×

Generalized vector equilibrium problems in generalized convex spaces. (English) Zbl 1100.90054

Summary: We introduce and study a class of abstract generalized vector equilibrium problems (AGVEP) in generalized convex spaces which includes most vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems, and generalized vector variational inequality problems as special cases. By using the generalized GKKM and generalized SKKM type theorems due to the first author, some new existence results of equilibrium points for the AGVEP are established in noncompact generalized convex spaces. As consequences, some recent results in the literature are obtained under much weaker assumptions.

MSC:

90C47 Minimax problems in mathematical programming
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
49J40 Variational inequalities
91B50 General equilibrium theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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, J. Wiley and Sons, New York, NY, pp. 151-186, 1980.
[2] Giannessi, F., Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, London, UK, 2000. · Zbl 0952.00009
[3] Ding, X. P., and Park, J. Y., Fixed Points and Generalized Vector Equilibria in General Topological Spaces, Acta Mathematic Nyirehaziensis (to appear). · Zbl 1045.91026
[4] Ding, X. P., and Park, J. Y., Fixed Points and Generalized Vector Equilibrium Problems in G-Convex Spaces, Indian Journal of Pure and Applied Mathematics (to appear). · Zbl 1043.49008
[5] Ansari, Q. H., Oettli, W., and Schläger, D., A Generalization of Vectorial Equilibria, Mathematical Methods of Operation Research, Vol. 46, pp. 147-152, 1997. · Zbl 0889.90155 · doi:10.1007/BF01217687
[6] Oettli, W., and Schläger, D., Generalized Vector Equilibria and Generalized Monotonicity, Functional Analysis with Current Applications in Science, Technology, and Industry, Edited by M. Brokate and A. H. Siddiqi, Pitman Research Notes in Mathematics, Longman, London, UK, Vol. 77, pp. 145-154, 1998. · Zbl 0904.90150
[7] Oettli, W., and Schläger, D., Existence of Equilibria for Monotone Multivalued Mappings, Mathematical Methods of Operations Research, Vol. 48, pp. 219-228, 1998. · Zbl 0930.90077 · doi:10.1007/s001860050024
[8] Oettli, W., and Schläger, D., Existence of Equilibria for g-Monotone Mappings, Nonlinear Analysis and Convex Analysis, Edited by W. Takahashi and T. Tanaka, World Scientific, Singapore, pp. 26-33, 1999. · Zbl 0998.49010
[9] Song, W., Vector Equilibrium Problems with Set-Valued Mappings, Vector Variational Inequalities and Vector Equilibria, F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 403-422, 2000. · Zbl 0993.49011
[10] Ansari, Q. H., and Yao, J. C., An Existence Result for the Generalized Vector Equilibrium Problem, Applied Mathematical Letters, Vol. 12, pp. 53-56, 1999. · Zbl 1014.49008 · doi:10.1016/S0893-9659(99)00121-4
[11] Bianchi, M., Hadjisavvas, N., and Schaible, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527-542, 1997. · Zbl 0878.49007 · doi:10.1023/A:1022603406244
[12] Oettli, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213-221, 1997. · Zbl 0914.90235
[13] Tan, N. X., and Tinh, P. N., On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Optimization, Vol. 19, pp. 141-156, 1998. · Zbl 0896.90161 · doi:10.1080/01630569808816820
[14] Lin, L. J., and Yu, Z. T., Fixed-Point Theorems and Equilibrium Problems, Nonlinear Analysis, Vol. 43, pp. 987-999, 2001. · Zbl 0989.47051 · doi:10.1016/S0362-546X(99)00202-3
[15] Ansari, Q. H., On Generalized Vector Variational-Like Inequalities, Annales des Sciences Mathématiques de Québec, Vol. 19, pp. 131-137, 1995. · Zbl 0847.49014
[16] Ansari, Q. H., Extended Generalized Vector Variational-Like Inequalities for Nonmonotone Set-Valued Maps, Annales des Sciences Mathématiques de Québec, Vol. 21, pp. 1-11, 1997. · Zbl 0894.49006
[17] Ansari, Q. H., A Note on Generalized Vector Variational-Like Inequalities, Optimization, Gordon and Breach, London, UK, Vol. 41, pp. 197-205, 1997. · Zbl 0913.49004
[18] Ansari, Q. H., Siddiqi, A. H., and Yao, J. C., Generalized Vector Variational-Like Inequalities and Their Scalarizations, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 17-37, 2000. · Zbl 0992.49013
[19] Ding, X. P., and Tarafdar, E., Generalized Vector Variational-Like Inequalities without Monotonicity, Vector Variational Inequalities and Vector Equilibria, F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 113-124, 2000. · Zbl 0991.49009
