Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 2, Pages 143-148

Partially relaxed cocoercive variational inequalities and auxiliary problem principle

Ram U. Verma

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

Received 30 May 2003; Revised 25 November 2003

Copyright © 2004 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 mapping from a nonempty closed convex subset K of a finite-dimensional Hilbert space H into H. Let f:K be proper, convex, and lower semicontinuous on K and let h:K be continuously Frećhet-differentiable on K with h (gradient of h), α-strongly monotone, and β-Lipschitz continuous on K. Then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x* of the variational inequality problem (VIP) described as follows: find an element x*K such that T(x*),xx*+f(x)f(x*)0 for all xK.