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Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these theorems do?

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Yes, there are constructive proofs for fixed point theorems including Brouwer. Also, the proof of the Banach fixed point theorem with which I am most familiar is constructive. In fact, here is a paper all about constructive methods for fixed point theorems:

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  • $\begingroup$ So that means we can find an algorithm for computing a fixed point? (for irrational numbers, standard interpretation of computing "reals" apply) $\endgroup$ – Kalami Jul 24 '14 at 13:37
  • $\begingroup$ This is not really a constructive proof of Brouwer's FPT. At least in the sense in which constructive mathematics is understood. $\endgroup$ – Rubi Shnol Sep 21 '15 at 20:12

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