Asymptotic to polynomials solutions for nonlinear differential equations Original Research Article

This article is concerned with solutions that behave asymptotically like polynomials for nth order (n>1)(n>1) nonlinear ordinary differential equations. For each given integer m with 1⩽m⩽n-11⩽m⩽n-1, sufficient conditions are presented in order that, for any real polynomial of degree at most m, there exists a solution which is asymptotic at ∞∞ to this polynomial. Conditions are also given, which are sufficient for every solution to be asymptotic at ∞∞ to a real polynomial of degree at most n-1n-1. The application of the results obtained to the special case of second order nonlinear differential equations leads to improved versions of the ones contained in the recent paper by Lipovan [Glasg. Math. J. 45 (2003) 179] and of other related results existing in the literature.