## A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem

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- by Naoki Shioji PDF
- Proc. Amer. Math. Soc.
**111**(1991), 187-195 Request permission

## 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

- 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** - 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** - Samuel Eilenberg and Deane Montgomery,
*Fixed point theorems for multi-valued transformations*, Amer. J. Math.**68**(1946), 214–222. MR**16676**, DOI 10.2307/2371832 - Samuel Eilenberg and Norman Steenrod,
*Foundations of algebraic topology*, Princeton University Press, Princeton, N.J., 1952. MR**0050886** - Ky Fan,
*A generalization of Tychonoff’s fixed point theorem*, Math. Ann.**142**(1960/61), 305–310. MR**131268**, DOI 10.1007/BF01353421 - 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** - Ky Fan,
*Some properties of convex sets related to fixed point theorems*, Math. Ann.**266**(1984), no. 4, 519–537. MR**735533**, DOI 10.1007/BF01458545 - L. Górniewicz,
*A Lefschetz-type fixed point theorem*, Fund. Math.**88**(1975), no. 2, 103–115. MR**391062**, DOI 10.4064/fm-88-2-103-115 - Lech Górniewicz,
*Homological methods in fixed-point theory of multi-valued maps*, Dissertationes Math. (Rozprawy Mat.)**129**(1976), 71. MR**394637**
A Granas, - Chung Wei Ha,
*On a minimax inequality of Ky Fan*, Proc. Amer. Math. Soc.**99**(1987), no. 4, 680–682. MR**877039**, DOI 10.1090/S0002-9939-1987-0877039-9
B. Knaster, C. Kuratowski, and S. Mazurkiewicz, - 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**, DOI 10.1016/0022-247X(83)90244-5 - 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**, DOI 10.1016/0022-247X(88)90162-X
S. Simons, - 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**, DOI 10.4310/pamq.2010.v6.n2.a14 - 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**

*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. C. W. Ha,

*Minimax and fixed point theorems*, Math. Ann.

**248**(1980), 73-77.

*Ein Beweis des Fixpunktsatzes für*$n$

*-dimensional simplexe*, Fund. Math.

**XIV**(1929), 132-137.

*Minimax and variational inequalities. Are they of fixed point or Hahn-Banach type?*, Game Theory and Mathematical Economics, North-Holland, 1981, pp. 379-387.

## Additional Information

- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**111**(1991), 187-195 - MSC: Primary 47H10; Secondary 47H19, 54H25, 58C06
- DOI: https://doi.org/10.1090/S0002-9939-1991-1045601-X
- MathSciNet review: 1045601