Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order

Asymptotic and oscillatory behaviours near x = 0 of all solutions y = y ( x ) of self-adjoint linear differential equation ( P p q ) : ( p y ′ ) ′ + q y = 0 on ( 0 , T ] , will be studied, where p = p ( x ) and q = q ( x ) satisfy the so-called Hartman–Wintner type condition. We show that the oscillatory behaviour near x = 0 of ( P p q ) is characterised by the nonintegrability of q / p on ( 0 , T ) . Moreover, under this condition, we show that the rectifiable (resp. unrectifiable) oscillations near x = 0 of ( P p q ) are characterised by the integrability (resp. nonintegrability) of q / p 3 4 on ( 0 , T ) . Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturmʼs comparison type theorem for the rectifiable oscillations is stated. Furthermore, previous results are used to establish such kind of oscillations for damped linear second-order differential equation y ″ + g ( x ) y ′ + f ( x ) y = 0 , and especially, the Bessel type damped linear differential equations are considered. Finally, some open questions are posed for the further study on this subject.