Detecting fixed points of nonexpansive maps by illuminating the unit ball

Lemmens, Bas, Lins, Brian, Nussbaum, Roger (2018) Detecting fixed points of nonexpansive maps by illuminating the unit ball. Israel Journal of Mathematics, 224 (1). pp. 231-262. ISSN 0021-2172. E-ISSN 1565-8511. (doi:10.1007/s11856-018-1641-0)

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Abstract

We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if f:V?V is a nonexpansive map on a finite dimensional normed space V , then the fixed point set of f is nonempty and bounded if and only if there exist w 1 ,…,w m in V such that {f(w i )?w i :i=1,…,m} illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.

Item Type: Article
DOI/Identification number: 10.1007/s11856-018-1641-0
Subjects: Q Science
Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Q Science > QA Mathematics (inc Computing science) > QA440 Geometry
Divisions: Central Services > Information Services
Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Central Services > Research Services
Depositing User: Bas Lemmens
Date Deposited: 08 Jul 2016 09:42 UTC
Last Modified: 29 May 2019 17:33 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/56242 (The current URI for this page, for reference purposes)
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