Browder’s Theorem through Brouwer’s Fixed Point Theorem

Eilon Solan*, Omri N. Solan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

A parametric version of Brouwer’s fixed point theorem, called Browder’s theorem, states that for every continuous mapping (Formula presented.), where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set (Formula presented.), has a connected component whose projection onto the first coordinate is (Formula presented.). Browder’s original proof relies on the theory of the fixed point index. We provide an alternative proof that uses Brouwer’s fixed point theorem and is valid whenever X is a nonempty, compact, and convex subset of a Hausdorff topological vector space.

Original languageEnglish
Pages (from-to)370-374
Number of pages5
JournalAmerican Mathematical Monthly
Volume130
Issue number4
DOIs
StatePublished - 2023

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