ISRN Applied Mathematics

Volume 2011 (2011), Article ID 276040, 9 pages

http://dx.doi.org/10.5402/2011/276040

## A Proof of Constructive Version of Brouwer's Fixed Point Theorem with Uniform Sequential Continuity

Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

Received 1 April 2011; Accepted 18 May 2011

Academic Editors: A. Cherouat and C. I. Siettos

Copyright © 2011 Yasuhito Tanaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is often said that Brouwer's fixed point theorem cannot be constructively proved. On the other hand, Sperner's lemma, which is used to
prove Brouwer's theorem, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. They, however, assume uniform continuity of functions. We consider uniform sequential continuity of functions. In classical mathematics, uniform continuity and uniform sequential continuity are equivalent. In constructive mathematics a la Bishop, however, uniform sequential continuity is weaker than uniform continuity. We will prove a constructive version of Brouwer's fixed point theorem in an