The following system of functional-differential equations with delay \[ \frac{dx(t)}{dt}=f(t,x_t),\quad f(t,0)\equiv 0, \tag{1} \] is considered, and the asymptotic stability of its solution \[ x(t)\equiv 0\tag{2} \] is studied. Let \(a, b\), and \(\omega\) be functions of Hahn’s class, and let \(p: \mathbb{R}_+\to \mathbb{R}_+\) be a function satisfying the condition \(\int_t^\infty p(s) ds=+\infty\) for all \(t>0\). The following theorem is the main result of the paper:
Suppose that for system (1) there exist a functional V and numbers \(K>1\) and \(H>0\) such that the following conditions are satisfied:
(i) \(a(|x(t)|)\leq V(t,x_t)\leq b(||x_t||)\) for \(t\in\mathbb{R}_+\), \(|x(t)|<H\);
(ii) \(\dot V(t,x_t)\leq-p(t)\omega(||x_t||)\) for \(t>t_1\) provides that \(KV(t,x_t)\geq V(t+\alpha,x_{t+\alpha})\) for all \(\alpha\in[-r(t),0]\).
Then solution (2) to system (1) is asymptotically stable. If, in addition, \(r(t)<r_0\) and \(p(t)>\varepsilon>0\) for all \(t\in \mathbb{R}_+\) and (iii) \(\dot V(t,x_t)\leq 0\) for \(t\in \mathbb{R}_+\), \(\Vvert x_t\|<H\), then solution (2) is uniformly asymptotically stable.