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Zbl 1117.34038
Fišnarová, Simona
Oscillatory properties of fourth order self-adjoint differential equations. (English)
[J] Arch. Math., Brno 40, No. 4, 457-469 (2004). ISSN 0044-8753; ISSN 1212-5059/e

Oscillation and nonoscillation criteria for the fourth order self-adjoint differential equation $$ (t^\alpha y^{\prime \prime })^{\prime \prime }- \frac {\gamma _{2,\alpha }}{t^{4-\alpha }}y=q(t)y,\quad \alpha \not \in \{1, 3\},\quad \gamma _{2,\alpha }=\frac {(\alpha -1)^2(\alpha -3)^2}{16}, \tag {*} $$ are established under no sign restriction on the continuous function $q$. In these criteria, equation (*) is viewed as a perturbation of the (nonoscillatory) equation $$(t^\alpha y^{\prime \prime })^{\prime \prime }- (\gamma _{2, \alpha }+ \tilde \gamma _{2, \alpha } \lg ^{-2} t ) t^{\alpha -4} y=0,\quad \tilde \gamma _{2, \alpha } = (\alpha ^2 -4\alpha +5)/8$$ and (non)oscillation criteria are formulated in terms of the difference $q(t) -\tilde \gamma _{2, \alpha } t^{\alpha -4} \lg ^{-2} t$. The main results of the paper are proved by using the variational principle coupled with the factorization of the differential operator on the left-hand side of (*).
[Ondřej Došlý (Brno)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: self-adjoint differential equation

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