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Maximal element theorems in product \(FC\)-spaces and generalized games. (English) Zbl 1120.91001

Summary: Let \(I\) be a finite or infinite index set, \(X\) be a topological space and \((Y_{i},\phi_{N_{i}})_{i \in I} \) be a family of finitely continuous topological spaces (\(FC\)-spaces, for short). Let \(A_{i}:X\to 2^{Y_{i}}\), for each \(i \in I\), be a set-valued mapping. Some existence theorems for maximal elements of the family \(\{A_{i}\}_{i\in I}\) are established when the \(FC\)-spaces are noncompact. As applications, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in noncompact \(FC\)-spaces. These theorems improve, unify and generalize many important results in the recent literature.

MSC:

91A07 Games with infinitely many players
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
91B50 General equilibrium theory
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[1] Aubin, J. P.; Ekeland, I., Applied Nonlinear Analysis (1984), John Wiley & Sons: John Wiley & Sons New York
[2] Ben-El-Mechaiekh, H.; Chebbi, S.; Florenzano, M.; Llinares, J., Abstract convexity and fixed points, J. Math. Anal. Appl., 222, 138-151 (1998) · Zbl 0986.54054
[3] Borglin, A.; Keiding, H., Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econom., 3, 313-316 (1876) · Zbl 0349.90157
[4] Chang, T. H.; Yen, C. L., KKM property and fixed point theorems, J. Math. Anal. Appl., 203, 224-235 (1996) · Zbl 0883.47067
[5] Debreu, G., A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA, 38, 121-126 (1952)
[6] Deguire, P.; Tan, K. K.; Yuan, X. Z., The study of maximal elements, fixed points for \(L_S\)-majorized mappings and their applications to minimax and variational inequalities in product topological spaces, Nonlinear Anal., 37, 933-951 (1999) · Zbl 0930.47024
[7] Ding, X. P., Coincidence theorems and equilibria of generalized games, Indian J. Pure Appl. Math., 27, 1057-1071 (1996) · Zbl 0870.54047
[8] Ding, X. P., Fixed points, minimax inequalities and equilibria of noncompact generalized games, Taiwanese J. Math., 2, 25-55 (1998) · Zbl 0911.47055
[9] Ding, X. P., Equilibria of noncompact generalized games with \(U\)-majorized preference correspondences, Appl. Math. Lett., 11, 115-119 (1998) · Zbl 0939.91013
[10] Ding, X. P., Maximal element principles on generalized convex spaces and their application, (Argawal, R. P., Set Valued Mappings with Applications in Nonlinear Analysis. Set Valued Mappings with Applications in Nonlinear Analysis, SIMMA, vol. 4 (2002)), 149-174 · Zbl 1018.47036
[11] Ding, X. P., Maximal elements of \(G_B\)-majorized mappings in product \(G\)-convex spaces (I), Appl. Math. Mech., 24, 659-672 (2003) · Zbl 1078.47006
[12] Ding, X. P., Maximal elements of \(G_B\)-majorized mappings in product \(G\)-convex spaces (II), Appl. Math. Mech., 24, 1017-1034 (2003) · Zbl 1078.47005
[13] Ding, X. P., New H-KKM theorems and their applications to geometric property, coincidence theorems, minimax inequality and maximal elements, Indian J. Pure Appl. Math., 26, 1-19 (1995) · Zbl 0830.49003
[14] Ding, X. P., Coincidence theorems in topological spaces and their applications, Appl. Math. Lett., 12, 99-105 (1999) · Zbl 0942.54030
[16] 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
[17] Ding, X. P.; Kim, W. K.; Tan, K. K., Equilibria of generalized games with \(L\)-majorized correspondences, Internat. J. Math. Math. Sci., 17, 783-790 (1994) · Zbl 0811.90146
[18] 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
[19] Ding, X. P.; Tan, K. K., On equilibria of noncompact generalized games, J. Math. Anal. Appl., 177, 226-238 (1993) · Zbl 0789.90008
