Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces

Lins, Brian

doi:10.1090/S0002-9939-08-09779-7

Asymptotic behavior of nonexpansive mappings in finite dimensional normed spaces
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Proc. Amer. Math. Soc. 137 (2009), 2387-2392 Request permission

Abstract:

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 $x \in C$.

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