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Oscillatory properties of fourth order self-adjoint differential equations

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*Simona Fisnarova*

**Address.**

Department of Mathematics, Masaryk University, Janackovo nam. 2a,
662 95 Brno, Czech Republic

**E-mail. **simona@math.muni.cz

**Abstract.**

Oscillation and nonoscillation criteria for the self-adjoint linear differential equation
$$
(t^\alpha y^{\prime\prime})''-\frac{\gamma_{2,\alpha}}{t^{4-\alpha}}y=q(t)y,\quad
\alpha \not \in \{1, 3\}\,,
$$
where
$$ \gamma_{2,\alpha}=\frac{(\alpha-1)^2(\alpha-3)^2}{16}$$ and $q$ is a real
and continuous function, are established.
It is proved, using these criteria, that the equation
$$\left(t^\alpha y''\right)''-\left(\frac{\gamma_{2,\alpha}}{t^{4-\alpha}}
+ \frac{\gamma}{t^{4-\alpha}\ln^2 t}\right)y = 0$$
is nonoscillatory if and only if
$\gamma \leq \frac{\alpha^2-4\alpha+5}{8}$.

**AMSclassification. **34C10.

**Keywords. **Self-adjoint differential equation, oscillation and nonoscillation criteria,
variational method, conditional oscillation.