Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 582792, 17 pages
http://dx.doi.org/10.1155/2012/582792
Research Article

Well-Posedness of Generalized Vector Quasivariational Inequality Problems

School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 28 October 2011; Accepted 14 December 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Jian-Wen Peng and Fang Liu. 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. F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J. L. Lions, Eds., pp. 151–186, John Wiley & Sons, New York, NY, USA, 1980. View at Google Scholar · View at Zentralblatt MATH
  2. G. Y. Chen and G. M. Cheng, “Vector variational inequalities and vector optimization,” in Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–456, 1987. View at Google Scholar
  3. G. Y. Chen, “Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem,” Journal of Optimization Theory and Applications, vol. 74, no. 3, pp. 445–456, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. G. Y. Chen and X. Q. Yang, “The vector complementary problem and its equivalences with the weak minimal element in ordered spaces,” Journal of Mathematical Analysis and Applications, vol. 153, no. 1, pp. 136–158, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. G.-Y. Chen and B. D. Craven, “Approximate dual and approximate vector variational inequality for multiobjective optimization,” Journal of the Australian Mathematical Society A, vol. 47, no. 3, pp. 418–423, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. G.-Y. Chen and B. D. Craven, “A vector variational inequality and optimization over an efficient set,” Zeitschrift für Operations Research. Mathematical Methods of Operations Research, vol. 34, no. 1, pp. 1–12, 1990. View at Google Scholar · View at Zentralblatt MATH
  7. A. H. Siddiqi, Q. H. Ansari, and A. Khaliq, “On vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 84, no. 1, pp. 171–180, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. Q. Yang, “Vector complementarity and minimal element problems,” Journal of Optimization Theory and Applications, vol. 77, no. 3, pp. 483–495, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. X. Q. Yang, “Vector variational inequality and its duality,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 21, no. 11, pp. 869–877, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. X. Q. Yang, “Generalized convex functions and vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, no. 3, pp. 563–580, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. J. Yu and J. C. Yao, “On vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 89, no. 3, pp. 749–769, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. G. M. Lee, D. S. Kim, B. S. Lee, and S. J. Cho, “Generalized vector variational inequality and fuzzy extension,” Applied Mathematics Letters, vol. 6, no. 6, pp. 47–51, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. G. M. Lee, D. S. Kim, and B. S. Lee, “Generalized vector variational inequality,” Applied Mathematics Letters, vol. 9, no. 1, pp. 39–42, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. K. L. Lin, D.-P. Yang, and J. C. Yao, “Generalized vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 92, no. 1, pp. 117–125, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. I. V. Konnov and J. C. Yao, “On the generalized vector variational inequality problem,” Journal of Mathematical Analysis and Applications, vol. 206, no. 1, pp. 42–58, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. A. Daniilidis and N. Hadjisavvas, “Existence theorems for vector variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 54, no. 3, pp. 473–481, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. Q. Yang and J. C. Yao, “Gap functions and existence of solutions to set-valued vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 115, no. 2, pp. 407–417, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. W. Oettli and D. Schläger, “Existence of equilibria for monotone multivalued mappings,” Mathematical Methods of Operations Research, vol. 48, no. 2, pp. 219–228, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. Y. Chen and S. J. Li, “Existence of solutions for a generalized vector quasivariational inequality,” Journal of Optimization Theory and Applications, vol. 90, no. 2, pp. 321–334, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. G. M. Lee, B. S. Lee, and S.-S. Chang, “On vector quasivariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 203, no. 3, pp. 626–638, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. A. N. Tykhonov, “On the stability of the functional optimization problem,” Computational Mathematics and Mathematical Physics, vol. 6, pp. 28–33, 1966. View at Publisher · View at Google Scholar
  22. E. S. Levitin and B. T. Polyak, “Convergence of minimizing sequences in conditional extremum problem,” Soviet Mathematics Doklady, vol. 7, pp. 764–767, 1966. View at Google Scholar
  23. A. S. Konsulova and J. P. Revalski, “Constrained convex optimization problems—well-posedness and stability,” Numerical Functional Analysis and Optimization, vol. 15, no. 7-8, pp. 889–907, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. X. X. Huang and X. Q. Yang, “Generalized Levitin-Polyak well-posedness in constrained optimization,” SIAM Journal on Optimization, vol. 17, no. 1, pp. 243–258, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. X. X. Huang and X. Q. Yang, “Levitin-Polyak well-posedness of constrained vector optimization problems,” Journal of Global Optimization, vol. 37, no. 2, pp. 287–304, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. X. X. Huang, X. Q. Yang, and D. L. Zhu, “Levitin-Polyak well-posedness of variational inequality problems with functional constraints,” Journal of Global Optimization, vol. 44, no. 2, pp. 159–174, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. X. X. Huang and X. Q. Yang, “Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints,” Journal of Industrial and Management Optimization, vol. 3, no. 4, pp. 671–684, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. B. Jiang, J. Zhang, and X. X. Huang, “Levitin-Polyak well-posedness of generalized quasivariational inequalities with functional constraints,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 4, pp. 1492–1503, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. Z. Xu, D. L. Zhu, and X. X. Huang, “Levitin-Polyak well-posedness in generalized vector variational inequality problem with functional constraints,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 505–524, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. J. Zhang, B. Jiang, and X. X. Huang, “Levitin-Polyak well-posedness in vector quasivariational inequality problems with functional constraints,” Fixed Point Theory and Applications, vol. 2010, Article ID 984074, 16 pages, 2010. View at Google Scholar · View at Zentralblatt MATH
  31. Y.-P. Fang, N.-J. Huang, and J.-C. Yao, “Well-posedness by perturbations of mixed variational inequalities in Banach spaces,” European Journal of Operational Research, vol. 201, no. 3, pp. 682–692, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. L. Q. Anh, P. Q. Khanh, D. T. M. Van, and J.-C. Yao, “Well-posedness for vector quasiequilibria,” Taiwanese Journal of Mathematics, vol. 13, no. 2, pp. 713–737, 2009. View at Google Scholar · View at Zentralblatt MATH
  33. L. C. Ceng, N. Hadjisavvas, S. Schaible, and J. C. Yao, “Well-posedness for mixed quasivariational-like inequalities,” Journal of Optimization Theory and Applications, vol. 139, no. 1, pp. 109–125, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. L.-H. Peng, C. Li, and J.-C. Yao, “Well-posedness of a class of perturbed optimization problems in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 346, no. 2, pp. 384–394, 2008. View at Publisher · View at Google Scholar
  35. L. C. Ceng and J. C. Yao, “Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 12, pp. 4585–4603, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. Y.-P. Fang, N.-J. Huang, and J.-C. Yao, “Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems,” Journal of Global Optimization, vol. 41, no. 1, pp. 117–133, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. A. Petruşel, I. A. Rus, and J.-C. Yao, “Well-posedness in the generalized sense of the fixed point problems for multivalued operators,” Taiwanese Journal of Mathematics, vol. 11, no. 3, pp. 903–914, 2007. View at Google Scholar · View at Zentralblatt MATH
  38. D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989.
  39. M. Furi and A. Vignoli, “About well-posed minimization problems for functionals in metric space,” Journal of Optimization Theory and Applications, vol. 5, pp. 225–229, 1970. View at Publisher · View at Google Scholar
  40. K. Kuratowski, Topology, vol. 1, Academic Press, New York, NY, USA, 1966.
  41. K. Kuratowski, Topology, vol. 2, Academic Press, New York, NY, USA, 1968.