Linear Perturbations of a Nonoscillatory Second Order Equation

It is shown that the equation <!-- MATH $(r(t)x')' + g(t)x = 0$ --> has solutions which behave asymptotically like those of a nonoscillatory equation <!-- MATH $(r(t)y')' + f(t)y = 0$ --> , provided that a certain integral involving converges (perhaps conditionally) and satisfies a second condition which has to do with its order of convergence. The result improves upon a theorem of Hartman and Wintner.