Stability of solutions of a nonstandard ordinary differential system by Lyapunov's second method (Q1180683)

The authors first prove certain existence results for systems of differential equations in nonstandard form, (1) \(x'=f(t,x,x')\), where \(f(t,x,z)\) is defined for \((t,x,z)\in[0,\infty)\times D\times S\), \(D\subset\mathbb{R}^ n\), \(S=\{z\in\mathbb{R}^ n: | z|\leq c\}\) and satisfies \(f(t,x,z)\in S\) and a Lipschitz condition \(| f(t,x_ 1,z_ 1)-f(t,x_ 2,z_ 2)|\leq k_ 1| x_ 1-x_ 2|+k_ 2| z_ 1-z_ 2|\) with \(k_ 2<1\). Then they present a generalization of the stability and asymptotic stability theorems of Lyapunov's direct method to systems (1) with \(f(t,0,0)=0\), using Lyapunov functions \(V(t,x,z)\) and calculating their derivative along a solution of (1) by the formula \[ V'(t,x,z)={\partial V\over\partial t}+\sum^ n_{j=1}\left({\partial V\over\partial y_ j}\right)f_ j+\sum^ n_{j=1}{\partial V\over\partial z_ j}\left(I-\left({\partial f_ i\over\partial z_ j}\right)\right)^{-1}\left({\partial f\over\partial t}+\left({\partial f_ i\over\partial y_ j}\right)f\right)_ j. \] As an illustrating example, they obtain a sufficient condition for uniform stability of the zero solution of the scalar second order equation \(y''+\omega^ 2y+\varepsilon(y')^ 3(y'')^ 2=0\).