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Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type. (English) Zbl 1126.34023
The paper deals with a new kind of oscillations, so-called rectifiable and unrectifiable oscillations. Besides the linear differential equations of Euler type $$y''+\frac{\lambda}{x^{\alpha}}y=0\text{ on }I=(0,b),\tag1$$ where $\lambda,\alpha,b\in\Bbb R$, $\lambda>0$, $\alpha\geq 2$, $b>0$; two boundary-layer conditions are separately considered: there are $c>0$ and $d\in I$ both depending on $y$ such that $$ \align \vert y'(t)\vert&\leq\frac{c}{x^{\alpha/4}}\quad\text{for all}\quad x\in(0,d), \tag"(2)"\\ \vert y(t)\vert&\leq cx^{\alpha/4}\quad\text{for all}\quad x\in(0,d). \tag"(3)" \endalign $$ The author presents sufficient conditions for the problem (1), (2) and/or the problem (1), (3) to be rectifiable (unrectifiable) oscillatory on $I$.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general
Full Text: DOI
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