Strong convergence theorems of relaxed hybrid steepest-descent methods for variational inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is eta-strongly monotone and n-Lipschitzian on a noneimpty 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 steep est-descent method which generates an iterative sequence {x(n)} from an arbitrary initial point x(0) is an element of H. The sequence {x(n)} is shown to converge in norm to the unique solution u* of the variational inequality < F(u*), v-u*> >= 0 for all(v) is an element of C under the conditions which are more general than those in Ref. 19. Applications to constrained generalized pseudoinverse are included.