Kakutani theorem
From Encyclopedia of Mathematics
Let be a non-empty compact subset of \mathbb{R}^{n} , let X^{*} be the set of its subsets, and let f: X \to X^{*} be an upper semi-continuous mapping such that for each x \in X , the set f(x) is non-empty, closed and convex. The theorem then states that f has a fixed point (i.e., there is a point x \in X such that x \in f(x) ). S. Kakutani showed in [1] that from his theorem, the minimax principle for finite games does follow.
References
[1] | S. Kakutani, “A generalization of Brouwer's fixed point theorem”, Duke Math. J., 8: 3 (1941), pp. 457–459. |
[2] | Ky Fan, “Fixed point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad. Sci. USA, 38 (1952), pp. 121–126, |
[3] | H. Nikaido, “Convex structures and economic theory”, Acad. Press (1968). |
References
[a1] | J. Dugundji, A. Granas, “Fixed point theory”, 1, PWN (1982). |
How to Cite This Entry:
Kakutani theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kakutani_theorem&oldid=40016
Kakutani theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kakutani_theorem&oldid=40016
This article was adapted from an original article by A.Ya. Kiruta (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article