We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.
R.Agarwal, S.Grace and D.O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Acad. Publ., Dordrecht, 2000.
2.
B.Baculíková and J.Džurina, Oscillation of third-order neutral differential equations, Math. Comput. Modelling52 (2010), 215–226. doi:10.1016/j.mcm.2010.02.011.
3.
B.Baculíková and J.Džurina, Oscillation theorems for second-order nonlinear neutral differential equations, Comput. Math. Appl.62 (2011), 4472–4478. doi:10.1016/j.camwa.2011.10.024.
4.
J.Džurina, Oscillation theorems for neutral differential equations of higher order, Czechoslovak Math. J.54(129) (2004), 107–117. doi:10.1023/B:CMAJ.0000027252.29549.bb.
5.
S.Fišnarová and R.Mařík, Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments, Math. Slovaca70(2) (2020), 389–400. doi:10.1515/ms-2017-0358.
6.
N.Fukagai and T.Kusano, Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat. Pura Appl. (4)136 (1984), 95–117. doi:10.1007/BF01773379.
7.
J.K.Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
8.
R.G.Koplatadze and T.A.Čanturija, On Oscillatory Properties of Differential Equations with Deviating Arguments, Tbilisi Univ. Press, Tbilisi, 1977, (Russian).
9.
T.Kusano, O.Akio and U.Hiroyuki, Oscillation theory for a class of second order differential equations with application to partial differential equations, Japan. J. Math.19 (1993), 131–147. doi:10.4099/math1924.19.131.
10.
T.Kusano and M.Naito, Nonlinear oscillation of second order differential equations with retarded argument, Ann. Mat. Pura Appl. (4)106 (1975), 171–185. doi:10.1007/BF02415027.
11.
T.Kusano and M.Naito, Nonlinear oscillation of fourth order differential equations, Can. J. Math.28(4) (1976), 840–852. doi:10.4153/CJM-1976-081-0.
12.
G.S.Ladde, V.Lakshmikantham and B.G.Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 110, Dekker, New York, 1987.
13.
T.Li and Y.V.Rogovchenko, On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations, Appl. Math. Lett.105 (2020), 106293. doi:10.1016/j.aml.2020.106293.
14.
Z.Nehari, Oscillation criteria for second order linear differential equations, Trans. Amer. Math. Soc.85 (1957), 428–445. doi:10.1090/S0002-9947-1957-0087816-8.
15.
C.Philos, On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay, Arch. Math. (Basel)36 (1981), 168–178. doi:10.1007/BF01223686.
16.
C.C.Travis, Oscillation theorems for second-order differential equations with functional arguments, Proc. Amer. Math. Soc.31 (1972), 199–202. doi:10.1090/S0002-9939-1972-0285789-X.
17.
C.Zhang, R.P.Agarwal, M.Bohner and T.Li, New results for oscillatory behavior of even-order half-linear delay differential equations, Appl. Math. Lett.26 (2013), 179–183. doi:10.1016/j.aml.2012.08.004.
18.
C.Zhang, T.Li and S.Saker, Oscillation of fourth-order delay differential equations, J. Math. Sci.201 (2014), 296–308. doi:10.1007/s10958-014-1990-0.
