Abstract
In this paper, we are concerned with the oscillation properties of the third order differential equation $$ \left( {b(t) \left( {[a(t)x'(t)'} \right)^\gamma } \right)^\prime + q(t)x^\gamma (t) = 0, \gamma > 0 $$. Some new sufficient conditions which insure that every solution oscillates or converges to zero are established. The obtained results extend the results known in the literature for γ = 1. Some examples are considered to illustrate our main results.
[1] Agarwal, R. P. —Grace, S. R. —O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations, Kluwer Acad. Publ., Drdrechet, 2000. 10.1007/978-94-015-9401-1Search in Google Scholar
[2] Bainov, D. D. —Mishev, D. P.: Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, New York, 1991. Search in Google Scholar
[3] Bartusek, M.: On oscillatory solutions of third order differential equations with quasiderivatives. In: Forth Mississippi Conf. Diff. Eqns and Comp. Simulation. Electron. J. Differential Equations 1999 (1999), 1–11. Search in Google Scholar
[4] Bartusek, M.: On oscillatory solutions of the system of differential equations with deviating arguments, Czechoslovak Math. J. 35(110) (1985), 443–448. 10.21136/CMJ.1985.102046Search in Google Scholar
[5] Dzurina, J.: Asymptotic properties of third order delay differential equations, Czechoslovak Math. J. 45(120) (1995), 443–448. 10.21136/CMJ.1995.128546Search in Google Scholar
[6] Dzurina, J.: Property (A) of the third order differential equations with deviating arguments, Math. Slovaca 45 (1995), 395–402. Search in Google Scholar
[7] Dzurina, J.: Comparison Theorems for Functional Differential Equations, EDIS, Zilina 2003. Search in Google Scholar
[8] Gera, M. —Graef, J. R. —Gregus, M.: On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations, Nonlinear Anal. 32 (1998), 417–425. http://dx.doi.org/10.1016/S0362-546X(97)00483-510.1016/S0362-546X(97)00483-5Search in Google Scholar
[9] Gregus, M. Jr.: Oscillation results on nonlinear third order differential equations. In: Proceedings of the Second World Congress of Nonlinear Analysts, Part 3 (Athens, 1996); Nonlinear Anal. 30 (1997), 1573–1581. http://dx.doi.org/10.1016/S0362-546X(97)00031-X10.1016/S0362-546X(97)00031-XSearch in Google Scholar
[10] Gregus, M. —Gregus, M. Jr.: On oscillatory properties of solutions of a certain nonlinear third-order differential equation, J. Math. Anal. Appl. 181 (1994), 575–585. http://dx.doi.org/10.1006/jmaa.1994.104510.1006/jmaa.1994.1045Search in Google Scholar
[11] Gregus, M. —Gregus, M. Jr.: On a perturbed nonlinear third order differential equation, Arch. Math. (Brno) 33 (1997), 121–126. Search in Google Scholar
[12] Gregus, M. Jr.— Graef, J. R. —Gera, M.: Oscillating nonlinear third order differential equations, Nonlinear Anal. 28 (1997), 1611–1622. http://dx.doi.org/10.1016/0362-546X(95)00239-R10.1016/0362-546X(95)00239-RSearch in Google Scholar
[13] Gregus, M. Jr.— Graef, J. R.: On a certain nonautonomous nonlinear third order differential equation, Appl. Anal. 58 (1995), 175–185. http://dx.doi.org/10.1080/0003681950884036910.1080/00036819508840369Search in Google Scholar
[14] Gregus, M. Jr.: On the oscillatory behavior of certain third order nonlinear differential equation, Arch. Math. (Brno) 28 (1992), 221–228. Search in Google Scholar
[15] Gyori, I. —Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. Search in Google Scholar
[16] Kiguradze, I. T. —Chaturia, T. A.: Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations, Kluwer Acad. Publ., Dordrecht, 1993. 10.1007/978-94-011-1808-8_1Search in Google Scholar
[17] Ladde, G. S. —Lakshmikantham, V. —Zhang, B. Z.: Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987. Search in Google Scholar
[18] Parhi, N. —Das, P.: Asymptotic property of solutions of a class of third-order differential equations, Proc. Amer. Math. Soc. 110 (1990), 387–393. http://dx.doi.org/10.2307/204808210.1090/S0002-9939-1990-1019279-4Search in Google Scholar
[19] Parhi, N. —Das, P.: Oscillation criteria for a class of nonlinear differential equations of third order, Ann. Polon. Math. 57 (1992), 219–229. 10.4064/ap-57-3-219-229Search in Google Scholar
[20] Parhi, N. —Das, P.: Oscillation and nonoscilation of nonhomogeneous third order differential equations, Czechoslovak Math. J. 44 (1994), 443–459. 10.21136/CMJ.1994.128483Search in Google Scholar
[21] Parhi, N. —Das, P.: On the oscillation of a class of linear homogeneous third order differential equations, Arch. Math. (Brno) 34 (1998), 435–443. Search in Google Scholar
[22] Parhi, N. —Das, P.: On Asymptotic behavior of delay-differential equations of third order, Nonlinear Anal. 34 (1998), 391–403. http://dx.doi.org/10.1016/S0362-546X(97)00600-710.1016/S0362-546X(97)00600-7Search in Google Scholar
[23] Parhi, N. —Das, P.: Asymptotic behavior of a class of third order delay differential equations, Math. Slovaca 50 (2000), 315–333. Search in Google Scholar
[24] Saker, S. H.: Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca 56 (2006), 433–450. Search in Google Scholar
[25] Skerlik, A.: An integral condition of oscillation for equation y‴(t) + p(t)y′(t) + q(t)y(t) = 0 with nonnegative coefficients, Arch. Math. (Brno) 31 (1995), 155–161. Search in Google Scholar
[26] Skerlik, A.: Integral criteria of oscillation for a third order linear differential equations, Math. Slovaca 45 (1995), 403–412. Search in Google Scholar
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