Abstract
In this paper, Stampacchia generalized vector quasiequilibrium problem and generalized vector loose saddle points for set-valued mappings are introduced. By using the scalarization method and the fixed-point theorem, existence theorems are established.
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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.
GIANNESSI, F., Editor, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, Netherlands, 2000.
GIANNESSI F., MASTROENI, G., and PELLEGRINI L., On the Theory of Vector Optimization and Variational Inequalities: Image Space Analysis and Separation, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 153–215, 2000.
CHEN, G. Y., and YANG, X. Q., The Vector Complementarity Problem and Its Equivalence with the Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990.
CHEN, G. Y., and CRAVEN, B. D., A Vector Variational Inequality and Optimization over an Efficient Set, Zeitschrift für Operations Research, Vol. 341, pp. 1–12, 1990.
YANG, X. Q., Vector Complementarity and Minimal Element Problems, Journal of Optimization Theorem and Applications, Vol. 77, pp. 483–495, 1993.
LEE, G. M., KIM, D. S., LEE B. S., and YEN, N. D., Vector Variational Inequality as a Tool for Studying Vector Optimization Problems, Nonlinear Analysis, Vol. 34, 745–765, 1998.
DANIILIDIS, A., and HADJISAVVAS, N., Existence Theorems for Vector Variational Inequalities, Bulletin of the Australian Mathematical Society, Vol. 54, pp. 473–481, 1996.
BLUM, E., and OETTLI, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123–145, 1994.
ANSARI, Q. H., KONNOV, I. V., and YAO, J.C., Characterizations of Solutions for Vector Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 113, pp. 435–447, 2002.
ANSARI, Q.H., OETTLI, W., and SCHLÄGER, D., A Generalization of Vector Equilibria, Mathematical Methods of Operations Research, Vol. 46, pp. 147–152, 1997.
BIANCHI, M., HADJISAVVAS, N., and SCHAIBLE, N., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527–542, 1997.
FU, J.Y., Simultaneous Vector Variational Inequalities and Vector Implicit Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 93, pp. 141–151, 1997.
FU, J. Y., and WAN, A. H., Generalized Vector Equilibrium Problems with Set-Valued Mappings, Mathematical Methods of Operations Research, Vol. 56, pp. 259–268, 2002.
LUC, D. T., and VARGAS, C., A Saddlepoint Theorem for Set-Valued Maps, Nonlinear Analysis, Vol. 18, pp. 1–7, 1992.
FERRO, F., A Minimax Theorem for Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 60, pp. 19–31, 1989.
TAN, K. K., YU, J., and YUAN, X. Z., Existence Theorems for Saddle Points of Vector-Valued Maps, Journal of Optimization Theory and Applications, Vol. 89, pp. 731–747, 1996.
YAO, J. C., The Generalized Quasivariational Inequality with Applications, Journal of Mathematical Analysis and Applications, Vol. 158, pp. 139–160, 1991.
AUBIN, J. P., and EKELAND, I., Aplied Nonlinear Analysis, Wiley, New York, NY, 1984.
TAN, N. X., Quasivariational Inequalities in Topological Linear Locally Convex Hausdorft Spaces, Mathematische Nachrichten, Vol. 122, pp. 231–245, 1985.
LIN, L. J., and YU, Z. T., On Some Equilibrium Problems for Multimaps, Journal of Computational and Applied Mathematics, Vol. 129, pp. 171–183, 2001.
JAHN, J., Mathematical Vector Optimization in Partially-Ordered Linear Spaces, Peter Lang, Frankfurt, Germany, 1986.
FAN, K., A Minimax Inequality and Applications, Inequalities III, Edited by O. Shisha, Academic Press, New York, NY, pp. 103–113, 1972.
GLICKSBERG, I., A Further Generalization of the Kakutani Fixed-Point Theorem with Application to Nash Equilibrium Points, Proceedings of the American Mathematical Society, Vol. 3, pp. 170–174, 1952.
CHEN, G. Y., YANG, X. Q., and, YU, H., A Nonlinear Scalarization Function and Generalized Quasivector Equilibrium Problems, Journal of Global Optimization (to appear).
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This work was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province, China. The author is grateful to Professor F. Giannessi and the referee for valuable comments and careful reading improving the original draft.
Communicated by F. Giannessi
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Fu, J.Y. Stampacchia Generalized Vector Quasiequilibrium Problems and Vector Saddle Points. J Optim Theory Appl 128, 605–619 (2006). https://doi.org/10.1007/s10957-006-9034-7
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DOI: https://doi.org/10.1007/s10957-006-9034-7