On the asymptotic stability for functional differential equations by Lyapunov functionals (Q1570930)

The author considers the nonautonomous system \[ x'(t)=F(t,x_t),\;F(t,0)\equiv 0,\tag{1} \] where \(C=C([-h,0];\mathbb{R}^n)\) denotes the space of continuous functions from \([-h,0]\) into \(\mathbb{R}^n,\;0<h=\text{const.}\); \(F:\mathbb{R}_{+}\times C\to\mathbb{R}^n\) is continuous and maps bounded sets into bounded sets. For any solution \(x:[t_0-h,T]\to\mathbb{R}^n\) and any \(t\in [t_0,T]\), the segment \(x_t\in C\) is defined by \(x_t(s):=x(t+s),\;-h\leq s\leq 0\). The paper is concerned with conditions of different kinds of stability of the zero solution to (1). The main tool in the stability investigations is Lyapunov's direct method.