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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 949141, 15 pages
Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems
1Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Received 21 September 2011; Revised 16 November 2011; Accepted 17 November 2011
Academic Editor: Khalida Inayat Noor
Copyright © 2012 Yeong-Cheng Liou 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.
- M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,” Mathematical Programming, vol. 53, no. 1, pp. 99–110, 1992.
- F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, New York, NY, USA, Plenum Press, 1995.
- F. Giannessi, A. Maugeri, and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic, Dodrecht, The Netherlands, 2001.
- R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, Berlin, Germany, 1984.
- R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8, North-Holland, Amsterdam, The Netherlands, 1981.
- R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1989.
- D. Han and H. K. Lo, “Two new self-adaptive projection methods for variational inequality problems,” Computers & Mathematics with Applications, vol. 43, no. 12, pp. 1529–1537, 2002.
- P. T. Harker and J.-S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications,” Mathematical Programming, vol. 48, no. 2, pp. 161–220, 1990.
- Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.
- B. He, “Inexact implicit methods for monotone general variational inequalities,” Mathematical Programming, vol. 86, no. 1, pp. 199–217, 1999.
- J.-S. Pang and J. C. Yao, “On a generalization of a normal map and equation,” SIAM Journal on Control and Optimization, vol. 33, no. 1, pp. 168–184, 1995.
- J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994.
- Y. Yao and S. Maruster, “Strong convergence of an iterative algorithm for variational inequalities in Banach spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 325–329, 2011.
- Y. Yao, Y. -C. Liou, S. M. Kang, and Y. Yu, “Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces,” Nonlinear Analysis, vol. 74, no. 17, pp. 6024–6034, 2011.
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem,” Fixed Point Theory and Applications, Article ID 383740, 19 pages, 2010.
- M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004.
- M. A. Noor, “Differentiable non-convex functions and general variational inequalities,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 623–630, 2008.
- M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988.
- G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964.
- M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000.
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, Article ID 208692, 13 pages, 2009.
- Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis, vol. 72, no. 7-8, pp. 3447–3456, 2010.
- M. A. Noor, “Some algorithms for general monotone mixed variational inequalities,” Mathematical and Computer Modelling, vol. 29, no. 7, pp. 1–9, 1999.
- M. A. Noor and K. I. Noor, “Self-adaptive projection algorithms for general variational inequalities,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 659–670, 2004.
- M. A. Noor, K. I. Noor, and T. M. Rassias, “Some aspects of variational inequalities,” Journal of Computational and Applied Mathematics, vol. 47, no. 3, pp. 285–312, 1993.
- Y. Yao, Y. C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization.
- Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triplehierarchical constrained optimization problem,” Mathematical and Computer Modelling.
- Y. Yao, M. Aslam Noor, and Y. C. Liou, “Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, p. 9, 2012.
- Y. Yao, Y. C. Liou, C. L. Li, and H. T. Lin, “Extended extra-gradient methods for generalized variational inequalities,” Journal of Applied Mathematics, p. 14, 2012.
- Y. Yao, M. A. Noor, Y. C. Liou, and S. M. Kang, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, p. 10, 2012.
- Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters.
- Y. Yao, Y.-C. Liou, and S. M. Kang, “Algorithms construction for variational inequalities,” Fixed Point Theory and Applications, Article ID 794203, 12 pages, 2011.
- V. Colao, G. L. Acedo, and G. Marino, “An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2708–2715, 2009.
- G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
- M. Aslam Noor and Z. Huang, “Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 504–510, 2007.
- K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
- T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103–123, 2005.
- H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
- L. J. Zhang, J. M. Chen, and Z. B. Hou, “Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities,” Acta Mathematica Sinica, vol. 53, no. 4, pp. 691–698, 2010.