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# Article

 Title: On property (B) of higher order delay differential equations (English) Author: Baculíková, Blanka Author: Džurina, Jozef Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 48 Issue: 4 Year: 2012 Pages: 301-309 Summary lang: English . Category: math . Summary: In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$-th order delay differential equations \begin{equation*} \big (r(t)\big [x^{(n-1)}(t)\big ]^{\gamma }\big )^{\prime }=q(t)f\big (x(\tau (t))\big )\,. \end{equation*} Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases $\int ^{\infty } r^{-1/\gamma }(t)\,{t}=\infty$ and $\int ^{\infty } r^{-1/\gamma }(t)\,{t}<\infty$ are discussed. (English) Keyword: $n$-th order differential equations Keyword: comparison theorem Keyword: oscillation Keyword: property (B) MSC: 34C10 MSC: 34K11 idMR: MR3007612 DOI: 10.5817/AM2012-4-301 . Date available: 2012-12-17T13:54:45Z Last updated: 2013-09-19 Stable URL: http://hdl.handle.net/10338.dmlcz/143104 . Reference: [1] Agarwal, R. P., Grace, S. R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations.Marcel Dekker, Kluwer Academic, Dordrecht, 2000. MR 1774732 Reference: [2] Agarwal, R. P., Grace, S. R., O’Regan, D.: Oscillation criteria for certain $n$–th order differential equations with deviating arguments.J. Math. Anal. Appl. 262 (2001), 601–622. Zbl 0997.34060, MR 1859327, 10.1006/jmaa.2001.7571 Reference: [3] Agarwal, R. P., Grace, S. R., O’Regan, D.: The oscillation of certain higher–order functional differential equations.Math. Comput. Modelling 37 (2003), 705–728. Zbl 1070.34083, MR 1981237, 10.1016/S0895-7177(03)00079-7 Reference: [4] Baculíková, B., Džurina, J.: Oscillation of third–order neutral differential equations.Math. Comput. Modelling 52 (2010), 215–226. Zbl 1201.34097, MR 2645933, 10.1016/j.mcm.2010.02.011 Reference: [5] Baculíková, B., Džurina, J., Graef, J. R.: On the oscillation of higher order delay differential equations.Nonlinear Oscillations 15 (2012), 13–24. Zbl 1267.34121, MR 2986592 Reference: [6] Bainov, D. D., Mishev, D. P.: Oscillation Theory for Nonlinear Differential Equations with Delay.Adam Hilger, Bristol, Philadelphia, New York, 1991. Reference: [7] Džurina, J.: Comparison theorems for nonlinear ODE’s.Math. Slovaca 42 (1992), 299–315. Zbl 0760.34030, MR 1182960 Reference: [8] Erbe, L. H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations.Marcel Dekker, New York, 1994. Zbl 0821.34067, MR 1309905 Reference: [9] Grace, S. R., Agarwal, R. P., Pavani, R., Thandapani, E.: On the oscillation of certain third order nonlinear functional differential equations.Appl. Math. Comput. 202 (2008), 102–112. Zbl 1154.34368, MR 2437140, 10.1016/j.amc.2008.01.025 Reference: [10] Grace, S. R., Lalli, B. S.: Oscillation of even order differential equations with deviating arguments.J. Math. Anal. Appl. 147 (1990), 569–579. Zbl 0711.34085, MR 1050228, 10.1016/0022-247X(90)90371-L Reference: [11] Kiguradze, I. T., Chaturia, T. A.: Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations.Kluwer Acad. Publ., Dordrecht, 1993. MR 1220223 Reference: [12] Kusano, T., Naito, M.: Comparison theorems for functional differential equations with deviating arguments.J. Math. Soc. Japan 3 (1981), 509–533. Zbl 0494.34049, MR 0620288, 10.2969/jmsj/03330509 Reference: [13] Ladde, G. S., Lakshmikantham, V., Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments.Marcel Dekker, New York, 1987. Zbl 0832.34071, MR 1017244 Reference: [14] Li, T., Thandapani, E.: Oscillation of solutions to odd–order nonlinear neutral functional differential equations.EJQTDE 2011 (2011), 1–12. Zbl 1211.34080, MR 2781058 Reference: [15] Li, T., Zhang, Ch., Baculíková, B., Džurina, J.: On the oscillation of third order quasi–linear delay differential equations.Tatra Mt. Math. Publ. 48 (2011), 1–7. Zbl 1265.34235 Reference: [16] Mahfoud, W. E.: Oscillation and asymptotic behavior of solutions of $n$–th order nonlinear delay differential equations.J. Differential Equations 24 (1977), 75–98. Zbl 0341.34065, MR 0457902, 10.1016/0022-0396(77)90171-1 Reference: [17] Philos, Ch. G.: On the existence of nonoscillatory solutions tending to zero at infinity for differential equations with positive delay.Arch. Math. (Brno) 36 (1981), 168–178. MR 0619435 Reference: [18] Philos, Ch. G.: Oscillation and asymptotic behavior of linear retarded differential equations of arbitrary order.Tech. Report 57, Univ. Ioannina, 1981. Reference: [19] Philos, Ch. G.: Some comparison criteria in oscillation theory.J. Austral. Math. Soc. 36 (1984), 176–186. Zbl 0541.34046, MR 0725744, 10.1017/S1446788700024630 Reference: [20] Shreve, W. E.: Oscillation in first order nonlinear retarded argument differential equations.Proc. Amer. Math. Soc. 41 (1973), 565–568. Zbl 0254.34075, MR 0372371, 10.1090/S0002-9939-1973-0372371-X Reference: [21] Tang, S., Li, T., Thandapani, E.: Oscillation of higher–order half–linear neutral differential equations.Demonstratio Math. (to appear). Reference: [22] Zhang, Ch., Li, T., Sun, B., Thandapani, E.: On the oscillation of higher–order half–linear delay differential equations.Appl. Math. Lett. 24 (2011), 1618–1621. Zbl 1223.34095, MR 2803721, 10.1016/j.aml.2011.04.015 Reference: [23] Zhang, Q., Yan, J., Gao, L.: Oscillation behavior of even order nonlinear neutral differential equations with variable coefficients.Comput. Math. Appl. 59 (2010), 426–430. Zbl 1189.34135, MR 2575529, 10.1016/j.camwa.2009.06.027 .

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