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Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. (English) Zbl 1137.47059
Summary: Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is $\eta$-strongly monotone and $\kappa$-Lipschitzian on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on $H$. We construct an iterative algorithm with variable parameters which generates a sequence ${x}_{n}$ from an arbitrary initial point ${x}_{0}H$. The sequence ${x}_{n}$ is shown to converge in norm to the unique solution ${u}^{}$ of the variational inequality $〈F\left({u}^{*}\right),v-{u}^{*}〉\ge 0,\phantom{\rule{1.em}{0ex}}\forall v\in C·$
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J20 Inequalities involving nonlinear operators 49J40 Variational methods including variational inequalities 65J15 Equations with nonlinear operators (numerical methods) 90C47 Minimax problems
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