This paper presents an analysis of the asymptotic stability of the system of differential equations
and the scalar functions
are piecewise continuous and nonnegative. A simple condition to ensure asymptotic stability is
. The paper works out several other results which use the milder assumption
, together with elaborate conditions of integral type. As these conditions are somewhat technical, the author presents several alternatives, which are less general but easier to use in applications. Comparison with known criteria are given. The method of proof is a combination of the method of Lyapunov functions and of the theory of differential inequalities.