标题： 通过照亮单位球检测非扩张映射的不动点 显示英文标题

标题： Detecting fixed points of nonexpansive maps by illuminating the unit ball

摘要： 我们给出了在有限维赋范空间上非扩张映射具有非空有界不动点集的充要条件。 在其他结果中，我们证明如果$f : V \rightarrow V$是有限维赋范空间$V$上的非扩张映射，那么$f$的不动点集非空且有界当且仅当存在$w_1, \ldots , w_m$在$V$中使得$\{f(w_i) - w_i : i = 1, \ldots, m \}$照亮单位球。 这提供了一种用于检测有限维空间上非扩张映射不动点的数值过程。 我们还讨论了该过程在博弈论和数学生物学中出现的某些非线性特征值问题中的应用。 显示英文摘要

摘要： 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.