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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 $$\left[a(t)(y'(t))^\sigma\right]'+q(t)f(y(t))=0,\tag*$$ where $\sigma>0$ is a quotient of odd integers, $a(t)>0$ and the nonlinearity $f$ satisfies the usual sign condition $yf(y)>0$ and $f'(y)>0$ for $y\ne 0$. A typical result is the following statement. Theorem. Suppose that $\int^\infty {ds\over a(s)^{1/\sigma}}=\infty$ and (i) $0<\int_{\varepsilon}^\infty (dy/f(y)^{1/\sigma}), \int_{-\varepsilon}^{-\infty} (dy/f(y)^{1/\sigma})<\infty$ for any $\varepsilon>0$; (ii) $\int^\infty q(s) ds$ exists and $\lim_{t\to\infty}\int^t(1/a(s)^{1/\sigma}) (\int_s^{\infty}q(u) du)^{1/\sigma} ds=\infty$. 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 $ {a(t)[y'(y)]^\sigma\over f(y(t))}$ satisfies certain Riccati-type differential equation. The results of the paper extend, among others, oscillation criteria of {\it P. J. Y. Wong} and {\it R. P. Agarwal} [J. Math. Anal. Appl. 198, No. 2, 337-354 (1996; Zbl 0855.34039)] and in the linear case $\sigma=1$, $f(y)\equiv y$ oscillation criteria of {\it H. J. Li} [J. Math. Anal. Appl. 194, No. 1, 217-234 (1995; Zbl 0836.34033)].
Reviewer: O.Došlý (Brno)

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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Full Text: DOI
References:
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