Constructive proofs of Tychonoff's and Schauder's fixed point theorems for sequentially locally non-constant functions

TL;DR

This paper provides a constructive proof of Tychonoff's and Schauder's fixed point theorems for sequentially locally non-constant functions within locally convex and Banach spaces, using Bishop-style constructive mathematics.

Contribution

It introduces a constructive proof approach for these fixed point theorems under the condition of sequential local non-constancy.

Findings

Constructive proof of Tychonoff's fixed point theorem

Extension to Schauder's fixed point theorem in Banach spaces

Applicable within Bishop-style constructive mathematics

Abstract

We present a constructive proof of Tychonoff's fixed point theorem in a locally convex space for sequentially locally non-constant functions, As a corollary to this theorem we also present Schauder's fixed point theorem in a Banach space for sequentially locally non-constant functions. We follow the Bishop style constructive mathematics.