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. 375-389.
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This paper extends the well-known 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|
|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|
|Depositing User:||Guoqiang Tian|
|Date Deposited:||13 Sep 2012 18:43|
|Last Modified:||27 Sep 2016 23:16|
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. 233-245, 1983.
3. BORDER, K. C., Fixed-Point 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 n-Demensionale Simpliexe, Fundamental Mathematica, Vol. 14, pp. 132-137, 1929.
5. FAN, K., Minimax Theorem, Proceedings of the National Academy of Sci- ences, Vol. 39, pp. 42-47, 1953.
6. FAN, K., A Generalization of Tychonoff s Fixed-Point Theorem, Mathematis- che Annalen, Vol. 142, pp. 305-310, 1962.
7. FAN, K., Some Properties of Convex Sets Related to Fixed-Points Theorems, Mathematische Annalen, Vol. 266, pp. 519-537, 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. 477-481, 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 Fixed-Point Theo- rems, Journal of the Mathematical Society of Japan, Vol. 28, pp. 477-481, 1976.
12. ZHOU, J., and CHEN, G., Diagonal Convexity Conditions for Problems in Convex Analysis and Quasi-Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 132, pp. 213-225, 1988.
13. BARDARO, C., and CEPPITELLI, R., Applications of the Generalized Knaster- Kuratowski-Mazurkiewicz Theorem to Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 137, pp. 46-58, 1989.
14. BARDARO, C., and CEPPITELLI, R., Some Further Generalizations of the Knaster-Kuratowski-Mazurkiewicz Theorem and Minimax Inequalities, Jour- nal of Mathematical Analysis and Applications, Vol. 132, pp. 484-490, 1988.
15. SHIH, M. H., and TAN, K. K., Generalized Quasi-Variational Inequalities in Locally Convex Topological Vector Spaces, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 333-343, 1985.
16. SHIH, M. H., and TAN, K. K., Browder-Hartman-Stampacchia Variational Inequalities for Multivalued Monotone Operators, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 333-343, 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. 457-471, 1992.
18. TIAN, G., Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations, Review of Economic Studies, Vol. 60, pp. 949-958, 1993.
19. TIAN, G., Generalized Quasi-Variational-Like Inequality, Mathematics of Oper- ations Research, Vol. 18, pp. 752-764, 1993.
20. TIAN, G., and ZHOU, J., Quasi-Variational Inequalities without Concavity Assumptions, Journal of Mathematical Analysis and Applications, Vol. 172, pp. 289-299, 1992.
21. MICHAEL, E., Continuous Selections, 1, Annals of Mathematics, Vol. 63, pp. 361-382, 1956.
22. SHAFER, W., and SONNENSCHEIN, H., Equilibrium in Abstract Economies without Ordered Preferences, Journal of Mathematical Economics, Vol. 2, pp. 345-348, 1975.
23. ALLEN, G., Variational Inequalities, Complementarity Problems, and Duality Theorems, Journal of Mathematical Analysis and Applications, Vol. 58, pp. 1-10, 1977.