## Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity.(English)Zbl 0767.49007

The paper presents a generalization of the FKKM theorem of Fan by introducing generalized closedness and continuity conditions. Then, this theorem is used to extend the Ky Fan minimax inequality by relaxing compactness, convexity and lower semicontinuity assumptions. Finally, some applications of these results to existence theorems in economics and optimization theory are given.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 54H25 Fixed-point and coincidence theorems (topological aspects) 91B50 General equilibrium theory 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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### References:

 [1] Allen, G., Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl., 58, 1-10 (1977) · Zbl 0383.49005 [2] Border, K. C., Fixed Point Theorems with Application to Economics and Game Theory (1985), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0558.47038 [3] Fan, K., Minimax theorem, (Proc. Nat. Acad. Sci. U.S.A., 39 (1953)), 42-47 · Zbl 0050.06501 [4] Fan, K., Fixed-point and related theorems for non-compact sets, (Moeschlin, O.; Pallaschke, D., Game Theory and Related Topics (1979), North-Holland: North-Holland Amsterdam), 151-156 · Zbl 0432.54040 [5] Fan, K., Some properties of convex sets related to fixed points theorems, Math. Ann., 266, 519-537 (1984) · Zbl 0515.47029 [6] Karamardian, S., Generalized complementarity problem, J. Optim. Theory Appl., 8, 416-427 (1971) · Zbl 0218.90052 [7] Kim, T.; Richter, M., Nontransitve-nontotal consumer theory, J. Econ. Theory, 38, 324-363 (1986) · Zbl 0598.90005 [8] Knaster, B.; Kuratowski, C.; Mazurkiewicz, S., Ein Beweis des Fixpunktsatze $$n$$-dimensionale Simpliexe, Fund. Math., 14, 132-137 (1929) [9] Mas-Colell, A., An equilibrium existence theorem without complete or transitive preferences, J. Math. Econom., 1, 237-246 (1974) · Zbl 0348.90033 [10] Gale, D.; Mas-Colell, A., An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econom., 2, 9-15 (1975) · Zbl 0324.90010 [11] Schmeidler, D., Competitive equilibrium in markets with a continuum of traders and incomplete preferences, Econometrica, 37, 578-585 (1969) · Zbl 0184.45201 [12] Shafer, W.; Sonnenschein, H., Equilibrium in abstract economies without ordered preferences, J. Math. Econom., 2, 345-348 (1975) · Zbl 0312.90062 [13] Sonnenschein, H., Demand theory without transitive preferences, with application to the theory of competitive equilibrium, (Chipman, J. S.; Hurwicz, L.; Richter, M. K.; Sonnenschein, H., Preferences, Utility, and Demand (1971), Harcourt Brace Jovanovich: Harcourt Brace Jovanovich New York) · Zbl 0277.90012 [14] Tian, G., Minimax inequalities equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorems (1989), mimeo · Zbl 0788.49015 [15] Tian, G., Generalizations of the Fan-Knaster-Kuratowski-Mazurkiewicz theorem and the Ky-Fan minimax inequality (1989), mimeo [16] Tian, G., Equilibrium in abstract economies with a non-compact infinite dimensional strategy space, an infinite number of agents and without ordered preferences, Econom. Lett., 33, 203-206 (1990) · Zbl 1375.91148 [17] Tian, G., Fixed points theorems for mappings with non-compact and non-convex domains, J. Math. Anal. Appl., 158, 161-167 (1991) · Zbl 0735.47037 [18] Tian, G.; Zhou, J., Transfer continuities, generalizations of the Weierstrass theorem and maximum theorems—A full characterization (1990), Texas A&M University, mimeo [19] Walker, M., On the existence of maximal elements, J. Econom. Theory, 16, 470-474 (1977) · Zbl 0421.54016 [20] Yannelis, N. C.; Prabhakar, N. D., Equilibria in abstract economies with an infinite number of agents, an infinite number of commodities and without ordered preferences, J. Math. Econom., 12, 223-245 (1983) [21] Zhou, J.; Chen, G., Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl., 132, 213-225 (1988) · Zbl 0649.49008
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