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Asymptotically periodic solutions of a class of second order nonlinear differential equations

Author: Zhong Chao Liang
Journal: Proc. Amer. Math. Soc. 99 (1987), 693-699
MSC: Primary 34C25
MathSciNet review: 877042
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Abstract: In this paper we give necessary and sufficient conditions for all solutions of the system

$\displaystyle ({\text{S}})\quad x' = y,\quad y' = - a(t)f(x)g(y)$

to be oscillatory or bounded, for all orbits of the system

$\displaystyle ({{\text{S}}_1})\quad x' = y,\quad y' = - \alpha f(x)g(y)$

to be periodic, where $ a(t) \to \alpha > 0$ as $ t \to \infty $, and for every orbit of (S) to approach a periodic orbit of (S$ _{1}$). The conditions assuring that every solution of (S) is asymptotically periodic are also established.

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Keywords: Oscillation, periodic orbit, asymptotically periodic solution
Article copyright: © Copyright 1987 American Mathematical Society