Browder's Theorem through Brouwer's Fixed Point Theorem

TL;DR

This paper offers an alternative proof of Browder's parametric fixed point theorem by leveraging Brouwer's Fixed Point Theorem, simplifying the original proof that used fixed point index theory.

Contribution

It introduces a new proof method for Browder's theorem based on Brouwer's Fixed Point Theorem, avoiding the fixed point index theory.

Findings

The set of fixed points has a connected component projecting onto [0,1]

The proof is simplified using Brouwer's Fixed Point Theorem

Provides an alternative approach to Browder's theorem

Abstract

One of the conclusions of Browder (1960) is a parametric version of Brouwer's Fixed Point Theorem, stating that for every continuous function $f : ([0,1] \times X) \to X$, where $X$ is a simplex in a Euclidean space, the set of fixed points of $f$, namely, the set $\{(t,x) \in [0,1] \times X \colon f(t,x) = x\}$, has a connected component whose projection on the first coordinate is $[0,1]$. Browder's (1960) proof relies on the theory of the fixed point index. We provide an alternative proof to Browder's result using Brouwer's Fixed Point Theorem.