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On the solution existence of pseudomonotone variational inequalities

  • B. T. Kien
  • J. -C. YaoEmail author
  • N. D. Yen
Article
  • 393 Downloads

Abstract

As shown by N. Thanh Hao (submitted data), the solution existence results established by F. Facchinei and J.-S. Pang [(vols. I, II, Springer, Berlin, 2003) Prop. 2.2.3 and Theorem 2.3.4] for variational inequalities in general and for pseudomonotone variational inequalities in particular, are very useful for studying the range of applicability of the Tikhonov regularization method. This paper proposes some extensions of these results of (Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II, Springer, 2003) to the case of generalized variational inequalities and of variational inequalities in infinite-dimensional reflexive Banach spaces. Various examples are given to analyze in detail the obtained results.

Keywords

Variational inequality Generalized variational inequality Pseudomonotone operator Solution existence Degree theory 

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References

  1. 1.
    Aussel D. and Hadjisavvas N. (2004). On quasimonotone variational inequalities. J. Optim. Theory Appl. 121: 445–450 CrossRefGoogle Scholar
  2. 2.
    Aubin J.-P. and Cellina A. (1984). Differential Inclusions. Springer, Berlin Google Scholar
  3. 3.
    Bianchi M., Hadjisavvas N. and Shaible S. (2004). Minimal Coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 122: 1–17 CrossRefGoogle Scholar
  4. 4.
    Crouzeix J.-P. (1997). Pseudomonotone varational inequality problems: existence of solutions. Math. Program. 78: 305–314 Google Scholar
  5. 5.
    Daniilidis A. and Hadjisavvas N. (1999). Coercivity conditions and variational inequalities. Math. Program. 86: 433–438 CrossRefGoogle Scholar
  6. 6.
    Deimling K. (1985). Nonlinear Functional Analysis. Springer, Berlin Google Scholar
  7. 7.
    Facchinei F. and Pang J.-S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, Berlin Google Scholar
  8. 8.
    Fang S.C. and Peterson E.L. (1982). Generalized variational inequalities. J. Optim. Theory Appl. 38: 363–383 CrossRefGoogle Scholar
  9. 9.
    Hartmann P. and Stampacchia G. (1966). On some nonlinear elliptic differential functional equations. Acta Math. 115: 153–188 CrossRefGoogle Scholar
  10. 10.
    Karamardian S. (1976). Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18: 445–454 CrossRefGoogle Scholar
  11. 11.
    Kinderlehrer D. and Stampacchia G. (1980). An Introduction to Variational Inequalities and Their Applications. Academic, New York Google Scholar
  12. 12.
    Konnov, I.V.: Generalized monotone equilibrium problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.), Handbook of Generalized Convexity and Generalized Monotonicity, pp. 559–618. Springer, Berlin (2005)Google Scholar
  13. 13.
    Konnov I.V., Ali M.S.S. and Mazurkevich E.O. (2006). Regularization of nonmonotone variational inequalities. Apl. Math. Optim. 53: 311–330 CrossRefGoogle Scholar
  14. 14.
    Konnov I.V. (2006). On the convergence of a regularization method for nonmonotone variational inequalities. Comp. Math. Math. Phys. 46: 541–547 CrossRefGoogle Scholar
  15. 15.
    Qi H.D. (1999). Tikhonov regularization methods for variational inequality problems. J. Optim. Theory Appl. 102: 193–201 CrossRefGoogle Scholar
  16. 16.
    Ricceri, B.: Basic existence theorems for generalization variational and quasi-variational inequalities. In: Giannessi, F., Maugeri, A. (eds.), Variational Inequalities and Network Equilibrium Problems, pp. 251–255. Plenum, New York (1995)Google Scholar
  17. 17.
    Yao J.C. (1994). Variational inequalities with generalized monotone operators. Math. Oper. Res. 19: 691–705 CrossRefGoogle Scholar
  18. 18.
    Yao J.C. (1994). Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 80: 63–74 CrossRefGoogle Scholar
  19. 19.
    Yao, J.C., Chadli, O.: Pseudomonotone complementarity problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.). Handbook of Generalized Convexity and Generalized Monotonicity, pp. 501–558. Springer, Berlin (2005)Google Scholar
  20. 20.
    Yen, N.D.: On a problem of B. Ricceri on variational inequalities. In: Cho, Y.J., Kim, J.K., Kang, S.M. (eds.), Fixed Point Theory and Applications, vol. 5, pp. 163–173. Nova Science Publishers, New York (2004)Google Scholar
  21. 21.
    Zeidler E. (1986). Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Springer, Berlin Google Scholar

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© Springer Science+Business Media LLC 2007

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan
  2. 2.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam

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