Nonoscillation theorems for second order nonlinear differential equations

Abstract:

We prove nonoscillation theorems for the second order Emden-Fowler equation (E): $y''+a(x)|y|^{\gamma -1}y=0$, $\gamma >0$, where $a(x)\in C(0,\infty )$ and $\gamma \not =1$. It is shown that when $x^{(\gamma +3)/2+\delta }a(x)$ is nondecreasing for any $\delta >0$ and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when $0<\gamma <1$ and $0<\delta <(1-\gamma )/2$.