Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 634921, 13 pages
Research Article

The Solvability of a Class of General Nonlinear Implicit Variational Inequalities Based on Perturbed Three-Step Iterative Processes with Errors

Zeqing Liu,1 Shin Min Kang,2 and Jeong Sheok Ume3

1Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaoning 116029, China
2Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, South Korea
3Department of Applied Mathematics, Changwon National University, Changwon 641-733, South Korea

Received 23 October 2007; Accepted 25 January 2008

Academic Editor: Mohammed Khamsi

Copyright © 2008 Zeqing Liu et al. 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.


We introduce and study a new class of general nonlinear implicit variational inequalities, which includes several classes of variational inequalities and variational inclusions as special cases. By applying the resolvent operator technique and fixed point theorem, we suggest a new perturbed three-step iterative algorithm with errors for solving the class of variational inequalities. Several existence and uniqueness results of solutions for the general nonlinear implicit variational inequalities, and convergence and stability results of the sequence generated by the algorithm are obtained. The results presented in this paper extend, improve, and unify a host of results in recent literatures.