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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 $\{T_{i}\}_{i=1}^\infty$, 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.

47J25Iterative procedures (nonlinear operator equations)
49J40Variational methods including variational inequalities
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
Full Text: DOI EuDML