Rectifiable oscillations in second-order linear differential equations

We study the linear differential equation , on I=(0,1), where the coefficient f(x) is strictly positive and continuous on I, and satisfies the Hartman–Wintner condition at x=0. The four main results of the paper are: (i) a criterion for rectifiable oscillations of (P), characterized by the integrability of on I; (ii) a stability result for rectifiable and unrectifiable oscillations of (P), in terms of a perturbation on f(x); (iii) the s-dimensional fractal oscillations (for which we assume also f(x)∼cx−α when x→0, α>2, and s=max{1,3/2−2/α}); and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman–Wintner condition on f(x). Explicit examples related to the above results are given.