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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 303569, 16 pages
doi:10.1155/2012/303569
Review Article

Some Aspects of Extended General Variational Inequalities

Muhammad Aslam Noor

Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

Received 1 December 2011; Accepted 23 December 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Muhammad Aslam Noor. 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. 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 · View at MathSciNet
  2. E. A. Youness, “E-convex sets, E-convex functions, and E-convex programming,” Journal of Optimization Theory and Applications, vol. 102, no. 2, pp. 439–450, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  3. M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. A. Noor, K. I. Noor, S. Zainab, and E. Al-Said, “Regularized mixed variational-like inequalities,” Journal of Applied Mathematics. In press.
  5. A. Bnouhachem and M. A. Noor, “Inexact proximal point method for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1195–1212, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. Cristescu and L. Lupşa, Non-Connected Convexities and Applications, vol. 68, Kluwer Academic, Dodrecht, The Netherlands, 2002.
  7. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 31, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2000.
  8. G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
  9. Q. Liu and J. Cao, “A recurrent neural network based on projection operator for extended general variational inequalities,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 3, Article ID 5339227, pp. 928–938, 2010. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
  10. Q. Liu and Y. Yang, “Global exponential system of projection neural networks for system of generalized variational inequalities and related nonlinear minimax problems,” Neurocomputing, vol. 73, no. 10–12, pp. 2069–2076, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Aslam 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 · View at MathSciNet
  12. M. A. Noor, “Iterative schemes for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 121, no. 2, pp. 385–395, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182–186, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. A. Noor, “Auxiliary principle technique for extended general variational inequalities,” Banach Journal of Mathematical Analysis, vol. 2, no. 1, pp. 33–39, 2008. View at Zentralblatt MATH
  15. M. A. Noor, “Sensitivity analysis of extended general variational inequalities,” Applied Mathematics E-Notes, vol. 9, pp. 17–26, 2009. View at Zentralblatt MATH
  16. M. A. Noor, “Projection iterative methods for extended general variational inequalities,” Journal of Applied Mathematics and Computing, vol. 32, no. 1, pp. 83–95, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. A. Noor, “Solvability of extended general mixed variational inequalities,” Albanian Journal of Mathematics, vol. 4, no. 1, pp. 13–17, 2010.
  18. M. A. Noor, “Extended general quasi-variational inequalities,” Nonlinear Analysis Forum, vol. 15, pp. 33–39, 2010.
  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, “Some iterative methods for general nonconvex variational inequalities,” Mathematical and Computer Modelling, vol. 21, pp. 87–96, 2010.
  21. M. A. Noor, “Some classes of general nonconvex variational inequalities,” Albanian Journal of Mathematics, vol. 3, no. 4, pp. 175–188, 2009. View at Zentralblatt MATH
  22. M. A. Noor, “Nonconvex quasi variational inequalities,” Journal of Advanced Mathematical Studies, vol. 3, no. 1, pp. 59–72, 2010. View at Zentralblatt MATH
  23. M. A. Noor, “Projection methods for nonconvex variational inequalities,” Optimization Letters, vol. 3, no. 3, pp. 411–418, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. M. A. Noor, “Implicit iterative methods for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 143, no. 3, pp. 619–624, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. M. A. Noor, “Iterative methods for general nonconvex variational inequalities,” Albanian Journal of Mathematics, vol. 3, no. 3, pp. 117–127, 2009. View at Zentralblatt MATH
  26. M. A. Noor, “An extragradient algorithm for solving general nonconvex variational inequalities,” Applied Mathematics Letters, vol. 23, no. 8, pp. 917–921, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. M. A. Noor, “Some iterative methods for general nonconvex variational inequalities,” Mathematical and Computer Modelling, vol. 54, pp. 2953–2961, 2011.
  28. M. A. Noor and K. I. Noor, “Some iterative methods for solving general bifunction variational inequalities,” Journal of Advanced Mathematical Studies, vol. 20, no. 6, 2012. View at Publisher · View at Google Scholar
  29. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative projection methods for general nonconvex variational inequalities,” Applied and Computational Mathematics, vol. 10, no. 2, pp. 309–320, 2011. View at Zentralblatt MATH
  31. M. A. Noor , K. I. Noor, and E. Al-Said, “On new proximal pint methods for solving the variational inequalities,” Journal of Applied Mathematics, vol. 2012, Article ID 412413, 7 pages, 2012. View at Publisher · View at Google Scholar
  32. M. Aslam Noor, S. Ullah, K. Inayat Noor, and E. Al-Said, “Iterative methods for solving extended general mixed variational inequalities,” Computers & Mathematics with Applications, vol. 62, no. 2, pp. 804–813, 2011. View at Publisher · View at Google Scholar
  33. M. A. Noor, K. I. Noor, Y. Z. Huang, and E. Al-Said\, “Implicit schemes for solving extended general nonconvex variational inequalities,” Journal of Applied Mathematics, vol. 2012, Article ID 646259, 10 pages, 2012. View at Publisher · View at Google Scholar
  34. M. A. Noor, K. I. Noor, and E. Al-Said, “Resolvent iterative methods for solving system of extended general variational inclusions,” Journal of Inequalities and Applications, vol. 2011, Article ID 371241, 10 pages, 2011. View at Publisher · View at Google Scholar
  35. M. A. Noor, K. I. Noor, and E. Al-Said, “Auxiliary principle technique for solving bifunction variational inequalities,” Journal of Optimization Theory and Applications, vol. 149, no. 2, pp. 441–445, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65–69, 2012.
  37. M. A. Noor, K. I. Noor, and E. Al-Said, “Some iterative methods for trifunction equilibrium variational inequalities,” International Journal of the Physical Sciences, vol. 6, no. 22, pp. 5223–5229, 2011.
  38. M. A. Noor, S. Zainab, K. I. Noor, and E. Al-Said, “Mixed equilibrium problems,” International Journal of Physical Sciences, vol. 6, no. 23, pp. 5412–5418, 2011.
  39. M. Sun, “Merit functions and equivalent differentiable optimization problems for the extended general variational inequalities,” International Journal of Pure and Applied Mathematics, vol. 63, no. 1, pp. 39–49, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus Mathematique, vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH
  41. Y. Yao, M. A. Noor, Y. C. Liou, and S. M. Kang, “Iterative algorithms for general multivalued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768272, 10 pages, 2012. View at Publisher · View at Google Scholar
  42. Y. Zhao and D. Sun, “Alternative theorems for nonlinear projection equations and applications to generalized complementarity problems,” Nonlinear Analysis, vol. 46, no. 6, pp. 853–868, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet