  In this note we show that if a continuous-time, nonlinear, time-invariant,
finite-dimensional system evolves on a compact subset of Rn and if the Jacobian
of the vector field is Hurwitz at each point of the compact set, then there is
a unique equilibrium on the set and solutions exponentially converge to it.
This shows that the Markus-Yamabe conjecture, which is false in general on Rn,
n>2, holds on compact sets. The results of this note can be viewed as an
application of Krasovskii's method for constructing Lyapunov functions and we
are able to similarly construct Lyapunov-like functions valid on the given
compact set. Examples are provided to illustrate the result.
