## Oscillation criteria for second order superlinear differential equations.(English)Zbl 0880.34033

Summary: A new oscillation criterion is given for general superlinear ordinary differential equations of second-order of the form $$x''(t)+ a(t)f(x(t))= 0$$, where $$a(t)\in C[t_0,\infty)$$, $$f(x)\in C(\mathbb{R})$$ and $$xf(x)>0$$, $$f'(x)\geq 0$$ for $$x\neq 0$$. Furthermore, $$f(x)$$ also satisfies a superlinear condition, which covers the prototype nonlinear function $$f(x)=|x|^\gamma\text{sgn }x$$ with $$\gamma>1$$ known as the Emden-Fowler case. The coefficient $$a(t)$$ is not assumed to be eventually nonnegative. The oscillation criterion involving integral averages of $$a(t)$$ gives a positive answer to a question asked by J. S. W. Wong [Can. J. Math. 45, No. 5, 1094-1103 (1993; Zbl 0797.34037)].

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C29 Averaging method for ordinary differential equations

Zbl 0797.34037