Browse > Article
http://dx.doi.org/10.4134/JKMS.2008.45.1.001

ELEMENTS OF THE KKM THEORY ON CONVEX SPACES  

Park, Se-Hie (The National Academy of Sciences, Department of Mathematical Sciences Seoul National University)
Publication Information
Journal of the Korean Mathematical Society / v.45, no.1, 2008 , pp. 1-27 More about this Journal
Abstract
We introduce a new concept of convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.
Keywords
abstract convex space; generalized convex space; KKM principle; multimap (map) classes K; KC; KD; coincidence; almost fixed point; map classes $A_c^{\kappa}$; B;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 23
Times Cited By SCOPUS : 19
연도 인용수 순위
1 S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31 (1994), no. 3, 493-519   과학기술학회마을
2 K. Fan, A minimax inequality and applications, Inequalities III (O. Shisha, ed.), 103-113, Academic Press, New York, 1972
3 S. Park, Elements of the KKM theory for generalized convex spaces, Korean J. Comp. Appl. Math. 7 (2000), no. 1, 1-28   과학기술학회마을
4 T.-H. Chang, Y.-Y. Huang, J.-C. Jeng, and K.-H. Kuo, On S-KKM property and related topics, J. Math. Anal. Appl. 229 (1999), no. 1, 212-227   DOI   ScienceOn
5 H. Kim and S. Park, Remarks on the KKM property for open-valued multimaps on generalized convex spaces, J. Korean Math. Soc. 42 (2005), no. 1, 101-110   과학기술학회마을   DOI   ScienceOn
6 P. S. Alexandroff, Combinatorial Topology, OGIZ, Moscow-Leningrad, 1947. [Russian]
7 P. Alexandroff and H. Hopf, Topologie I, Springer, Berlin-Heidelberg-New York, 1974
8 P. Alexandroff and B. Pasynkoff, Elementary proof of the essentiality of the identical mapping of a simplex, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5 (77), 175-179. [Russian]
9 A. Amini, M. Fakhar, and J. Zafarani, Fixed point theorems for the class S-KKM mappings in abstract convex spaces, Nonlinear Anal. 66 (2006), 14-21   DOI   ScienceOn
10 T.-H. Chang and C.-L. Yen, KKM property and fixed point theorems, J. Math. Anal. Appl. 203 (1996), no. 1, 224-235   DOI   ScienceOn
11 K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310   DOI
12 H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, Points fixes et coincidences pour les applications multivoques (applications de Ky Fan), C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), no. 4, 337-340
13 H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, Points fixes et coincidences pour les fonctions multivoques II (applications de type ${\varphi}$ et ${\varphi}^{\ast}$), C. R. Acad. Sci. Paris 295 (1982), no. 5, 381-384
14 A. Borglin and H. Keiding, Existence of equilibrium actions and of equilibrium, J. Math. Econom. 3 (1976), no. 3, 313-316   DOI   ScienceOn
15 F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301   DOI
16 C. D. Horvath, Some results on multivalued mappings and inequalities without convexity, Nonlinear and Convex Analysis- Proc. in honor of Ky Fan (B. L. Lin and S. Simons, eds.), 99-106, Marcel Dekker, New York, 1987
17 K. Fan, A further generalization of Shapley's generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Game Theory and Mathematical Economics (O. Moeschlin and D. Palaschke, ed.), 275-279, North-Holland, Amsterdam, 1981
18 K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519-537   DOI
19 C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205-207   DOI
20 C. D. Horvath, Convexite generalisee et applications, Sem. Math. Super. 110, 79-99, Press. Univ. Montreal, 1990
21 C. D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991), no. 2, 341-357   DOI
22 C. D. Horvath, Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 2, 253-269   DOI
23 J. Nash, Non-cooperative games, Ann. of Math. (2) 54 (1951), 286-293   DOI
24 S. Park, Generalizations of Ky Fan's matching theorems and their applications, J. Math. Anal. Appl. 141 (1989), no. 1, 164-176   DOI
25 S. Park, Generalized matching theorems for closed coverings of convex sets, Numer. Funct. Anal. Optim. 11 (1990), no. 1-2, 101-110   DOI   ScienceOn
26 K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234-240   DOI
27 Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed Point Theory and Applications (K.-K. Tan, ed.), 248-277, World Sci. Publ., River Edge, NJ, 1992
28 K. Fan, Sur un theoreme minimax, C. R. Acad. Sci. Paris 259 (1964), 3925-3928
29 K. Fan, Applications of a theorem concerning sets with convex sections, Math. Ann. 163 (1966), 189-203   DOI
30 S. Park and K. S. Jeong, Fixed point and non-retract theorems - Classical circular tours, Taiwanese J. Math. 5 (2001), no. 1, 97-108   DOI
31 H. Kim, KKM property, S-KKM property and fixed point theorems, Nonlinear Anal. 63 (2005), e1877-e1884   DOI   ScienceOn
32 B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-Dimensionale Simplexe, Fund. Math. 14 (1929), 132-137   DOI
33 M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), no. 1, 151-201   DOI
34 S. Park, A unified approach to generalizations of the KKM-type theorems related to acyclic maps, Numer. Funct. Anal. Optim. 15 (1994), no. 1-2, 105-119   DOI
35 S. Park, Coincidence theorems for the better admissible multimaps and their applications, Nonlinear Anal. 30 (1997), no. 7, 4183-4191   DOI   ScienceOn
36 S. Park, A unified fixed point theory of multimaps on topological vector spaces, J. Korean Math. Soc. 35 (1998), no. 4, 803-829   과학기술학회마을
37 S. Park, New topological versions of the Fan-Browder fixed point theorem, Nonlinear Anal. 47 (2001), no. 1, 595-606   DOI   ScienceOn
38 S. Park, Corrections, J. Korean Math. Soc. 36 (1999), no. 4, 829-832   과학기술학회마을
39 S. Park, Ninety years of the Brouwer fixed point theorem, Vetnam J. Math. 27 (1999), no. 3, 187-222
40 S. Park, Remarks on topologies of generalized convex spaces, Nonlinear Funct. Anal. Appl. 5 (2000), no. 2, 67-79
41 S. Park, Fixed point theorems in locally G-convex spaces, Nonlinear Anal. 48 (2002), no. 6, Ser. A: theory Methods, 869-879   DOI   ScienceOn
42 S. Park, Coincidence, almost fixed point, and minimax theorems on generalized convex spaces, J. Nonlinear Convex Anal. 4 (2003), no. 1, 151-164
43 S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11 (2006), no. 1, 67-77
44 S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci., Seoul Nat. Univ. 18 (1993), 1-21
45 S. Park and H. Kim, Coincidences of composites of u.s.c. maps on H-spaces and applications, J. Korean Math. Soc. 32 (1995), no. 2, 251-264   과학기술학회마을
46 S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996), no. 1, 173-187   DOI   ScienceOn
47 S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl. 209 (1997), no. 2, 551-571   DOI   ScienceOn
48 J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), no. 1, 295-320   DOI
49 M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171-176   DOI
50 B. P. Sortan, Introduction to Axiomatic Theory of Convexity, Kishyneff, 1984. [Russian with English summary.]
51 J. von Neumann, Uber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, Ergeb. Math. Kolloq. 8 (1937), 73-83
52 N. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983), no. 3, 233-245   DOI   ScienceOn