M\to M\) be a continuous onto and proper map such that \(g^{-1}(z)\) is an acyclic subset of M for every \(z\in M\). Then there exists \(x\in M\) so that \(\| g(x)-f(x)\| d(f(x),M).\) Some interesting consequences are also pointed out.
scientific article; zbMATH DE number 4205069
scientific article