Browder's Theorem: from One-Dimensional Parameter Space to General Parameter Space

TL;DR

This paper extends Browder's Theorem to general parameter spaces, providing an elementary proof that leverages reduction to the one-dimensional case, broadening its applicability in topological fixed point theory.

Contribution

The paper offers a simplified proof of a parametric fixed point theorem for general spaces, generalizing Browder's Theorem from one-dimensional to arbitrary parameter spaces.

Findings

Connected component of fixed points projects onto the entire parameter space

Elementary proof reduces the problem to the case X = [0,1]

Generalizes Browder's Theorem to broader settings

Abstract

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff topological space and $Y$ is a nonempty, convex, and compact subset of a locally-convex topological vector space, the set of fixed points of $f$, defined by $C_f := \{ (x,y) \in X \times Y \colon f(x,y)=y\}$, has a connected component whose projection onto the first coordinate is $X$. In this note we provide an elementary proof for this result, using its reduction to the case $X = [0,1]$.