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Fixed point theorems for asymptotically nonexpansive mappings. (English) Zbl 0812.47058

This article deals with a mapping $T$ of a nonempty set $C$ of a Banach space $𝕏$ into itself for which the inequalities

$\parallel {T}^{n}x-{T}^{n}y\parallel \le {k}_{n}\parallel x-y\parallel \phantom{\rule{1.em}{0ex}}\left(x,y\in C,\phantom{\rule{4pt}{0ex}}n=1,2,\cdots \right)$

hold. The following four results are presented:

(1) if $𝕏$ has uniformly normal structure, $C$ is a bounded set and $sup{k}_{n} ($N\left(𝕏\right)=inf\left\{\text{diam}\phantom{\rule{4.pt}{0ex}}E/\text{rad}\phantom{\rule{4.pt}{0ex}}E$: $E$ is bounded closed convex set of $𝕏\right\}$) and there exists a nonempty bounded closed convex set $E$ containing weak $\omega$-limit set of $T$ at $E$ then $T$ has a fixed point in $E$;

(2) if $𝕏$ is uniformly smooth, ${k}_{n}\to 1$, ${x}_{n}$ $\left(n=1,2,\cdots \right)$ is a fixed point of

${S}_{n}x=\left(1-{k}_{n}^{-1}{t}_{n}\right)x+{k}_{n}^{-1}{t}_{n}Tx,\phantom{\rule{1.em}{0ex}}\left({k}_{n}-1\right)/\left({k}_{n}-{t}_{n}\right)\to 0,$

${x}_{n}-T{x}_{n}\to 0$ then ${x}_{n}$ converges strongly to a fixed point of $T$;

(3) if $𝕏$ is a Banach space with a weakly continuous duality map, $C$ is a weakly compact convex subset, ${k}_{n}\to 1$ then $T$ has a fixed point and moreover if $T$ is weakly asymptotically regular at some $x\in C$ then ${T}^{n}x$ converges weakly to a fixed point of $T$;

(4) if the Maluta constant $D\left(𝕏\right)<1$, $C$ is a closed bounded convex set, ${k}_{n}\to 1$, $T$ is weakly asymptotically regular on $C$ then $T$ has a fixed point.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties