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I need some nice application of fixed point theorem for some kind of divulgative seminar. About Banach fixed point theorem a nice application is the definition of self-similar fractals (and I could mention Cauchy Theorem and local Invertibility Theorem).

Have you some nice application of Shauder or Brouwer fixed point theorems to point out?

Emanuele Paolini
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You could show that the Brouwer fixed point theorem implies Sperner's Lemma (and then perhaps some combinatorial applications thereof) or prove the existence of Nash Equilibrium in any game. In the same realm but not exactly implications are borsuk-ulam and the ham sandwich theorem, which are interesting in their own right.

Bob Krueger
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