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

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→∞ under some reasonable conditions on the function f. Examples are given to illustrate the results.