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Journal of Function Spaces
Volume 2015, Article ID 980352, 22 pages
Research Article

Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

1Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
4Department of Mathematics, College of Science, University of Jeddah, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 28 October 2014; Accepted 5 January 2015

Academic Editor: Mohamed-Aziz Taoudi

Copyright © 2015 L. C. Ceng 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 analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.