A new class of iterative algorithms for approximation-solvability of nonlinear variational inequalities
Consider the convergence of the projection methods based on a new iterative algorithm for the approximation-solvability of the following class of nonlinear variational inequality (NVI) problems: find an element x* is an element of K such that [T(x*), x - x*] greater than or equal to 0, for all x is an element of K, where T: K --> H is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. The new iterative procedure is characterized as a nonlinear variational inequality (for any arbitrarily chosen initial point x(0) is an element of K, and for constants rho > 0 and beta > 0) [rhoT (y(k)) + x(k+1) - y(k), x - x(k+1)] greater than or equal to 0; for all x is an element of K, and for k greater than or equal to 0, where [betaT (x(k)) + y(k) - x(k), x - y(k)] greater than or equal to 0, for all x is an element of K. This nonlinear Variational inequality type algorithm has an equivalent projection formula x(k+1) = P-K [y(k) - rhoT (y(k))], where y(k) = P-K [x(k) - betaT (x(k))], for the projection P-K onto K. (C) 2001 Elsevier Science Ltd. All rights reserved.
iterative algorithms nonlinear variational inequalities solvability