Asymptotics for higher order differential equations with a middle term

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 ⩾ 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.