So just thinking out loud here but like in the 1d case... The set will be an interval [a,b] Let g(x) = f(x) - x
g(a) >= 0 and g(b) <= 0. Now we can use the intermediate value value theorem and binary search to find (get arbitrary close) a point with g(x) = 0 (i.e. fix point of f).
Not possible to do such searches in higher dimensions?
This is useful for practical applications, but it's not the same as knowing the fixed point for mathematical applications.
E.g. you may want to prove that some function preserves the irrational number pi. But you may not be able to prove it, only show that computations make your conjecture more and more likely.
I don't think you necessarily have g(a) >= 0 and g(b) <= 0. Finding the starting points for binary search might be essentially as hard as finding the fixed point.
Hence a <= f(a), so 0 <= f(a) - a = g(a)