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Keywords:
third-order differential equation; oscillation; nonoscillation; disconjugacy
third-order differential equation; oscillation; nonoscillation; disconjugacy
Summary:
In this paper we consider the third-order nonlinear delay differential equation (*) $$ ( a(t)\left ( x''(t)\right ) ^{\gamma })' +q(t)x^{\gamma }(\tau (t))=0,\quad t\geq t_0, $$ where $a(t)$, $q(t)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau (t)\leq t$ satisfies $\lim _{t\rightarrow infty }\tau (t)=infty $. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.
References:
[1] Bainov, D. D., Mishev, D. P.: Oscillation Theory for Neutral Differential Equations with Delay. Adam Hilger, New York (1991). MR 1147908 | Zbl 0747.34037
[2] Barrett, J. H.: Oscillation theory of ordinary linear differential equations. Adv. Math. (1969), 445-504. MR 0257462 | Zbl 0213.10801
[3] Candan, T., Dahiya, R. S.: Oscillation of third order functional differential equations with delay. Fifth Mississippi Conf. Diff. Eqns. and Comp. Simulation, Electron. J. Diff. Equations Conf. 10 (2003), 39-88. MR 1983096 | Zbl 1028.34061
[4] Džurina, J.: Asymptotic properties of third order delay differential equations. Czech. Math. J. 45 (1995), 443-448. MR 1344509
[5] Džurina, J.: Asymptotic properties of the third order delay differential equations. Nonlinear Anal., Theory Methods Appl. 26 (1996), 33-34. DOI 10.1016/0362-546X(94)00239-E | MR 1354789
[6] Erbe, L. H., Kong, Q., Zhang, B. G.: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York (1994). MR 1309905 | Zbl 0821.34067
[7] Grace, S. R., Agarwal, R. P., Pavani, R., Thandapani, E.: On the oscillation of certain third order nonlinear functional differential equations. Appl. Math. Comp. (2008). MR 2437140 | Zbl 1154.34368
[8] Greguš, M.: Third Order Linear Differential Equations. Reidel, Dordrecht (1982).
[9] Gyŏri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford (1991). MR 1168471
[10] Hanan, M.: Oscillation criteria for third order differential equations. Pacific J. Math. 11 (1961), 919-944. DOI 10.2140/pjm.1961.11.919 | MR 0145160
[11] Kiguradze, I. T., Chanturia, T. A.: Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations. Kluwer Acad. Publ., Dordrecht (1993). MR 1220223
[12] Kusano, T., Naito, M.: Comparison theorems for functional differential equations with deviating arguments. J. Math. Soc. Japan 3 (1981), 509-533. DOI 10.2969/jmsj/03330509 | MR 0620288 | Zbl 0494.34049
[13] Lacková, D.: The Asymptotic Properties of the Solutions of the n-th Order Functional Neutral Differential Equations. Comput. Appl. Math. 146 (2003), 385-392. DOI 10.1016/S0096-3003(02)00590-8 | MR 2006078 | Zbl 1035.34087
[14] Ladde, G. S., Lakshmikantham, V., Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York (1987). MR 1017244 | Zbl 0832.34071
[15] Lazer, A. C.: The behavior of solutions of the differential equation $x'''(t)+P(t)x'(t)+q(t) x(t)=0$. Pacific J. Math. 17 (1966), 435-466. DOI 10.2140/pjm.1966.17.435 | MR 0193332
[16] Mehri, B.: On the conditions for the oscillation of solutions of nonlinear third order differential equations. Čas. Pěst Mat. 101 (1976), 124-124. MR 0442366
[17] Parhi, N., Das, P.: Asymptotic behavior of a class of third order delay-differential equations. Proc. Am. Math. Soc. 110 387-393 (1990). DOI 10.1090/S0002-9939-1990-1019279-4 | MR 1019279
[18] Parhi, N., Padhi, S.: On asymptotic behavior of delay-differential equations of third order. Nonlinear Anal., Theory Methods Appl. 34 (1998), 391-403. DOI 10.1016/S0362-546X(97)00600-7 | MR 1635717 | Zbl 0935.34063
[19] Parhi, N., Padhi, S.: Asymptotic behavior of solutions of third order delay-differential equations. Indian J. Pure Appl. Math. 33 (2002), 1609-1620. MR 1941081
[20] Saker, S. H.: Oscillation criteria of certain class of third-order nonlinear delay differential equations. Math. Slovaca 56 (2006), 433-450. MR 2267765
[21] Swanson, C. A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York (1968). MR 0463570 | Zbl 0191.09904
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