Journal of Applied Mathematics and Stochastic Analysis, :1 (2002), 89-90.15 SHORT REPORTS AND COMMUNICATIONS
Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company 89
A GENERALIZATION OF AUXILIARY PROBLEM PRINCIPLE WITH APPLICATIONS
TO VARIATIONAL INEQUALITIES
RAM U. VERMA
University of Toledo Department of Mathematics
Toledo, Ohio 43606 USA E-mail: [email protected]
We announce the approximation-solvability of the following class of nonlinear variational inequality (NVI) problems based on a new generalized auxiliary problem principle:
Find an element such that
for all
where is a mapping from a nonempty closed convex subset of a real Hilbert space into , and is a continuous convex functional on .
The generalized auxiliary problem principle is described as follows: for a given iterate and, for constants and , compute such that
, for all and for , where
for all where is twice Frechet-differential functional on .
Theorem: Let be a real Hilbert space and a - -partially relaxed mono- tone mapping from a nonempty closed convex subset of into . Let be twice continuous Frechet-differentiable on with the following assumptions
and
Then for any fixed solution of the NVI problem, the sequence ! is bounded and converges to for
" " # .
R
eferences
[1] Cohen, G., Auxiliary problem principle extended to variational inequalities, J. Optim.
Theo. Appl. :2 (1988), 325-333.59
90 RAM U. VERMA
[2] Dunn, J.C., Convexity, monotonicity and gradient processes in Hilbert spaces, J. Math.
Anal. Appl. (1976), 145-158.53
[3] Verma, R.U., Nonlinear variational and constrained hemivariational inequalities in- volving relaxed operators, ZAMM :5 (1997), 387-391.77
[4] Verma, R.U., Approximation-solvability of nonlinear variational inequalities involving partially relaxed monotone (prm) mappings, Adv. Nonl. Variat. Ineq. :2 (1999), 137-2 148.
