Fixed point theorem in subsets of topological vector spaces
Let \(E\) be a topological vector space and \(X\) a compact convex subset of \(E\) possessing the simplicial approximation property. Then, in their principal result, the authors show that a multivalued map \(T\) on \(X\) has a fixed point whenever the values of \(T\) have some contractible small neighbourhoods. The map considered by the authors is a generalization of Kakutani maps, viz., upper semicontinuous multivalued maps with convex compact values. The convexity assumption in the values of the map \(T\) is reduced by using some ideas from \textit{Charles D. Horvath} [J. Math. Anal. Appl. 156, 341--357 (1991; Zbl 0733.54011)].
scientific article; zbMATH DE number 2182045