Skip to main content
SpringerLink
Log in

On vector variational inequalities

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study vector variational inequalities. The concept of weaklyC-pseudomonotone operator is introduced. By employing the Fan lemma, we establish several existence results. The new results extend and unify existence results of vector variational inequalities for monotone operators under a Banach space setting. In particular, existence results for the generalized vector complementarity problem with weaklyC-pseudomonotone operators in Banach space are obtained.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Giannessi, F.,Theorem of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, New York, New York, pp. 151–186, 1980.

    Google Scholar 

  2. Chen, G. Y., andYang, X. Q.,Vector Complementarity Problem and Its Equivalence with Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990.

    Article  Google Scholar 

  3. Chen, G. Y., andCheng, G. M.,Vector Variational Inequalities and Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Heidelberg, Germany, Vol. 258, 1987.

    Google Scholar 

  4. Chen, G. Y., andCraven, B. D.,A Vector Variational Inequality and Optimization over an Efficient Set, Zeitschrift für Operations Research, Vol. 3, pp. 1–12, 1990.

    Google Scholar 

  5. Chen, G. Y.,Existence of Solution for a Vector Variational Inequality: An Extension of the Hartmann-Stampacchia Theorem, Journal of Opimization Theory and Applications, Vol. 74, pp. 445–456, 1992.

    Article  Google Scholar 

  6. Yang, X. Q.,Vector Variational Inequality and Its Duality, Nonlinear Analysis: Theory, Methods, and Analysis, Vol. 21, pp. 869–877, 1993.

    Google Scholar 

  7. Hartmann, G. J., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.

    Google Scholar 

  8. Fan, K.,A Generalization of Tychonoff's Fixed-Point Theorem Mathematische Annalen, Vol. 142, pp. 305–310, 1961.

    Article  Google Scholar 

  9. Knaster, B., Kurotowski, C., andMazukiewicz, S.,Ein Beweis des Fixpunktsatzes für N-Dimensionale Simplexe, Fundamental Mathematica, Vol. 14, pp. 132–137, 1929.

    Google Scholar 

  10. Karmardian, S.,Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.

    Article  Google Scholar 

  11. Lee, G. M., Kim, D. S., Lee, B. S., andCho, S. J.,Generalized Vector Variational Inequality and Fuzzy Extension, Applied Mathematics Letters, Vol. 6, pp. 47–51, 1993.

    Article  Google Scholar 

  12. Minty, G.,Monotone Nonlinear Operators in Hilbert Space, Duke Mathematical Journal, Vol. 29, pp. 341–346, 1962.

    Article  Google Scholar 

  13. Yao, J. C.,Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994.

    Google Scholar 

  14. Yu, L. P.,Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, New York, 1985.

    Google Scholar 

  15. Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, New York, New York, 1985.

    Google Scholar 

  16. Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.

    Article  Google Scholar 

  17. Browder, F. E.,Nonlinear Monotone Operators and Convex Sets in Banach Space, Bulletin of the American Mathematical Society, Vol. 71, pp. 780–785, 1965.

    Google Scholar 

  18. Opial, Z.,Nonexpansive Monotone Mapping in Banach Spaces, Technical Report 67-1, Department of Mathematics, Brown University, Providence, Rhode Island, 1967.

    Google Scholar 

  19. Stampacchia, G.,Variational Inequalities Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, 1969.

    Google Scholar 

  20. Karamardian, S.,The Nonlinear Complementarity Problem, with Applications, Parts 1 and 2, Journal of Optimization Theory and Applications, Vol. 4, pp. 87–98, 1969 and Vol. 4, pp. 167–181, 1969.

    Article  Google Scholar 

  21. Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.

    Article  Google Scholar 

  22. Rheinboldt, W. C.,On M-Functions and Their Application to Nonlinear Gauss-Seidel Iterations and Network Flows, Journal of Mathematical Analysis and Applications, Vol. 32, pp. 274–307, 1971.

    Article  Google Scholar 

  23. Schaible, S., andYao, J. C.,On the Equivalence of Nonlinear Complementarity Problems, Mathematical Programming, 1995.

  24. Yao, J. C.,Multi-Valued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 445–454, 1994.

    Article  Google Scholar 

  25. Conway, J. B.,A Course in Functional Analysis, 2nd Edition, Springer Verlag New York, New York, 1990.

    Google Scholar 

  26. Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.

    Article  Google Scholar 

  27. Théra, M.,Existence Results for the Nonlinear Complementarity Problem and Applications to Nonlinear Analysis, Journal of Mathematical Analysis and Applications, Vol. 154, pp. 572–584, 1991.

    Article  Google Scholar 

  28. Harker, P. T., andPang, J. S.,Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48B, pp. 161–220, 1990.

    Article  Google Scholar 

  29. Nanda, S.,Nonlinear Complementarity Problem of Mathematical Programming in Banach Space, Indian Journal of Pure Applied Mathematics, Vol. 18, pp. 215–218, 1987.

    Google Scholar 

  30. Isac, G., andThéra, M.,Complementarity Problem and the Existence of the Post-Critical Equilibrium State of a Thin Elastic Plate, Journal of Optimization Theory and Applications, Vol. 58, pp. 241–257, 1988.

    Article  Google Scholar 

  31. Dash, A. T., andNanda, S.,A Complementarity Problem in Mathematical Programming in Banach Space, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 328–331, 1984.

    Article  Google Scholar 

  32. Borwein, J. M.,Generalized Linear Complementarity Problems Treated without Fixed-Point Theory, Journal of Optimization Theory and Applications, Vol. 43, pp. 445–454, 1984.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC

    S. J. Yu (Graduate Student) & J. C. Yao (Professor)

Authors
  1. S. J. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by F. Giannessi

This research was partially supported by the National Science Council of the Republic of China under Contract NSC 84-2121-M-110-008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, S.J., Yao, J.C. On vector variational inequalities. J Optim Theory Appl 89, 749–769 (1996). https://doi.org/10.1007/BF02275358

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02275358

Key Words

Search

Navigation