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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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