Browder's Theorem through Brouwer's Fixed Point Theorem

2021/07/06 by Eilon Solan, Solan, Eilon, Omri N. Solan +1 · 1 citation
Mathematics · Computer Science · #Advanced Optimization Algorithms Research #Mathematics and Applications #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2107.02428

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] × X) → X, where X is a simplex in a Euclidean space, the set of fixed points of f, namely, the set \(t,x) ∈ [0,1] × 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.

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