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 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 develop a relaxed hybrid steepest-descent method which generates an iterative sequence {x_n} from an arbitrary initial point x₀ ∈ 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 ∀v ∈ C under the conditions which are more general than those in Ref. 19. Applications to constrained generalized pseudoinverse are included.