Detecting fixed points of nonexpansive maps by illuminating the unit ball

TL;DR

This paper characterizes fixed points of nonexpansive maps in finite-dimensional normed spaces using illumination of the unit ball, providing a numerical method for detecting fixed points with applications in game theory and biology.

Contribution

It introduces a necessary and sufficient condition for fixed points based on illuminating the unit ball, leading to a practical numerical detection method.

Findings

Fixed point set is nonempty and bounded iff the unit ball is illuminated by certain vectors.

Provides a numerical procedure for fixed point detection in finite-dimensional spaces.

Applications demonstrated in nonlinear eigenvalue problems in game theory and biology.

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 $f : V \rightarrow V$ 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 $w_1, \ldots , w_m$ in $V$ such that $\{f(w_i) - w_i : i = 1, \ldots, m \}$ 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.