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Difference between revisions of "Kakutani theorem"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550901.png" /> be a non-empty compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550902.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550903.png" /> be the set of its subsets and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550904.png" /> be an upper [[Semi-continuous mapping|semi-continuous mapping]] such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550905.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550906.png" /> is non-empty, closed and convex; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550907.png" /> has a fixed point (i.e. there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550908.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550909.png" />). S. Kakutani showed [[#References|[1]]] that from his theorem the [[Minimax principle|minimax principle]] for finite games does follow.
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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>
 
  
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<table>
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<TR><TD valign="top">[1]</TD><TD valign="top">
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S. Kakutani, “A generalization of Brouwer's fixed point theorem”, ''Duke Math. J.'', '''8''': 3 (1941), pp. 457–459.</TD></TR>
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<TR><TD valign="top">[2]</TD><TD valign="top">
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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>
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<TR><TD valign="top">[3]</TD><TD valign="top">
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H. Nikaido, “Convex structures and economic theory”, Acad. Press (1968).</TD></TR>
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</table>
  
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====References====
  
====Comments====
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<table>
 
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<TR><TD valign="top">[a1]</TD><TD valign="top">
 
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J. Dugundji, A. Granas, “Fixed point theory”, '''1''', PWN (1982).</TD></TR>
====References====
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</table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Dugundji,   A. Granas,   "Fixed point theory" , '''1''' , PWN (1982)</TD></TR></table>
 

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=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