Strong convergence of an iterative method for hierarchical fixed-point problems. (English) Zbl 1158.47057

Maingé, Paul-Emile; Moudafi, Abdellatif

Pac. J. Optim. 3, No. 3, 529-538 (2007).

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 \(T\).

Reviewer: Vasile Berinde (Baia Mare)

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

Keywords:

Hilbert space; nonexpansive mapping; fixed point; variational inequality; viscosity-like method; convergence theorem