[20] Ding, X. P., and Tarafdar, E., Generalized Vector Variational-Like Inequalities with C x -{\(\eta\)}-Pseudomonotone Set-Valued Mappings, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 125-140, 2000. · Zbl 0991.49010
[21] Ding, X. P., Generalized Vector Quasivariational-Like Inequalities, Computers and Mathematics with Applications, Vol. 37, pp. 57-67, 1999. · Zbl 0944.47040 · doi:10.1016/S0898-1221(99)00076-0
[22] Chang, S. S., Thompson, H. B., and Yuan, G. X. Z., Existence of Solutions for Generalized Vector Variational-Like Inequalities, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 39-53, 2000. · Zbl 0997.49005
[23] Luo, Q., Generalized Vector Variational-Like Inequalities, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 363-369, 2000. · Zbl 0992.49014
[24] Ding, X. P., New HKKM Theorems and Their Applications to Geometric Property, Coincidence Theorems, Minimax Inequality, and Maximal Elements, Indian Journal of Pure and Applied Mathematics, Vol. 26, pp. 1-19, 1995. · Zbl 0830.49003
[25] Ding, X. P., Generalized Variational Inequalities and Equilibrium Problems in Generalized Convex Spaces, Computers and Mathematics with Applications, Vol. 38, pp. 189-197, 1999. · Zbl 0946.49003 · doi:10.1016/S0898-1221(99)00249-7
[26] Tian, G., Generalization of the FKKM Theorem and the Ky Fan Minimax Inequality with Applications to Maximal Elements, Price Equilibrium, and Complementarity, Journal of Mathematical Analysis and Applications, Vol. 170, pp. 457-471, 1992. · Zbl 0767.49007 · doi:10.1016/0022-247X(92)90030-H
[27] Park, S., and Kim, H., Coincidence Theorems for Admissible Multifunctions on Generalized Convex Spaces, Journal of Mathematical Analysis and Applications, Vol. 197, pp. 173-187, 1996. · Zbl 0851.54039 · doi:10.1006/jmaa.1996.0014
[28] Park, S., and Kim, H., Foundations of the KKM Theory on Generalized Convex Spaces, Journal of Mathematical Analysis and Applications, Vol. 209, pp. 551-571, 1997. · Zbl 0873.54048 · doi:10.1006/jmaa.1997.5388
[29] Lin, L. J., and Park, S., On Some Generalized Quasi-Equilibrium Problems, Journal of Mathematical Analysis and Applications, Vol. 224, pp. 167-181, 1998. · Zbl 0924.49008 · doi:10.1006/jmaa.1998.5964
[30] Park, S., Remarks on the Fixed-Point Problem of Ben El-Mechaiekh, Nonlinear Analysis and Convex Analysis, Edited by W. Takahashi and T. Tanaka, World Scientific, Singapore, pp. 79-86, 1999. · Zbl 0997.47046
[31] Ding, X. P., Generalized GKKM Theorems in Generalized Convex Spaces and Their Applications, Journal of Mathematical Analysis and Applications, Vol. 226, pp. 21-37, 2002. · Zbl 1006.47041 · doi:10.1006/jmaa.2000.7207
[32] Tan, K. K., GKKM Theorem, Minimax Inequalities, and Saddle Points, Nonlinear Analysis, Vol. 30, pp. 4151-4160, 1997. · Zbl 0953.49009 · doi:10.1016/S0362-546X(96)00129-0
[33] Chang, S. S., and Zhang, Y., Generalized KKM Theorem and Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 159, pp. 208-223, 1991. · Zbl 0739.47026 · doi:10.1016/0022-247X(91)90231-N
[34] Lin, L. J., and Chang, T. H., SKKM Theorems, Saddle Points, and Minimax Inequalities, Nonlinear Analysis, Vol. 34, pp. 73-86, 1998. · Zbl 1010.47042 · doi:10.1016/S0362-546X(97)00684-6
[35] Aubin, J. P., and Ekeland, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, NY, 1984. · Zbl 0641.47066
[36] Fan, K., Some Properties of Convex Sets Related to Fixed-Point Theorems, Mathematische Annalen, Vol. 226, pp. 519-537, 1984. · Zbl 0515.47029 · doi:10.1007/BF01458545
[37] Lassonde, M., On the Use of KKM Multifunctions in Fixed-Point Theory and Related Topics, Journal of Mathematical Analysis and Applications, Vol. 9, pp. 151-201, 1983. · Zbl 0527.47037 · doi:10.1016/0022-247X(83)90244-5
[38] Chang, S. Y., A Generalization of the KKM Principle and Its Applications, Soochow Journal of Mathematics, Vol. 15, pp. 7-17, 1989. · Zbl 0747.47033
[39] Ansari, Q. H., Lin, Y. C., and Yao, J. C., General KKM Theorem with Applications to Minimax and Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 104, pp. 41-57, 2000. · Zbl 0956.49004 · doi:10.1023/A:1004620620928
[40] Ansari, A. H., Vector Equilibrium Problems and Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, UK, pp. 1-15, 2000. · Zbl 0992.49012
[41] Lee, G. M., and Kum, S. H., On Implicit Vector Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 104, pp. 409-425, 2000. · Zbl 0970.47052 · doi:10.1023/A:1004617914993
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.