Strong Ergodic Theorems for Non-Lipschitzian Mappings of Asymptotically Nonexpansive Type in Uniformly Convex Banach Spaces
Isao Miyadera · Tokyo Journal of Mathematics · 1999
In this paper we establish strong ergodic theorems for non-Lipschitzian mappings of asymptotically nonexpansive type in uniformly convex Banach spaces. Introduction.Throughout this paper $X$ denotes a uniformly convex Banach space, $C$ a nonempty bounded closed convex subset of $X$ , and $T$ a mapping from $C$ into itself.The asymptotic behavior of asymptotically nonexpansive mappings has been studied by many authors.There appear in the literature the following three definitions of an asymptotically nonexpansive mapping:(Goebel and Kirk [3]) There exists a sequence $\{a_{k}\}$ with $\lim_{k\rightarrow\infty}a_{k}=1$ such that $\Vert T^{k}u-T^{k}v\Vert\leq a_{k}\Vert u-v\Vert$ for $u,$ $v\in C$ and integers $k\geq 0$ .In this case we say that $T$ is asymptotically nonexpansive in the strong sense.$(c_{2})$ (Kirk [4]) $T^{K}$ is continuous for some positive integer $K$ and (0.1) $\varlimsup_{k\rightarrow\infty}\sup_{v\in C}(\Vert T^{k}u-T^{k}v\Vert-\Vert u-v\Vert)\leq 0$ for $u\in C$ .In this case we say that $T$ is asymptotically nonexpansive in the weak sense. $(c_{3})$(Bruck, Kuczumow and Reich [2]) $T$ is called asymptotically nonexpansive in the intermediate sense if $T^{K}$ is continuous for some positive integer $K$ and (0.2) $\varlimsup_{k\rightarrow\infty}\sup_{u,v\in C}(\Vert T^{k}u-T^{k}v\Vert-\Vert u-v\Vert)\leq 0$ .DEFINITION 0.1.A sequence $\{x_{n}\}_{n\geq 0}$ in $X$ is said to be strongly almost convergent to an element $x$ in $X$ if (the strong limit) $\lim_{n\rightarrow\infty}(1/n)\sum_{i=0}^{n-1}x_{i+k}=x$ uniformly in $k=0,1,2,$ $\cdots$ The purpose of this paper is to prove the following strong ergodic theorems.