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

Abstract:

In this paper we give necessary and sufficient conditions for all solutions of the system \[ ({\text {S}})\quad x’ = y,\quad y’ = - a(t)f(x)g(y)\] to be oscillatory or bounded, for all orbits of the system \[ ({{\text {S}}_1})\quad x’ = y,\quad y’ = - \alpha f(x)g(y)\] 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.