Electron. J. Diff. Eqns., Vol. 2003(2003), No. 115, pp. 1-21.

Oscillation and nonoscillation of solutions to even order self-adjoint differential equations

Ondrej Dosly & Simona Fisnarova

Abstract:
We establish oscillation and nonoscilation criteria for the linear differential equation
$$ (-1)^n\big(t^\alpha y^{(n)}\big)^{(n)}- \frac{\gamma_{n,\alpha}}{t^{2n-\alpha}}y=q(t)y,\quad \alpha \not\in \{1, 3, \dots , 2n-1\}, $$
where
$$ \gamma_{n,\alpha}=\frac{1}{4^n}\prod_{k=1}^n(2k-1-\alpha)^2 $$
and $q$ is a real-valued continuous function. It is proved, using these criteria, that the equation
$$ (-1)^n\big(t^\alpha y^{(n)}\big)^{(n)} -\big(\frac{\gamma_{n,\alpha}}{t^{2n-\alpha}} + \frac{\gamma}{t^{2n-\alpha}\lg^2 t}\big)y = 0 $$
is nonoscillatory if and only if
$$ \gamma \leq \tilde \gamma_{n,\alpha}:= \frac{1}{4^n}\prod_{k=1}^n(2k-1-\alpha)^2 \sum_{k=1}^n\frac{1}{(2k-1-\alpha)^2}. $$

Submitted September 30, 2003. Published November 25, 2003.
Math Subject Classifications: 34C10
Key Words: Self-adjoint differential equation, variational method, oscillation and nonoscillation criteria, conditional oscillation.

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Ondrej Dosly
Masaryk University, Janackovo nam. 2a
CZ-662 95 Brno, Czech Republic
email: dosly@math.muni.cz
Simona Fisnarova
Masaryk University, Janackovo nam. 2a
CZ-662 95 Brno, Czech Republic
email: simona@math.muni.cz

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