I have not given this an extreme amount of thought, but my first guess is to consider

. We first can note that since

that

and so since

is increasing we have that

and so

and thus by induction

. Thus,

is a bounded monotone sequence and so by the monotone convergence theorem it is convergent with limit

. So, I have absolutely no proof (yet) but it may be true that

is the fixed point. If

were continuous this would obviously be true since

. But, from there I'm not sure yet.

That said, I feel that can't be right. The fact that

is compact seems like it should come into play.

Just some input.