“Markov–Kakutani fixed-point theorem”的意思、由来-开放百科全书

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Markov–Kakutani fixed-point theorem

释义

Statement
Proof for a single affine self-mapping
Proof of theorem
References

In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point.

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 +(1 – t)y) = tT(x) + (1 – t)T(y) for t in [0,1] and x, y in C. Then the mappings have a common fixed point in 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

Since C is compact, there is a convergent subnet in C:

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

Taking N = N_i and passing to the limit as i goes to infinity, it follows that

Hence

Proof of theorem

The set of fixed points of a single affine mapping T is a non-empty compact convex set C^T by the result for a single mapping. The other mappings in the family S commute with T so leave C^T 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^T as T ranges over the subset. From the compactness of C it follows that the set

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}}

1 : Fixed-point theorems

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