Asymptotic stability of non-autonomous functional differential equations with distributed delays

Abstract

We consider the integro differential equation $$ x'(t)=-a(t)x(t)+b(t)\int^t_{t-h} \lambda(s)x(s)\,ds,\quad o\leq a(t),\; 0\le t<\infty, $$ where $a,b:\mathbb{R}_+\to\mathbb{R}$, $\lambda:[-h,\infty)\to \mathbb{R}$ are piecewise continuous functions and $h$ is a positive constant. We establish sufficient conditions guaranteeing either asymptotic stability or uniform asymptotic stability for the zero solution. These conditions state that the instantaneous stabilizing term on the right-hand side dominates in some sense the perturbation term with delays. Our conditions not require $a$ being bounded from above. The results are based on the method of Lyapunov functionals and Razumikhin functions.

Keywords

• Annulus argument
• uniform asymptotic stability