Une généralisation de la conjecture de point fixe de Schauder

Abstract

We prove the following generalisation of Schauder's fixed point conjecture: Let $C_1,...,C_n$ be convex subsets of a Hausdorff topological vector space. Suppose that the $C_i$ are closed in $C=C_1\cup ...\cup C_n$. If $f:C\to C$ is a continuous function whose image is contained in a compact subset of $C$, then its Lefschetz number $\Lambda (f)$ is defined. If $\Lambda (f)\ne 0$, then $f$ has a fixed point.