Abstract
In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces. As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem.
Similar content being viewed by others
References
Ansari, Q.H. (1995), On generalized vector variational-like inequalities, Ann. Sci. Math. Québec 19(2): 131–137.
Ansari, Q.H., Oetli, W. and Schlager, D. (1997), A generalization of vectorial equilibria, Math. Meth. Oper. Res. 46: 147–152.
Baiochi, C. and Capelo, A. (1984), Variational and quasivariational inequalities, Application to Free-Boundary Problems, J. Wiley & Sons.
Bianchi, M. and Schaible, S. (1996), Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90: 31–43.
Bianchi, M., Hadjisavvas, N. and Schaible, S. (1997), Vector equilibrium problems with generalized monotone function, J. Optim. Theory Appl. 92: 527–542.
Blum, E. and Oetli, W. (1993), From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63: 1–23.
Chadli, O., Chbani, Z. and Riahi, H. (1999), Equilibrium problems with generalized monotone functions and applications to variational inequalities, Séminaire d'Analyse Convexe, Montpellier, exposé No 20, 1996; J. Optim. Theory Appl. (to appear).
Chen, G.Y. (1992), Existence of solutions for a vector variational inequality: An extension of Hartman-Stampacchia theorem, J. Optim. Theory Appl. 74: 445–456.
Fan, K. (1961), A generalization of Tychonoff's fixed point theorem, Matematische Annalen, 142: 305–310.
Hadjisavvas, N. and Schaible, S. (1996), Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl. 90, 95–111.
Hlavaĉek, I., Haslinger, J., Neĉas, J. and Laviŝek, I. (1988), Solution of Variational Inequalities in mechanics, Springer, Berlin.
Karamardian, S. and Schaible, S. (1990), Seven kinds of monotone maps, J. Optim. Theory Appl. 66: 37–46.
Lee, G.M., Kim, D.S. and Lee, B.S. (1996), On noncooperative vector equilibrium, Indian J. Pure Appl. Math. 278: 735–739.
Lee, G.M., Kim, D.S., Lee, B.S. and Cho, S.J. (1993), Generalized vector variational inequality and fuzzy extension, Appl. Math. Lett. 6: 47–51.
Luc, D.T. (1989), Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer Verlag.
Oettli, W. (1997), A remark on vector-valued equilibria and generalized monotonicity, Acta Math. Vietnamica, 22: 213–221.
Parida, J., Shoo, M. and Kumar, A. (1989), A variational-like inequality problem, Bull. Austral. Math. Soc. 39: 225–231.
Siddiqi, A.H., Ansari, Q.H. and Ahmad, R. (1997), On vector variational-like inequalities, Indian J. Pure Appl. Math., 28(8): 1009–1016.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chadli, O., Riahi, H. On Generalized Vector Equilibrium Problems. Journal of Global Optimization 16, 69–75 (2000). https://doi.org/10.1023/A:1008393715273
Issue Date:
DOI: https://doi.org/10.1023/A:1008393715273