Stability of Delay Differential Equations with Oscillating Coefficients

We study the solutions to the delay differential equation equation ẋ(t) = −a(t)x(t− h), where the coefficient a(t) is not necessarily positive. It is proved that this equation is exponentially stable provided that a(t) = b+ c(t) for some positive constant b less than π/(2h), and the integral R t 0 c(s)ds is sufficiently small for all t > 0. In this case the 3/2-stability theorem is improved.