Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 2, Pages 143-148
doi:10.1155/S1048953304305010
Partially relaxed cocoercive variational inequalities and auxiliary problem principle
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.
Abstract
Let T:K→H be a mapping from a nonempty closed convex subset K of a finite-dimensional Hilbert space H into H. Let f:K→ℝ be proper, convex, and lower semicontinuous on K and let h:K→ℝ 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*),x−x*〉+f(x)−f(x*)≥0 for all x∈K.