Suppose we have a family $B(\mathbb R^n)$ of real valued functions e.g. $f:\mathbb R^n \to \mathbb R$.
If $\mathcal{T}$ is a contraction mapping on $B(\mathbb R^n)$ then there exists a constant $\alpha \in (0,1)$ such that $$ | \mathcal{T} f'(x) - \mathcal{T} f''(x)| \leq \alpha | f'(x) - f''(x) | $$ for all $f',f''\in B(\mathbb R^n)$ and $x \in \mathbb R^n$.
Are there any sufficient conditions such that $\mathcal{T}$ is a contraction on $B(\mathbb R^n)$ when $B(\mathbb R^n)$ contains unbounded functions?
I've learnt in class about Blackwell's sufficient conditions. However, these conditions require $B(\mathbb R^n)$ to be a family of bounded functions.