×

On vector variational inequalities. (English) Zbl 0827.47050

Summary: We introduce a general form of a vector variational inequality and prove the existence of its solution with and without convexity assumptions.

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Giannessi, F.,Theorems of the Alternative, Quadratic Programs, and Complementary Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, New York, pp. 151–186, 1980. · Zbl 0484.90081
[2] Chen, Y., andCheng, G. M.,Vector Variational Inequalities and Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 285, pp. 408–416, 1987.
[3] Chen, G. Y., andYang, X. Q.,Vector Complementarity Problem and Its Equivalence with a Weak Minimal Element in Ordered Space, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990. · Zbl 0719.90078 · doi:10.1016/0022-247X(90)90223-3
[4] Chen, G. Y.,Existence of Solutions for a Vector Variational Inequality: An Extension of the Hartmann-Stampacchia Theorem, Journal of Optimization Theory and Applications, Vol. 74, pp. 445–456, 1992. · Zbl 0795.49010 · doi:10.1007/BF00940320
[5] Isac, G.:A Special Variational Inequality and the Implicit Complementarity Problem, Journal of the Faculty of Sciences of the University of Tokyo, Vol. 37, pp. 109–127, 1990. · Zbl 0702.49008
[6] Noor, M. A.,General Variational Inequality, Applied Mathematical Letters, Vol. 1, pp. 119–122, 1988. · Zbl 0655.49005 · doi:10.1016/0893-9659(88)90054-7
[7] Hartman, P., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966. · Zbl 0142.38102 · doi:10.1007/BF02392210
[8] 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
[9] Fan, K.,A Minimax Inequality and Applications, Inequalities III, Edited by O. Shisha, Academic Press, New York, New York, pp. 103–113, 1972.
[10] Bardaro, C., andCeppitelli, R.,Some Further Generalizations of the Knaster-Kuratowski-Mazurkiewicz Theorem and Minimax Inequalities, Journal of Mathematical Analysis and Applications, Vol. 132, pp. 484–490, 1988. · Zbl 0667.49016 · doi:10.1016/0022-247X(88)90076-5
[11] Jameson, G.,Ordered Linear Spaces, Springer Verlag, Heidelberg, Germany, 1970. · Zbl 0196.13401
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.