Linear perturbations of a nonoscillatory second order equation

Abstract

It is shown that the equation ( r ( t ) x ′ ) ′ + g ( t ) x = 0 (r(t)x’)’ + g(t)x = 0 has solutions which behave asymptotically like those of a nonoscillatory equation ( r ( t ) y ′ ) ′ + f ( t ) y = 0 (r(t)y’)’ + f(t)y = 0 , provided that a certain integral involving f − g f - g 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.