Skip to main content

Advertisement

Log in

An Erratum to this article was published on 24 October 2007

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Aussel D. and Hadjisavvas N. (2004). On quasimonotone variational inequalities. J. Optim. Theory Appl. 121: 445–450

    Article  Google Scholar 

  2. Aubin J.-P. and Cellina A. (1984). Differential Inclusions. Springer, Berlin

    Google Scholar 

  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

    Article  Google Scholar 

  4. Crouzeix J.-P. (1997). Pseudomonotone varational inequality problems: existence of solutions. Math. Program. 78: 305–314

    Google Scholar 

  5. Daniilidis A. and Hadjisavvas N. (1999). Coercivity conditions and variational inequalities. Math. Program. 86: 433–438

    Article  Google Scholar 

  6. Deimling K. (1985). Nonlinear Functional Analysis. Springer, Berlin

    Google Scholar 

  7. Facchinei F. and Pang J.-S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, Berlin

    Google Scholar 

  8. Fang S.C. and Peterson E.L. (1982). Generalized variational inequalities. J. Optim. Theory Appl. 38: 363–383

    Article  Google Scholar 

  9. Hartmann P. and Stampacchia G. (1966). On some nonlinear elliptic differential functional equations. Acta Math. 115: 153–188

    Article  Google Scholar 

  10. Karamardian S. (1976). Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18: 445–454

    Article  Google Scholar 

  11. Kinderlehrer D. and Stampacchia G. (1980). An Introduction to Variational Inequalities and Their Applications. Academic, New York

    Google Scholar 

  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)

  13. Konnov I.V., Ali M.S.S. and Mazurkevich E.O. (2006). Regularization of nonmonotone variational inequalities. Apl. Math. Optim. 53: 311–330

    Article  Google Scholar 

  14. Konnov I.V. (2006). On the convergence of a regularization method for nonmonotone variational inequalities. Comp. Math. Math. Phys. 46: 541–547

    Article  Google Scholar 

  15. Qi H.D. (1999). Tikhonov regularization methods for variational inequality problems. J. Optim. Theory Appl. 102: 193–201

    Article  Google Scholar 

  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)

  17. Yao J.C. (1994). Variational inequalities with generalized monotone operators. Math. Oper. Res. 19: 691–705

    Article  Google Scholar 

  18. Yao J.C. (1994). Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 80: 63–74

    Article  Google Scholar 

  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)

  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)

  21. Zeidler E. (1986). Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Springer, Berlin

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. -C. Yao.

Additional information

B.T. Kien – on leave from the Hanoi University of Civil Engineering.

An erratum to this article is available at http://dx.doi.org/10.1007/s10898-007-9251-0.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kien, B.T., Yao, J.C. & Yen, N.D. On the solution existence of pseudomonotone variational inequalities. J Glob Optim (2007). https://doi.org/10.1007/s10898-007-9170-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10898-007-9170-0

Keywords

Navigation