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Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings. (English) Zbl 1073.47059

Let \(E\) be a real Banach space, \(K\) be a nonempty closed convex and bounded subset of \(E\) and \(T:K \to K\) be an asymptotically nonexpansive operator. Let \(u \in K\) and \(\alpha _n \in (0,1), \;n \in \mathbb N\). The authors give conditions on \(E\), on \(\alpha _n\) and \(T\) so that the sequence \((z_n)_{n\in \mathbb N}\) defined by \(z_{n+1}= (1-\alpha _n)u+ \alpha _n T^n z_n\), \(n\in \mathbb N\), converges strongly to a fixed point of \(T\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
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