My assumptions are:
My space of interest is $\mathbb{R}$ as Banach space.
My sequence of interest $X_{n+1}=\beta+\alpha X_n$ converges in my space $\mathbb{R}$. (It is already proved). Here $\beta\in\mathbb{R}\setminus \{0\}$ and $X_0\in\mathbb{R}\setminus\{0\}$.
The $\alpha$ is defined as $\alpha=\underbrace{v}_{\in (0,1)}\underbrace{\sqrt{\frac{1}{w^2}-1}}_{>0}$. Note, for every $\alpha$ one could chose a small $v$ such that the whole $\alpha$ is less than 1.
In my Banach space of interest that is $\mathbb{R}$, my sequence$X_{n+1}=\beta+\alpha X_n$ is Cauchy since it is convergent.
I define my mapping as $T(X)=\beta+\alpha X$.
Now, could I conclude based on above reasons that my map $T:[1,\infty)\mapsto [1,\infty)$ is a Contraction map? if this is concluded, my contractor map $T$ admits its fixed point inside my Banach space $\mathbb{R}$ and based on the fixed-point theorem.
One observation here: $\underbrace{[1,\infty)}_{R_T}\subseteq \underbrace{[1,\infty)}_{D_T}$. It simply means $T$ always fires inside its domain.