- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 907105, 19 pages
Hybrid Steepest Descent Viscosity Method 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, Aligarh Muslim University, Aligarh 202 002, India
3Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
Received 21 June 2012; Accepted 26 July 2012
Academic Editor: Jen-Chih Yao
Copyright © 2012 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.
- C. Baiocchi and A. Capelo, Variational and Quasi-variational Inequalities: Applications to Free Boundary Problems, John Wiley & Sons, Chichester, UK, 1984.
- F. Faccchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1, Springer, Berlin, Germany, 2003.
- F. Faccchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 2, Springer, Berlin, Germany, 2003.
- R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, The Netherlands, 1981.
- P. Jaillet, D. Lamberton, and B. Lapeyre, “Variational inequalities and the pricing of American options,” Acta Applicandae Mathematicae, vol. 21, no. 3, pp. 263–289, 1990.
- D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, NY, USA, 1980.
- I. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, Germany, 2001.
- J. T. Oden, Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.
- M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic, Dodrecht, The Netherlands, 1999.
- E. Zeidler, Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Applications, Springer, New York, NY, USA, 1985.
- A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
- A. Moudafi and P.-E. Maingé, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 95453, pp. 1–10, 2006.
- H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
- H.-K. Xu, “Viscosity method for hierarchical fixed point approach to variational inequalities,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 463–478, 2010.
- I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
- H. Iiduka, “Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem,” Nonlinear Analysis, vol. 71, no. 12, pp. e1292–e1297, 2009.
- H. Iiduka, “Iterative algorithm for solving triple-hierarchical constrained optimization problem,” Journal of Optimization Theory and Applications, vol. 148, no. 3, pp. 580–592, 2011.
- K. Geobel and W. A. Kirk, Topics on Metric Fixed-Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990.
- H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.