Asymptotic integration of second-order nonlinear delay differential equations

We study the asymptotic integration problem for second-order nonlinear delay differential equations of the form (p(t)x' (t))' q(t)x(t) = f (t, x(g(t))). It is shown that if a and v are principal and nonprincipal solutions of equation (p(t)x')' q(t)x = 0, then there are solutions x(1)(t) and x(2) (t) of the above nonlinear equation such that x(1)(t) = au(t) o(u(t)), t -> infinity and x(2)(t) = bv(t) o(v(t)), t -> infinity. (C) 2015 Elsevier Ltd. All rights reserved.