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Strong convergence of the iterative methods for hierarchical fixed point problems of an infinite family of strictly nonself pseudocontractions. (English) Zbl 1253.65091
Summary: We deals with a new iterative algorithm for solving hierarchical fixed point problems of an infinite family of pseudocontractions in Hilbert spaces by ${y}_{n}={\beta }_{n}S{x}_{n}+\left(1-{\beta }_{n}\right){x}_{n},{x}_{n+1}={P}_{C}\left[{\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right){\sum }_{i=1}^{\infty }{\mu }_{i}^{\left(n\right)}{T}_{i}{y}_{n}\right]$, and $\forall n\ge 0$, where ${T}_{i}:C↦H$ is a nonself ${k}_{i}$-strictly pseudocontraction. Under certain approximate conditions, the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${x}^{*}\in {\bigcap }_{i=1}^{\infty }F\left({T}_{i}\right)$, which solves some variational inequality. The results here improve and extend some recent results.
MSC:
 65J15 Equations with nonlinear operators (numerical methods) 49J40 Variational methods including variational inequalities 47H10 Fixed point theorems for nonlinear operators on topological linear spaces