The method of Lyapunov functionals in stability analysis of functional-differential equations (Q5941988)

The authors give an extension of the method of limit equations and limit Lyapunov functionals (see, for instance, \textit{A. S. Andreev} and \textit{D. Kh. Khusanov} [Differ. Equations 34, No.~7, 876-885 (1998; Zbl 0958.34061)]) for nonautonomous functional-differential equations of neutral type \[ \frac{d}{dt}\left[x(t)-G(t,x_{t})\right]=F(t,x_{t}), \tag{1} \] where \(F,G:{\mathbb{R}}^{+}\times C_{H}\to{\mathbb{R}}^{n}\) are continuous mappings, \({\mathbb{R}}^{+}=[0,\infty)\), \(C_{H}=\{\phi\in C_{[-h,0]}: \|\phi\|<H\}\), \(C_{[-h,0]}\) is the Banach space of continuous functions \(\phi:[-h,0]\to{\mathbb{R}}^{n}\) equipped with the supremum norm \(\|\phi\|\), \(x_{t}\) is an element of \(C_{[-h,0]}\) defined as \(x_{t}(s)=x(t+s)\) for \(-h\leq s\leq 0\). For these equations, the authors introduce the notion of a limit equation \[ \frac{d}{dt}\left[x(t)-G^{*}(t,x_{t})\right]=F^{*}(t,x_{t}), \tag{2} \] where \(G^{*}(t,x_{t})\) and \(F^{*}(t,x_{t})\) are such that there exists a sequence \(t_{n}\to\infty\), \(n\to\infty\), such that \(G(t+t_{n},\phi)\) uniformly converges to \(G^{*}(t,x_{t})\) and \(F(t+t_{n},\phi)\) uniformly converges to \(F^{*}(t,x_{t})\) on every set of the form \([0,n]\times K\), where \(n=1,2,\ldots\), and \(K\) is a compact subset of \(C_{H}\). Using a Lyapunov functional of the form \(V(t,x_{t},x(t)-G(t,x_{t}))\) with derivative of constant sign along the solutions to (1) and some properties of the limit equation (2), the authors consider the localizing of the positive limit sets of solutions to (1) and obtain several theorems on sufficient conditions for asymptotic stability and instability of the zero solution to equation (1). An illustrative example is given.