On the Lyapunov theorem for semi-dynamical systems

The paper deals with the stability problem of semi-dynamical systems on an arbitrary metric space. Variants of theorems of Lyapunov's second method are presented in the form of sufficient conditions for non-asymptotic stability of compact positively invariant sets in the class of constant-sign auxiliary functions. A comparative analysis of the results of the second Lyapunov method is given, depending on the requirements regarding the zero-level set of Lyapunov functions. The relative non-asymptotic stability is studied when the Lyapunov function's zero level set consists of fixed points of the system. Illustrative examples are given.