Linear perturbations of a nonoscillatory second order differential equation II

Let $y_1$ and $y_2$ be principal and nonprincipal solutions of the nonoscillatory differential equation $(r(t)y')'+f(t)y=0$. In an earlier paper we showed that if $\int^\infty(f-g)y_1y_2\,dt$ converges (perhaps conditionally), and a related improper integral converges absolutely and sufficently rapidly, then the differential equation $(r(t)x')'+g(t)x=0$ has solutions $x_1$ and $x_2$ that behave asymptotically like $y_1$ and $y_2$. Here we consider the case where $\int^\infty(f-g)y_2^2\,dt$ converges (perhaps conditionally) without any additional assumption requiring absolute convergence.