Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 90295, 12 pages

Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

Ram U. Verma

Department of Mathematics, University of Toledo, Toledo 43606, OH, USA

Received 30 April 2004; Accepted 29 August 2004

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 T:KH be a nonlinear mapping from a nonempty closed invex subset K of an infinite-dimensional Hilbert space H into H. Let f:KR be proper, invex, and lower semicontinuous on K and let h:KR 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, r0, ρ(a<p<α) such that for all t[0,1] and for all xK, the set S defined by S={(h,η):h(x+t(xx))(xx)h(x+tη(x,x)),η(x,x)} is nonempty, where K={xK:xxr} and η:K×KH 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 yK, map xη(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 xK.