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A professor of mine has suggested to me to look at this theorem and to find a problem related to it to explain in a future class. I found an understandable proof in "Linear operators" by Dunford-Schwartz and I think I studied it, so now I know how to probe Brouwer's Theorem. Now I was thinking of some interesting related problem I could try to solve, do you have any suggestion of something not too hard (I am a second year undergraduate student!) ?

Thank you very much!

EDIT: I put PDE in the tags as this professor is mainly interested in this area so if you have any idea on that then even better (I guess most of the applications to PDEs will be very hard though!! )

Moritzplatz
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  • If I get your question correctly, it can be used to prove Period three implies chaos. –  Feb 18 '13 at 17:52
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    See my answer here: http://math.stackexchange.com/questions/119891/finite-state-markov-chain-stationary-distribution –  Feb 18 '13 at 18:12

2 Answers2

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How about Ky Fan's inequality? The one from his 1972 paper, "A minimax inequality and its applications" (there are several inequalities frequently called Ky Fan inequality). This Ky Fan Inequality is used to establish the existence of equilibria in various games studied in economics.

For the applications to PDEs that I know you will need infinite-dimensional generalisations of Brouwer's fixed point theorem.

Maybe the Brouwer invariance of domain theorem is more accessible. It is discussed in Terry Tao's blog, see http://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/.

UwF
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  • thank you! is the inequality you refer to this one at page 1? http://www.cs.elte.hu/~illes/Tantargyak/Szakmai%20anyagok/Cikk6/Kassay-Pales-1999.pdf – Moritzplatz Feb 18 '13 at 18:02
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You can prove Nash's theorem that every symmetric game with two players has a mixed strategy Nash equilibrium. This can be done using differential equations (ordinary though) and Brouwer's theorem. A very accessible exposition is given in this book.

Artem
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