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.