International Journal of Mathematics and Mathematical Sciences 
Volume 2003 (2003), Issue 43, Pages 2759-2770
doi:10.1155/S0161171203210462

Forward-backward resolvent splitting methods for general mixed variational inequalities

Muhammad Aslam Noor,1 Muzaffar Akhter,1 and Khalida Inayat Noor2

 1Etisalat College of Engineering, P.O. Box 980, Sharjah, United Arab Emirates
2Department of Mathematics and Computer Science, College of Science, United Arab Emirates University, P.O. Box 17551, Al Ain, United Arab Emirates
Received 27 October 2002

Abstract

We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known three-step forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for solving various classes of variational inequalities and complementarity problems. Since general mixed variational inequalities include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.