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.
Abstract
Let T:K→H be a nonlinear mapping from a nonempty
closed invex subset K of an infinite-dimensional Hilbert space
H into H. Let f:K→R be proper, invex, and lower
semicontinuous on K and let h:K→R be continuously Fréchet-differentiable on K with h′, the gradient of h, (η,α)-strongly monotone, and (η,β)-Lipschitz continuous on K. Suppose that
there exist an x*∈K, and numbers a>0, r≥0, ρ(a<p<α) such that for all t∈[0,1]
and for all x∈K∗, the set S∗ defined by S∗={(h,η):h′(x∗+t(x−x∗))(x−x∗)≥〈h′(x∗+tη(x,x∗)),η(x,x∗)〉} is nonempty, where K∗={x∈K:‖x−x∗‖≤r} and η:K×K→H is (λ)-Lipschitz continuous with the following assumptions. (i)
η(x,y)+η(y,x)=0,η(x,y)=η(x,z)+η(z,y), and ‖η(x,y)‖≤r. (ii) For each fixed y∈K, map x→η(y,x) is sequentially continuous from the weak
topology to the weak topology. If, in addition, h′ is continuous from H equipped with weak topology to H equipped with strong topology, then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x∗ of the variational inequality problem (VIP): 〈T(x∗),η(x,x∗)〉+f(x)−f(x∗)≥0 for all x∈K.