[20] Ding, X. P.; Tarafdar, E., Fixed point theorems and existence of equilibrium points of noncompact abstract economies, Nonlinear World, 1, 319-340 (1994) · Zbl 0813.47069
[21] Ding, X. P.; Xia, F. Q., Equilibria of nonparacompact generalized games with \(L_{F_c}\)-majorized correspondence in \(G\)-convex spaces, Nonlinear Anal., 56, 831-849 (2004) · Zbl 1087.91007
[22] Ding, X. P.; Yao, J. C.; Lin, L. J., Solutions of system of generalized vector quasi-equilibrium problems in locally \(G\)-convex uniform spaces, J. Math. Anal. Appl., 292, 398-410 (2004) · Zbl 1072.49005
[23] Ding, X. P.; Yuan, G. X.-Z., The study of existence of equilibria for generalized games without lower semicontinuity in locally convex topological vector spaces, J. Math. Anal. Appl., 227, 420-438 (1998) · Zbl 0917.90301
[24] Horvath, C. D., Contractibility and general convexity, J. Math. Anal. Appl., 156, 341-357 (1991) · Zbl 0733.54011
[25] Kim, W. K.; Tan, K. K., New existence theorems of equilibria and applications, Nonlinear Anal., 47, 531-542 (2001) · Zbl 1042.47534
[26] Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037
[27] Lin, L. J.; Yu, Z. T.; Ansari, Q. H.; Lai, L. P., Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities, J. Math. Anal. Appl., 284, 656-671 (2003) · Zbl 1033.47039
[28] Park, S., Fixed points of admissible maps on generalized convex spaces, J. Korean Math. Soc., 37, 885-899 (2000) · Zbl 0967.47039
[29] Park, S., Coincidence theorems for the better admissible multimaps and their applications, Nonlinear Anal., 30, 4183-4191 (1977) · Zbl 0922.47052
[30] Park, S., Continuous selection theorems for admissible multifunctions on generalized convex spaces, Numer. Funct. Anal. Optim., 25, 567-583 (1999) · Zbl 0931.54017
[31] Park, S.; Kim, H., Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl., 209, 551-571 (1997) · Zbl 0873.54048
[32] Shen, Z. F., Maximal element theorems of \(H\)-majorized correspondence and existence of equilibrium for abstract economies, J. Math. Anal. Appl., 256, 67-79 (2001) · Zbl 0994.91009
[33] Singh, S. P.; Tarafdar, E.; Watson, B., A generalized fixed point theorem and equilibrium point of an abstract economy, J. Comput. Appl. Math., 113, 65-71 (2000) · Zbl 0947.47048
[34] Tan, K. K.; Yuan, X. Z., Existence of equilibrium for abstract economies, J. Math. Econom., 23, 243-251 (1994) · Zbl 0821.90022
[35] Tan, K. K.; Yuan, X. Z., Approximation method and equilibria of abstract economies, Proc. Amer. Math. Soc., 122, 503-510 (1994) · Zbl 0842.47033
[36] Tan, K. K.; Zhang, X. L., Fixed point theorems on \(G\)-convex spaces and applications, Proc. Nonlinear Funct. Anal. Appl., 1, 1-19 (1996)
[37] Tarafdar, E., A fixed point theorem and equilibrium point of an abstract economy, J. Math. Econom., 20, 211-218 (1991) · Zbl 0718.90014
[38] 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
[39] Toussaint, S., On the existence of equilibria in economies with infinite commodities and without ordered preferences, J. Econom. Theory, 33, 98-115 (1984) · Zbl 0543.90016
[41] Tulcea, C. I., On the approximation of upper semi-continuous correspondences and the equilibriums of generalized games, J. Math. Anal. Appl., 136, 267-289 (1988) · Zbl 0685.90100
[42] 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
[43] Yuan, G. X.-Z., The study of minimax inequalities and applications to economies and variational inequalities, Mem. Amer. Math. Soc., 132 (1998)
[44] Yuan, G. X.-Z., KKM Theory and Application in Nonlinear Analysis (1999), Dekker: Dekker New York
[45] Yuan, G. X.-Z., The existence of equilibria for noncompact generalized games, Appl. Math. Lett., 13, 57-63 (2000) · Zbl 1028.91515
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