## Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps.(English)Zbl 1126.65054

Let $$E$$ be a real Banach space and $$K$$ be a nonempty closed convex part of it. The map $$T:K\to K$$ is called generalized asymptotically quasi-nonexpansive if $$F(T)\neq \emptyset$$ and
$\| T^nx-x^*\| \leq (1+u_n)\| x-x^*\| +c_n,\quad n\geq 1,\;x\in K,\;x^*\in F(T),$ where $$(u_n)$$ and $$(c_n)$$ are sequences converging to zero. Let $$(T_i; i=1,\dots,N)$$ be a family of such maps. Sufficient conditions are given so that the implicit iterative scheme $x_n=\alpha_nx_{n-1}+(1-\alpha_n)T_i^kx_n, n\geq 1;\;x_0\in K$
where $$n=(k-1)N+i$$, $$T_n=T_{n\pmod N}=T_i$$, $$i\in \{1,\dots,N\}$$, should strongly converge to a common fixed point of the family $$(T_i; i=1,\dots,N)$$.
This, in particular, answers affirmatively a question raised by H.-K. Xu and R. G. Ori [Numer. Funct. Anal. Optim. 22, 767–773 (2001; Zbl 0999.47043)].

### MSC:

 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.

Zbl 0999.47043
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### References:

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