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Über die Trichotomie von Lösungen einer nichtlinearen Vektordifferentialgleichung zweiter Ordnung. (German) Zbl 0707.34030
Consider the vector differential system \(x''=F(t,x)\) on \({\mathbb{R}}_ 0^+\times {\mathbb{R}}^ n\) with F of class \(C^ 1\), bounded on bounded x- sets and such that \[ \liminf_{| x_ i| \to \infty}f_ i(t,x_ i,z_ i)sign x_ i>0,\quad 1\leq i\leq n, \] uniformly in \((t,z_ i)\) with \(z_ i=(x_ 1,...,x_{i-1},x_{i+1}...,x_ n)\). It is proved that for each \(T>0\), all the solutions of the periodic problem on [0,T] such that \(\| x(0)\| \leq R\) have their norm bounded by the same constant R for all \(t\in [0,T]\). This remark is used to give conditions on the solution of the above system which imply that \(\| x(t)\| \to \infty\) when \(t\to \infty\).
Reviewer: J.Mawhin
MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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