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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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