Banach's fixed how to find q

Question

I am looking at Banach's fixed point theorem in $\Bbb R$ and I am wondering what is $q$? I have seen that it is $$\max_{ x \in [a,b] }|f'(x)|, $$ can anyone confirm this?

Answer 1

If $A$ is a closed intervall in $ \mathbb R$ and if $f:A \to A$ is a differentiable function with $q:= \sup \{|f(x)| : x \in A\} < \infty$, then we get by the mean value theorem:

$$|f(x)-f(y)| \le q|x-y|$$

for all $x,y \in A$. If $q<1$ then, by Banach's theorem, there is a unique $x_0 \in A$ such that $f(x_0)=x_0$.