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.

You get as progression of the distance to the fixed point
$$
\frac{x_{n+1}-1}{x_n-1}=c(x_n^2+x_n)-1.
$$
To get convergence you need the right side to be smaller than $1$ in absolute value. This gives
$$
0<c(x^2+x)\le 2\iff \frac14<\left(x+\frac12\right)^2<\frac2c+\frac14
$$
To get $x=α=1$ inside the contracting interval one needs $0<c<1$. The interval is then
$$
0<x<\sqrt{\frac2c+\frac14}-\frac12=1+\frac{4(1-c)}{\sqrt{8c+c^2}+3c}
$$
