Stability with respect to nonautonomous first approximation (Q1603100)

The author considers the system \(x'=A(t)x+R(t,x)\), where \(x\) is a vector with the Euclidean norm \(\|x\|\), and \(\|R(t,x)\|\leq c\|x\|^\sigma\) with \(c>0\), \(\sigma >1\). A sufficient condition for the asymptotic stability of the zero solution is given, provided that the first approximation admits a Lyapunov function satisfying the inequalities \[ a_1\|x\|^2\leq V(t,x)\leq a_2\|x\|^2,\quad V'(t,x)\leq -\lambda (t)\|x\|^2. \] The method is generalized to the case of homogeneous (nonlinear) first approximation.