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Journal of Global Optimization

, Volume 41, Issue 1, pp 135–145 | Cite as

On the solution existence of pseudomonotone variational inequalities

  • B. T. Kien
  • J.-C. Yao
  • N. D. Yen
Erratum

Abstract

As shown by Thanh Hao [Acta Math. Vietnam 31, 283–289, 2006], the solution existence results established by Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I (Springer, Berlin, 2003) Prop. 2.2.3 and Theorem 2.3.4] for variational inequalities (VIs) in general and for pseudomonotone VIs 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 Facchinei and Pang to the case of generalized variational inequalities (GVI) 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 

References

  1. 1.
    Aubin, J.-P., Cellina, A.: Differential Inclusions. Springer-Verlag (1984)Google Scholar
  2. 2.
    Aussel D. and Hadjisavvas N. (2004). On quasimonotone variational inequalities. J. Optim. Theory Appl. 121: 445–450 CrossRefGoogle 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 variational 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.: Nonlinear Functional Analysis. Springer-Verlag (1985)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 Press, New York Google Scholar
  12. 12.
    Konnov I.V.  (2005). Generalized monotone equilibrium problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S. and Schaible, S. (eds) Handbook of Generalized Convexity and Generalized Monotonicity, pp 559–618. Springer, Berlin CrossRefGoogle Scholar
  13. 13.
    Konnov I.V. (2006). On the convergence of a regularization method for nonmonotone variational inequalities. Comput. Math. Math. Phys. 46: 541–547 CrossRefGoogle Scholar
  14. 14.
    Konnov I.V., Ali M.S.S. and Mazurkevich E.O. (2006). Regularization of nonmonotone variational inequalities. Appl. Math. Optim. 53: 311–330 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. (1995). Basic existence theorems for generalized variational and quasi-variational inequalities. In: Giannessi, F. and Maugeri, A. (eds) Variational Inequalities and Network Equilibrium Problems, pp 251–255. Plenum, New York Google Scholar
  17. 17.
    Thanh Hao N. (2006). Tikhonov regularization algorithm for pseudomonotone variational inequalities. Acta Math. Vietnam 31: 283–289 Google Scholar
  18. 18.
    Yao J.C. (1994). Variational inequalities with generalized monotone operators. Math. Oper. Res. 19: 691–705 CrossRefGoogle Scholar
  19. 19.
    Yao J.C. (1994). Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 80: 63–74 CrossRefGoogle Scholar
  20. 20.
    Yao J.C. and Chadli O.  (2005). Pseudomonotone complementarity problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S. and Schaible, S. (eds)  Handbook of Generalized Convexity and Generalized Monotonicity, pp 501–558. Springer, Berlin CrossRefGoogle Scholar
  21. 21.
    Yen N.D. (2004). On a problem of B. Ricceri on variational inequalities. In: Cho, Y.J., Kim, J.K. and Kang, S.M. (eds) Fixed Point Theory and Applications, vol. 5, pp 163–173. Nova Science Publishers, New York Google Scholar
  22. 22.
    Zeidler E. (1986). Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Springer, Berlin Google Scholar

Copyright information

© 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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