본문 바로가기
KERD
논문KCI 등재

Elements of the KKM theory on abstract convex spaces

박세희
대한수학회 (The Korean Mathematical Society)|2008.01|대한수학회지 (Journal of the KMS)|제45권 제1호|pp. 1-27

1

조회

0

다운로드

7

인용

0

참고문헌

키워드

coincidenceB.abstract convex spaceKCgeneralized convex spaceKKM principlemultimap (map) classes KKOalmost fixed pointmap classes A·kc
주제분류수학

초록

We introduce a new concept of abstract convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an abstract convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of abstract 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.

Abstract

We introduce a new concept of abstract convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an abstract convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of abstract 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.

참고문헌 (53)

  1. P. S. Alexandroff, "Combinatorial Topology, OGIZ, Moscow-Leningrad", ~ (1947)
  2. P. Alexandroff, "Topologie I" (1974), Springer
  3. P. Alexandroff, "Elementary proof of the essentiality of the identical mapping of a simplex", Uspekhi Matematicheskikh Nauk (N.S.), 12(5), pp.175~179 (1957)
  4. A. Amini, "Fixed point theorems for the class S-KKM mappings in abstract convex spaces", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 66, pp.14~21 (2006)
  5. H. Ben-El-Mechaiekh, "Points fixes et coincidences pour les applications multivoques (applications de Ky Fan)", Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 295(4), pp.337~340 (1982)
  6. H. Ben-El-Mechaiekh, "Points fixes et coincidences pour les fonctions multivoques II (applications de type ' et '¤)", Comptes Rendus de l'Académie des Sciences, 295(5), pp.381~384 (1982)
  7. A. Borglin, "Existence of equilibrium actions and of equilibrium", Journal of Mathematical Economics, 3(3), pp.313~316 (1976)
  8. F. E. Browder, "The fixed point theory of multi-valued mappings in topological vector spaces", Mathematische Annalen, 177, pp.283~301 (1968)
  9. T.-H. Chang, "KKM property and fixed point theorems", Journal of Mathematical Analysis and Applications, 203(1), pp.224~235 (1996)
  10. T.-H. Chang, "On S-KKM property and related topics", Journal of Mathematical Analysis and Applications, 229(1), pp.212~227 (1999)
  11. K. Fan, "A generalization of Tychonoff’s fixed point theorem", Mathematische Annalen, 142, pp.305~310 (1961)
  12. K. Fan, "Sur un th´eor`eme minimax", Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 259, pp.3925~3928 (1964)
  13. K. Fan, "Applications of a theorem concerning sets with convex sections", Mathematische Annalen, 203, pp.189~293 (1966)
  14. K. Fan, "Extensions of two fixed point theorems of F. E. Browder", Mathematische Zeitschrift, 112, pp.23~240 (1969)
  15. K. Fan, "A minimax inequality and applications, Inequalities III", pp.103~113 (1972), Academic Press
  16. K. Fan, "A further generalization of Shapley’s generalization of the Knaster-Kuratowski- Mazurkiewicz theorem, Game Theory and Mathematical Economics", pp.275~279 (1981)
  17. K. Fan, "Some properties of convex sets related to fixed point theorems", Mathematische Annalen, 266(4), pp.519~537 (1984)
  18. C. J. Himmelberg, "Fixed points of compact multifunctions", Journal of Mathematical Analysis and Applications, 38, pp.205~207 (1972)
  19. C. D. Horvath, "Some results on multivalued mappings and inequalities without convexity, Nonlinear and Convex Analysis", Proc. in honor of Ky Fan, pp.99~106 (1987), New York
  20. C. D. Horvath, "Convexite generalisee et applications, Sem. Math. Super", pp.79~99 (1990), Press. Univ. Montreal
  21. C. D. Horvath, "Contractibility and generalized convexity", Journal of Mathematical Analysis and Applications, 156(2), pp.341~357 (1991)
  22. C. D. Horvath, "Extension and selection theorems in topological spaces with a generalized convexity structure", Annales de la Faculté des Sciences de Toulouse (Mathématiques), 2(2), pp.253~269 (1993)
  23. H. Kim, "KKM property, S-KKM property and fixed point theorems", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 63, pp.e1877~e1884 (2005)
  24. Hoonjoo Kim, "Remarks on the KKM property for open-valued multimaps on generalized convex spaces", 대한수학회보, 42(1), pp.101~110 (2005), 대한수학회
  25. B. Knaster, "Ein Beweis des Fixpunktsatzes f¨urn-Dimensionale Simplexe", Fundamenta Mathematicae, 14, pp.132~137 (1929)
  26. M. Lassonde, "On the use of KKM multifunctions in fixed point theory and related topics", Journal of Mathematical Analysis and Applications, 97(1), pp.151~201 (1983)
  27. J. Nash, "Nash, Non-cooperative games", Annals of Mathematics, 54(2), pp.286~293 (1951)
  28. S. Park, "Generalizations of Ky Fan’s matching theorems and their applications", Journal of Mathematical Analysis and Applications, 141(1), pp.164~176 (1989)
  29. S. Park, "Generalized matching theorems for closed coverings of convex sets", Numerical Functional Analysis and Optimization, 11(1), pp.101~110 (1990)
  30. S. Park, "Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed Point Theory and Applications", pp.248~277 (1992), World Scientific Publishing
  31. S. Park, "Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps", Journal of the Korean Mathematical Society, 31(3), pp.493~519 (1994)
  32. S. Park, "A unified approach to generalizations of the KKM-type theorems related to acyclic maps", Numerical Functional Analysis and Optimization, 15(1), pp.105~119 (1994)
  33. S. Park, "Coincidence theorems for the better admissible multimaps and their applications", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 30(7), pp.4183~4191 (1997)
  34. S. Park, "A unified fixed point theory of multimaps on topological vector spaces", Journal of the Korean Mathematical Society, 35(4), pp.803~832 (1998), 829–832
  35. S. Park, "Corrections", Journal of the Korean Mathematical Society, 36(4), pp.829~832 (1999)
  36. S. Park, "Ninety years of the Brouwer fixed point theorem", Vietnam Journal of Mathematics, 27(3), pp.187~222 (1999)
  37. S. Park, "Elements of the KKM theory for generalized convex spaces", Journal of Applied Mathematics and Computing, 7(1), pp.1~28 (2000)
  38. S. Park, "Remarks on topologies of generalized convex spaces", Nonlinear functional analysis and its applications, 5(2), pp.67~79 (2000)
  39. S. Park, "New topological versions of the Fan-Browder fixed point theorem", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 47(1), pp.595~606 (2001)
  40. S. Park, "Fixed point theorems in locally G-convex spaces", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 4(6), pp.869~879 (2002)
  41. S. Park, "Coincidence, almost fixed point, and minimax theorems on generalized convex spaces", Journal of Nonlinear and Convex Analysis, 4(1), pp.151~164 (2003)
  42. S. Park, "On generalizations of the KKM principle on abstract convex spaces", Nonlinear Analysis Forum, 11(1), pp.67~77 (2006)
  43. S. Park, "A unified fixed point theory in generalized convex spaces", Acta Mathematica Sinica English Series, 23(8), pp.1509~1536 (2007)
  44. S. Park, "Fixed point and non-retract theorems – Classical circular tours", Taiwanese Journal of Mathematics, 5(1), pp.97~108 (2001)
  45. S. Park, "Admissible classes of multifunctions on generalized convex spaces", Proceedings of College of Natural Sciences, 통권 : 18, pp.1~21 (1993)
  46. S. Park, "Coincidences of composites of u.s.c. maps on H-spaces and applications", Journal of the Korean Mathematical Society, 32(2), pp.251~264 (1995)
  47. S. Park, "Coincidence theorems for admissible multifunctions on generalized convex spaces", Journal of Mathematical Analysis and Applications, 197(1), pp.173~187 (1996)
  48. S. Park, "Foundations of the KKM theory on generalized convex spaces", Journal of Mathematical Analysis and Applications, 209(2), pp.551~571 (1997)
  49. M. Sion, "On general minimax theorems", Pacific Journal of Mathematics, 8, pp.171~176 (1958)
  50. B. P. Sortan, "Introduction to Axiomatic Theory of Convexity" (1984), Kishyneff
  51. J. von Neumann, "Zur Theorie der Gesellschaftsspiele", Mathematische Annalen, 100(1), pp.295~320 (1928)
  52. J. von Neumann, "Ubereinokonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes", Ergebnisse eines mathematischen Kolloquiums, 8, pp.73~83 (1937)
  53. N. Yannelis, "Existence of maximal elements and equilibria in linear topological spaces", Journal of Mathematical Economics, 12(3), pp.233~245 (1983)

