Fixed point theorems in constructive mathematics

Matthew Hendtlass

doi:10.4115/jla.2012.4.10

Abstract

This paper gives the beginnings of a development of the theory of fixed point theorems within Bishop's constructive analysis. We begin with a construc- tive proof of a result, due to Borwein, which characterises when some sets have the contraction mapping property. A review of the constructive content of Brouwer's fixed point theorem follows, before we turn our attention to Schauder's general- isation of Brouwer's fixed point theorem. As an application of our constructive Schauder's fixed point theorem we give an approximate version of Peano's theorem on the existence of solutions of differential equations. Other fixed point theorems are mentioned in passing. 2010 Mathematics Subject Classification 03F55, 03F60, 46S30 (primary); 34A30, 47H10 (secondary)

Highlights

Fixed-point theorems are a major tool in both functional analysis and mathematical economics1 and are used to prove the existence of solutions to differential equations and the existence of Nash equilibria among other things
(ii) The non-constructive nature of Brouwer’s fixed point theorem, and the subsequent rejection of this theorem by Brouwer, is well known, and a constructive approximate version for simplices is part of the folklore
It should be noted that the constructive mathematician is interested only in what can in theory be computed, and is not concerned with questions of practicality

Summary

Introduction

Fixed-point theorems are a major tool in both functional analysis and mathematical economics and are used to prove the existence of solutions to differential equations and the existence of Nash equilibria among other things. Recently has a fully constructive proof of the approximate version of Brouwer’s fixed point theorem, for simplices, been presented [23]. In the first we consider the fixed point theorems of Banach and Brouwer; we prove that spaces with a strong connectedness property are complete if and only if every contraction mapping has a fixed point, and we give an approximate version of Brouwer’s fixed point theorem for uniformly sequentially continuous functions on totally bounded subsets of Rn with convex closures. In the final section we give an application of our constructive Schuader’s fixed point theorem: we prove an approximate version of Peano’s theorem on the existence of solutions of differential equations. A totally bounded subset of X is located [4, page 95, Propostion (4.4)]

Banach’s fixed point theorem

Brouwer’s fixed point theorem

Schauder’s fixed point theorem

An application