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Oscillation and nonoscillation of solutions to even order self-adjoint differential equations. (English) Zbl 1039.34024
The authors establish oscillation and nonoscillation criteria for the linear differential equation \[ (-1)^n (t^{\alpha}y^{(n)})^{(n)}-\frac{\gamma_{n,\alpha}}{t^{2n-\alpha}}y=q(t)y, \quad \alpha \not \in \{1,3,\dots,2n-1\}, \] where \[ \gamma_{n,\alpha}=\frac{1}{4^n}\prod_{k=1}^n (2k-1-\alpha)^2 \] and \(q\) is a real-valued continuous function. Using these criteria, the authors prove that the equation \[ (-1)^n (t^{\alpha}y^{(n)})^{(n)}- \left(\frac{\gamma_{n,\alpha}} {t^{2n-\alpha}}+ \frac{\gamma}{t^{2n-\alpha}\lg^2t} \right)y=0 \] is nonoscillatory if and only if \[ \gamma\leq\frac{1}{4^n}\prod_{k=1}^n(2k-1-\alpha)^2\sum_{k=1}^n\frac{1}{(2k-\alpha-1)^2}. \]
MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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