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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 $$\langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0, \quad \forall v \in C.$$

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47J20 Variational and other types of inequalities involving nonlinear operators (general) 49J40 Variational inequalities 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 90C47 Minimax problems in mathematical programming
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