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Oscillation of certain second-order nonlinear differential equations. (English) Zbl 0893.34023

The author investigates oscillation properties of solutions of the nonlinear differential equation

${\left[a\left(t\right){\left({y}^{\text{'}}\left(t\right)\right)}^{\sigma }\right]}^{\text{'}}+q\left(t\right)f\left(y\left(t\right)\right)=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $\sigma >0$ is a quotient of odd integers, $a\left(t\right)>0$ and the nonlinearity $f$ satisfies the usual sign condition $yf\left(y\right)>0$ and ${f}^{\text{'}}\left(y\right)>0$ for $y\ne 0$. A typical result is the following statement.

Theorem. Suppose that ${\int }^{\infty }\frac{ds}{a{\left(s\right)}^{1/\sigma }}=\infty$ and

(i) $0<{\int }_{\epsilon }^{\infty }\left(dy/f{\left(y\right)}^{1/\sigma }\right),{\int }_{-\epsilon }^{-\infty }\left(dy/f{\left(y\right)}^{1/\sigma }\right)<\infty$ for any $\epsilon >0$;

(ii) ${\int }^{\infty }q\left(s\right)ds$ exists and ${lim}_{t\to \infty }{\int }^{t}\left(1/a{\left(s\right)}^{1/\sigma }\right){\left({\int }_{s}^{\infty }q\left(u\right)du\right)}^{1/\sigma }ds=\infty$.

Then every solution of (*) is oscillatory.

Proofs of the results presented are essentially based on the generalized Riccati technique consisting in the fact that the quotient $\frac{a\left(t\right){\left[{y}^{\text{'}}\left(y\right)\right]}^{\sigma }}{f\left(y\left(t\right)\right)}$ satisfies certain Riccati-type differential equation.

The results of the paper extend, among others, oscillation criteria of P. J. Y. Wong and R. P. Agarwal [J. Math. Anal. Appl. 198, No. 2, 337-354 (1996; Zbl 0855.34039)] and in the linear case $\sigma =1$, $f\left(y\right)\equiv y$ oscillation criteria of H. J. Li [J. Math. Anal. Appl. 194, No. 1, 217-234 (1995; Zbl 0836.34033)].

Reviewer: O.Došlý (Brno)

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