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New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform

  • Kannan Suresh , Ganesh Purushothaman , Ethiraju Thandapani and Ercan Tunç EMAIL logo
Published/Copyright: May 7, 2025
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Abstract

In this paper, the third-order delay differential equation of the form

(a2(t)((a1(t)x(t)))α)+b(t)xα(σ(t))=0

under the conditions t01a21/α(t)dt= and t01a1(t)dt< is studied. Our newly proposed approach allow us to reduce the number of conditions required for all solutions of the considered equation to be oscillatory. The results established here are new, improve and complement some oscillation results known for the studied equation. Examples are provided to show the strength and novelty of the results obtained.

MSC 2010: 34C10; 34K11
  1. (Communicated by Jozef Džurina)

References

[1] Agarwal, R. P.—Bohner, M.—Li, T.—Zhang, C.: Oscillation of third-order nonlinear delay differential equations, Taiwanese J. Math. 17 (2013), 545–558.10.11650/tjm.17.2013.2095Search in Google Scholar

[2] Agarwal, R. P.—Bohner, M.—Li, T.—Zhang, C.: A Philos-type theorem for third-order nonlinear retarded dynamic equations, Appl. Math. Comput. 249 (2014), 527–531.10.1016/j.amc.2014.08.109Search in Google Scholar

[3] Agarwal, R. P.—Bohner, M.—Li, W. T.: Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, New York, 2004.10.1201/9780203025741Search in Google Scholar

[4] Baculíková, B.: Oscillation of strongly noncanonical equations, J. Nonlinear Sci. Appl. 11 (2018), 1124–1128.10.22436/jnsa.011.10.02Search in Google Scholar

[5] Baculíková, B.—Džurina, J.: Oscillation of the third order Euler differential equation with delay, Math. Bohem. 139 (2014), 649–655.10.21136/MB.2014.144141Search in Google Scholar

[6] Baculíková, B.—Džurina, J.: Oscillation of third-order nonlinear differential equations, Appl. Math. Lett. 24 (2011), 466–470.10.1016/j.aml.2010.10.043Search in Google Scholar

[7] Baculíková, B.—Džurina, J.: Oscillation of third-order functional differential equations, Electron. J. Qual. Theory Differ. Equ. 2010 (2010), Art. No. 43.10.14232/ejqtde.2010.1.43Search in Google Scholar

[8] Baculíková, B.—Džurina, J.—Jadlovská, I.: On asymptotic properties of solutions to third-order delay differential equations, Electron. J. Qual. Theory Differ. Equ. 2019 (2019), Art. No. 7.10.14232/ejqtde.2019.1.7Search in Google Scholar

[9] Chatzarakis, G. E.—Džurina, J.—Jadlovská, I.: Oscillatory and asymptotic properties of third-order quasilinear delay differential equations, J. Inequal. Appl. 2019 (2019), Art. No. 23.10.1186/s13660-019-1967-0Search in Google Scholar

[10] Džurina, J.—Jadlovská, I.: Oscillation of third-order differential equations with noncanonical operators, Appl. Math. Comput. 336 (2018), 394–402.10.1016/j.amc.2018.04.043Search in Google Scholar

[11] Džurina, J.—Baculíková, B.—Jadlovská, I: Integral oscillation criteria for third-order differential equations with delay argument, Int. J. Pure Appl. Math. 108 (2016), 169–183.10.12732/ijpam.v108i1.15Search in Google Scholar

[12] Elabbasy, E. M.—Hassan, T. S.—Elmatary, B. M.: Oscillation criteria for third order delay nonlinear differential equations, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), Art. No. 5.10.14232/ejqtde.2012.1.5Search in Google Scholar

[13] Grace, S. R.—Agarwal, R. P.—Pavani, R.—Thandapani, E.: On the oscillation of certain third order nonlinear functional differential equations, Appl. Math. Comput. 202 (2008), 102–112.10.1016/j.amc.2008.01.025Search in Google Scholar

[14] Grace, S. R.—Graef, J. R.—Tunç, E.: On oscillatory behavior of third order half-linear delay differential equations, Math. Slovaca 73 (2023), 729–736.10.1515/ms-2023-0053Search in Google Scholar

[15] Grace, S. R.—Graef, J. R.—Tunç, E.: On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term, Math. Slovaca 67 (2017), 501–508.10.1515/ms-2016-0284Search in Google Scholar

[16] Grace, S. R.—Jadlovská, I.—Zafer, A.: On oscillation of third-order noncanonical delay differential equations, Appl. Math. Comput. 362 (2019), Art. ID 124530.10.1016/j.amc.2019.06.044Search in Google Scholar

[17] 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/1368217950Search in Google Scholar

