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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
##### Keywords:
bounded solutions
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##### References:
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