Tian, Guoqiang (1990): Generalizations of the FKKM Theorem and KyFan Minimax Inequality, with Applications to Maximal Elements, Price Equilibrium, and Complementarity. Published in: Journal of Mathematical Analysis and Applications , Vol. 170, (1992): pp. 457471.

PDF
MPRA_paper_41225.pdf Download (735kB)  Preview 
Abstract
This paper generalizes the FanKnasterKuratowskiMazurkiewicz (FKKM) theorem of Ky Fan (“Game Theory and Related Topics,” pp. 151–156, NorthHolland, Amsterdam, 1979; and Math. Ann.266, 1984, 519–537) and the Ky Fan minimax inequality by introducing a class of the generalized closedness and continuity conditions, which are called the transfer closedness and transfer continuities. We then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions. We also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions. Thus our results generalize many of the existence theorems in the literature.
Item Type:  MPRA Paper 

Original Title:  Generalizations of the FKKM Theorem and KyFan Minimax Inequality, with Applications to Maximal Elements, Price Equilibrium, and Complementarity 
Language:  English 
Keywords:  FKKM Theorem; Inequality; Maximal Elements; Price Equilibrium; Complementarity 
Subjects:  D  Microeconomics > D6  Welfare Economics > D63  Equity, Justice, Inequality, and Other Normative Criteria and Measurement 
Item ID:  41225 
Depositing User:  Guoqiang Tian 
Date Deposited:  13. Sep 2012 18:35 
Last Modified:  22. Aug 2015 14:04 
References:  G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58 (1977), 110. K. C. BORBER, "Fixed Point Theorems with Application to Economics and Game Theory" Cambridge University Press, London, 1985. K. Fan, A minimax inequality and applications, in “inequalities” (O. Shisha, Ed.), Vol.3, pp. 103113, Academic Press, New York, 1972. K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519537. S. Karamardian, Generized complementary problem. J. Optim. Theory Appl. 8 1971, 416427. T. Kim and M. Richter, Nontransivenontotal consumer theory, J. Econ. Theory 38, 198324363. B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunksatze n=dimensionale Simpliexe, Fund. Math. 14 1929, 132137. E. Tarafdar, A fix point theorem equivalent to the FanKnasterKuratowskiMazurkiewicz theorem, J. Math. Anal. Appl. 128 (1987), 475479. G. Tian, Minimax inequalities equivalent to the FanKnasterKuratowskiMazurkiewicz theorems, to appear. G. Tian, Equilibrium in abstract economies with a noncompact infinite dimensional strategy space, an infinite number of agents and without ordered preferences, Econom. Lett., in press. G. Tian, Fixed points theorems for mappings with noncompact and nonconvex domains, J. Math. Anal. Appl., in press. G. Tian and J. Zhou, Transfer continuities, generalizations of the Weierstrass theorem and maximum theorems A full characterization, Texas A &M University, mimeo, 1990. M. Walker, On the existence of maximal elements, J. Econom. Theory 16, 1977, 470474 N. C. Yannelis and N. D. Prabhakar, Equilibria in abstract economies with infinite number of agents, an infinite number of conditions and without ordered preferences, J. Math. Econom. 12, 1983, 223245. J.X. Zhou and G. Chen, Diagonal convexity conditions for problems in convex analysis and quasivariational inequalities, J. Math. Anal. Appl. 132 (1988), 213225. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/41225 