On the \(\phi{}_ 0\)-stability of comparison differential systems
Combining and generalizing results of \textit{V. Lakshmikantham} and \textit{S. Leela} [Nonlinear Anal., Theory Methods Appl. 1, 215-222 (1977; Zbl 0357.34051)] and \textit{G. S. Ladde} [Trans. Am. Math. Soc. 221, 391-402 (1976; Zbl 0308.34046)], the authors consider a situation where, using vector Lyapunov functions, the stability problem for a differential system (1) \(x'=f(t,x)\), \(f\in C[\mathbb{R}_ +\times\mathbb{R}^ N,\mathbb{R}^ N]\), can be reduced to the investigation of a comparison system (2) \(u'=g(t,u)\) where \(g\in C[\mathbb{R}_ +\times K,\mathbb{R}^ n]\), and \(K\) is a cone in \(\mathbb{R}^ n\), \(n\leq N\). The trivial solution of (2) is said to be \(\varphi_ 0\)-equistable if, given \(\varepsilon>0\), there exists \(\delta=\delta(t_ 0,\varepsilon)\), continuous in \(t_ 0\) for each \(\varepsilon\), such that the inequality \((\varphi_ 0,u_ 0)<\delta\) implies \((\varphi_ 0,r(t))<\varepsilon\), \(t\geq t_ 0\), where \(\varphi_ 0\in K^*_ 0\). Here \(r(t)\) denotes the maximal solution of (2), \(u(t_ 0)=u_ 0\), relative to be cone \(K\), and \(K^*_ 0\) is the adjoint cone of \(K_ 0=K-\{0\}\). The authors present a series of theorems giving sufficient and/or necessary conditions for various types of \(\varphi_ 0\)-stability, based on the method of cone-valued Lyapunov functions.
scientific article; zbMATH DE number 62039