Oscillatory properties of fourth order self-adjoint differential equations

Simona Fisnarova

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

E-mail. simona@math.muni.cz

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.