Difference between revisions of "Kakutani theorem"
From Encyclopedia of Mathematics
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| − | Let | + | Let $ X $ 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|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 [[#References|[1]]] that from his theorem, the [[Minimax principle|minimax principle]] for finite games does follow. |
====References==== | ====References==== | ||
| − | |||
| + | <table> | ||
| + | <TR><TD valign="top">[1]</TD><TD valign="top"> | ||
| + | S. Kakutani, “A generalization of Brouwer's fixed point theorem”, ''Duke Math. J.'', '''8''': 3 (1941), pp. 457–459.</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD><TD valign="top"> | ||
| + | Ky Fan, “Fixed point and minimax theorems in locally convex topological linear spaces”, ''Proc. Nat. Acad. Sci. USA'', '''38''' (1952), pp. 121–126,</TD></TR> | ||
| + | <TR><TD valign="top">[3]</TD><TD valign="top"> | ||
| + | H. Nikaido, “Convex structures and economic theory”, Acad. Press (1968).</TD></TR> | ||
| + | </table> | ||
| + | ====References==== | ||
| − | + | <table> | |
| − | + | <TR><TD valign="top">[a1]</TD><TD valign="top"> | |
| − | + | J. Dugundji, A. Granas, “Fixed point theory”, '''1''', PWN (1982).</TD></TR> | |
| − | + | </table> | |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | ||
Latest revision as of 16:18, 15 December 2016
Let $ X $ 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=16664
Kakutani theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kakutani_theorem&oldid=16664
This article was adapted from an original article by A.Ya. Kiruta (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article