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Oscillatory behavior of solutions of certain second order nonlinear differential equations. (English) Zbl 0855.34039

The authors study oscillatory behavior of solutions of the nonlinear second order differential equation (*) ${\left[a\left(t\right){\left({y}^{\text{'}}\right)}^{\sigma }\right]}^{\text{'}}+q\left(t\right)f\left(y\right)=r\left(t\right)$, where $a$ is an eventually positive function, the nonlinearity $f$ satisfies $uf\left(u\right)>0$, ${f}^{\text{'}}\left(u\right)$ for all $u\ne 0$, and the power $\sigma$ is a positive ratio of the type (odd/odd) or (even/odd). A typical result is the following statement:

Theorem: Let $\sigma$ be the quotient of two odd integers and suppose that the following assumptions are satisfied: ${\int }^{\infty }|r\left(s\right)|ds<\infty$, $-\infty {\int }^{\infty }q\left(s\right)ds<\infty$, there exist $0<\mu \le \nu$ such that $\mu \le {f}^{\text{'}}\left(u\right)\le \nu$ and

${\int }^{\infty }\frac{ds}{a{\left(s\right)}^{1/\sigma }}=\infty ={\int }^{\infty }\frac{ds}{a\left(s\right)}·$

If $y$ is a nonoscillatory solution of (*) such that ${lim inf}_{t\to \infty }|y\left(t\right)|>0$ and there exists $L>0$ so that $|{y}^{\text{'}}\left(t\right)|\le {L}^{1/\left(\sigma -1\right)}$, then

${\int }^{\infty }\frac{a\left(s\right){\left[{y}^{\text{'}}\left(s\right)\right]}^{\sigma +1}{f}^{\text{'}}\left(y\left(s\right)\right)}{{\left[f\left(y\left(s\right)\right)\right]}^{2}}ds<\infty ,\phantom{\rule{1.em}{0ex}}\underset{t\to \infty }{lim}\frac{a\left(t\right){\left[{y}^{\text{'}}\left(t\right)\right]}^{\sigma }}{f\left(y\left(t\right)\right)}=0$

and

$\frac{a\left(t\right){\left[{y}^{\text{'}}\left(t\right)\right]}^{\sigma }}{f\left(y\left(t\right)\right)}+{\int }_{t}^{\infty }\frac{a\left(s\right){\left[{y}^{\text{'}}\left(s\right)\right]}^{\sigma +1}{f}^{\text{'}}\left(y\left(s\right)\right)}{{\left[f\left(y\left(s\right)\right)\right]}^{2}}ds+{\int }_{t}^{\infty }\left[q\left(s\right)-\frac{r\left(s\right)}{f\left(y\left(s\right)\right)}\right]ds$

for $t$ sufficiently large.

Reviewer: O.Došlý (Brno)

MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory