Skip to main content
SpringerLink
Log in

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

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

Abstract

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.

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. BAIOCCHI, C., and CAPELO, A., Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, John Wiley and Sons, New York, NY, 1984.

    Google Scholar 

  2. BLUM, E., and OETTLI, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123-145, 1994.

    Google Scholar 

  3. BEN-EL-MECHAIEKH, H., DEGUIRE, P., and GRANAS, A., Une Alternative Non-Lineaire en Analyse Convexe et Applications, Comptes Rendus de l'Academie des Sciences de Paris, Vol. 295, pp. 257-259, 1982.

    Google Scholar 

  4. BLUM, E., and OETTLI, W., Variational Principle for Equilibrium Problems, 11th International Meeting on Mathematical Programming, Matrafured, Hungary, 1992.

  5. BIANCHI, M., Una Classe di Funzioni Monotone Generalizzate, Rivista di Matematica per le Scienze Economiche e Sociali, Vol. 16, pp. 17-32, 1993.

    Google Scholar 

  6. 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 

  7. SCHAIBLE, S., Generalized Monotonicity: A Survey, Proceeding of the Symposium on Generalized Convexity, Edited by S. Komlosi, T. Rapcsak, and S. Schaible, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 405, pp. 229-249, 1994.

    Google Scholar 

  8. SCHAIBLE, S., Generalized Monotonicity: Concepts and Uses, Proceedings of the Meeting on Variational Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A. Maugeri, Plenum Publishing Corporation, New York, NY, pp. 289-299, 1995.

    Google Scholar 

  9. DING, X. P., and TARAFDAR, E., Monotone Generalized Variational Inequalities and Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 88, pp. 107-122, 1996.

    Google Scholar 

  10. COTTLE, R. W., and YAO, J. C., Pseudomonotone Complementarity Problems in Hilbert Spaces, Journal of Optimization Theory and Applications, Vol. 75, pp. 281-295, 1992.

    Google Scholar 

  11. GIUSTI, E., Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Basel, Switzerland, 1984.

    Google Scholar 

  12. ATTOUCH, H., Viscosity Solutions of Minimization Problems, SIAM Journal on Optimization, Vol. 6, pp. 769-806, 1996.

    Google Scholar 

  13. COTTLE, R. W., and PANG, J. S., A Least Element Theory of Solving Linear Complementarity Problems as Linear Programs, Mathematics of Operations Research, Vol. 3, pp. 155-170, 1978.

    Google Scholar 

  14. SCHAIBLE, S., and YAO, J. C., On the Equivalence of Nonlinear Complementarity Problems and Least-Element Problems, Mathematical Programming, Vol. 70, pp. 191-200, 1995.

    Google Scholar 

  15. BRÉZIS, H., NIRENBERG, L., and STAMPACCHIA, G., A Remark on Ky Fan's Minimax Principle, Bollettino della Unione Matematica Italiana, Vol. 6, pp. 293-300, 1972.

    Google Scholar 

  16. KARAMARDIAN, S., and SCHAIBLE, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37-46, 1990.

    Google Scholar 

  17. SAIGAL, R., Extension of the Generalized Complementarity Problem, Mathematics of Operations Research, Vol. 1, pp. 260-266, 1976.

    Google Scholar 

  18. KARAMARDIAN, S., Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445-454, 1976.

    Google Scholar 

  19. RIDDELL, R. C., Equivalence of Nonlinear Complementarity Problems and Least-Element Problems in Banach Lattices, Mathematics of Operations Research, Vol. 6, pp. 462-474, 1981.

    Google Scholar 

  20. HASSOUNI, A., Sous-Differentiels des Fonctions Quasi-Convexes, Thèse du 3e Cycle, Université Paul Sabatier, Toulouse, France, 1983.

    Google Scholar 

  21. HASSOUNI, A., Opérateurs Quasimonotones: Application à Certains Problèmes Variationnels, Thèse d'Etat, Université Paul Sabatier, Toulouse, France, 1993.

    Google Scholar 

  22. CORREA, R., JOFRÉ, A., and THIBAULT, L., Subdifferential Monotonicity as Characterization of Convex Functions, Numerical Functional Analysis and Optimization, Vol. 15, pp. 531-535, 1994.

