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An implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1172.47058
The authors consider a nonempty closed convex subset $K$ of a reflexive Banach space $E$ with a weakly continuous dual mapping, and ${\left\{{T}_{i}\right\}}_{i=1}^{\infty }$, an infinite family of asymptotically nonexpansive mappings with the sequence $\left\{{k}_{in}\right\}$ satisfying ${k}_{in}\ge 1$ for each $i=1,2\cdots$, $n=1,2,\cdots$, and ${lim}_{n\to \infty }{k}_{in}=1$ for each $i=1,2,\cdots$. They introduce a new implicit iterative scheme generated by ${\left\{{T}_{i}\right\}}_{i=1}^{\infty }$ and prove that the scheme converges strongly to a common fixed point of ${\left\{{T}_{i}\right\}}_{i=1}^{\infty }$, which solves a certain variational inequality.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 49J40 Variational methods including variational inequalities 47H05 Monotone operators (with respect to duality) and generalizations 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J20 Inequalities involving nonlinear operators
##### References:
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