Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

We study the asymptotic integration problem for a general class of n-th order nonlinear delay differential equations of the form L(n)x(t) = f (t, x(g(t))), where L(n)x = x((n)) + a(n-1)(t) x((n-1)) + . . . + a(0)(t) x. It is shown that if {u(1), u(2), . . . , u(n)} is a set of principal solutions for L(n)x = 0, then the n-th order nonlinear delay differential equation above has solutions x(k) with the property that x(k)(t) = b(1)u(1)(t) + . . . + b(k)u(k)(t) + o(u(k)(t)), t -> infinity for k = 1,2, . . . , n.