## 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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### References:

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