Constructive proof of Brouwer's fixed point theorem for sequentially locally non-constant functions

Abstract

We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions implies Sperner's lemma for a simplex. Since the existence of approximate fixed points is derived from Sperner's lemma, our Brouwer's fixed point theorem is equivalent to Sperner's lemma.

Cited by in corpus (2)

• Constructive proofs of Tychonoff's and Schauder's fixed point theorems for sequentially locally non-constant functions
• Constructive proof of Brouwer's fixed point theorem for sequentially locally non-constant and uniformly sequentially continuous functions by Sperner's lemma