Asymptotics for higher order differential equations with a middle term

作者：Bartusek M; Cecchi M; Dosla Z; Marini M^*

来源：Journal of Mathematical Analysis and Applications, 2012, 388(2): 1130-1140.

DOI：10.1016/j.jmaa.2011.10.059

• Higher order nonlinear differential equation
• Oscillatory solution
• Bounded solution
• Asymptotic linear solution
• Monotone solution
• Non-vanishing at infinity solutions

We study the higher order differential equations with a middle term x((n))(t) + q(t)x((n-2))(t) + r(t)f(x(t)) = 0, n %26gt;= 3, (*) %26lt;br%26gt;as a perturbation of the linear equation %26lt;br%26gt;y((n))(t) + q(t)y((n-2)) = 0. (**) %26lt;br%26gt;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.

• 出版日期2012-4-15