Consider a nonlinear dynamical system S of the form: $\dot{x}=f(x)$
The function $f$ (and the system S) is said to be Lipschitz if : $\forall x,y \in \mathbb{R}, \| f(x) - f(y) \| \leq K \|x - y \| $
The smallest $K >0$ is called the Lipschitz constant of the function $f$ (or the system S).
My question :
How can one relate this constant $K$ to the analysis of stability of the system S ?
My idea is that if $K<1$, then the function $f$ defines a contraction mapping.
Is it correct ?
If yes, I was thinking about using the Banach fixed-point theorem to proove the existence of an equilibrium point of S. But it is not clear how to study stability of this point ...
I would really appreciate any idea or remark about this question ! Thanks in advance.