Inequalities and stability for a linear scalar functional differential equation

AbstractThere have been a lot of investigations about stability of the linear scalar functional differential equation x′(t)=a(t)x(t)+b(t)x(t−h), where a,b:R+→R continuous and h>0 a constant. However, almost all investigations require a(t)⩽0 for asymptotic stability and a(t)⩾0 for instability. In this paper, we investigate Wazewski inequalities of solutions of the equation. As a consequence, we offer some sufficient conditions for asymptotic stability if a(t)⩾0 and instability if a(t)⩽0. In the case that a(t) and b(t) are constant, we offer a region showing uniform asymptotic stability and instability of the zero solution of the equation. This region is different from J. Hale's (1977)