Generalized monotone bifunctions and equilibrium problems. (English) Zbl 0903.49006

An equilibrium problem is studied in this paper which includes, e.g., the variational inequality as a special case. The existence of a solution for the equilibrium problem is derived using the quasimonotone condition. Moreover, characterizations, such as convexity and compactness, of the solution set for the equilibrium problem are also given.


49J40 Variational inequalities
Full Text: DOI


[1] Blum, E., andOettli, W.,From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student, Vol. 63, pp. 123–145, 1994. · Zbl 0888.49007
[2] Hadjisavvas, N., andSchaible, S.,On Strong Pseudomonotonicity and (Semi) Strict Quasimonotonicity, Journal of Optimization Theory and Applications, Vol. 79, pp. 139–155, 1993. · Zbl 0792.90068 · doi:10.1007/BF00941891
[3] Karamardian, S.,Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimizatio Theory and Applications, Vol 18, pp. 445–454, 1976. · Zbl 0304.49026 · doi:10.1007/BF00932654
[4] Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990. · Zbl 0679.90055 · doi:10.1007/BF00940531
[5] Schaible, S.,Generalized Monotonicity: A Survey,Proceedings of the Symposium on Generalized Convexity, Pécs, Hungary, 1992; Edited by S. Komlosi, T. Rapcsak, and S. Chaible, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin Germany, Vol. 405, pp. 229–249, 1994.
[6] Schaible, S.,Generalized Monotonicity: Concepts and Uses, Proceedings of the Meeting on Variational Inequalities and Network Equilibrium Problems, Erice, Italy, 1994; Edited by F. Giannessi and A. Maugeri, Plenum Publishing Corporation, New York, New York, pp. 289–299, 1995. · Zbl 0847.49013
[7] Bianchi, M.,Una Classe di Funzioni Monotone Generalizzate, Rivista di Matematica per le Scienze Economiche e Sociali, Vol. 16, pp. 17–32, 1993. · Zbl 0859.90110 · doi:10.1007/BF02086760
[8] Komlosi, S.,Generalized Monotomicity and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 84, pp. 361–376, 1995. · Zbl 0824.90124 · doi:10.1007/BF02192119
[9] Cottle, R. W., andYao, J. C.,Pseudomonotone Complementarity Problems in Hilbert Space, Journal of Optimization Theory and Applications, Vol. 75, pp. 281–295, 1992. · Zbl 0795.90071 · doi:10.1007/BF00941468
[10] Hadjisavvas, N., andSchaible, S.,Quasimonotone Variational Inequalities in Bannach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 95–111, 1996. · Zbl 0904.49005 · doi:10.1007/BF02192248
[11] Harker, P. T., andPang, J. S.,Finite-Dimensional Variational Inequalities and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161–220 1990. · Zbl 0734.90098 · doi:10.1007/BF01582255
[12] Schaible, S., andYao, J. C.,On the Equivalence of Nonlinear Complementarity Problems and Least-Element Problems, Mathematical Program, Vol. 70, pp. 191–200, 1995. · Zbl 0847.90133 · doi:10.1007/BF01585936
[13] Yao, J. C.,Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994. · Zbl 0813.49010 · doi:10.1287/moor.19.3.691
[14] Yao, J. C.,Multivalued Variational Inequalities with K-Monotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994. · Zbl 0812.47055 · doi:10.1007/BF02190064
[15] Blum, E., andOettli, W.,Variational Principle for Equilibrium Problems, 11th International Meeting on Mathematical Programming, Matrafured, Hungary, 1992.
[16] Brezis, H., Niermberg, L., andStampacchia, G.,A Remark on Fan’s Minimax Principle, Bollettino della Unione Matematica Italiana, Vol. 6, pp. 293–300, 1972.
[17] Avriel, M., Diewert, W. E., Schaible, S., andZiemba, W. T.,Introduction to Concave and Generalized Concave Functions, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, pp. 21–50, 1981. · Zbl 0539.90087
[18] Fan, K.,A Generalization of Tychonoff’s Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961. · Zbl 0093.36701 · doi:10.1007/BF01353421
[19] Baiocchi, C., andCapelo, A.,Disequazioni Variazionali e Quasivariazionali: Applicationi a Problemi di Frontiera Libera, Vols. 1 and 2, Pitagora Editrice, Bologna, Italy, 1978.
[20] Mosco, U.,Implicit Variational Problems and Quasivariational Inequalities, Lecture Notes in Mathematics, Springer, Berlin, Germany, Vol. 543, pp. 83–156, 1976. · Zbl 0346.49003
[21] Avriel, M., Diewert, W. E., Schaible, S., andZang, I.,Generalized Concavity, Plenum Press, New York, New York, 1988.
[22] Köthe, G.,Topological Vector Spaces, Vol. 1 Springer Verlag, Berlin, Germany, 1969. · Zbl 0179.17001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.