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Towards viscosity approximations of hierarchical fixed-point problems. (English) Zbl 1143.47305
Summary: We introduce methods which seem to be a new and promising tool in hierarchical fixed-point problems. The goal of this note is to analyze the convergence properties of these new types of approximating methods for fixed-point problems. The limit attained by these curves is the solution of the general variational inequality $$0\in (I - Q)x_{\infty }+N_{\text{Fix}\,P}(x_{\infty })$$, where $$N_{\text{Fix\,}P}$$ denotes the normal cone to the set of fixed point of the original nonexpansive mapping $$P$$ and $$Q$$ a suitable nonexpansive mapping criterion. The link with other approximation schemes in this field is also made.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J20 Variational and other types of inequalities involving nonlinear operators (general)
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