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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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