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Zbl 1172.47058
Wang, Shenghua; Yu, Lanxiang; Guo, Baohua
(Wang, Sheng-hua; Yu, Lan-xiang; Guo, Bao-hua)
An implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces.
(English)
[J] Fixed Point Theory Appl. 2008, Article ID 350483, 10 p. (2008). ISSN 1687-1812/e

The authors consider a nonempty closed convex subset $K$ of a reflexive Banach space $E$ with a weakly continuous dual mapping, and $\{T_{i}\}_{i=1}^\infty$, an infinite family of asymptotically nonexpansive mappings with the sequence $\{k_{in}\}$ satisfying $k_{in}\geq 1$ for each $i=1,2\dots$, $n=1,2,\dots$, and $\lim _{n\to\infty}k_{in}=1$ for each $i=1,2,\dots$. They introduce a new implicit iterative scheme generated by $\{ T_{i}\}_{i=1}^{\infty}$ and prove that the scheme converges strongly to a common fixed point of $\{ T_{i}\}_{i=1}^{\infty}$, which solves a certain variational inequality.
[Edward Prempeh (Kumasi)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
49J40 Variational methods including variational inequalities
47H05 Monotone operators (with respect to duality)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J20 Inequalities involving nonlinear operators

Keywords: reflexive Banach space; weakly continuous dual map; asymptotically nonexpansive mappings; implicit iterative scheme; common fixed point; variational inequality

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