Asymptotic integration of second-order nonlinear differential equations via principal and nonprincipal solutions

Let u and v denote respectively the principal and nonprincipal solutions of the second-order linear equation (p(t)x')' + q(t)x = 0 defined on some half-line of the form [t(*), infinity). It is shown that for any given real numbers a and b the nonlinear equation (p(t)x')' + q(t)x = f(t, x) has a solution x(t) asymptotic to av(t) + bu(t) as t -> infinity under some reasonable conditions on the function f. Examples are given to illustrate the results. (C) 2012 Elsevier Inc. All rights reserved.