Journal of Inequalities and Applications
Volume 5 (2000), Issue 4, Pages 407-418
doi:10.1155/S1025583400000229
A new approach to the extragradient method for nonlinear variational inequalities
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.
Abstract
The approximation-solvability of the following nonlinear variational inequality (NVI) problem is presented:
Determine an element x∗∈K such that
〈T(x∗),x−x∗〉≥0
for all
x∈K,
where T:K→H 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 x0∈K)
〈ρT(PK[xk−ρT(xk)])+xk+1−xk,x−xk+1〉≥0for all
x∈K
and for
k≥0,
which is equivalent to a double projection formula
xk+1=PK[xk−ρT(PK[xk−ρT(xk)])],
where PK denotes the projection of H onto K.