×

Oscillation theorems for certain nonlinear differential equations of second order. (English) Zbl 1057.34019

The authors are concerned with the general second-order nonlinear differential equation \[ [ a(t)\Psi(x(t))k(x^{\prime}(t))] ^{\prime} +q(t)f(x(t))=r(t),\tag{1} \] with \(q,r\in C[ [t_{0},\infty),\mathbb{R}] ,\) \(a\in C[ [t_{0},\infty),\mathbb{R}^{+}] ,\) \(\Psi\in C[ \mathbb{R} ,\mathbb{R}^{+}] ,\) \(k\in C[ \mathbb{R},\mathbb{R}] ,\) and \(f\in C^{1}[ \mathbb{R},\mathbb{R}] .\) The research follows closely the recent paper by J. V. Manojlovic [Acta Sci. Math. 65, No. 3–4, 515–527 (1999; Zbl 0957.34037)], where under certain assumptions on the coefficients it has been shown that solutions of the nonlinear differential equation \[ [ p(t)g(x(t))x^{\prime}(t)] ^{\prime}+q(t)f(x(t))=r(t)\tag{2} \] are either oscillatory or satisfy \[ \liminf_{t\to\infty}| x(t)| =0. \] The main goal of the paper under review, according to the authors, is to present new oscillation criteria and to show that some of Manojlovic’s results contain the superfluous condition. Results of the paper are proved using the so-called integral averaging technique similar to that developed by H. J. Li [J. Math. Anal. Appl. 194, No. 1, 217–234 (1995; Zbl 0836.34033)]. However, it has been pointed out by O. Došlý in his review on the paper by J. V. Manojlovic [Acta Sci. Math. 65, No. 3–4, 515–527 (1999; Zbl 0957.34037)] that a certain assumption in the oscillation criteria of Li is not necessary as observed by Yu. V. Rogovchenko [J. Math. Anal. Appl. 203, No. 2, 560–563 (1996; Zbl 0862.34024)]. The criteria of the reviewed paper contain a similar assumption and it seems that this assumption can be removed without affecting the results of the paper.
The authors just repeat these comments on several occasions referring to the assumption \[ \int_{t_{0}}^{t}a(s)h^{2}(t,s)\,ds<\infty\qquad\text{for }t\geq t_{0}. \] It is not clear whether the study of equation (1) has been motivated by any real-world applications since all examples in the paper are artificial, while the choice of the functions which appear in all examples illustrating the main results reduces to trivial: \(\Phi(s)=1,\) \(R(t)=0\) (examples 1, 2 and 4) or \(\Phi(s)=s,\) \(R(t)=0\) (example 3). The authors could not exhibit any exact oscillatory solution to illustrative examples. All these facts make the usefulness of extension from equation (2) to equation (1) questionable. We conclude by noticing that new oscillation criteria extending Theorems 6-9 in the paper under review have been reported for a nonlinear differential equation with damping \[ [ a(t)\Psi(x(t))k(x^{\prime}(t))] ^{\prime}+p(t)k(x^{\prime }(t))+q(t)f(x(t))=0 \] in the recent paper by Q.-R. Wang [Acta Math. Hung. 102, No. 1–2, 117–139 (2004; Zbl 1052.34040)].

