TY - JOUR

T1 - Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order

AU - Pašic, Mervan

AU - Tanaka, Satoshi

PY - 2011/9/1

Y1 - 2011/9/1

N2 - Asymptotic and oscillatory behaviours near x=0 of all solutions y=y(x) of self-adjoint linear differential equation (Ppq): (py')'+qy=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 (Ppq) 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 (Ppq) are characterised by the integrability (resp. nonintegrability) of q/p34 on (0,T). Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturm's 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.

AB - Asymptotic and oscillatory behaviours near x=0 of all solutions y=y(x) of self-adjoint linear differential equation (Ppq): (py')'+qy=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 (Ppq) 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 (Ppq) are characterised by the integrability (resp. nonintegrability) of q/p34 on (0,T). Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturm's 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.

KW - Asymptotic behaviour of solutions

KW - Bessel equation

KW - Comparison of solutions

KW - Euler equation

KW - Graph

KW - Linear equations

KW - Liouville transformation

KW - Oscillations

KW - Rectifiability

KW - Sturm's comparison

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U2 - 10.1016/j.jmaa.2011.03.051

DO - 10.1016/j.jmaa.2011.03.051

M3 - Article

AN - SCOPUS:79955464793

VL - 381

SP - 27

EP - 42

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -