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Strong convergence of an iterative method for hierarchical fixed-point problems. (English) Zbl 1158.47057
The authors introduce a viscosity type iterative algorithm for finding hierarchically a solution (which is generally nonunique) of the variational problem to find $$\tilde{x} \in \operatorname{Fix}(T)$$ so that $$\langle \tilde{x}-P(\tilde{x}), x-\tilde{x})\rangle \geq 0$$ for all $$x\in \operatorname{Fix}(T)$$ by using its equivalent fixed point formulation to find $$\tilde{x}\in D$$ so that $$\tilde{x}= \operatorname{proj}_{\operatorname{Fix}(T)}\circ P(\tilde{x})$$, in the case where $$H$$ is a Hilbert space, $$P$$ and $$T$$ are two nonexpansive mappings on a closed convex subset $$D$$ of $$H$$, and $$\operatorname{proj}_{\operatorname{Fix}(T)}$$ denotes the metric projection on the set of fixed points of $$T$$.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 65J15 Numerical solutions to equations with nonlinear operators
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