Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are converging towards ech other:

$d(x(t),y(t)) \leq e^{-\alpha t}d(x(0),y(0))$

for some $\alpha > 0$. Must there be an equilibrium point of this system?

In the discrete time setting this is what the contraction mapping theorem is saying: If we define the trajectories of the system by $x_{n+1} = T^{n+1}(x)$, then the condition $d(x_{n},y_{n}) \leq e^{-\alpha n}d(x,y)$ is sufficient to prove the existence of (and convergence to) an equillibrium point. I am just wondering when this criteria can be carried over to the continuous setting.