Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 115, pp. 1-21.

Title: Oscillation and nonoscillation of solutions to even order 
       self-adjoint differential equations

Authors: Ondrej Dosly (Masaryk University, Czech Republic)
 Simona Fisnarova (Masaryk University, Czech Republic)

Abstract: 
 We establish oscillation and nonoscilation criteria for the linear
 differential equation
 $$
 (-1)^n\big(t^\alpha y^{(n)}\big)^{(n)}-
 \frac{\gamma_{n,\alpha}}{t^{2n-\alpha}}y=q(t)y,\quad
 \alpha \not\in \{1, 3, \dots , 2n-1\},
 $$
 where
 $$
 \gamma_{n,\alpha}=\frac{1}{4^n}\prod_{k=1}^n(2k-1-\alpha)^2
 $$
 and $q$ is a real-valued continuous function.
 It is proved, using these criteria, that the equation
 $$
 (-1)^n\big(t^\alpha y^{(n)}\big)^{(n)}
 -\big(\frac{\gamma_{n,\alpha}}{t^{2n-\alpha}}
 + \frac{\gamma}{t^{2n-\alpha}\lg^2 t}\big)y = 0
 $$
 is nonoscillatory if and only if
 $$
 \gamma \leq \tilde \gamma_{n,\alpha}:=
 \frac{1}{4^n}\prod_{k=1}^n(2k-1-\alpha)^2
 \sum_{k=1}^n\frac{1}{(2k-1-\alpha)^2}.
 $$

Submitted September 30, 2003. Published November 25, 2003.
Math Subject Classifications: 34C10
Key Words: Self-adjoint differential equation; variational method; 
    oscillation and nonoscillation criteria; conditional oscillation.