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

摘要

Assume that F is a nonlinear operator on a real Hilbert space H which is strongly monotone and Lipschitzian with constants 77 > 0 and k > 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) is an element of H. We characterize the weak convergence of {u(n)} to the unique solution u* of the variational inequality: < F(u*), v - u*> >= 0 for all v is an element of C. Applications to constrained generalized pseudoinverse are included.