Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces
Brian Lins · Proceedings of the American Mathematical Society · 2008
If $X$ is a finite dimensional real normed space, $C$ is a closed convex subset of $X$ and $f:C \rightarrow C$ is nonexpansive with respect to the norm on $X$, then we show that either $f$ has a fixed point in $C$ or there is a linear functional $\varphi \in X^*$ such that $\lim _{k \rightarrow \infty } \varphi (f^k(x)) = \infty$ for all $x \in C$.