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 n-dimensional simplex for uniformly sequentially continuous functions. We follow the Bishop style constructive mathematics.
Highlights
IntroductionReference 1 provided a constructive proof of Brouwer’s fixed point theorem
It is often said that Brouwer’s fixed point theorem cannot be constructively proved
The existence of an exact fixed point of a function which satisfies some property of local non-constancy may be constructively proved
Summary
Reference 1 provided a constructive proof of Brouwer’s fixed point theorem. Brouwer’s fixed point theorem can be constructively, in the sense of constructive mathematics a la Bishop, proved only approximately. Some authors have presented a constructive or an approximate version of Brouwer’s fixed point theorem using Sperner’s lemma. We consider uniform sequential continuity of ISRN Applied Mathematics functions according to 5. In classical mathematics uniform continuity and uniform sequential continuity are equivalent. In constructive mathematics a la Bishop, uniform sequential continuity is weaker than uniform continuity. We will prove a constructive version of Brouwer’s fixed point theorem in an n-dimensional simplex for uniformly sequentially continuous functions. We follow the Bishop style constructive mathematics according to 2, 8, 9
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