I'm having a problem understanding this:
Show that if is a continuous map such that is a strict contraction for some fixed integer , then has a unique fixed point.
I'm not sure I understand what they mean by .
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?.
That is the classic method--but if it's compact there is an eaiser way by considering the map . It's easy to show that this mapping is continuous since and are both continuous (the first obviously and the second since each coordinate map is continuous) and . Thus, since is compact one has that has a minimum point. Use the -fold composition contraction fact then to show (by contradiction) that this point must be a fixed point--for uniqueness see Tinyboss's response.
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?