ELEMENTS OF THE KKM THEORY ON CONVEX SPACES

Title & Authors
ELEMENTS OF THE KKM THEORY ON CONVEX SPACES
Park, Se-Hie;

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 $\small{A_c^{\kappa}}$;B;
Language
English
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10.
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COINCIDENCE THEOREMS FOR NONCOMPACT ℜℭ-MAPS IN ABSTRACT CONVEX SPACES WITH APPLICATIONS, Bulletin of the Korean Mathematical Society, 2012, 49, 6, 1147
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References
1.
P. S. Alexandroff, Combinatorial Topology, OGIZ, Moscow-Leningrad, 1947. [Russian]

2.
P. Alexandroff and H. Hopf, Topologie I, Springer, Berlin-Heidelberg-New York, 1974

3.
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]

4.
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

5.
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

6.
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

7.
A. Borglin and H. Keiding, Existence of equilibrium actions and of equilibrium, J. Math. Econom. 3 (1976), no. 3, 313-316

8.
F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301

9.
T.-H. Chang and C.-L. Yen, KKM property and fixed point theorems, J. Math. Anal. Appl. 203 (1996), no. 1, 224-235

10.
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

11.
K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310

12.
K. Fan, Sur un theoreme minimax, C. R. Acad. Sci. Paris 259 (1964), 3925-3928

13.
K. Fan, Applications of a theorem concerning sets with convex sections, Math. Ann. 163 (1966), 189-203

14.
K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234-240

15.
K. Fan, A minimax inequality and applications, Inequalities III (O. Shisha, ed.), 103-113, Academic Press, New York, 1972

16.
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

17.
K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519-537

18.
C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205-207

19.
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

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

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

23.
H. Kim, KKM property, S-KKM property and fixed point theorems, Nonlinear Anal. 63 (2005), e1877-e1884

24.
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

25.
B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-Dimensionale Simplexe, Fund. Math. 14 (1929), 132-137

26.
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

27.
J. Nash, Non-cooperative games, Ann. of Math. (2) 54 (1951), 286-293

28.
S. Park, Generalizations of Ky Fan's matching theorems and their applications, J. Math. Anal. Appl. 141 (1989), no. 1, 164-176

29.
S. Park, Generalized matching theorems for closed coverings of convex sets, Numer. Funct. Anal. Optim. 11 (1990), no. 1-2, 101-110

30.
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

31.
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

32.
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

33.
S. Park, Coincidence theorems for the better admissible multimaps and their applications, Nonlinear Anal. 30 (1997), no. 7, 4183-4191

34.
S. Park, A unified fixed point theory of multimaps on topological vector spaces, J. Korean Math. Soc. 35 (1998), no. 4, 803-829

35.
S. Park, Corrections, J. Korean Math. Soc. 36 (1999), no. 4, 829-832

36.
S. Park, Ninety years of the Brouwer fixed point theorem, Vetnam J. Math. 27 (1999), no. 3, 187-222

37.
S. Park, Elements of the KKM theory for generalized convex spaces, Korean J. Comp. Appl. Math. 7 (2000), no. 1, 1-28

38.
S. Park, Remarks on topologies of generalized convex spaces, Nonlinear Funct. Anal. Appl. 5 (2000), no. 2, 67-79

39.
S. Park, New topological versions of the Fan-Browder fixed point theorem, Nonlinear Anal. 47 (2001), no. 1, 595-606

40.
S. Park, Fixed point theorems in locally G-convex spaces, Nonlinear Anal. 48 (2002), no. 6, Ser. A: theory Methods, 869-879

41.
S. Park, Coincidence, almost fixed point, and minimax theorems on generalized convex spaces, J. Nonlinear Convex Anal. 4 (2003), no. 1, 151-164

42.
S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11 (2006), no. 1, 67-77

43.
S. Park and K. S. Jeong, Fixed point and non-retract theorems - Classical circular tours, Taiwanese J. Math. 5 (2001), no. 1, 97-108

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

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

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

48.
M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171-176

49.
B. P. Sortan, Introduction to Axiomatic Theory of Convexity, Kishyneff, 1984. [Russian with English summary.]

50.
J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), no. 1, 295-320

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