Inequalities and exponential decay in time varying delay differential equations

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential decay of solutions to zero of the time varying delay differential equation x′(t)=b(t)x(t)−a(t)x(t−h(t)).x′(t)=b(t)x(t)−a(t)x(t−h(t)). The highlights of the paper are allowing b(t)b(t) to change signs and the delay to vary with time. In addition, we obtain a criterion for the instability of the zero solution. Moreover, by comparison to existing literature we show effectiveness of our results.