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New generalizations of basic theorems in the KKM theory. (English) Zbl 1225.47060
Some new KKM type theorems are stated for intersectionally closed-valued KKM maps. Section 2 deals with some history of the basic theorems and their pioneering applications due to Fan, Browder and others; Section 3 introduces the basic concepts of abstract spaces and KKM spaces. Section 4 deals with general KKM type theorems on abstract convex spaces satisfying a partial KKM principle for KKM maps having intersectionally closed values. In Section 5, the Fan type whole intersection theorem, the Fan-Browder type fixed point theorems, and maximal element theorems for abstract convex spaces having a partial KKM principle are deduced. Finally, Section 6 deals with a Fan type matching theorem, the Fan minimax inequality, and some variational inequalities on abstract convex spaces.

47H04Set-valued operators
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
58E35Variational inequalities (global problems)
54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
[1] Park, S.: Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed point theory and applications, 248-277 (1992)
[2] Park, S.: Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27, 187-222 (1999) · Zbl 0938.54039
[3] Park, S.: The KKM principle in abstract convex spaces: equivalent formulations and applications, Nonlinear anal. 73, 1028-1042 (2010) · Zbl 1214.47042 · doi:10.1016/j.na.2010.04.029
[4] Luc, D. T.; Sarabi, E.; Soubeyran, A.: Existence of solutions in variational relation problems without convexity, J. math. Anal. appl. 364, 544-555 (2010) · Zbl 1183.49017 · doi:10.1016/j.jmaa.2009.10.040
[5] Fan, K.: A generalization of tychonoff’s fixed point theorem, Math. ann. 142, 305-310 (1961) · Zbl 0093.36701 · doi:10.1007/BF01353421
[6] Fan, K.: Invariant subspaces of certain linear operators, Bull. amer. Math. soc. 69, 773-777 (1963) · Zbl 0118.10803 · doi:10.1090/S0002-9904-1963-11028-9
[7] Fan, K.: Sur un théorème minimax, C. R. Acad. sci. Paris sér. I. math. 259, 3925-3928 (1964) · Zbl 0138.37304
[8] Fan, K.: Applications of a theorem concerning sets with convex sections, Math. ann. 163, 189-203 (1966) · Zbl 0138.37401 · doi:10.1007/BF02052284
[9] Fan, K.: Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112, 234-240 (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225
[10] Fan, K.: A minimax inequality and applications, Inequalities III, 103-113 (1972)
[11] Fan, K.: Fixed-point and related theorems for noncompact convex sets, Game theory and related topics, 151-156 (1979) · Zbl 0432.54040
[12] Fan, K.: Some properties of convex sets related to fixed point theorems, Math. ann. 266, 519-537 (1984) · Zbl 0515.47029 · doi:10.1007/BF01458545
[13] Browder, F. E.: The fixed point theory of multi-valued mappings in topological vector spaces, Math. ann. 177, 283-301 (1968) · Zbl 0176.45204 · doi:10.1007/BF01350721
[14] S. Park, Generalized Fan--Browder fixed point theorems and their applications, Collec. of Papers Dedicated to J.G. Park, Chonbook Nat. Univ., 1989, pp. 51--77.
[15] Park, S.: New topological versions of the Fan--Browder fixed point theorem, Nonlinear anal. 47, 595-606 (2001) · Zbl 1042.47519 · doi:10.1016/S0362-546X(01)00204-8
[16] Park, S.: Elements of the KKM theory on abstract convex spaces, J. korean math. Soc. 45, No. 1, 1-27 (2008) · Zbl 1149.47040 · doi:10.4134/JKMS.2008.45.1.001
[17] Rockafellar, R. T.: Convex analysis, (1970) · Zbl 0193.18401
[18] Park, S.: New foundations of the KKM theory, J. nonlinear convex anal. 9, No. 3, 331-350 (2008) · Zbl 1167.47041 · http://www.ybook.co.jp/online/jncae/vol9/p331.html
[19] Park, S.; Kim, H.: Foundations of the KKM theory on generalized convex spaces, J. math. Anal. appl. 209, 551-571 (1997) · Zbl 0873.54048 · doi:10.1006/jmaa.1997.5388
[20] Kim, H.; Park, S.: Generalized KKM maps, maximal elements, and almost fixed points, J. korean math. Soc. 44, No. 2, 393-406 (2007) · Zbl 1148.47038 · doi:10.4134/JKMS.2007.44.2.393
[21] Tian, G.: Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity, J. math. Anal. appl. 170, 457-471 (1992) · Zbl 0767.49007 · doi:10.1016/0022-247X(92)90030-H
[22] Park, S.: Remarks on some basic concepts in the KKM theory, Nonlinear anal. (2010)