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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 \[ x_{n+ 1}= t_ n T^ n x_ n+ (1- t_ n) x_ n\quad\text{and}\quad x_{n+ 1}= t_ n T^ n (s_ n T^ n x_ n+ (1- s_ n) x_ n)+ (1- t_ n) x_ n, \] respectively, converge weakly to a fixed point of \(T\).

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
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