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Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping. (English) Zbl 1064.34021

The authors are concerned with the oscillatory behavior of the second-order nonlinear differential equation with a nonlinear damping term \[ \left[ r(t)k_{1}(x,x^{\prime})\right] ^{\prime}+p(t)k_{2}(x,x^{\prime })x^{\prime}+q(t)f(x)=0,\qquad t\geq t_{0}\geq0,\tag{b} \] with \(p,q:[t_{0},\infty)\to\mathbb{R},\) \(r:[t_{0},\infty )\to(0,\infty),\) \(f:\mathbb{R}\to\mathbb{R},\) \(k_{1} ,k_{2}:\mathbb{R}^{2}\to\mathbb{R}.\) It is also assumed that \[ k_{1}^{2}(u,v)\leq\alpha_{1}k_{1}(u,v), \] for some \(\alpha_{1}>0\) and for all \((u,v)\in\mathbb{R}^{2}.\) Two cases are considered:
(a) \(f(x)\) is differentiable, \(xf(x)\neq0\) and \(f^{\prime} (x)\geq\mu_{1}\) for some \(\mu_{1}>0\) and all \(x\neq0,\) and \[ vf(u)k_{2}(u,v)\geq\alpha_{2}k_{1}^{2}(u,v)\tag{b1} \] for some \(\alpha_{2}>0\) and for all \((u,v)\in\mathbb{R}^{2};\)
(b) \(f(x)\) is not necessarily differentiable, \(f(x)/x\geq\mu_{2}\) for some \(\mu_{2}>0\) and all \(x\neq0,\) and \[ vuk_{2}(u,v)\geq\alpha_{3}k_{1}^{2}(u,v)\tag{b2} \] for some \(\alpha_{1}>0\) and for all \((u,v)\in\mathbb{R}^{2}.\) Using standard integral averaging technique, several interval oscillation criteria are obtained which require information on the behavior of the coefficients in equation ({b}) on a sequence of intervals \((a_{n},b_{n})\) such that \(a_{n}\to\infty\) as \(n\to\infty\). Unfortunately, rather specific assumptions ({b1}) and ({b2}) significantly restrict possible the applicability of the theorems.
The statement of the fundamental Lemma 1.1 should be corrected as follows: “If there exists an interval \((a,b)\subset[ t_{0},\infty)\) such that (1.2) holds, then, for all \(c\in(a,b),\) (1.3) is satisfied for every \(H\in\mathcal{P}\)” instead of the incorrect formulation “If there exist an interval \((a,b)\subset[ t_{0},\infty)\) and a \(c\in(a,b)\) such that (1.2) holds, then (1.3) is satisfied for every \(H\in\mathcal{P}.\)” The statement of Theorem 3.1 should be corrected by adding the phrase “and there exists a \(c\in(a,b)\) such that (3.1) holds”.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Baker, J. W., Oscillation theorems for a second order damped nonlinear equation, SIAM J. Appl. Math., 25, 37-40 (1973) · Zbl 0239.34015
[2] Butler, G. J.; Erbe, L. H.; Mingarelli, A. B., Riccati techniques and variational principles in oscillation theory for linear systems, Trans. Amer. Math. Soc., 303, 263-282 (1987) · Zbl 0648.34031
[3] El-Sayed, M. A., An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc., 118, 813-817 (1993) · Zbl 0777.34023
[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., Oscillation criteria for second order nonlinear differential equations with damping, J. Austral. Math. Soc., 49A, 43-54 (1990) · Zbl 0725.34030
[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] Grace, S. R.; Lalli, B. S., Oscillation theorems for second order superlinear differential equations with damping, J. Austral. Math. Soc., 53A, 156-175 (1992) · Zbl 0762.34012
[8] Grace, S. R.; Lalli, B. S.; Yeh, C. C., AddendumOscillation theorems for nonlinear second order differential equations with a nonlinear damping term, SIAM J. Math. Anal., 19, 1252-1253 (1988) · Zbl 0651.34028
[9] Huang, C. C., Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl., 210, 712-723 (1997) · Zbl 0880.34034
[10] Kamenev, I. V., Integral criteria of linear differential of second order, Math. Zametki, 23, 249-251 (1978) · Zbl 0386.34032
[11] Kong, Q., Interval criteria for oscillation of second-order linear ordinary differential equations, J. Math. Anal. Appl., 229, 258-270 (1999) · Zbl 0924.34026
[12] Kwong, M. K.; Zettl, A., Integral inequalities and second order linear oscillation, J. Differential Equations, 45, 16-33 (1982) · Zbl 0498.34022
[13] Kwong, M. K.; Wong, J. S.W., Oscillation and nonoscillation of Hill’s equation with periodic damping, J. Math. Anal. Appl., 288, 15-19 (2003) · Zbl 1039.34026
[14] Li, W. T.; Zhong, C. K., Integral averages and interval oscillation of second-order nonlinear differential equations, Math. Nachr., 246/247, 156-169 (2002) · Zbl 1045.34011
[15] 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
[16] Li, W. T.; Agarwal, R. P., Interval oscillation criteria for second-order nonlinear differential equations with damping, Comput. Math. Appl., 40, 217-230 (2000) · Zbl 0959.34026
[17] Li, W. T.; Agarwal, R. P., Interval oscillation criteria for a forced second order nonlinear ordinary differential equations, Ukranian Math. J., 53, 1391-1406 (2001)
[18] Li, H. J., Oscillation criteria for second order linear differential equations, J. Math. Anal. Appl., 194, 217-234 (1995) · Zbl 0836.34033
[19] Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel), 53, 482-492 (1989) · Zbl 0661.34030
[20] Rogovchenko, S. P.; Rogovchenko, Yu. V., Oscillation theorems for differential equations with a nonlinear damping, J. Math. Anal. Appl., 279, 121-134 (2003) · Zbl 1027.34040
[21] Rogovchenko, Yu. V., Oscillation theorems for second order differential equations with damping, Nonlinear Anal., 41, 1005-1028 (2000) · Zbl 0972.34022
[22] Sun, Y. G., New Kamenev-type oscillation criteria for second order nonlinear differential equations with damping, J. Math. Anal. Appl., 291, 341-351 (2004) · Zbl 1039.34027
[23] Tiryaki, A.; Zafer, A., Oscillation of second-order nonlinear differential equations with nonlinear damping, Math. Comput. Model., 39, 197-208 (2004) · Zbl 1049.34040
[24] Wong, J. S.W., Second order nonlinear forced equations, SIAM J. Math. Anal., 19, 667-675 (1988) · Zbl 0655.34023
[25] Wong, J. S.W., Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl., 231, 235-240 (1999) · Zbl 0922.34029
[26] Wong, J. S.W., Oscillation criteria for second order nonlinear differential equations involving general means, J. Math. Anal. Appl., 247, 489-505 (2000) · Zbl 0964.34028
[27] Wong, J. S.W., On Kamenev-type oscillation theorems for second order differential equations with damping, J. Math. Anal. Appl., 248, 244-257 (2001) · Zbl 0987.34024
[28] Yan, J., Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc., 98, 276-282 (1986) · Zbl 0622.34027
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