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 \[ ( S ) x ′ = y , y ′ = − a ( t ) f ( x ) g ( y ) ({\text {S}})\quad x’ = y,\quad y’ = - a(t)f(x)g(y) \] to be oscillatory or bounded, for all orbits of the system \[ ( S 1 ) x ′ = y , y ′ = − α f ( x ) g ( y ) ({{\text {S}}_1})\quad x’ = y,\quad y’ = - \alpha f(x)g(y) \] to be periodic, where a ( t ) → α > 0 a(t) \to \alpha > 0 as t → ∞ t \to \infty , and for every orbit of (S) to approach a periodic orbit of (S 1 _{1} ). The conditions assuring that every solution of (S) is asymptotically periodic are also established.