Inequalities and stability for a linear scalar functional differential equation

There 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) less than or equal to 0 for asymptotic stability and a(t) greater than or equal to 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) greater than or equal to 0 and instability if a(t) less than or equal to 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). (C) 2004 Elsevier Inc. All rights reserved.