Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-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 devise an iterative algorithm which generates a sequence (xn) from an arbitrary initial point x0∈H. The sequence (xn) is shown to converge in norm to the unique solution u* of the variational inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\langle {F(u*),\user1{v} - u*} \right\rangle \geqslant 0$$ \end{document}Applications to constrained pseudoinverse are included.