Stability and boundedness properties of certain second-order differential equations

In this paper, we investigated the differential equation X ¨ + A ( t ) F ( X ) = P ( t , X , X ˙ ) in two cases; (a) P ≡ 0 and (b) P ≠ 0 . For the case (a), the stability of the solution X = 0 and the uniform boundedness of all solutions of this equation are investigated; in the case (b) the boundedness of all solutions of the same equation is discussed.