Strong Weak Convergence Theorems of Implicit Hybrid Steepest-descent Methods for Variational Inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is strongly monotone and Lipschitzian with constants η > 0 and κ > 0, respectively 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 an implicit hybrid steepest-descent method which generates an iterative sequence {un} from an arbitrary initial point u0 ∈ H . We characterize the weak convergence of {un} to the unique solution u∗ of the variational inequality: 〈F (u∗), v − u∗〉 ≥ 0 ∀v ∈ C. Applications to constrained generalized pseudoinverse are included.