STRONG WEAK CONVERGENCE THEOREMS OF IMPLICIT HYBRID STEEPEST-DESCENT METHODS FOR VARIATIONAL INEQUALITIES

Abstract

Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is strongly monotone and Lipschitzian with constants $\eta \gt 0$ and $\kappa \gt 0$, 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 $\{u_n\}$ from an arbitrary initial point $u_0\in H$. We characterize the weak convergence of $\{u_n\}$ to the unique solution $u^*$ of the variational inequality: $$\langle F(u^*),v-u^*\rangle\geq0\quad\forall v\in C.$$ Applications to constrained generalized pseudoinverse are included.