Journal of Inequalities and Applications
Volume 5 (2000), Issue 4, Pages 407-418

A new approach to the extragradient method for nonlinear variational inequalities

Ram U. Verma

International Publications, Mathematical Sciences Division, 12046 Coed Drive, Suite A-29, Orlando, FL 32826, USA

Received 30 May 1999; Revised 22 July 1999

Copyright © 2000 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.


The approximation-solvability of the following nonlinear variational inequality (NVI) problem is presented:

Determine an element xK such that T(x),xx0forallxK, where T:KH is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. The iterative procedure is characterized as a nonlinear variational inequality (for any arbitrarily chosen initial point x0K) ρT(PK[xkρT(xk)])+xk+1xk,xxk+10forallxKandfork0, which is equivalent to a double projection formula xk+1=PK[xkρT(PK[xkρT(xk)])], where PK denotes the projection of H onto K.