On some problems of instability in semi-dynamical systems

The problem of instability of a closed positively invariant set $M$ of a semi-dynamical system on an arbitrary metric space $X$ is considered. The Lyapunov’s direct method for such problems has been developed quite completely in the case when $M$ is compact and $X$ is locally compact. In this article, we obtain sufficient conditions for instability in terms of Lyapunov functions in two situations: $M$ has a neighbourhood of positive Lagrange stable semi-trajectories; the space $X$ is asymptotically compact in some neighbourhood of $M$.