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Keywords:
self-adjoint differential equation; oscillation and nonoscillation criteria; variational method; conditional oscillation.
Summary:
Oscillation and nonoscillation criteria for the self-adjoint linear differential equation $(t^\alpha y^{\prime \prime })^{\prime \prime }-\frac{\gamma _{2,\alpha }}{t^{4-\alpha }}y=q(t)y,\quad \alpha \notin \lbrace 1, 3\rbrace \,,$ 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 $\left(t^\alpha y^{\prime \prime }\right)^{\prime \prime }-\left(\frac{\gamma _{2,\alpha }}{t^{4-\alpha }} + \frac{\gamma }{t^{4-\alpha }\ln ^2 t}\right)y = 0$ is nonoscillatory if and only if $\gamma \le \frac{\alpha ^2-4\alpha +5}{8}$.
References:
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