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

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
49J40Variational methods including variational inequalities
65J15Equations with nonlinear operators (numerical methods)
90C47Minimax problems
Full Text: DOI
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