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Zbl 1062.34034
Došlý, Ondřej; Osička, Jan
Oscillatory properties of higher order Sturm-Liouville differential equations.
(English)
[J] Stud. Univ. Žilina, Math. Ser. 15, No. 1, 25-40 (2002). ISSN 1336-149X

The authors investigate oscillation properties of the higher order Sturm--Liouville differential equation $$(-1)^n\left(t^\alpha y^{(n)}\right)^{(n)}=q(t)y, \tag{E}$$ with $\alpha\in I:=\{1,3,\dots,2n-1\}$. This equation is viewed as a perturbation of the nonoscillatory differential equation $$(-1)^n\left(t^\alpha y^{(n)}\right)^{(n)}={\gamma_{n,\alpha}\over t^{2n-\alpha}\ln^2t}\ y,$$ with $$\gamma_{n,\alpha}:={[m!(n-m-1)!]^2\over 4},\quad m:={2n-1-\alpha\over 2}.$$ Using the variational characterization of the corresponding linear Hamiltonian system, the authors derive sufficient conditions involving the integral of the difference $\displaystyle \left(q(t)-{\gamma_{n,\alpha}\over t^{2n-\alpha}\ln^2t}\right)$, which ensures that equation \thetag{E} becomes oscillatory (remains nonoscillatory). In view of the fact that there are several similar recent results concerning equation \thetag{E} with $\alpha\not\in I$, the paper resolves the remaining and more complicated case $\alpha\in I$.
[Robert Mař\'ik (Brno)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: higher-order Sturm-Liouville differential equation; oscillation criteria; nonoscillation criteria, variational method; conditional oscillation

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