Abstract and Applied Analysis
Volume 2013 (2013), Article ID 718624, 17 pages
http://dx.doi.org/10.1155/2013/718624
Research Article

Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

Zhao-Rong Kong,1 Lu-Chuan Ceng,2 Qamrul Hasan Ansari,3 and Chin-Tzong Pang4

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Department of Mathematics, Aligarh Muslim University, Aligarh 202 00, India
4Department of Information Management, Yuan Ze University, Chung Li 32003, Taiwan

Received 26 December 2012; Accepted 25 January 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Zhao-Rong Kong 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.

Abstract

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.