That's what n-fold composition is, yes. And I believe it's completeness, not compactness, that you need for this result. To get a fixed point, make a sequence and show that it's Cauchy (this is where we need the space to be complete). For uniqueness, suppose there are two distinct fixed points and look at their images under U--do they get closer together?.
Thanks for your replies! Compactness can not be assumed. I'm not sure, but I think completeness can be assumed. Anyway, lets assume is complete. Then since is a strict contraction, by Banach fix point theorem there exists a unique fixed point s.t . Performing on both sides we get
This means that is a fixed point to as well.
Uniqueness implies that . Thus is a fixed point to as well. As for uniqueness, assume that and both are fixed points to
because of that is a strict contraction.
Is this correct?