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Strong convergence theorems of relaxed hybrid steepest-descent methods for variational inequalities. (English) Zbl 1092.49013
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 develop a relaxed hybrid steepest-descent method which generates an iterative sequence $$\{x_n\}$$ from an arbitrary initial point $$x_0\in H$$. The sequence $$\{x_n\}$$ is shown to converge in norm to the unique solution $$u^*$$ of the variational inequality $\bigl\langle F(u^*),v-u^*\bigr\rangle\geq 0\quad\forall v\in C$ under the conditions which are more general than those in [H. K. Xu and T. H. Kim, J. Optimization Theory Appl. 119, No. 1, 185–201 (2003; Zbl 1045.49018)]. Applications to constrained generalized pseudoinverses are included.

##### MSC:
 49J40 Variational inequalities 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems
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