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On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. (English) Zbl 1086.47044

The authors study the weak and strong convergence of the implicit iteration sequences \(\{ x_{n}\} \) defined by: \[ \begin{aligned} x_{n}&= \alpha _{n}x_{n-1} + (1-\alpha _{n})T^{k(n)}_{i(n)}x_{n} + u_{n}, \quad n \geq 1\quad\text{and}\\ x_{n}&=\alpha _{n}x_{n-1} + (1- \alpha _{n})T^{k(n)}_{i(n)}x_{n},\quad n \geq 1\end{aligned} \] to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in real uniformly convex Banach spaces. Their results extend and improve results of several other authors.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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References:

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