NEW GENERALIZED MINTY'S LEMMA

  • Published : 2009.05.31

Abstract

In this paper, we introduce new pseudomonotonicity and proper quasimonotonicity with respect to a given function, and show some existence results for strong implicit vector variational inequalities by considering new generalized Minty's lemma. Our results generalize and extend some results in [1].

Keywords

References

  1. Y. P. Fang and N. J. Huang, Existence results for system of stromg implicit vector variational inequalities, Acta Math. Hungar., 103(4) (2004), 265–277
  2. P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966), 271–310 https://doi.org/10.1007/BF02392210
  3. G. Minty, Monotone (nonlinear) operators in Hilbert spaces, Duke Math. J. 29 (1962), 341–346 https://doi.org/10.1215/S0012-7094-62-02933-2
  4. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, John Wiley and Sons, New york(1984)
  5. D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities, Academic Press, New York(1980)
  6. A. Behera and G.K. Panda, A generalization of Minty's lemma, Indian J. Pure Appl. Math.28(1997), 897–903
  7. G. Kassay, J. Kolumban and Z. Pales, Factorization of Minty and Stampacchia variational inequality system, European J. Oper. Res., 143(2) (2002), 377-389 https://doi.org/10.1016/S0377-2217(02)00290-4
  8. Q.H. Ansari, A.H. Siddiqi and J.C. Yao, Generalized vector variational-like inequalities and their scalarizations, In vector variational inequalities and vector equilibria,(Edited by F. Giannessi), Kluwer Academic, Boston (2000), 17–37
  9. G.Y. Chen, Existence of solutions for a vector variational inequality: an extension of Hartman-Stampacchia theorem, J. Optim. Theory and Appl., 74 (1992), 445–456 https://doi.org/10.1007/BF00940320
  10. G.Y. Chen and X.Q. Yang, The vector complementarity problem and its equivalences with the weak minimal element in ordered spaces, J. Math. Anal. Appl., 153 (1990), 136–158 https://doi.org/10.1016/0022-247X(90)90270-P
  11. F. Giannessi, On Minty variational principle, in new trends in mathematical programming, (Edited by F. Giannessi, S. Komlosi and T. Rapcsak), Kluwer Academic, Dordrecht, 1998
  12. I.V. Konnov and J.C. Yao, On the generalized vector variational inequality problems, J. Math. Anal. Appl., 206 (1997), 42–58 https://doi.org/10.1006/jmaa.1997.5192
  13. B.S. Lee and G.M. Lee, A vector version of Minty's lemma and application, Appl. Math. Lett 12 (1999), 43–50
  14. B.S. Lee and S.J. Lee, A vector extension to Bhera and Panda's generalization of Minty's lemma, Indian J. Pure Math. 31 (2000), 1483–1489
  15. G.M Lee And S. Kum, Vector variational inequalities in a Hausdorff topological vactor space In vector variational inequalities and vector equilibria,(Edited by F. Giannessi), Kluwer Academic, Boston (2000), 307–320
  16. G. Mastroeni, On Minty vector variational inequality, In vector variational inequalities and vector equilibria,(Edited by F. Giannessi), Kluwer Academic, Boston (2000), 351–361
  17. S.J. Yu and J.C. Yao, On vector variational inequalities, J. Optim. Theory and Appl., 89 (1996), 749–769 https://doi.org/10.1007/BF02275358
  18. B.S. Lee, G.M. Lee and D.S. Kim, Generalized vector variational-like inequalities on locally convex Hausdorff topological vector spaces, Indian J. Pure Appl, Math. 28(1) (1997), 33–41
  19. A.H. Siddiqi, Q.H. Ansari and A. Khaliq, On vector variational inequalities, J. Optim. Theory and Appl., 84 (1995), 171–180 https://doi.org/10.1007/BF02191741
  20. F. Giannessi, Theorems of alterative, quadratic programs and complementarity problems, in: Variational Inequalities and Complementarity Problems (Edited by R.W. Cottle, F. Giannessi, and J.L. Lions), John Wiley and Sons, New York, 1980
  21. B.S. Lee, S.S. Chang, J.S. Jung and S.J. Lee, Generalized vector version of Minty's lemma and applications, Comp. Math. with Appl. 45 (2003), 647–653 https://doi.org/10.1016/S0898-1221(03)00024-5
  22. M.F. Khan and Salahuddin, On generalized vector variational-like inequalities, Nonlinear Analysis 59 (2004), 879–889
  23. Y. Zhao and Z. Xia, On the existence of solutions to generalized vector variational-like inequalities, Nonlinear Analysis 64 (2006), 2075–2083 https://doi.org/10.1016/j.na.2005.08.003
  24. K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519–537 https://doi.org/10.1007/BF01458545
  25. P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Velag, New York Inc.(1982)