[18] Jadlovská, I.—Chatzarakis, G. E.—Džurina, J.—Grace, S. R.: On sharp oscillation criteria for general third-order delay differential equations, Mathematics 9 (2021), Art. No. 1675.10.3390/math9141675Search in Google Scholar

[19] 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).Search in Google Scholar

[20] Kuramoto, Y.—Yamada, T.: Turbulent state in chemical reactions, Prog. Theor. Phys. 56 (1976), 679–681.10.1143/PTP.56.679Search in Google Scholar

[21] Li, T.—Frassu, S.—Viglialoro, G.: Combining effects ensuring boundedness in an attraction-repulsion chemotaxis model with production and consumption, Z. Angew. Math. Phys. 74 (2023), Art. No. 109.10.1007/s00033-023-01976-0Search in Google Scholar

[22] Li, T.—Pintus, N.—Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys. 70 (2019), Art. No. 86.10.1007/s00033-019-1130-2Search in Google Scholar

[23] Li, T.—Viglialoro, G.: Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differential Integral Equations 34 (2021), 315–336.10.57262/die034-0506-315Search in Google Scholar

[24] Li, T.—Zhang, C.—Baculíková, B.—Džurina, J.: On the oscillation of third-order quasi-linear delay differential equations, Tatra Mt. Math. Publ. 48 (2011), 117–123.10.2478/v10127-011-0011-7Search in Google Scholar

[25] Li, T.—Han, Z.—Sun, S.—Zhao, Y.: Oscillation results for third order nonlinear delay dynamic equations on time scales, Bull. Malays. Math. Sci. Soc. 34 (2011), 639–648.Search in Google Scholar

[26] Li, T.—Rogovchenko, Y. V.: On asymptotic behavior of solutions to higher-order sublinear Emden-Fowler delay differential equations, Appl. Math. Lett. 67 (2017), 53–59.10.1016/j.aml.2016.11.007Search in Google Scholar

[27] Li, T.—Rogovchenko, Y. V.: On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations, Appl. Math. Lett. 105 (2020), Art. ID 106293.10.1016/j.aml.2020.106293Search in Google Scholar

[28] Michelson, D.: Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D 19 (1986), 89–111.10.1016/0167-2789(86)90055-2Search in Google Scholar

[29] Omar, N.—Moaaz, O.—Alnemer, G.—Elabbasy, E. M.: New results on the oscillation of solutions of third-order differential equations with multiple delays, Symmetry 15 (2023), Art. No. 1920.10.3390/sym15101920Search in Google Scholar

[30] Padhi, S.—Pati, S.: Theory of Third-Order Differential Equations, Springer, New Delhi, 2014.10.1007/978-81-322-1614-8Search in Google Scholar

[31] Parhi, N.—Padhi, S.: On asymptotic behavior of delay-differential equations of third order, Nonlinear Anal. 34 (1998), 391–403.10.1016/S0362-546X(97)00600-7Search in Google Scholar

[32] Philos, C. G.: On the existence of nonoscillatory solutions tending to zero at 1 for differential equations with positive delays, Arch. Math. (Basel) 36 (1981), 168–178.10.1007/BF01223686Search in Google Scholar

[33] Prabaharan, N.—Thandapani, E.—Tunç, E.: Asymptotic behavior of semi-canonical third-order delay differential equations with a superlinear neutral term, Palest. J. Math. 12 (2023), 473–483.Search in Google Scholar

[34] Saker, S. H.—Džurina, J.: On the oscillation of certain class of third-order nonlinear delay differential equations, Math. Bohem. 135 (2010), 225–237.10.21136/MB.2010.140700Search in Google Scholar

[35] Saranya, K.—Piramanantham, V.—Thandapani, E.—Tunç, E.: Asymptotic behavior of semi-canonical third-order nonlinear functional differential equations, Palest. J. Math. 11 (2022), 433–442.Search in Google Scholar

[36] Srinivasan, R.—Saravanan, S.—Thandapani, E.—Tunç, E.: Oscillation of noncanonical third-order delay differential equations via canonical transform, Appl. Math. E-Notes 23 (2023), 265–273.Search in Google Scholar

[37] Thandapani, E.—Goktürk, B.—Özdemir, O.—Tunç, E.: Oscillatory behavior of semi-canonical nonlinear neutral differential equations of third-order via comparison principles, Qual. Theory Dyn. Syst. 22 (2023), Art. No. 30.10.1007/s12346-022-00731-6Search in Google Scholar

[38] Trench, W. F.: Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 189 (1974), 319–327.10.1090/S0002-9947-1974-0330632-XSearch in Google Scholar

Received: 2024-02-15
Accepted: 2024-10-30
Published Online: 2025-05-07
Published in Print: 2025-04-28

© 2025 Mathematical Institute Slovak Academy of Sciences

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