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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

T n x-T n yk n x-y(x,yC,n=1,2,)

hold. The following four results are presented:

(1) if 𝕏 has uniformly normal structure, C is a bounded set and supk n <N(𝕏) 1/2 (N(𝕏)=inf{diamE/radE: E is bounded closed convex set of 𝕏}) and there exists a nonempty bounded closed convex set E containing weak ω-limit set of T at E then T has a fixed point in E;

(2) if 𝕏 is uniformly smooth, k n 1, x n (n=1,2,) is a fixed point of

S n x=(1-k n -1 t n )x+k n -1 t n Tx,(k n -1)/(k n -t n )0,

x n -Tx n 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 1 then T has a fixed point and moreover if T is weakly asymptotically regular at some xC then T n x converges weakly to a fixed point of T;

(4) if the Maluta constant D(𝕏)<1, C is a closed bounded convex set, k n 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