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Zbl 0971.47038
Osilike, M.O.; Aniagbosor, S.C.
Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings.
(English)
[J] Math. Comput. Modelling 32, No.10, 1181-1191 (2000). ISSN 0895-7177

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.\par 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:\par 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:\par (i) $a_n+ b_n+ c_n= a_n'+ b_n'+ c_n'= 1$, $\forall n\ge 1$;\par (ii) $a< a_n< b_n'< b< 1$, $\forall n\ge 1$;\par (iii) $\lim b_n= 0$;\par (iv) $\sum^\infty_{n=1} e_{n}<\infty$, $\sum^\infty_{n=1} c_n'< \infty$.\par 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 $T$.\par 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 $T$.
[V.Popa (Bacau)]
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J25 Methods for solving nonlinear operator equations (general)
46B20 Geometry and structure of normed spaces

Keywords: uniformly convex Banach space; asymptotically nonexpansive mapping; iterative approximation of fixed points; Opial's condition

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