## 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 $$y\neq 0$$. 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 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 $$\sigma=1$$, $$f(y)\equiv y$$ 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 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Citations:

Zbl 0855.34039; Zbl 0836.34033
Full Text:

### References:

 [1] Graef, J. R.; Spikes, P. W., On the oscillatory behavior of solutions of second order non-linear differential equations, Czechoslovak Math. J., 36, 275-284 (1986) · Zbl 0627.34034 [2] Kamenev, I. V., An integral criterion for oscillation of linear differential equations of second order, Mat. Zametki, 23, 249-251 (1978) · Zbl 0386.34032 [3] Kartsatos, A. G., Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, Stability of Dynamical Systems: Theory and Applications. Stability of Dynamical Systems: Theory and Applications, Lecture Notes in Pure and Appl. Math., 28 (1977), p. 17-72 [4] Kwong, M. K.; Wong, J. S.W., An application of integral inequality to second order non-linear oscillation, J. Differential Equations, 46, 63-77 (1992) [5] Li, H. J., Oscillation criteria for second order linear differential equations, J. Math. Anal. Appl., 194, 217-234 (1995) · Zbl 0836.34033 [6] Li, W. T.; Yan, J. R., An oscillation criterion for second order superlinear differential equations, Indian J. Pure Appl. Math., 28, 735-740 (1997) · Zbl 0880.34033 [7] Philos, C. G., On a Kamenev’s integral criterion for oscillation of linear differential equations of second order, Utilitas Math., 24, 277-289 (1983) · Zbl 0528.34035 [8] Philos, C. G., Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel), 53, 482-492 (1989) · Zbl 0661.34030 [9] Philos, Ch. P.; Purnaras, I. K., Oscillation in superlinear differential equations of second order, J. Math. Anal. Appl., 165, 1-11 (1992) · Zbl 0756.34036 [10] Rogovchenko, Y. R., Note on “Oscillation criteria for second order linear differential equations”, J. Math. Anal. Appl., 203, 560-563 (1996) · Zbl 0862.34024 [11] Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations (1968), Academic Press: Academic Press New York · Zbl 0191.09904 [12] Thandapani, E.; Gyori, I.; Lalli, B. S., An application of discrete inequality to second order nonlinear oscillation, J. Math. Anal. Appl., 186, 200-208 (1994) · Zbl 0823.39004 [13] Thandapani, E.; Pandian, S., On the oscillatory behavior of solutions of second order nonlinear difference equations, Z. Anal. Anwendungent, 13, 347-358 (1994) · Zbl 0803.39004 [14] Wong, J. S., On the second order nonlinear oscillations, Funkcial. Ekvac., 11, 207-234 (1968) [15] Wong, J. S., An oscillation criterion for second order nonlinear differential equations, Proc. Amer. Math. Soc., 98, 109-112 (1986) · Zbl 0603.34025 [16] Wong, J. S., Oscillation theorems for second order nonlinear differential equations, Proc. Amer. Math. Soc., 106, 1069-1077 (1989) · Zbl 0694.34027 [17] Wong, J. S., Oscillation criteria for second order nonlinear differential equations with integrable coefficients, Proc. Amer. Math. Soc., 115, 389-395 (1992) · Zbl 0760.34032 [18] Wong, J. S., Oscillation criteria for second order nonlinear differential equations involving integral averages, Canad. J. Math., 45, 1094-1103 (1993) · Zbl 0797.34037 [19] Wong, P. J.Y.; Agarwal, R. P., Oscillatory behavior of solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl., 198, 337-354 (1996) · Zbl 0855.34039 [20] Yan, J. R., Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc., 98, 276-282 (1986) · Zbl 0622.34027 [21] Yu, Y. H., Leighton type oscillation criterion and Sturm type comparison theorem, Math. Nachr., 153, 137-143 (1991) · Zbl 0795.34025
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