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Zbl 0709.47051
Schu, J.
Weak and strong convergence to fixed points of asymptotically nonexpansive mappings.
(English)
[J] Bull. Aust. Math. Soc. 43, No.1, 153-159 (1991). ISSN 0004-9727

One of the main results is: Let E be a uniformly convex Banach space satisfying Opial's condition, $\emptyset \ne A\subset E$ closed bounded and convex and T: $A\to A$ asymptotically nonexpansive with sequence $(k\sb n)\in [1,\infty)\sp N$ for which $\sum\sp{\infty}\sb{n=1}(k\sb n- 1)<\infty$. Suppose that $x\sb 1\in A$ and $(\alpha\sb n)\in [0,1]\sp N$ is bounded away. Then the sequence $(x\sb n)$ given by $x\sb{n+1}=\alpha\sb nT\sp n(x\sb n)+(1-\alpha\sb n)x\sb n$ converges weakly to some fixed point of T. \par Two similar results are also obtained concerning the strong convergence of the sequence $(x\sb n)$ to a fixed point of T.
[S.L.Singh]
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties
47J25 Methods for solving nonlinear operator equations (general)

Keywords: uniformly convex Banach space satisfying Opial's condition; asymptotically nonexpansive

Cited in: Zbl 1117.47055 Zbl 0971.47038

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