Fixed point theorem of Kakutani

The fixed point theorem of Kakutani is a mathematical theorem that can be assigned to the field of functional analysis and goes back to a work by the Japanese mathematician Shizuo Kakutani from 1938. The theorem is based on properties of convex sets in Hausdorff locally convex vector spaces and gives a sufficient condition for the existence of common fixed points for certain groups of homeomorphisms of such sets. It gave rise to numerous follow-up investigations and is closely linked to other important theorems of functional analysis, such as the fixed point theorem by Ryll-Nardzewski . The fixed point theorem of Kakutani implies the existence of Haarsch measures on compact groups. Hausdorff's maximal chain theorem or Zorn's lemma (and thus the axiom of choice ) is required for its proof .

Formulation of the sentence

Kakutani's fixed point theorem can be represented as follows:

Let a Hausdorff locally convex space be given and in it a non-empty , compact and convex subset together with a group of linear automorphisms which leave invariant, i.e. in which all automorphisms satisfy the subset relation. ${\displaystyle X}$ ${\displaystyle C\subset X}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle g\colon X\to X}$ ${\displaystyle C}$ ${\displaystyle g\in {\mathcal {G}}}$ ${\displaystyle g(C)\subseteq C}$

Let the group be uniformly steady . ${\displaystyle {\mathcal {G}}}$

Then:

${\displaystyle {\mathcal {G}}}$has a common fixed point, ie: there is one with for everyone . ${\displaystyle C}$ ${\displaystyle c_{0}\in C}$ ${\displaystyle g(c_{0})=c_{0}}$ ${\displaystyle g\in {\mathcal {G}}}$

Related result: Markov's Theorem

The Russian mathematician Markov has presented a set in the year 1936 and prior to the publication of kakutanischen fixed-point theorem, which is very similar to this in question and statement, the Markov set essentially differs in that it the requirement of even-gleichgradigen continuity through a Vertauschbarkeitsbedingung replaced :

Let a Hausdorff locally convex space and a non-empty compact convex subset be given . ${\displaystyle X}$ ${\displaystyle C\subset X}$

A family of continuous affine mappings is also given , which should be interchangeable in pairs with regard to the sequential execution . ${\displaystyle {\mathcal {F}}=(f_{i})_{i\in I}}$ ${\displaystyle f_{i}\colon C\to C}$

Then:

${\displaystyle {\mathcal {F}}}$has a common fixed point, ie: there is one with for everyone . ${\displaystyle C}$ ${\displaystyle c_{0}\in C}$ ${\displaystyle f_{i}(c_{0})=c_{0}}$ ${\displaystyle i\in I}$

additive

The statement of the theorem of Markov is particularly the case that - other things being equal - than Abelian group of continuous linear automorphisms with is assumed . This modified set is also called the fixed point theorem of Kakutani Markov ( English Kakutani Markov fixed point theorem ) ${\displaystyle {\mathcal {F}}}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle f(C)\subseteq C}$

Explanations

• The uniformly-Equicontinuity ( English Equicontinuity ) of the above image group is to by the - environment system of given uniform structure to reflect. In this context one calls - in full generality - a family of linear mappings between two topological vector spaces and uniformly uniformly continuous if and only if the following applies: ${\displaystyle {\mathcal {G}}}$ ${\displaystyle 0}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}=(f_{i})_{i\in I}}$ ${\displaystyle f_{i}\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

For every environment there is an environment that satisfies the condition . ${\displaystyle 0}$ ${\displaystyle U_{Y}(0)\subseteq Y}$ ${\displaystyle 0}$ ${\displaystyle U_{X}(0)\subseteq X}$ ${\displaystyle \bigcup _{i\in I}{f_{i}\left(U_{X}(0)\right)}\subseteq U_{Y}(0)}$

• A mapping of the convex set is called affine if the equation is always fulfilled for every two points and every real number . ${\displaystyle f\colon C\to C}$ ${\displaystyle C\subseteq X}$ ${\displaystyle x,y\in C}$ ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle f\left(\lambda x+(1-\lambda )y\right)=\lambda f(x)+(1-\lambda )f(y)}$

literature

• Shizuo Kakutani: On the uniqueness of Haar's measure . In: Proceedings of the Imperial Academy . tape 14 , 1938, pp. 27-31 ( MR1568492 ).
• Shizuo Kakutani: Two fixed-point theorems concerning bicompact convex sets . In: Proceedings of the Imperial Academy . tape 14 , 1938, pp. 242-245 ( MR1568507 ).
• Vasile I. Istrățescu : Fixed Point Theory . An Introduction. With a Preface by Michiel Hazewinkel (=  Mathematics and its Application . Volume 7 ). D. Reidel Publishing Company , Dordrecht, Bosto, London 1981, ISBN 90-277-1224-7 ( MR0620639 ).
• AA Markov : Quelques théorèmes sur les ensembles abeliens . In: Doklady Akad. Nauk. SSSR . tape 10 , 1936, pp. 311-314 .
• Barbara Przebieracz : A proof of the Mazur-Orlicz theorem via the Markov-Kakutani common fixed point theorem, and vice versa . In: Fixed Point Theory and Applications . 2015, doi : 10.1186 / s13663-014-0257-2 ( MR3304965 ).
• Walter Rudin : Functional Analysis (=  International Series in Pure and Applied Mathematics ). 2nd Edition. McGraw-Hill , Boston (et al.) 1991, ISBN 0-07-054236-8 ( MR1157815 ).
• Dirk Werner : A proof of the Markov-Kakutani fixed point theorem via the Hahn-Banach theorem . In: Extracta Mathematicae . tape 8 , 1993, pp. 37-38 ( MR1270326 ).
• Robert J. Zimmer : Essential Results of Functional Analysis (=  Chicago Lectures in Mathematics ). The University of Chicago Press, Chicago, London 1990, ISBN 0-226-98337-4 ( MR1045444 ).

Individual references and notes

1. ^ Walter Rudin: Functional Analysis. 1991, pp. 120 ff, 377, 393
2. ^ Vasile I. Istrățescu: Fixed Point Theory. 1987, p. 276 ff
3. ^ Robert J. Zimmer: Essential Results of Functional Analysis. 1990, p. 38 ff
4. ↑ Rudin, op.cit., P. 120
5. ↑ Istrățescu, op.cit., P. 277
6. ↑ Zimmer, op.cit., P. 39
7. ↑ Rudin, op.cit., P. 43
8. ↑ Istrățescu, op.cit., P. 276