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On a new proof of the key step in the proof of Brouwer's fixed point theorem

N. V. Krylov

Abstract

We present a solution of Exercise 1.2.1 of [2] which yields a short new proof of a key step in one of proofs of Brouwer's fixed point theorem, 1910. A few people asked the author about the details of the solution and they might be interesting to a broader audience. Our approach is absolutely different from the ones using algebraic or differential topology or differential calculus and is based on a simple observation which somehow escaped many authors treating this theorem in the past.

On a new proof of the key step in the proof of Brouwer's fixed point theorem

Abstract

We present a solution of Exercise 1.2.1 of [2] which yields a short new proof of a key step in one of proofs of Brouwer's fixed point theorem, 1910. A few people asked the author about the details of the solution and they might be interesting to a broader audience. Our approach is absolutely different from the ones using algebraic or differential topology or differential calculus and is based on a simple observation which somehow escaped many authors treating this theorem in the past.

Paper Structure

This paper contains 3 theorems, 18 equations.

Key Result

Lemma 1

Let $\Omega$ be a (connected) bounded domain in $\mathbb{R}^{d}$ with $C^{1}$ boundary and let $F, G:\bar{\Omega}\to\mathbb{R}^{d}$ be $C^{1}(\bar{\Omega} )$ mappings such that Then

Theorems & Definitions (5)

  • Lemma 1
  • Remark 2
  • Remark 3
  • Corollary 4
  • Theorem 5