James S. W. Wong, C. H. Ou

Oscillation and Nonoscillation Theorems for Superlinear Emden-Fowler Equations of Even Order

We study the existence of oscillatory solutions of the even order superlinear ordinary differential equations
\[(E_1^n)\qquad y^{(n)}+p(t)|y|^\alpha \sgn y=0,\]
\[(E_{2}^{n})\qquad y^{(n)}-p(t)|y|^{\alpha }\sgn y=0,\]
where $n$ is an even integer $\geq 2$, $\alpha >1$ and $p(t)\in C[t_{0},\infty )$, $t_{0}>0$ and $p(t)>0.$ When $n=2$, our results reduce to those of Jasny and Kurzweil, and Erbe and Muldowney for ($E_{1}^{2}$). When $n=4$, our result becomes that of Kura for equation ($E_{2}^{4}$). We present here new techniques suitable for the study of oscillation and nonoscillation of solutions of the general equations of even order ($E_{1}^{n}$) and ($E_{2}^{n}$).

Ordinary differential equation, oscillation, nonoscillation, even order, superlinear.

MSC 2000: 34C10, 34C15