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Über die Existenz einer begrenzten und periodischen Lösung der nichtlinearisierten Jacobischen Gleichung mit negativ definitivem Träger. (On the existence of a bounded and periodic solution to the nonlinear Jacobi equation with negative definite support). (German) Zbl 0706.34031

Under consideration is the equation \(x''+q(t)x=p(t,x)\) where p and q are continuous functions, bounded with respect to t. A new condition is established for this equation to have a bounded solution. This condition is of the type: \(\limsup (\partial p/\partial x)<q_ 0,\) as \(| x| \to \infty\). According to a classical theorem of Massera, the existence of a bounded solution then implies the existence of a periodic solution. It is also shown here that, if q(t)-\(\partial p/\partial x\leq 0\) for all t and all \(| x| \leq\) some D, then this periodic solution is unstable.
Reviewer: E.O.Roxin

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
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References:

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