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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 768272, 10 pages
http://dx.doi.org/10.1155/2012/768272
Research Article

Iterative Algorithms for General Multivalued Variational Inequalities

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Mathematics Department, College of Science, King Saud University, Riyadh 1145, Saudi Arabia
3Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
4Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
5Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 21 October 2011; Accepted 1 November 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Yonghong Yao 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.

Linked References

  1. L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112–1123, 2011. View at Publisher · View at Google Scholar
  2. L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. S. Chang, H. W. J. Lee, C. K. Chan, and J. A. Liu, “A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6830–6837, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. 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, vol. 2009, Article ID 208692, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1-2 of Springer Series in Operations Research, Springer, New York, NY, USA, 2003.
  6. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984.
  7. B. S. He, “A new method for a class of linear variational inequalities,” Mathematical Programming, vol. 66, no. 2, pp. 137–144, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. N. Iusem and B. F. Svaiter, “A variant of Korpelevich's method for variational inequalities with a new search strategy,” Optimization, vol. 42, no. 4, pp. 309–321, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. E. N. Khobotov, “Modification of the extra-gradient method for solving variational inequalities and certain optimization problems,” USSR Computational Mathematics and Mathematical Physics, vol. 27, no. 5, pp. 120–127, 1987.
  11. G. M. Korpelevich, “An extragradient method for finding saddle points and for other problems,” Ekonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
  12. P. Kumam, N. Petrot, and R. Wangkeeree, “Existence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7496–7503, 2011. View at Publisher · View at Google Scholar
  13. X. Lu, H. K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1032–1041, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. Marino, L. Muglia, and Y. Yao, “Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points,” Nonlinear Analysis, Theory, Methods and Applications. In press. View at Publisher · View at Google Scholar
  15. M. A. Noor, On Variational Inequalities, Brunel University, London, UK, 1975.
  16. M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. A. Noor, “Wiener-Hopf equations and variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, no. 1, pp. 197–206, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. M. A. Noor, “Differentiable non-convex functions and general variational inequalities,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 623–630, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. A. Noor and E. A. Al-Said, “Wiener-Hopf equations technique for quasimonotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 103, no. 3, pp. 705–714, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. A. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. P. Shi, “Equivalence of variational inequalities with Wiener-Hopf equations,” Proceedings of the American Mathematical Society, vol. 111, no. 2, pp. 339–346, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,” SIAM Journal on Control and Optimization, vol. 37, no. 3, pp. 765–776, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Academie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH
  25. R. U. Verma, “Projection methods, algorithms, and a new system of nonlinear variational inequalities,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025–1031, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. Y. Yao, R. Chen, and H. K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3447–3456, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. 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. In press. View at Publisher · View at Google Scholar
  30. Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extra-gradient method to the 4 minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar
  31. Y. Yao, M. A. Noor, K. I. Noor, Y. C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211–1224, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert 5 spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 79, 2011.
  33. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters. In press. View at Publisher · View at Google Scholar