Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem

Author: Naoki Shioji
Journal: Proc. Amer. Math. Soc. 111 (1991), 187-195
MSC: Primary 47H10; Secondary 47H19, 54H25, 58C06
MathSciNet review: 1045601
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Granas and Dugundji obtained the following generalization of the Knaster-Kuratowski-Mazurkiewicz theorem.

Let $ X$ be a subset of a topological vector space $ E$ and let $ G$ be a set-valued map from $ X$ into $ E$ such that for each finite subset $ \{ {x_1}, \ldots ,{x_n}\} $ of $ X,co\{ {x_1}, \ldots ,{x_n}\} \subset \cup _{i = 1}^nG{x_i}$ and for each $ x \in X,Gx$ is finitely closed, i.e., for any finite-dimensional subspace $ L$ of $ E,Gx \cap L$ is closed in the Euclidean topology of $ L$. Then $ \{ Gx:x \in X\} $ has the finite intersection property.

By relaxing, among others, the condition that $ X$ is a subset of $ E$, we obtain a further generalization of the theorem and show some of its applications.

References [Enhancements On Off] (What's this?)

  • [1] Hichem Ben-El-Mechaiekh, Paul Deguire, and Andrzej Granas, Une alternative non linéaire en analyse convexe et applications, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 3, 257–259 (French, with English summary). MR 681592
  • [2] J. Dugundji and A. Granas, KKM maps and variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 4, 679–682. MR 519889
  • [3] Samuel Eilenberg and Deane Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222. MR 0016676
  • [4] Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952. MR 0050886
  • [5] Ky Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1960/1961), 305–310. MR 0131268
  • [6] Ky Fan, A minimax inequality and applications, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 103–113. MR 0341029
  • [7] Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519–537. MR 735533, 10.1007/BF01458545
  • [8] L. Górniewicz, A Lefschetz-type fixed point theorem, Fund. Math. 88 (1975), no. 2, 103–115. MR 0391062
  • [9] Lech Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. (Rozprawy Mat.) 129 (1976), 71. MR 0394637
  • [10] A Granas, KKM-maps and their applications to nonlinear problems, The Scottish Book: Mathematics from the Scottish Café (R. D. Mauldin, ed.), Birkhäuser, Basel, Boston, 1982, pp. 45-61.
  • [11] C. W. Ha, Minimax and fixed point theorems, Math. Ann. 248 (1980), 73-77.
  • [12] Chung Wei Ha, On a minimax inequality of Ky Fan, Proc. Amer. Math. Soc. 99 (1987), no. 4, 680–682. MR 877039, 10.1090/S0002-9939-1987-0877039-9
  • [13] B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für $ n$-dimensional simplexe, Fund. Math. XIV (1929), 132-137.
  • [14] Marc Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), no. 1, 151–201. MR 721236, 10.1016/0022-247X(83)90244-5
  • [15] Naoki Shioji and Wataru Takahashi, Fan’s theorem concerning systems of convex inequalities and its applications, J. Math. Anal. Appl. 135 (1988), no. 2, 383–398. MR 967217, 10.1016/0022-247X(88)90162-X
  • [16] S. Simons, Minimax and variational inequalities. Are they of fixed point or Hahn-Banach type?, Game Theory and Mathematical Economics, North-Holland, 1981, pp. 379-387.
  • [17] S. Simons, Two-function minimax theorems and variational inequalities for functions on compact and noncompact sets, with some comments on fixed-point theorems, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 377–392. MR 843623
  • [18] Wataru Takahashi, Fixed point, minimax, and Hahn-Banach theorems, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 419–427. MR 843628

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H10, 47H19, 54H25, 58C06

Retrieve articles in all journals with MSC: 47H10, 47H19, 54H25, 58C06

Additional Information

Keywords: Fixed point, KKM-map, minimax theorem
Article copyright: © Copyright 1991 American Mathematical Society