관련 논문AI추천

COMMENTS ON GENERALIZED R-KKM TYPE THEOREMS박세희대한수학회 · 대한수학회논문집2010 · 제25권 제2호COINCIDENCE THEOREMS FOR NONCOMPACT RC-MAPS IN ABSTRACT CONVEX SPACES WITH APPLICATIONSMing-ge Yang, Nan-Jing Huang대한수학회 · 대한수학회보2012 · 제49권 제6호A SIMPLE PROOF OF THE SION MINIMAX THEOREM박세희대한수학회 · 대한수학회보2010 · 제47권 제5호Applications of results on abstract convex spaces to topological ordered spaces김훈주대한수학회 · 대한수학회보2013 · 제50권 제1호Remarks on the KKM property for open-valued multimaps on generalized convex spacesHoonjoo Kim, Sehie Park대한수학회 · 대한수학회지2005 · 제42권 제1호Generalized KKM-type theorems for best proximity pointsHoonjoo Kim대한수학회 · 대한수학회보2016 · 제53권 제5호

이 간행물의 다른 논문 (17건)

Strong convergence for three classes of uniformly equi-continuous and asymptotically quasi-nonexpansive mappingsXiaolong Qin, Meijuan Shang, Yongfu Supp. 29-40Lorentzian surfaces with constant curvatures and transformations in the 3-dimensional Lorentzian space박준상pp. 41-61Holomorphic functions on the mixed norm spaces on the polydiscStevo Stevicpp. 63-78The initial-boundary-value problem of a generalized Boussinesq equation on the half lineRuying Xuepp. 79-95Integral representations in electrical impedance tomography using boundary integral operators권기운pp. 97-119
원문 보기