Journal of Optimization Theory and Applications

, Volume 107, Issue 3, pp 547–557

System of Vector Equilibrium Problems and Its Applications

  • Q. H. Ansari
  • S. Schaible
  • J. C. Yao
Article

Abstract

In this paper, we introduce a system of vector equilibrium problems andprove the existence of a solution. As an application, we derive someexistence results for the system of vector variational inequalities. We alsoestablish some existence results for the system of vector optimizationproblems, which includes the Nash equilibrium problem as a special case.

system of vector equilibrium problems system of vector variational inequalities system of vector optimization problems Nash equilibrium problem fixed points 

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References

  1. 1.
    Giannessi, F., Theorems of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, NY, pp. 151–186, 1980.Google Scholar
  2. 2.
    Giannessi, F., Editor, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.Google Scholar
  3. 3.
    Bianchi, M., and Schaible, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996.Google Scholar
  4. 4.
    Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student, Vol. 63, pp. 123–145, 1994.Google Scholar
  5. 5.
    Ansari, Q. H., Vector Equilibrium Problems and Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.Google Scholar
  6. 6.
    Ansari, Q. H., Oettli, W., and SchlÄger, D., A Generalization of Vectorial Equilibria, Mathematical Methods of Operations Research, Vol. 46, pp. 147–152, 1997.Google Scholar
  7. 7.
    Bianchi, M., Hadjisavvas, N., and Schaible, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527–542, 1997.Google Scholar
  8. 8.
    Hadjisavvas, N., and Schaible, S., From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297–309, 1998.Google Scholar
  9. 9.
    Hadjisavvas, N., and Schaible, S., Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 257–275, 1998.Google Scholar
  10. 10.
    Lee, G. M., Kim, D. S., and Lee, B. S., On Noncooperative Vector Equilibrium, Indian Journal of Pure and Applied Mathematics, Vol. 27, pp. 735–739, 1996.Google Scholar
  11. 11.
    Oettli, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213–221, 1997.Google Scholar
  12. 12.
    Schaible, S., From Generalized Convexity to Generalized Monotonicity, Operations Research and Its Applications, Proceedings of the 2nd International Symposium, ISORA'96, Guilin, PRC; Edited by D. Z. Du, X. S. Zhang, and K. Cheng, Beijing World Publishing Corporation, Beijing, PRC, pp. 134–143, 1996.Google Scholar
  13. 13.
    Tan, N. X., and Tinh, P. N., On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Optimization, Vol. 19, pp. 141–156, 1998.Google Scholar
  14. 14.
    Pang, J. S., Asymmetric Variational Inequality Problems over Product Sets: Applications and Iterative Methods, Mathematical Programming, Vol. 31, pp. 206–219, 1985.Google Scholar
  15. 15.
    Cohen, G., and Chaplais, F., Nested Monotony for Variational Inequalities over a Product of Spaces and Convergence of Iterative Algorithms, Journal of Optimization Theory and Applications, Vol. 59, pp. 360–390, 1988.Google Scholar
  16. 16.
    Bianchi, M., Pseudo P-Monotone Operators and Variational Inequalities, Report 6, Istituto di Econometria e Matematica per le Decisioni Economiche, Università Cattolica del Sacro Cuore, Milan, Italy, 1993.Google Scholar
  17. 17.
    Luc, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 319, 1989.Google Scholar
  18. 18.
    Luc, D. T., and Vargas, C. A., A Saddle-Point Theorem for Set-Valued Maps, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 18, pp. 1–7, 1992.Google Scholar
  19. 19.
    Li, Z. F., and Wang, S. Y., A Type of Minimax Inequality for Vector-Valued Mappings, Journal of Mathematical Analysis and Applications, Vol. 227, pp. 68–80, 1998.Google Scholar
  20. 20.
    Yu, J., Essential Equilibria of n-Person Noncooperative Games, Journal of Mathematical Economics, Vol. 31, pp. 361–372, 1999.Google Scholar
  21. 21.
    Ansari, Q. H., and Yao, J. C., A Fixed-Point Theorem and Its Applications to the System of Variational Inequalities, Bulletin of the Australian Mathematical Society, Vol. 59, pp. 433–442, 1999.Google Scholar
  22. 22.
    Ding, X. P., Generalized Vector Quasivariational-Like Inequalities, Computers and Mathematics with Applications, Vol. 37, pp. 57–67, 1999.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Q. H. Ansari
    • 1
  • S. Schaible
    • 2
  • J. C. Yao
    • 3
  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.A. G. Anderson Graduate School of ManagementUniversity of CaliforniaRiverside
  3. 3.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan, ROC

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