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Fixed point iteration processes for asymptotically nonexpansive mappings. (English) Zbl 0820.47071
Summary: Let $X$ be a uniformly convex Banach space which satisfies Opial’s condition or has a Fréchet differentiable norm, $C$ a bounded closed convex subset of $X$, and $T: C\to C$ an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by $$x\sb{n+ 1}= t\sb n T\sp n x\sb n+ (1- t\sb n) x\sb n\quad\text{and}\quad x\sb{n+ 1}= t\sb n T\sp n (s\sb n T\sp n x\sb n+ (1- s\sb n) x\sb n)+ (1- t\sb n) x\sb n,$$ respectively, converge weakly to a fixed point of $T$.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
46B20Geometry and structure of normed linear spaces
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