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Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. (English) Zbl 0971.47038
Let $X$ be a uniformly convex Banach space and $K$ a nonempty subset of $X$. A mapping $T: K\to K$ is said to be asymptotically nonexpansive mapping if there exists a sequence $\{k_n\}$ with $k_n\ge 1$ and $\lim_{n\to\infty} k_n=1$ such that $\|T^nx- T^ny\|\le k_n\|x-y\|$ for all $x,y\in K$ and for all $n\in\bbfN$. In this paper, if $K$ is a nonempty closed convex subset of $X$ and $T: K\to K$ is a nonexpansive asymptotically mapping with a nonempty fixed point set, weak and strong convergence theorems for the iterative approximation of fixed points of $T$ are proved. Furthermore, the results by this paper show that the boundedness requirement imposed on the subset $K$ in recent results by {\it Z. Huang} [Comput. Math. Appl. 37, No. 3, 1-7 (1999; Zbl 0942.47046)]; {\it B. E. Rhoades} [J. Mat. Anal. Appl. 183, No. 1, 118-120 (1994; Zbl 0807.47045)]; {\it J. Schu} [J. Math. Anal. Appl. 158, No. 2, 407-413 (1991; Zbl 0734.47036); Bull. Aust. Math. Soc. 43, No. 1, 153-159 (1991; Zbl 0709.47051)], can be dropped. The main results are the following: Theorem 1: Let $E$ be a uniformly convex Banach space sastisfying Opial’s condition and let $K$ be a nonempty closed convex subset of $E$. Let $T: K\to K$ be an asymptotically nonexpansive mapping with $F(T)\ne \emptyset$ and sequence $\{k_n\}\subset [1,\infty)$ such that $\lim k_n= 1$ and $\sum^\infty_{n=1} (k_n-1)< \infty$. Let $\{u_n\}$ and $\{v_n\}$ be bounded sequences in $K$ and let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a_n'\}$, $\{b_n'\}$ and $\{c_n'\}$ be real sequence in $[0,1]$ satisfying the conditions: (i) $a_n+ b_n+ c_n= a_n'+ b_n'+ c_n'= 1$, $\forall n\ge 1$; (ii) $a< a_n< b_n'< b< 1$, $\forall n\ge 1$; (iii) $\lim b_n= 0$; (iv) $\sum^\infty_{n=1} e_{n}<\infty$, $\sum^\infty_{n=1} c_n'< \infty$. Then the sequence generated from an arbitrary $x_1\subset K$ by $y_n= a_n x_n+ b_n T^n x_n+ c_n u_n$, $n\ge 1$, $x_{n+1}= a_n' x_n+ b_n'T^n y_n+ c_n'v_n$, $n\ge 1$ converges weakly to some fixed point of $T$. Theorem 2. Let $E$ be a uniformly convex Banach space and $K$ a nonempty closed subset of $E$. Let $T: K\to K$ be an asymptotically nonexpansive mapping with $F(T)\ne \emptyset$ and sequence $\{k_n\}\subset [1,\infty)$ such that $\lim k_n=1$ and $\sum^\infty_{n=1} (k_n-1)<\infty$. Suppose $T^n$ is compact for some $m\in\bbfN$. Let $\{u_n\}$ and $\{v_n\}$ be bounded sequence in $K$ and let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a_n'\}$, $\{b_n'\}$ and $\{c_n'\}$ be as in Theorem 1. Then the sequence $\{x_n\}$ generate from an arbitrary $x_1\in K$ as in Theorem 1 converges strongly to some fixed point of $T$.
Reviewer: V.Popa (Bacau)

##### MSC:
 47H09 Mappings defined by “shrinking” properties 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations) 46B20 Geometry and structure of normed linear spaces
Full Text:
##### References:
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