Detecting fixed points of nonexpansive maps by illuminating the unit ball |
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Authors: | Bas Lemmens Brian Lins Roger Nussbaum |
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Institution: | 1.School of Mathematics, Statistics, and Actuarial Science,University of Kent, Canterbury,Kent,England;2.Department of Mathematics and Computer Science,Hampden-Sydney College,Hampden-Sydney,USA;3.Rutgers University, Hill Center for the Mathematical Sciences,Piscataway,USA |
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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 w1,..., 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. |
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