  This paper deals with asymptotic stability of a class of dynamical systems in
terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov
results for differential inclusions cannot be applied to this class of
dynamical systems. Following an abstract approach we put an assumption on the
trajectories of the dynamical systems which demands for any trajectory the
existence of a neighboring trajectory such that their difference grows linearly
in time and distance of the starting points. Under this assumption, we prove
the existence of a $C^\infty$-smooth Lyapunov pair. We also show that this
assumption is satisfied by differential inclusions defined by Lipschitz
continuous set-valued maps taking nonempty, compact and convex values.
