The global stability of nonautonomous systems. II (Q1387266)

It is considered the system of differential equations (1) \( \dot{x} = \varphi(t,x,y)\), \(\dot{y} = \psi(t,x,y)\), with \(t\in \mathbb{R}_{+}\), \(x\in \mathbb{R}\), \(y\in \mathbb{R}\), the continuous functions \(\varphi,\psi\) are satisfying in every finite domain Lipschitz conditions with respect to \(x,y\) with constant independent of the time \(t\); \(\varphi(0,x,y)\equiv 0\), \(\psi(0,x,y)\equiv 0 \). Let: \( H_{1}(t,x,y,t_{0})= (\varphi(t,x,y) -\varphi(t_{0},0,y))/x\), \(x\neq 0\); \(H_{4}(t,x,y,t_{0})= (\psi(t,x,y) -\psi(t_{0},x,0))/y\), \(y\neq 0\); \(h_{2}(t_{0},y)= \varphi(t_{0},0,y)/y\), \(y\neq 0\); \(h_{3}(t_{0},x)= \psi(t_{0},x,0)/x\), \(x\neq 0\). The generalized Routh-Hurwitz conditions are: \( H_{1} + H_{2}\leq -\alpha_{1}(x,y) < 0\), \(H_{1} H_{4}- h_{2} h_{3}\geq\alpha_{2}(x,y) > 0\) as \(t\geq 0\), \(x\neq 0\), \(y\neq 0\). One of the main results is the theorem: Let be fulfilled the generalized Routh-Hurwitz conditions for system (1) with a certain \(t_0\geq 0\) and, in addition, the conditions: 1. \(\beta_{1}(t_{0},x,y)=\sup_{t\geq 0}\{H_{1}(t,x,y,t_{0})\}\leq 0\) for all \(x\neq 0\), \(y\in \mathbb{R}\), \(\beta_{2}(t_{0},x,y)=\sup_{t\geq 0}\{H_{4}(t,x,y,t_{0})\}\leq 0\) for all \(y\neq 0\), \(x\in \mathbb{R}\); 2. if \(\beta_{1}(t_{0},x,0)=0\) for a certain \(x\neq 0\), then \(| \psi(t,x,0)| \geq\gamma_{1}(x)>0\) for all \(t\geq 0\); if \(\beta_{2}(t_{0},0,y)=0\) for a certain \(y\neq 0\), then \(| \varphi(t,0,y)| \geq\gamma_{2}(y)>0\) for all \(t\geq 0;\) 3. \(\int_{0}^{\pm\infty}| \varphi(t_{0},0,y)| \text{sign}(y) dy=+\infty\), \(\int_{0}^{\pm\infty}| \psi(t_{0},x,0)| \text{sign}(x) dx=+\infty\). Then the zero solution is uniformly asymptotically stable in the whole. If the solutions to (1) are uniformly bounded, then condition 3 can be omitted. [For part I see the author, Russ. Math. 41, No. 7, 28-35 (1997); translation from Izv. Vyzssh. Uchebn. Zaved., Math. 1997, No. 7(422), 26-32 (1997; Zbl 0909.34042)].