Boundedness and Convergence of Solutions of a Second-Order Nonlinear Differential System1

Consider the second-order nonlinear differential system 1 ˙x hŽ y . FŽ x . , aŽ x . ˙y aŽ x . g Ž x . eŽt . , where a is a positive and continuous function on R Ž , .; h, F, and g are continuous functions on R; and eŽt. is a continuous function on I 0, .. We obtain sufficient and necessary conditions for all solutions to be bounded and to converge to zero. Our results can be applied to the well-known equation ¨x f1Ž x . ˙x f2 Ž x . ˙x2 g Ž x . eŽt ., which substantially extends and improves important results in the literature.