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ayanlar, B.; Tiryaki, A., Oscillation theorems for nonlinear second order differential equations with damping, Acta Math. Hungar., 89, 1-2, 1-13 (2000) · Zbl 0973.34021
[2] Butler, G. J., The oscillatory behaviour of second order nonlinear differential equation with damping, J. Math. Anal. Appl., 179, 14-27 (1977) · Zbl 0348.34022
[3] El-Sheikh, M. M.A., Oscillation and nonoscillation criteria for second order nonlinear differential equations, J. Math. Anal. Appl., 57, 273-289 (1993) · Zbl 0804.34035
[4] Grace, S. R., Oscillation theorems for second order nonlinear differential equations with damping, Math. Nachr., 141, 117-127 (1989) · Zbl 0673.34041
[5] Grace, S. R.; Lalli, B. S., Integral averaging technique for the oscillation of second order nonlinear differential equations, J. Math. Anal. Appl., 149, 277-311 (1990) · Zbl 0697.34040
[6] Grace, S. R., Oscillation theorems for nonlinear differential equations of second order, J. Math. Anal. Appl., 171, 220-241 (1992) · Zbl 0767.34017
[7] Graef, J. R.; Spikes, P. W., On the oscillatory behavior of solutions of second order nonlinear differential equations, Czechoslovak Math. J., 36, 275-284 (1986) · Zbl 0627.34034
[8] Hartman, P., On nonoscillatory linear differential equation of second order, Amer. J. Math., 74, 389-400 (1952) · Zbl 0048.06602
[9] Kamenev, I. V., An integral criterion for oscillation of linear differential equation of second order, Mat. Zametki, 23, 249-251 (1978) · Zbl 0386.34032
[10] Li, H. J., Oscillation criteria for second order linear differential equations, J. Math. Anal. Appl., 194, 217-234 (1995) · Zbl 0836.34033
[11] Li, W. T., Oscillation of certain second-order nonlinear differential equations, J. Math. Anal. Appl., 217, 1-14 (1998) · Zbl 0893.34023
[12] Li, W. T.; Agarwal, R. P., Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl., 245, 171-188 (2000) · Zbl 0983.34020
[13] Li, W. T.; Agarwal, R. P., Interval oscillation criteria for second order nonlinear equations with damping, Computers Math. Applic., 40, 217-230 (2000) · Zbl 0959.34026
[14] Manojlovic, J. V., Oscillation theorems for nonlinear second order differential equations, Acta Sci. Math., 65, 515-527 (1999) · Zbl 0957.34037
[15] Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel), 53, 483-492 (1989) · Zbl 0661.34030
[16] Rogovchenko, Y. V., Oscillation theorems for second order perturbed differential equations, J. Math. Anal. Appl., 215, 354-357 (1997) · Zbl 0892.34031
[17] Rogovchenko, Y. V., Oscillation theorems for second order equations with damping, Nonlinear Analysis, 41, 1005-1028 (2000) · Zbl 0972.34022
[18] Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations (1968), Academic Press: Academic Press New York · Zbl 0191.09904
[19] Tiryaki, A.; Zafer, A., Oscillation criteria for second order nonlinear differential equations with damping, Turk. J. Math., 24, 185-196 (2000) · Zbl 0977.34027
[20] Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801
[21] Wong, J. S.W., A second order nonlinear oscillation, Funckcial. Ekvac., 11, 207-234 (1968) · Zbl 0157.14802
[22] Wong, J. S.W., Oscillation criteria for second order nonlinear differential equations with integrable coefficients, (Proc. Amer. Math. Soc., 115 (1992)), 389-395 · Zbl 0760.34032
[23] Wong, J. S.W., Oscillation criteria for second order nonlinear differential equations involving integral averages, Canad. J. Math., 45, 1094-1103 (1993) · Zbl 0797.34037
[24] Wong, P. J.Y.; Agarwal, R. P., Oscillatory behaviour of solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl., 198, 337-354 (1996) · Zbl 0855.34039
[25] Yan, J., Oscillation theorems for second order linear differential equations with damping, (Proc. Amer. Math. Soc., 98 (1986)), 276-282 · Zbl 0622.34027
[26] Yeh, C. C., An oscillation criterion for second order nonlinear differential equation with functional arguments, J. Math. Anal. Appl., 76, 72-76 (1980) · Zbl 0465.34043
[27] Yeh, C. C., Oscillation theorems for nonlinear second order differential equation with damped term, (Proc. Amer. Math. Soc., 84 (1982)), 397-402 · Zbl 0498.34023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.