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Ding, Xie Ping; Wang, Lei

Fixed points, minimax inequalities and equilibria of noncompact abstract economies in \(FC\)-spaces. (English) Zbl 1157.47037

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 2, 730-746 (2008).
The authors first present a fixed point theorem in the setting of noncompact \(FC\)-spaces. Then they give several equivalent results to this fixed point theorem. As usual application of such kind of fixed point theorem, the authors derive a minimax theorem. Several maximal element theorems are also presented. Once again, as usual application of the maximal element theorems, several equilibrium existence results for abstract economies are derived.
Reviewer: Qamrul Hasan Ansari (Aligarh)

Cited in 6 Documents

MSC:

47H10 Fixed-point theorems
49K35 Optimality conditions for minimax problems
91A10 Noncooperative games
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics

Keywords:

fixed points; minimax inequalities; maximal elements; abstract economies; \(FC\)-spaces
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\textit{X. P. Ding} and \textit{L. Wang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 2, 730--746 (2008; Zbl 1157.47037)
Full Text: DOI

References:

[1] Ding, X. P.; Tan, K. K., Fixed point theorems and equilibria of noncompact generalized games, (Tan, K. K., Fixed Point Theory and Applications (1992), World Sci. Pub.: World Sci. Pub. Singarpo), 80-96 · Zbl 1404.47007
[2] Ding, X. P.; Tan, K. K., A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math., 63, 233-247 (1992) · Zbl 0833.49009
[3] Ding, X. P.; Tan, K. K., On equilibria of noncompact generalized games, J. Math. Anal. Appl., 177, 226-238 (1993) · Zbl 0789.90008
[4] Ding, X. P.; Kim, W. K.; Tan, K. K., Equilibria of noncompact generalized games with \(L^\ast \)-majorized preference correspondences, J. Math. Anal. Appl., 162, 508-517 (1992) · Zbl 0765.90092
[5] Ding, X. P.; Kim, W. K.; Tan, K. K., A selection theorem and its applications, Bull. Austral. Math., 46, 205-212 (1992) · Zbl 0762.47030
[6] Yuan, G. X.Z., KKM Theory and Applications in Nonlinear Analysis (1999), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York
[7] Borglin, A.; Keiding, H., Existence of equilibrium action and of equilibrium: A note on the ‘new’ existence theorems, J. Math. Econom., 3, 313-316 (1973) · Zbl 0349.90157
[8] Yannelis, N. C.; Prabhakar, N. D., Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom., 12, 233-245 (1983) · Zbl 0536.90019
[9] Toussaint, S., On the existence of equilibria in economics with infinite many commodities and without ordered preferences, J. Econom. Theory, 33, 98-115 (1984) · Zbl 0543.90016
[11] Tarafdar, E., A fixed point theorem equivalent to Fan-Knaster-Kuratowski-Mazurkiewicz theorem, J. Math. Anal. Appl., 128, 229-240 (1987) · Zbl 0644.47050
[12] Tan, K. K.; Yuan, X. Z., A minimax inequality with applications to existence of equilibrium points, Bull. Austral. Math., 47, 483-503 (1993) · Zbl 0803.47059
[13] Tarafdar, E., Fixed point theorems in \(H\)-spaces and equilibrium points of abstract economies, J. Austral. Math. Soc. Ser. A, 53, 252-260 (1992) · Zbl 0761.47041
[14] Ding, X. P., Equilibrium existence theorems of abstract economics in \(H\)-spaces, Indian J. Pure Appl. Math., 25, 309-317 (1994) · Zbl 0811.90147
[15] Ding, X. P., Fixed points, minimax inequalities and equilibria of noncompact abstract economies, Taiwanese J. Math., 2, 25-55 (1998) · Zbl 0911.47055
[16] Ding, X. P., Maximal element theorems in product \(F C\)-spaces and generalized games, J. Math. Anal. Appl., 305, 29-42 (2005) · Zbl 1120.91001
[17] Yannelis, N. C., Maximal elements over noncompact subsets of linear topological spaces, Econom. Lett., 17, 133-136 (1985) · Zbl 1273.91184
[18] Ding, X. P.; Tarafdar, E., Fixed point theorems and existence of equilibrium points of noncompact abstract economics, Nonlinear World, 1, 319-340 (1994) · Zbl 0813.47069
[20] Horvath, C. D., Contractibility and general convexity, J. Math. Anal. Appl., 156, 341-357 (1991) · Zbl 0733.54011
[21] Park, S.; Kim, H., Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl., 209, 551-571 (1997) · Zbl 0873.54048
[22] Ben-El-Mechaiekh, H.; Chebbi, S.; Flornzano, M.; Llinares, J. V., Abstract convexity and fixed points, J. Math. Anal. Appl., 222, 138-150 (1998) · Zbl 0986.54054
[23] Deng, L.; Xia, X., Generalized \(R\)-KKM theorems in topological space and their applications, J. Math. Anal. Appl., 285, 679-690 (2003) · Zbl 1039.54011
[25] Dugundji, J., Topology (1966), Allyn and Bacon Inc. · Zbl 0144.21501
[26] Gale, D.; Mas-Colell, A., On the role of complete, transitive preferences in equilibrium theory, (Schwodiauer, G., Equilibrium and Disequilibrium in Economic Theory (1978), Reider: Reider Pordrecht), 7-14 · Zbl 0377.90011
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