Tian, Guoqiang (1994): Generalized KKM theorem, minimax inequalities and their applications. Published in: Journal of Optimization Theory and Applications , Vol. 83, No. 2 (November 1994): pp. 375389.

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Abstract
This paper extends the wellknown KKM theorem and variational inequalities by relaxing the closedness of values of a correspondence and lower semicontinuity of a function. The approach adopted is based on Michael's continuous selection theorem. As applications, we provide theorems for the existence of maximum elements of a binary relation, a price equilibrium, and the complementarity problem. Thus our theorems, which do not require the openness of lower sections of the preference correspondences and the lower semicontinuity of the excess demand functions, generalize many of the existence theorems such as those in Sonnenschein (Ref. 1), Yannelis and Prabhakar (Ref. 2), and Border (Ref. 3).
Item Type:  MPRA Paper 

Original Title:  Generalized KKM theorem, minimax inequalities and their applications 
Language:  English 
Keywords:  KKM theorem; Variational inequalities; Complementarity problem; Price equilibrium; Maximal elements ; Binary relations 
Subjects:  D  Microeconomics > D6  Welfare Economics > D63  Equity, Justice, Inequality, and Other Normative Criteria and Measurement 
Item ID:  41217 
Depositing User:  Guoqiang Tian 
Date Deposited:  13 Sep 2012 18:43 
Last Modified:  07 Aug 2016 05:15 
References:  1. SONNENSCHEIN, H., Demand Theory without Transitive Preferences, with Application to the Theory of Competitive Equilibrium, Preferences, Utility, and Demand, Edited by J. S. Chipman, L. Hurwicz, M. K. Richter, and H. Sonnenschein, Harcourt Brace Jovanovich, New York, New York, 1971. 2. YANNELIS, N. C., and PRABHAKAR, N. D., Existence of Maximal Elements and Equilibria in Linear Topological Spaces, Journal of Mathematical Economics, Vol. 12, pp. 233245, 1983. 3. BORDER, K. C., FixedPoint Theorems with Application to Economics and Game Theory, Cambridge University Press, Cambridge, England, 1985. 4. KNASTER, B., KURATOWSKI, C., and MAZURKIEWICZ, S., Ein Beweis des Fixpunktsatze nDemensionale Simpliexe, Fundamental Mathematica, Vol. 14, pp. 132137, 1929. 5. FAN, K., Minimax Theorem, Proceedings of the National Academy of Sci ences, Vol. 39, pp. 4247, 1953. 6. FAN, K., A Generalization of Tychonoff s FixedPoint Theorem, Mathematis che Annalen, Vol. 142, pp. 305310, 1962. 7. FAN, K., Some Properties of Convex Sets Related to FixedPoints Theorems, Mathematische Annalen, Vol. 266, pp. 519537, 1984. 8. AUBIN, J. P., Mathematical Methods of Game and Economic Theory, North Holland, Amsterdam, Holland, 1979. 9. YEN, C. L., A Minimax Inequality and lts Applications to Variational In equalities, Pacific Journal of Mathematics, Vol. 132, pp. 477481, 1981. 10. AUBIN, J. P., and EKELAND, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, New York, 1984. 11. TAKAHASHI, W., Nonlinear Variational Inequalities and FixedPoint Theo rems, Journal of the Mathematical Society of Japan, Vol. 28, pp. 477481, 1976. 12. ZHOU, J., and CHEN, G., Diagonal Convexity Conditions for Problems in Convex Analysis and QuasiVariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 132, pp. 213225, 1988. 13. BARDARO, C., and CEPPITELLI, R., Applications of the Generalized Knaster KuratowskiMazurkiewicz Theorem to Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 137, pp. 4658, 1989. 14. BARDARO, C., and CEPPITELLI, R., Some Further Generalizations of the KnasterKuratowskiMazurkiewicz Theorem and Minimax Inequalities, Jour nal of Mathematical Analysis and Applications, Vol. 132, pp. 484490, 1988. 15. SHIH, M. H., and TAN, K. K., Generalized QuasiVariational Inequalities in Locally Convex Topological Vector Spaces, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 333343, 1985. 16. SHIH, M. H., and TAN, K. K., BrowderHartmanStampacchia Variational Inequalities for Multivalued Monotone Operators, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 333343, 1985. 17. TIAN, G., Generalizations 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. 457471, 1992. 18. TIAN, G., Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations, Review of Economic Studies, Vol. 60, pp. 949958, 1993. 19. TIAN, G., Generalized QuasiVariationalLike Inequality, Mathematics of Oper ations Research, Vol. 18, pp. 752764, 1993. 20. TIAN, G., and ZHOU, J., QuasiVariational Inequalities without Concavity Assumptions, Journal of Mathematical Analysis and Applications, Vol. 172, pp. 289299, 1992. 21. MICHAEL, E., Continuous Selections, 1, Annals of Mathematics, Vol. 63, pp. 361382, 1956. 22. SHAFER, W., and SONNENSCHEIN, H., Equilibrium in Abstract Economies without Ordered Preferences, Journal of Mathematical Economics, Vol. 2, pp. 345348, 1975. 23. ALLEN, G., Variational Inequalities, Complementarity Problems, and Duality Theorems, Journal of Mathematical Analysis and Applications, Vol. 58, pp. 110, 1977. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/41217 