Une g\'en\'eralisation de la conjecture de point fixe de Schauder

TL;DR

This paper generalizes Schauder's fixed point conjecture to convex subsets in topological vector spaces, establishing conditions under which a continuous function has a fixed point based on Lefschetz number.

Contribution

It extends Schauder's fixed point theorem to unions of convex sets in topological vector spaces with a new fixed point criterion involving Lefschetz number.

Findings

Lefschetz number is well-defined for the class of functions considered.

Non-zero Lefschetz number guarantees the existence of a fixed point.

The generalization applies to convex sets in Hausdorff topological vector spaces.

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 $\Lambda(f)\ne0$, then $f$ has a fixed point.