When does contraction in sup norm implies pointwise contraction?

Question

I was wondering about the following question :

Suppose I have a contraction mapping $\tau$ for a bounded continuous function in supremum norm.

$$ \beta \cdot ||u-v||_{\infty}\geq ||\tau(u)-\tau(v)||_{\infty} $$

for $\beta \in (0,1)$. This is easily attained if I use Blackwell's sufficient condition. However, what I'm interested in is if

$$ \beta \cdot |u(x)-v(x)| \geq |(\tau(u)(x)-\tau(v)(x)| $$

for any $x \in [0,1]$. In this case, is there any sufficient conditions that I can use to show this pointwise contraction?