Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type (Q2642145)

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