TITLE: Can the proof of fixed point theorems ever be constructive?
QUESTION [7 upvotes]: 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?

REPLY [10 votes]: 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:

Hendtlass, Matthew. Fixed point theorems in constructive mathematics. Journal of Logic and Analysis 4:10 (2012) 1–20. ISSN 1759-9008.