Oscillation of certain second-order nonlinear differential equations
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 \(y\neq 0\). 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 \textit{P. J. Y. Wong} and \textit{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 \textit{H. J. Li} [J. Math. Anal. Appl. 194, No. 1, 217-234 (1995; Zbl 0836.34033)].
scientific article; zbMATH DE number 1115456
scientific article