Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 194509, 38 pages
http://dx.doi.org/10.1155/2012/194509
Research Article

Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

1Department of Mathematics, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China
2Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Received 24 September 2011; Accepted 3 November 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Lu-Chuan Ceng and Ching-Feng Wen. 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. A. N. Tikhonov, “On the stability of the functional optimization problem,” USSR Computational Mathematics and Mathematical Physics, vol. 6, no. 4, pp. 631–634, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. T. Zolezzi, “Well-posedness criteria in optimization with application to the calculus of variations,” Nonlinear Analysis, vol. 25, no. 5, pp. 437–453, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. T. Zolezzi, “Extended well-posedness of optimization problems,” Journal of Optimization Theory and Applications, vol. 91, no. 1, pp. 257–266, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. R. Lucchetti and F. Patrone, “A characterization of Tikhonov well-posedness for minimum problems, with applications to variational inequalities,” Numerical Functional Analysis and Optimization, vol. 3, no. 4, pp. 461–476, 1981. View at Publisher · View at Google Scholar
  5. E. Bednarczuk and J. P. Penot, “Metrically well-set minimization problems,” Applied Mathematics & Optimization, vol. 26, no. 3, pp. 273–285, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. A.L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, vol. 1543 of Lecture Notes in Math, Springer, Berlin, Germany, 1993.
  7. X. X. Huang, “Extended and strongly extended well-posedness of set-valued optimization problems,” Mathematical Methods of Operations Research, vol. 53, no. 1, pp. 101–116, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. 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 · View at Scopus
  9. 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 · View at Scopus
  10. 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 · View at Scopus
  11. L. C. Ceng and J. C. Yao, “Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 12, pp. 4585–4603, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. M. B. Lignola and J. Morgan, “Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution,” Journal of Global Optimization, vol. 16, no. 1, pp. 57–67, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. I. Del Prete, M. B. Lignola, and J. Morgan, “New concepts of well-posedness for optimization problems with variational inequality constraints,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 5, 2003. View at Google Scholar · View at Zentralblatt MATH
  14. M. B. Lignola and J. Morgan, “Approximating solutions and well-posedness for variational inequalities and Nash equilibria,” in Decision and Control in Management Science, pp. 367–378, Kluwer Academic, 2002. View at Google Scholar
  15. E. Cavazzuti and J. Morgan, “Well-posed saddle point problems,” in Optimization, Theory and Algorithms, J. B. Hirriart-Urruty, W. Oettli, and J. Stoer, Eds., pp. 61–76, Marcel Dekker, New York, NY, USA, 1983. View at Google Scholar · View at Zentralblatt MATH
  16. J. Morgan, “Approximations and well-posedness in multicriteria games,” Annals of Operations Research, vol. 137, no. 1, pp. 257–268, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. R. Lucchetti and J. Revalski, Eds., Recent Developments in Well-Posed Variational Problems, Kluwer Academic, Dodrecht, The Netherlands, 1995.
  18. M. Margiocco, F. Patrone, and L. Pusillo, “On the Tikhonov well-posedness of concave games and Cournot oligopoly games,” Journal of Optimization Theory and Applications, vol. 112, no. 2, pp. 361–379, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. Y. P. Fang, R. Hu, and N. J. Huang, “Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints,” Computers and Mathematics with Applications, vol. 55, no. 1, pp. 89–100, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. B. Lemaire, C. Ould Ahmed Salem, and J. P. Revalski, “Well-posedness by perturbations of variational problems,” Journal of Optimization Theory and Applications, vol. 115, no. 2, pp. 345–368, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. B. Lemaire, “Well-posedness, conditioning, and regularization of minimization, inclusion, and fixed point problems,” Pliska Studia Mathematica Bulgarica, vol. 12, pp. 71–84, 1998. View at Google Scholar
  22. H. Yang and J. Yu, “Unified approaches to well-posedness with some applications,” Journal of Global Optimization, vol. 31, no. 3, pp. 371–381, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. S. Schaible, J. C. Yao, and L. C. Zeng, “Iterative method for set-valued mixed quasi-variational inequalities in a Banach space,” Journal of Optimization Theory and Applications, vol. 129, no. 3, pp. 425–436, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. L. C. Zeng and J. C. Yao, “Existence of solutions of generalized vector variational inequalities in reflexive banach spaces,” Journal of Global Optimization, vol. 36, no. 4, pp. 483–497, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. L.C. Zeng, “Perturbed proximal point algorithm for generalized nonlinear set-valued mixed quasi-variational inclusions,” Acta Mathematica Sinica, vol. 47, no. 1, pp. 11–18, 2004. View at Google Scholar · View at Zentralblatt MATH
  26. H. Brezis, Operateurs Maximaux Monotone et Semigroups de Contractions dans les Es-Paces de Hilbert, North-Holland, Amsterdam, The Netherlands, 1973. View at Zentralblatt MATH
  27. S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969. View at Google Scholar
  28. K. Kuratowski, Topology, vol. 1-2, Academic Press, New York, NY, USA, 1968.
  29. E. Zeidler, Nonlinear Functional Analysis and Its Applications II: Monotone Operators, Springer, Berlin, Germany, 1985.
  30. S. Adly, E. Ernst, and M. Thera, “Well-positioned closed convex sets and well-positioned closed convex functions,” Journal of Global Optimization, vol. 29, no. 4, pp. 337–351, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  31. Y. R. He, X. Z. Mao, and M. Zhou, “Strict feasibility of variational inequalities in reflexive Banach spaces,” Acta Mathematica Sinica, vol. 23, no. 3, pp. 563–570, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. A. Brondsted and R. T. Rockafellar, “On the subdifferentiability of convex functions,,” Proceedings of the American Mathematical Society, vol. 16, no. 4, pp. 605–611, 1965. View at Google Scholar
  33. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, 1990.
  34. Z. B. Xu and G. F. Roach, “On the uniform continuity of metric projections in Banach spaces,” Approximation Theory and its Applications, vol. 8, no. 3, pp. 11–20, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus