I wish to find the values for which the logistic map behaves as a contraction map
i.e, I wish to find for which $r$, the mapping above admits a unique fixed point $F(x^*;r)=x^*$ using the contraction map theorem.
It is easy to find the fixed points, and it exists in any text book, as well as Banach's fixed point theorem, however I have no idea on how to apply this theorem over the example of the logistic map.
The most helpful reference I've found is in here, but I still can't figure out what is required.
Can any one please suggest how to approach this problem?