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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 \(\mathbb{X}\) into itself for which the inequalities \[ \| T^ n x- T^ n y\|\leq k_ n\| x- y\|\quad (x,y\in C,\;n= 1,2,\dots) \] hold. The following four results are presented:
(1) if \(\mathbb{X}\) has uniformly normal structure, \(C\) is a bounded set and \(\sup k_ n< N(\mathbb{X})^{1/2}\) (\(N(\mathbb{X})= \inf\{\text{diam }E/\text{rad }E\): \(E\) is bounded closed convex set of \(\mathbb{X}\}\)) 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 \(\mathbb{X}\) is uniformly smooth, \(k_ n\to 1\), \(x_ n\) \((n= 1,2,\dots)\) is a fixed point of \[ S_ n x= (1- k_ n^{-1} t_ n) x+ k^{-1}_ n t_ n Tx,\quad (k_ n- 1)/(k_ n- t_ n)\to 0, \] \(x_ n- Tx_ n\to 0\) then \(x_ n\) converges strongly to a fixed point of \(T\);
(3) if \(\mathbb{X}\) 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 \(D(\mathbb{X})< 1\), \(C\) is a closed bounded convex set, \(k_ n\to 1\), \(T\) is weakly asymptotically regular on \(C\) then \(T\) has a fixed point.
Reviewer: P.Zabreiko (Minsk)

MSC:
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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