The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces

Tan, Kok-Keong; Xu, Hong Kun

doi:10.1090/S0002-9939-1992-1068133-2

The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces
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Proc. Amer. Math. Soc. 114 (1992), 399-404 Request permission

Abstract:

Let $X$ be a uniformly convex Banach space with a Frechet differentiable norm, $C$ a bounded closed convex subset of $X$, and $T:C \to C$ an asymptotically nonexpansive mapping. It is shown that for each $x$ in $C$, the sequence $\{ {T^n}x\}$ is weakly almost-convergent to a fixed point $y$ of $T$, i.e., $(1/n)\sum \nolimits _{i = 0}^{n - 1} {{T^{k + i}}x \to y}$ weakly as $n$ tends to infinity uniformly in $k = 0,1,2, \ldots$

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