Abstract
In this paper, we introduce systems of simultaneous generalized vector equilibrium problems and prove the existence of their solutions. As application of our results, we derive the existence theorems for solutions of systems of vector saddle–point problems. Consequently, we prove the existence of a solution of systems of generalized minimax inequalities. Further application of our results is also given to establish the existence of a solution of a Debreu-type equilibrium problem for vector-valued functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Husain E. Tarafdar (1994) ArticleTitleSimultaneous Variational Inequalities, Minimization Problems, and Related Results Mathematica Japonica 39 221–231
J. Y. Fu (1997) ArticleTitleSimultaneous Vector Variational Inequalities and Vector Implicit Complementarity Problems Journal of Optimization Theory and Applications 93 141–151 Occurrence Handle10.1023/A:1022653918733
Lin, L. J., Existence Theorems of Simultaneous Equilibrium Problems and Generalized Vector Saddle Points, Journal of Global Optimization 2004.
Q. H. Ansari W. K. Chan X. Q. Yang (2004) ArticleTitleThe System of Vector Equilibrium Problems with Applications Journal of Global Optimization 29 45–57 Occurrence Handle10.1023/B:JOGO.0000035018.46514.ca
Q.H. Ansari Z. Khan (2004) System of Generalized Vector Quasiequilibrium Problems with Applications S. Nanda G.P. Rajsekhar (Eds) Mathematical Analysis and Applications Narosa Publishing House New Delhi, India 1–13
Q. H. Ansari S. Schaible J. C. Yao (2000) ArticleTitleSystem of Vector Equilibrium Problems and Its Applications Journal of Optimization Theory and Applications 107 547–557 Occurrence Handle10.1023/A:1026495115191
Q. H. Ansari S. Schaible J. C. Yao (2003) ArticleTitleSystems of Generalized Vector Equilibrium Problems with Applications Journal of Global Optimization 22 3–16 Occurrence Handle10.1023/A:1013857924393
Ansari, Q. H., Schaible, S., and Yao, J. C., Generalized Vector Quasivariational Inequality Problems over Product Sets, Journal of Global Optimization, 2004.
Q. H. Ansari J. C. Yao (2000) ArticleTitleSystem of Generalized Variational Inequalities and Their Applications Applicable Analysis 76 203–217
G. X. Z. Yuan (1999) KKM Theory and Applications in Nonlinear Analysis Marcel Dekker New York, NY
Lin, L. J., and Tasi, Y. L., On Vector Quasisaddle Points of Set-Valued Maps, Generalized Convexity/Monotonicity, Edited by D. T. Luc and N. Hadjisavvas, Kluwer Academic Publishers, Dordrecht, Netherlands.
T. Tanaka (1999) ArticleTitleGeneralized Semicontinuity and Existence Theorems for Cone Saddle Points Applied Mathematics and Optimization 36 313–322 Occurrence Handle10.1007/s002459900065
J. Y. Fu (2000) ArticleTitleGeneralized Vector Equilibrium Problems. Mathematical Methods of Operations Research 52 57–64 Occurrence Handle10.1007/s001860000058
Q. H. Ansari A. Idzik J. C. Yao (2000) ArticleTitleCoincidence and Fixed-Point Theorems with Applications Topological Methods in Nonlinear Analysis 15 191–202
V. Jeyakumar W. Oettli (1993) ArticleTitleA Solvability Theorem for a Class of Quasiconvex Mappings with Applications to Optimization Journal of Mathematical Analysis and Applications 197 537–546 Occurrence Handle10.1006/jmaa.1993.1368
F. Ferro (1989) ArticleTitleA Minimax Theorem for Vector-Valued Functions Journal of Optimization Theory and Applications 60 19–31 Occurrence Handle10.1007/BF00938796
C. Berge (1963) Topological Spaces Oliver and Byod Edinburgh, Scotland
L. J. Lin Z. T. Yu (2001) ArticleTitleOn Some Equilibrium Problems for Multimaps Journal of Computational and Applied Mathematics 129 171–183
S. Kakutani (1941) ArticleTitleA Generalization of Brouwer’s Fixed-Point Theorem Duke Mathematical Journal 8 457–459 Occurrence Handle10.1215/S0012-7094-41-00838-4
J. P. Aubin A. Cellina (1994) Differential Inclusions Springer Verlag Berlin, Germany
G. Kothe (1983) Topological Vector Spaces Springer Verlag Berlin, Germany
K. Fan (1952) ArticleTitleFixed Point and Minimax Theorems in Locally Convex Topological Linear Spaces Proceedings of National Academy of Sciences 38 121–126
X.P. Ding E. Tarafdar (2000) Generalized Vector Variational-Like Inequalities without Monotonicity F. Giannessi (Eds) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories Kluwer Academic Publishers Dordrecht, Netherlands 113–124
G. Debreu (1952) ArticleTitleA Social Equilibrium Existence Theorem Proceedings of the National Academy of Sciences 38 886–893
Author information
Authors and Affiliations
Additional information
The first author thanks the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The second and third authors were supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable suggestions and comments.
Rights and permissions
About this article
Cite this article
Ansari, Q.H., Lin, L.J. & Su, L.B. Systems of Simultaneous Generalized Vector Quasiequilibrium Problems and their Applications. J Optim Theory Appl 127, 27–44 (2005). https://doi.org/10.1007/s10957-005-6391-6
Issue Date:
DOI: https://doi.org/10.1007/s10957-005-6391-6