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Zbl 1095.47046
Sun, Zhao-hong
Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings.
(English)
[J] J. Math. Anal. Appl. 286, No. 1, 351-358 (2003). ISSN 0022-247X

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.
[Vasile Berinde (Baia Mare)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: uniformly convex Banach space; asymptotically quasi-nonexpansive mapping; common fixed point; strong convergence; implicit iterative process

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