Copyright © 2006 Ram U. Verma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a nonlinear mapping from a nonempty
closed invex subset of an infinite-dimensional Hilbert space
into . Let be proper, invex, and lower
semicontinuous on and let be continuously Fréchet-differentiable on with , the gradient of , -strongly monotone, and -Lipschitz continuous on . Suppose that
there exist an , and numbers , , such that for all
and for all , the set defined by is nonempty, where and is -Lipschitz continuous with the following assumptions. (i)
, and . (ii) For each fixed , map is sequentially continuous from the weak
topology to the weak topology. If, in addition, is continuous from equipped with weak topology to equipped with strong topology, then the sequence generated by the general auxiliary problem principle converges to a solution of the variational inequality problem (VIP): for all .