Let $x\in[0,1]$，$f(x)\in[0,1]$ be a continuous function. Is there a polynomial-time method to find a (approximate) fixed point that $f(x)=x$?
If the answer to the above question is positve, how about the 2-dimensional case that $x\in [0,1]^2$ and $f(x)\in[0,1]^2$?
There is no such polynomial time method, unfortunately. Finding a fixed point is known to be at least exponential, see "Exponential lower bounds for finding Brouwer fixed points" by Hirsch, et al. (link here)
It might also be worth checking out Sperner's Lemma, which gives a combinatorial proof of Brouwer's Theorem, and thus a constructive proof. Plus the paper linked above gives some good references for what algorithms people use in practice.
If all else fails, there is some good discussion on MO here, though they don't say much past what I've summarized here.
I hope this helps ^_^