Oscillation and nonoscillation of perturbed higher order Euler-type differential equations

Abstract

Oscillatory properties of even order self-adjoint linear differential equations in the form $$ \sum_{k=0}^{n} (-1)^k\nu_k\left(\frac{y^{(k)}}{t^{2n-2k-\alpha}}\right)^{(k)} =(-1)^m\left(q_m(t)y^{(m)}\right)^{(m)},\; \nu_n:=1, $$ where $m \in \{0, 1\}$, $\alpha \not\in \{1, 3, \dots , 2n-1\}$ and $\nu_0, \dots, \nu_{n-1},$ are real constants satisfying certain conditions, are investigated. In particular, the case when $q_m(t)=\frac{\beta}{t^{2n-2m-\alpha}\ln^2 t }$ is studied.