## Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications.(English)Zbl 1113.47055

Let $$C$$ be a nonempty closed convex subset of a uniformly convex Banach space $$E$$. A mapping $$T:C\rightarrow C$$ is called uniformly $$L$$-Lipschitzian on $$C$$ if, for some $$L>0$$, $\left\| T^n x-T^n y\right\| \leq L \left\| x-y\right\|$ for all $$x,y \in C$$ and for all $$n=1,2,3\dots$$. Denote by $$\text{Fix}(T)$$ the set of all fixed points of $$T$$. If $$\text{Fix}(T)\neq \varnothing$$, then $$T$$ is called asymptotically quasi-nonexpansive if, for a sequence $$\{k_n\}\subset [0,\infty)$$ with $$\lim\limits_{n\rightarrow \infty} k_n=0$$, we have $\left\| T^n x-p\right\| \leq (1+k_n) \left\| x-p\right\|$ for all $$x \in C$$, $$p \in \text{Fix}(T)$$ and for all $$n=1,2,3\dots$$. It is proved (Theorem 2) that if $$S,T: C\rightarrow C$$ are uniformly $$L$$-Lipschitzian and asymptotically quasi-nonexpansive mappings which satisfy a certain condition (A’), then a two-step iteration process of Ishikawa type associated with $$S$$ and $$T$$ converges to a common fixed point of $$S$$ and $$T$$.

### 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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