Startseite Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms

  • Said R. Grace , Tongxing Li und Gokula Nanda Chhatria EMAIL logo
Veröffentlicht/Copyright: 14. August 2024
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

This paper deals with the oscillation and asymptotic behaviour of even order nonlinear differential equations with mixed nonlinear neutral terms. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first order advanced and/or delay differential equation. We provide novel oscillation criteria that improve, extend, and simplify previously published ones. The results are illustrated by two examples.

Acknowledgement

We would like to thank the anonymous reviewers for the helpful suggestions that allowed us to improved the presentation of our manuscript.

  1. Communicated by Jozef Džurina

References

[1] Agarwal, R. P.—Grace, S. R.—O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations, Kluwer, Dordrecht, 2000.10.1007/978-94-015-9401-1Suche in Google Scholar

[2] Agarwal, R. P.—Grace, S. R.—O’Regan, D.: Oscillation criteria for certain nth order differential equations with deviating arguments, J. Math. Anal. Appl. 262 (2001), 601–622.10.1006/jmaa.2001.7571Suche in Google Scholar

[3] Agarwal, R. P.—Grace, S. R.—O’Regan, D.: Oscillation Theory for Second Order Linear, Half Linear, Superlinear and Sublinear Dynamic Equations, Kluwer, Dordrecht, 2002.10.1007/978-94-017-2515-6Suche in Google Scholar

[4] Agarwal, R. P.—Grace, S. R.: Oscillation theorems for certain functional differential equations of higher order, Math. Comput. Modelling 39 (2004), 1185–1194.10.1016/S0895-7177(04)90539-0Suche in Google Scholar

[5] Agarwal, R. P.—Grace, S. R.—Dontha, S.: On the oscillation of certain functional differential equations, Commun. Korean Math. Soc. 19 (2004), 307–319.10.4134/CKMS.2004.19.2.307Suche in Google Scholar

[6] Agarwal, R. P.—Bohner, M.—Li, T.—Zhang, C.: A new approach in the study of oscillatory behavior of even-order neutral delay differential equations, Appl. Math. Comput. 225 (2013), 787–794.10.1016/j.amc.2013.09.037Suche in Google Scholar

[7] Agarwal, R. P.—Bohner, M.—Li, T.—Zhang, C.: Oscillation of second order differential equations with a sublinear neutral term, Carpathian J. Math. 30 (2014), 1–6.10.37193/CJM.2014.01.01Suche in Google Scholar

[8] Alzabut, J.—Grace, S. R.—Chhatria, G. N.: New oscillation results for higher order nonlinear differential equations with a nonlinear neutral terms, J. Math. Computer Sci. 28 (2023), 294–305.10.22436/jmcs.028.03.07Suche in Google Scholar

[9] Bainov, D. D.—Mishev, D. P.: Oscillation Theory of Neutral Differential Equations with Delay, Adam Hilger Ltd. Bristol, 1991.Suche in Google Scholar

[10] Bazighifan, O.—Ragusa, M. A.: Nonlinear equations of fourth-order with p-Laplacian like operators: Oscillation, methods and applications, Proc. Amer. Math. Soc. 150 (2022), 1009–1020.10.1090/proc/15794Suche in Google Scholar

[11] Dong, J. G.: Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments, Comput. Math. Appl. 59 (2010), 3710–3717.10.1016/j.camwa.2010.04.004Suche in Google Scholar

[12] Džurina, J.—Kotorová, R.: Properties of the third order trinomial differential equations with argument, Nonlinear Anal. 71 (2009), 1995–2002.10.1016/j.na.2009.01.070Suche in Google Scholar

[13] Erbe, L. H.—Kong, Q.—Zhang, B. G.: Oscillation Theory for Functional Differential equations, Marcel Dekker Inc. New York, 1995.Suche in Google Scholar

[14] Grace, S. R.—Agarwal, R. P.—O’Regan, D.: A selection of oscillation criteria for second order differential inclusions, Appl. Math. Lett. 22 (2009), 153–158.10.1016/j.aml.2008.01.006Suche in Google Scholar

[15] Grace, S. R.—Bohner, M.—Agarwal, R. P.: On the oscillation of second order half-linear dynamic equations, J. Difference Eqn. Appl. 15 (2009), 451–460.10.1080/10236190802125371Suche in Google Scholar

[16] Grace, S. R.—Agarwal, R. P.—Bohner, M.—O’Regan, D.: Oscillation of second order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Stimul. 14 (2009), 3463–3471.10.1016/j.cnsns.2009.01.003Suche in Google Scholar

[17] Grace, S. R—Agarwal, R. P—Kaymakalan, B.—Sae-Jie, W.: Oscillation theorems for second order nonlinear dynamic equations, Appl. Math. Comput. 32 (2010), 205–218.10.1007/s12190-009-0244-7Suche in Google Scholar

