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Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. (English) Zbl 1095.47046

Let \(E\) be a uniformly convex Banach space, \(C\) a closed convex subset of \(E\) and \(\{T_{i}\}_{i=1,N}\) a finite family of uniformly \(L\)-Lipschitzian asymptotically quasi-nonexpansive self mappings of \(C\). Under some additional assumptions, it is proven that the sequence \(\{x_{n}\}\) defined by \(x_{n}=\alpha_{n}x_{n-1}+(1-\alpha_{n})T_{i}^{k}x_{n},\;n\geq 1\), where \(n=(k-1)N+i\), \(i\in \{1,2,...,N\}\), and \(\{\alpha _{n}\}\) is a real sequence in \((0,1)\), converges strongly to a common fixed point of the mappings \(\{T_{i}\}_{i=1,N}\) provided that one mapping in the family is semi-compact. Other related results are also considered.

MSC:

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