    Google Scholar 

  23. AUSSEL, D., CORVELLEC, J. N., and LASONDE, M., Subdifferential Characterization of Quasiconvexity and Convexity, Journal of Convex Analysis, Vol. 1, pp. 195-201, 1994.

    Google Scholar 

  24. LUC, D. T., Characterizations of Quasiconvex Functions, Bulletin of the Australian Mathematical Society, Vol. 48, pp. 393-406, 1993.

    Google Scholar 

  25. POLIQUIN, R. A., Subgradient Monotonicity and Convex Functions, Nonlinear Analysis, Vol. 14, pp. 305-317, 1990.

    Google Scholar 

  26. HADJISAVVAS, N., and SCHAIBLE, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 55-111, 1996.

    Google Scholar 

  27. DING, X. P., Browder—Hartman—Stampacchia Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 173, pp. 577-587, 1993.

    Google Scholar 

  28. KOMLOSI, S., Generalized Monotonicity and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 84, pp. 361-376, 1995.

    Google Scholar 

  29. KARAMARDIAN, S., SCHAIBLE, S., and CROUZEIX, J. P., First Characterization of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399-413, 1993.

    Google Scholar 

  30. OETTLI, W., and THÉRA, M., On Maximal Monotonicity of Perturbed Mappings, Bollettino della Unione Matematica Italiana, Vol. 7, pp. 47-55, 1995.

    Google Scholar 

  31. OETTLI, W., and RIAHI, H., On Maximal α—Monotonicity of Sums of Operators, Communications in Applied Nonlinear Analysis, Vol. 5, pp. 1-17, 1998.

    Google Scholar 

  32. FAN, K., A Generalization of Tychonoff 's Fixed-Point Theorem, Mathematishe Annalen, Vol. 142, pp. 305-310, 1961.

    Google Scholar 

  33. TAN, K. K., and YUAN, Z. Z., A Minimax Inequality with Applications to the Existence of Equilibrium Points, Bulletin of the Australian Mathematical Society, Vol. 47, pp. 483-503, 1993.

    Google Scholar 

  34. BOURBAKI, N., Espaces Vectoriels Topologiques, Chapters 1–5, Masson, Paris, France.

  35. AUBIN, J. P., Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer Verlag, Berlin, Germany, 1993.

    Google Scholar 

  36. BERGE, C., Espaces Topologiques: Fonctions Multivoques, Dunod, Paris, France, 1959.

    Google Scholar 

  37. SION, M., On General Minimax Theorems, Pacific Journal of Mathematics, Vol. 8, pp. 171-176, 1958.

    Google Scholar 

  38. SIMONS, S., You Cannot Generalize the Minimax Theorem Too Much, Rendiconti del Seminario Matematico e Fisico di Milano, Vol. 59, pp. 59-64, 1989.

    Google Scholar 

  39. YAO, J. C., Multivalued Variational Inequalities with K-Monotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391-403, 1994.

    Google Scholar 

  40. ATTOUCH, H., and BRÉZIS, H., Duality for Sum of Functions in General Banach Spaces, As Respects of Mathematics and Its Applications, Edited by J. Barroso, North Holland, Amsterdam, Holland, pp. 125-133, 1986.

  41. PHELPS, R. R., Convex Functions, Monotone Operators, and Differentiability, 2nd Edition, Springer Verlag, New York, NY, 1993.

    Google Scholar 

  42. RUDIN, W., Functional Analysis, McGraw-Hill Book Company, New York, NY, 1973.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Semlalia Faculty of Sciences, Mathematics, University Cadi Ayyad, Marrakech, Morocco

    O. Chaldi (Doctor in Mathematics)

  2. Semlalia Faculty of Sciences, Mathematics, University Cadi Ayyad, Marrakech, Morocco

    Z. Chbani (Assistant Professor)

  3. Semlalia Faculty of Sciences, Mathematics, University Cadi Ayyad, Marrakech, Morocco

    H. Riahi (Professor)

Authors
  1. O. Chaldi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. Chbani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Riahi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chaldi, O., Chbani, Z. & Riahi, H. Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities. Journal of Optimization Theory and Applications 105, 299–323 (2000). https://doi.org/10.1023/A:1004657817758

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004657817758

Search

Navigation