functional iteration convergence 
functional iteration sequence $x_{n+1} = 2 - (1+c)x_n + cx_{n}^3$ will converge for some values of c to $ \alpha = 1$ for what values of c this sequence will converge?

My attempt to solve this was using contractive mapping theorem First finding a closed set $ D = [1-a, 1+b] $ and show that $ f: D \rightarrow D $ and also $ |f^{'}(x)| < 1$ then by the contractive mapping theorem this series will converge to a unique fixed point in $D$ which is 1.
$f^{'}(x) = -1 - c + 3cx^2$ is for some values of c always less than 1 but for some values not. I'm not sure how to define c value.

Except when $c=0$, the convergence depends largely on the initial value $x_0$.
For example, if $c \ne 0$ and $|x_0|> \max(100/|c|,5)$, you can prove easily that by induction that:
$$|x_{n+1}|>2|x_n|$$
which means the sequence doesn't converge.
