On the sign definiteness of Liapunov functionals and stability of a linear delay equation

Abstract

In this paper we give a new definition of the positive-definiteness of the Liapunov functional involved in the stability and asymptotic stability investigation. Using this notion we prove Liapunov type theorems and apply these results to the scalar equation $\dot x(t)=b(t)x(t-r(t))$, where $b(t)$ and $r(t)$ may be unbounded.