[18] Grace, S. R.—Graef, J. R.—El-Beltagy, M. A.: On the oscillation of third order neutral delay dynamic equations on time scales, Comput. Math. Appl. 63 (2012), 775–782.10.1016/j.camwa.2011.11.042Suche in Google Scholar

[19] Grace, S. R.—Akin, E.—Dikmen, M.: On the oscillation of second order nonlinear neutral dynamic equations with distributed deviating arguments on time scales, Dyn. Syst. Appl. 23 (2014), 735–748.Suche in Google Scholar

[20] Grace, S. R.—Jadlovská, I.—Zafer. A.: Oscillatory behavior of n-th order nonlinear delay differential equations with a nonpositive neutral term, Hacet. J. Math. Stat. 49 (2020), 766–776.10.15672/hujms.471023Suche in Google Scholar

[21] Graef, J. R.—Saker, S. H.: Oscillation theory of third-order nonlinear functional differential equations, Hiroshima Math. J. 43 (2013), 49–72.10.32917/hmj/1368217950Suche in Google Scholar

[22] Graef, J. R.—Grace, S. R.—Tunc, E.: Oscillation of even order advanced functional differential equations, Publ. Math. Debrecen 93 (2018), 445–455.10.5486/PMD.2018.8205Suche in Google Scholar

[23] Graef, J. R.—Grace, S. R.—Tunc, E.: Oscillation of even-order nonlinear differential equations with sublinear and superlinear neutral terms, Publ. Math. Debrecen 96 (2020), 195–206.10.5486/PMD.2020.8648Suche in Google Scholar

[24] Hale, J. K.—Lunel, S. M. V.: Introduction to Functional Differential Equations, Springer, New York, 1993.10.1007/978-1-4612-4342-7Suche in Google Scholar

[25] Hardy, G. H.—Littlewood, I. E.—Polya, G.: Inequalities, Cambridge University Press, Cambridge, Mass, USA, 1959.Suche in Google Scholar

[26] Karpuz, B.—Ocalan, O.—Ozturk, S.: Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations, Glasgow Math. J. 52 (2010), 107–114.10.1017/S0017089509990188Suche in Google Scholar

[27] Kiguradze, I. T.: On the oscillatory character of solutions of the equation dmu/dtm + a(t)|u| n sign u = 0, Mat. Sb. (N.S.) 65 (1964), 172–187 (in Russian).Suche in Google Scholar

[28] Kiguradze, I. T.—Chanturia, T. A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic Publishers, Dordrecht, 1993; Translated from the 1985 Russian original.10.1007/978-94-011-1808-8Suche in Google Scholar

[29] Kitamura, Y.—Kusano, T.: Oscillation of first order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc. 78 (1980), 64–68.10.1090/S0002-9939-1980-0548086-5Suche in Google Scholar

[30] Koplatadze, R. G.—Chanturiya, T. A.: Oscillating and monotone solutions of first-order differential equations with deviating argument, Differ. Uravn. 18 (1982), 1463–1465 (in Russian).Suche in Google Scholar

[31] Ladas, G.—Stavroulakis, I. P.: Oscillation caused by several retarded and advanced arguments, J. Differ. Equ. 44 (1982), 134–152.10.1016/0022-0396(82)90029-8Suche in Google Scholar

[32] Li, T.—Rogovchenko, Y.—Zhang, C.: Oscillation results for second-order nonlinear neutral differential equations, Adv. Difference Equ. 2013 (2013), Art. ID 336.10.1186/1687-1847-2013-336Suche in Google Scholar

[33] Li, T.—Rogovchenko, Y.: Oscillation criteria for even-order neutral differential equations, Appl. Math. Lett. 61 (2016), 35–41.10.1016/j.aml.2016.04.012Suche in Google Scholar

[34] Philos, C. G.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays, Arch. Math. (Basel) 36 (1981), 168–178.10.1007/BF01223686Suche in Google Scholar

[35] Qin, H.—Shang, N.—Lu, Y.: A note on oscillation criteria of second order nonlinear neutral delay differential equations, Comput. Math. Appl. 56 (2008), 2987–2992.10.1016/j.camwa.2008.09.004Suche in Google Scholar

[36] Staikos, V. A.—Sficas, Y. G.: Oscillatory and asymptotic behavior of functional differential equations, J. Differ. Equ. 12 (1972), 426–437.10.1016/0022-0396(72)90016-2Suche in Google Scholar

[37] Zafer, A.: Oscillation criteria for even order neutral differential equations, Appl. Math. Lett. 11 (1998), 21–25.10.1016/S0893-9659(98)00028-7Suche in Google Scholar

Received: 2023-08-24
Accepted: 2024-03-09
Published Online: 2024-08-14
Published in Print: 2024-08-27

© 2024 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 20.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0068/html
Button zum nach oben scrollen