Oscillatory and asymptotic behavior of third-order nonlinear differential equations with a superlinear neutral term

Sufficient conditions are derived for all solutions of a class of third-order nonlinear differential equations with a superlinear neutral term to be either oscillatory or convergent to zero asymptotically. Examples illustrating the results are included and some suggestions for further research are indicated.

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  • [1] Agarwal RP, Grace SR, O’Regan D. Oscillation theory for difference and functional differential equations. Dordrecht, Netherlands: Kluwer Academic Publishers, 2010.
  • [2] Baculíková B, Džurina J. Oscillation of third-order neutral differential equations. Mathematical and Computer Modelling 2010; 52 (1-2): 215-226. doi:10.1016/j.mcm.2010.02.011
  • [3] Chatzarakis GE, Džurina J, Jadlovská I. Oscillatory properties of third-order neutral delay differential equations with noncanonical operators. Mathematics 2019; 7(12): 1-12. doi:10.3390/math7121
  • [4] Chatzarakis GE, Grace SR, I. Jadlovská, Li T, Tunç E. Oscillation criteria for third-order Emden–Fowler differential equations with unbounded neutral coefficients. Complexity 2019; 2019: 1-7. doi: 10.1155/2019/5691758
  • [5] Chen DX, Liu JC. Asymptotic behavior and oscilation of solutions of third-order nonlinear neutral delay dynamic equations on time scales. Canadian Applied Mathematcs Quarterly 2008; 16 (1): 19-43.
  • [6] Das P. Oscillation criteria for odd order neutral equations. Journal of Mathematical Analysis and Applications 1994; 188 (1): 245-257. doi: 10.1006/jmaa.1994.1425
  • [7] Džurina J, Grace SR, Jadlovská I. On nonexistence of Kneser solutions of third-order neutral delay differential equations. Appllied Mathematics Letters 2019; 88: 193-200. doi: 10.1016/j.aml.2018.08.016
  • [8] Došlá Z, Liška P. Comparison theorems for third-order neutral differential equations. Electroninc Journal of Differential Equations 2016; 2016 (38): 1-13.
  • [9] Grace SR, Graef JR, El-Beltagy MA. On the oscillation of third order neutral delay dynamic equations on time scales. Computers & Mathematics with Applications 2012; 63 (4): 775-782. doi: 10.1016/j.camwa.2011.11.042
  • [10] Grace SR, Graef JR, Tunç E. Oscillatory behaviour of third order nonlinear differential equations with a nonlinear nonpositive neutral term. Journal of Taibah University for Science 2019; 13 (1): 704-710. doi: 10.1080/16583655.2019.1622847
  • [11] Graef JR, Savithri R, Thandapani E. Oscillatory properties of third order neutral delay differential equations. In: Proceedings of the fourth international conference on dynamical systems and differential equations; Wilmington, NC, USA; 2002. pp. 342-350.
  • [12] Graef JR, Tunç E, Grace SR. Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation. Opuscula Mathematica 2017; 37 (6): 839-852. doi: 10.7494/OpMath.2017.37.6.839
  • [13] Graef JR, Spikes WP, Grammatikopoulos MK. Asymptotic behavior of nonoscillatory solutions of neutral delay differential equations of arbitrary order. Nonlinear Analysis 1993; 21 (1): 23-42. doi: 10.1016/0362-546X(93)90175- R
  • [14] Hale JK, Verduyn Lunel SM. Introduction to Functional Differential Equations. New York, NY, USA: Springer, 1993.
  • [15] Jiang Y, Jiang C, Li T. Oscillatory behavior of third-order nonlinear neutral delay differential equations. Advances in Difference Equations 2016; 2016 (171): 1-12.
  • [16] Kiguradze IT. On the oscillatory character of solutions of the equation d mu/dtm +a(t)|u| n sign u = 0. Matematicheskii Sbornik 1964; 65: 172-187.
  • [17] Koplatadze RG, Chanturiya TA. Oscillating and monotone solutions of first-order differential equations with deviating argument. Differentsial’nye Uravneniya 1982; 18: 1463-1465.
  • [18] Li T, Rogovchenko YV. Asymptotic behavior of higher-order quasilinear neutral differential equations. Abstract and Applied Analysis 2014; 2014: 1-11. doi: 10.1155/2014/395368
  • [19] Li T, Rogovchenko YV. On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appllied Mathematics Letters 2020; 105: 1-7. doi: 10.1016/j.aml.2020.106293
  • [20] Li T, Rogovchenko YV. Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations. Monatshefte für Mathematik 2017; 184: 489-500. doi: 10.1007/s00605-017-1039-9
  • [21] Mihalíková B, Kostiková E. Boundedness and oscillation of third order neutral differential equations. Tatra Mountains Mathematical Publications 2009; 43: 137-144. doi: 10.2478/v10127-009-0033-6
  • [22] Philos CG. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Archiv der Mathematik 1981; 36 (1): 168-178. doi: 10.1007/BF01223686
  • [23] Saker SH, Graef JR. Oscillation of third-order nonlinear neutral functional dynamic equations on time scales. Dynamic Systems and Applications 2012; 21 (4): 583-606.
  • [24] Sun Y, Hassan TS. Comparison criteria for odd order forced nonlinear functional neutral dynamic equations. Applied Mathematics and Computation 2015; 251: 387-395. doi: 10.1016/j.amc.2014.11.095
  • [25] Sun Y, Zhao Y. Oscillatory behavior of third-order neutral delay differential equations with distributed deviating arguments. Journal of Inequalities and Applications 2019; 2019 (207): 1-16. doi: 10.1186/s13660-019-2161-0
  • [26] Thandapani E, Li T. On the oscillation of third-order quasi-linear neutral functional differential equations. Archivum Mathematicum (Brno) 2011; 47 (3): 181-199.
  • [27] Thandapani E, Padmavathy S, Pinelas S. Oscillation criteria for odd-order nonlinear differential equations with advanced and delayed arguments. Electronic Journal of Differential Equations 2014; 2014 (174): 1-13.
  • [28] Tunç E, Grace SR. Oscillatory behavior of solutions to third-order nonlinear differential equations with a superlinear neutral term. Electronic Journal of Differential Equations 2020; 2020 (32): 1-11.
  • [29] Tunç E. Oscillatory and asymptotic behavior of third-order neutral diferential equations with distributed deviating arguments. Electroninc Journal of Differential Equations 2017; 2017 (16): 1-12.