Unique solutions

✍ Scribed by Peter Schuster

• Book ID: 102485055
• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 124 KB
• Volume: 52
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200610012

✦ Synopsis

Abstract

It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non‐uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)