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Completeness and the contraction principle

Borwein, Jonathan M. (1983) Completeness and the contraction principle. Proceedings of the American Mathematical Society, 87 . pp. 246-250. ISSN 0002-9939

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Abstract

We prove (something more general than) the result that a convex subset of a Banach space is closed if and only if every contraction of the space leaving the convex set invariant has a fixed point in that subset. This implies that a normed space is complete if and only if every contraction on the space has a fixed point. We also show that these results fail if "convex" is replaced by "Lipschitz-connected" or "starshaped".

Item Type: Article
Subjects: UNSPECIFIED
Faculty: UNSPECIFIED
Depositing User: Mrs Naghmana Tehseen
Date Deposited: 23 Feb 2015 13:55
Last Modified: 23 Feb 2015 13:55
URI: https://docserver.carma.newcastle.edu.au/id/eprint/1637

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