×

zbMATH — the first resource for mathematics

From scalar to vector equilibrium problems in the quasimonotone case. (English) Zbl 0903.90141
Summary: In a unified approach, existence results for quasimonotone vector equilibrium problems and quasimonotone (multivalued) vector variational inequality problems are derived from an existence result for a scalar equilibrium problem involving two (rather than one) quasimonotone bifunctions. The results in the vector case are not only obtained in a new way, but they are also stronger versions of earlier existence results.

MSC:
90C29 Multi-objective and goal programming
49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Brezis, H., Niremberg, L., and Stampacchia, G., A Remark on Ky Fan’s Minimax Principle, Bollettino della Unione Matematica Italiana, Vol. 6, pp. 293–300, 1972. · Zbl 0264.49013
[2] Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student, Vol. 63, pp. 1–23, 1993.
[3] Bianchi, M., and Schaible, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996. · Zbl 0903.49006
[4] Bianchi, M., Hadjisavvas, N., and Schaible, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 531–546, 1997. · Zbl 0878.49007
[5] Hadjisavvas, N., and Schaible, S., Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems, Generalized Convexity and Monotonicity: Recent Developments, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer Academic Publishers, Dordrecht, Netherlands (to appear). · Zbl 0946.49005
[6] Oettli, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Festschrift for H. Tuy on the Occasion of His 70th birthday, Edited by D. T. Luc and N. V. Trung, Acta Mathematica Vietnamica, (to appear).
[7] Daniilidis, A., and Hadjisavvas, N., Existence Theorems for Vector Variational Inequalities, Bulletin of the Australian Mathematical Society, Vol. 54, pp. 473–481, 1996. · Zbl 0887.49004
[8] Avriel, M., Diewert, W. E., Schaible, S., and Zang, I., Generalized Concavity, Plenum Publishing Corporation, New York, New York, 1988.
[9] Berge, C., Topological Spaces, Including a Treatment of Multivalued Functions, Vector Spaces, and Convexity, Macmillan, New York, New York, 1963. · Zbl 0114.38602
[10] Jameson, G., Ordered Linear Spaces, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 141, 1970. · Zbl 0196.13401
[11] Granas, A., Sur Quelques Méthodes Topologiques en Analyse Convexe, Méthodes Topologiques en Analyse Convexe, Edited by A. Granas, Les Presses de l’Université de Montréal, Montréal, Québec, Canada, 1990.
[12] Fan, K., A Generalization of Tychonoff’s Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961. · Zbl 0093.36701
[13] Chen, G. Y., Existence of Solutions for a Vector Variational Inequality: An Extension of the Hartman-Stampacchia Theorem, Journal of Optimization Theory and Applications, Vol. 74, pp. 445–456, 1992. · Zbl 0795.49010
[14] Lin, K. L., Yang, D. P., and Yao, J. C., Generalized Vector Variational Inequalities, Journal of Optimization Theory and Its Applications, Vol. 92, pp. 117–125, 1997. · Zbl 0886.90157
[15] Konnov, I. V., and Yao, J. C., On the Generalized Vector Variational Inequality Problem, Journal of Mathematical Analysis and Applications, Vol. 206, pp. 42–58, 1997. · Zbl 0878.49006
[16] Lee, G. M., Kim, D. S., Lee, B. S., and Cho, S. J., Generalized Vector Variational Inequality and Fuzzy Extensions, Applied Mathematical Letters, Vol. 6, pp. 47–51, 1993. · Zbl 0804.49004
[17] Chen, G. Y., and Craven, B. D., A Vector Variational Inequality and Optimization over the Efficient Set, Zeitschrift für Operations Research, Vol. 34, pp. 1–12, 1990. · Zbl 0693.90091
[18] Chen, G. Y., and Y ANG, X. Q., The Vector Complementarity Problem and Its Equivalence with the Weak Minimal Element in Ordered Space, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990. · Zbl 0719.90078
[19] 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, Wiley, New York, New York, pp. 151–186, 1980.
[20] Diestel, J., and Uhl, J. J., Vector Measures, American Mathematical Society, Providence, Rhode Island, 1970.
[21] KÖthe, G., Topological Vector Spaces, Vol. 2, Springer Verlag, Berlin, Germany, 1969. · Zbl 0179.17001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.