Asymptotics for higher order differential equations with a middle term

摘要

We study the higher order differential equations with a middle term x ~n (t)+q(t)x ~((n-2))(t)+f(x(t))=0, n > 3, as a perturbation of the linear equation y ~((n))(t)+q(t)y ~((n-2))=0. Using an iterative method, we show that for every solution y of (**), there exists a solution x of (*) such that x ~((i))-y ~((i)) (i=0,..., n-1) have bounded variation in a neighborhood of infinity and tend to zero. The existence of monotone solutions and bounded solutions for (*) is also examined. The cases n=3, 4 are considered in detail and there are given conditions for the existence of bounded oscillatory solutions of (*) with an analogous asymptotic behavior to corresponding oscillatory solutions of (**). Our results are new also in the linear case.