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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 $\bbfX$ into itself for which the inequalities $$\Vert T\sp n x- T\sp n y\Vert\le k\sb n\Vert x- y\Vert\quad (x,y\in C,\ n= 1,2,\dots)$$ hold. The following four results are presented: (1) if $\bbfX$ has uniformly normal structure, $C$ is a bounded set and $\sup k\sb n< N(\bbfX)\sp{1/2}$ ($N(\bbfX)= \inf\{\text{diam }E/\text{rad }E$: $E$ is bounded closed convex set of $\bbfX\}$) 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 $\bbfX$ is uniformly smooth, $k\sb n\to 1$, $x\sb n$ $(n= 1,2,\dots)$ is a fixed point of $$S\sb n x= (1- k\sb n\sp{-1} t\sb n) x+ k\sp{-1}\sb n t\sb n Tx,\quad (k\sb n- 1)/(k\sb n- t\sb n)\to 0,$$ $x\sb n- Tx\sb n\to 0$ then $x\sb n$ converges strongly to a fixed point of $T$; (3) if $\bbfX$ is a Banach space with a weakly continuous duality map, $C$ is a weakly compact convex subset, $k\sb n\to 1$ then $T$ has a fixed point and moreover if $T$ is weakly asymptotically regular at some $x\in C$ then $T\sp n x$ converges weakly to a fixed point of $T$; (4) if the Maluta constant $D(\bbfX)< 1$, $C$ is a closed bounded convex set, $k\sb n\to 1$, $T$ is weakly asymptotically regular on $C$ then $T$ has a fixed point.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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