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In [[mathematics]], the '''Markov–Kakutani fixed-point theorem''', named after [[Andrey Markov, Jr.|Andrey Markov]] and [[Shizuo Kakutani]], states that a commuting family of continuous [[affine transformation|affine self-mappings]] of a [[convex set|compact convex subset]] in a [[locally convex topological vector space]] has a common fixed point.
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==Statement==
Let ''E'' be a locally convex topological vector space. Let ''C'' be a compact convex subset of ''E''.  
Let ''S'' be a commuting family of self-mappings ''T'' of ''C'' which are continuous and affine, i.e.
''T''(''tx''&nbsp;+(1&nbsp;–&nbsp;''t'')''y'') =&nbsp;''tT''(''x'')&nbsp;+&nbsp;(1&nbsp;–&nbsp;''t'')''T''(''y'') for ''t'' in [0,1] and ''x'', ''y'' in ''C''. Then the mappings have a common fixed point in&nbsp;''C''.
 
==Proof for a single affine self-mapping==
 
Let ''T'' be a continuous affine self-mapping of ''C''.
 
For ''x'' in ''C'' define other elements of ''C'' by
 
:<math> x(N)={1\over N+1}\sum_{n=0}^N T^n(x). </math>
 
Since ''C'' is compact, there is a convergent subsequence in ''C'':
 
:<math> x(N_i)\rightarrow y. \,</math>
 
To prove that ''y'' is a fixed point, it suffices to show that ''f''(''Ty'') = ''f''(''y'') for every ''f'' in the dual of ''E''. Since ''C'' is compact, |''f''| is bounded on ''C'' by a positive constant ''M''. On the other hand
 
:<math> |f(Tx(N))-f(x(N))|={1\over N+1} |f(T^{N+1}x)-f(x)|\le {2M\over N+1}. </math>
 
Taking ''N'' = ''N''<sub>''i''</sub> and passing to the limit as ''i'' goes to infinity, it follows that
 
:<math> f(Ty) = f(y). \, </math>
 
Hence
 
:<math>Ty = y. \, </math>
 
==Proof of theorem==
The set of fixed points of a single affine mapping ''T'' is a non-empty compact convex set ''C''<sup>''T''</sup> by the result for a single mapping. The other mappings in the family ''S'' commute with ''T'' so leave ''C''<sup>''T''</sup> invariant. Applying the result for a single mapping successively, it follows that any finite subset of ''S'' has a non-empty fixed point set given as the intersection of the compact convex sets ''C''<sup>''T''</sup> as ''T'' ranges over the subset. From the [[finite intersection property|compactness]] of ''C'' it follows that the set
 
:<math> C^S=\{y\in C|Ty=y, \, y\in S\}=\bigcap_{T\in S} C^T \,</math>
 
is non-empty (and compact and convex).
 
==References==
*{{citation|first=A.|last=Markov|title=Quelques théorèmes sur les ensembles abéliens|journal=Dokl. Akad. Nauk. SSSR|year=1936|volume=10|pages=311–314}}
*{{citation|first=S.|last=Kakutani|title=Two fixed point theorems concerning bicompact convex sets|year=1938|volume=14|pages=242–245|journal=Proc. Imp. Akad. Tokyo}}
*{{citation|first=M.|last=Reed|first2=B.|last2=Simon|title=Functional Analysis|series=Methods of Mathematical Physics|volume=1| edition=2nd revised|publisher=Academic Press|year=1980|isbn=0-12-585050-6|page=152}}
 
{{DEFAULTSORT:Markov-Kakutani fixed-point theorem}}
[[Category:Fixed-point theorems]]
[[Category:Functional analysis]]
[[Category:Mathematical theorems]]

Latest revision as of 14:10, 14 May 2014

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