Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators
Abstract
:1. Introduction
- (I)
- is the ratio of positive odd integers, and
- (II)
- with for large values, and is not identical to zero for large values;
- (III)
- , is strictly increasing and
2. Some Lemmas and Auxiliary Notations
- (i)
- and
- (ii)
- and
3. Main Results
- (1)
- for
- (2)
- has the partial derivative on H such that is locally integrable with respect to in H and
- (1)
- We note that the approach used in [25] requires the condition , which is a harsh condition on the delay functions. In our results, we do not require this condition.
- (2)
- (3)
4. Conclusions
- (1)
- (2)
- The oscillation behavior of the higher-order DE
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- El-Gaber, A.A. On the oscillatory behavior of solutions of canonical and noncanonical even-order neutral differential equations with distributed deviating arguments. J. Nonlinear Sci. Appl. 2024, 17, 82–92. [Google Scholar] [CrossRef]
- Ladde, G.S.; Lakshmikantham, V.; Zhang, B.G. Oscillation Theory of Differential Equations with Deviating Arguments; Marcel Dekker: New York, NY, USA, 1987. [Google Scholar]
- Wu, Y.; Yu, Y.; Zhang, J.; Xiao, J. Oscillation criteria for second order Emden–Fowler functional differential equations of neutral type. J. Inequal. Appl. 2016, 2016, 328. [Google Scholar] [CrossRef]
- Santra, S.S.; Khedher, K.M.; Moaaz, O.; Muhib, A.; Yao, S.-W. Second-order impulsive delay differential systems: Necessary and sufficient conditions for oscillatory or asymptotic behavior. Symmetry 2021, 13, 722. [Google Scholar] [CrossRef]
- Masood, F.; Moaaz, O.; Santra, S.S.; Fernandez-Gamiz, U.; El-Metwally, H.A. Oscillation theorems for fourth-order quasi-linear delay differential equations. AIMS Math. 2023, 8, 16291–16307. [Google Scholar] [CrossRef]
- Lackova, D. The asymptotic properties of the solutions of the n-th order functional neutral differential equations. Comput. Appl. Math. 2003, 146, 385–392. [Google Scholar]
- Muhib, A.; Abdeljawad, T.; Moaaz, O.; Elabbasy, E.M. Oscillatory properties of odd-order delay differential equations with distribution deviating arguments. Appl. Sci. 2020, 10, 5952. [Google Scholar] [CrossRef]
- Ramos, H.; Moaaz, O.; Muhib, A.; Awrejcewicz, J. More effective results for testing oscillation of non-canonical neutral delay differential equations. Mathematics 2021, 9, 1114. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Bohner, M.; Li, T.; Zhang, C. Even-order half-linear advanced differential equations: Improved criteria in oscillatory and asymptotic properties. Appl. Math. Comput. 2015, 266, 481–490. [Google Scholar] [CrossRef]
- Grace, S.R.; Akin, E. Oscillation Criteria for Fourth Order Nonlinear Positive Delay Differential Equations with a Middle Term. Dyn. Syst. Appl. 2016, 25, 431–438. [Google Scholar]
- Saker, S.H.; Dzurina, J. On the oscillation of certain class of third-order nonlinear delay differential equations. Math. Bohem. 2010, 135, 225–237. [Google Scholar] [CrossRef]
- Chatzarakis, G.E.; Grace, S.R.; Jadlovska, I. Oscillation criteria for third-order delay differential equations. Adv. Differ. Equ. 2017, 2017, 330. [Google Scholar] [CrossRef]
- Moaaz, O.; Muhib, A.; Ahmad, H.; Muhsin, W. Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator. Math. Slovaca 2023, 73, 703–712. [Google Scholar] [CrossRef]
- Li, T.; Zhang, C.; Baculíková, B.; Dzurina, J. On the oscillation of third-order quasi-linear delay differential equations. Tatra Mt. Math. Publ. 2011, 48, 117–123. [Google Scholar] [CrossRef]
- Chatzarakis, G.E.; Dzurina, J.; Jadlovska, I. Oscillatory and asymptotic properties of third-order quasilinear delay differential equations. J. Inequal. Appl. 2019, 2019, 23. [Google Scholar] [CrossRef]
- Astashova, I.V. Asymptotic Behavior of Singular Solutions of Emden-Fowler Type Equations. Diff. Equ. 2019, 55, 581–590. [Google Scholar] [CrossRef]
- Astashova, I.V. On qualitative properties and asymptotic behavior of solutions to higher-order nonlinear differential equations. Wseas Trans. Math. 2017, 16, 39–47. [Google Scholar]
- Hassan, T.S.; El-Matary, B.M. Asymptotic Behavior and Oscillation of Third-Order Nonlinear Neutral Differential Equations with Mixed Nonlinearities. Mathematics 2023, 11, 424. [Google Scholar] [CrossRef]
- Thandapani, E.; Li, T. On the oscillation of third-order quasi-linear neutral functional differential equations. Arch. Math. 2011, 47, 181–199. [Google Scholar]
- Moaaz, O.; Dassios, I.; Muhsin, W.; Muhib, A. Oscillation Theory for Non-Linear Neutral Delay Differential Equations of Third Order. Appl. Sci. 2020, 10, 4855. [Google Scholar] [CrossRef]
- Dosla, Z.; Liska, P. Oscillation of third-order nonlinear neutral differential equations. Appl. Math. Lett. 2016, 56, 42–48. [Google Scholar] [CrossRef]
- Yang, L.; Xu, Z. Oscillation of certain third-order quasilinear neutral differential equations. Math. Slovaca 2014, 64, 85–100. [Google Scholar] [CrossRef]
- Jiang, Y.; Li, T. Asymptotic behavior of a third-order nonlinear neutral delay differential equation. J. Inequal. Appl. 2014, 2014, 512. [Google Scholar] [CrossRef]
- Graef, J.R.; Tunç, E.; Grace, S.R. Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation. Opusc. Math. 2017, 37, 839–852. [Google Scholar] [CrossRef]
- Qaraad, B.; Moaaz, O.; Baleanu, D.; Santra, S.S.; Ali, R.; Elabbasy, E.M. Third-order neutral differential equations of the mixed type: Oscillatory and asymptotic behavior. MBE 2022, 19, 1649–1658. [Google Scholar] [CrossRef] [PubMed]
- Kiguradze, I.T.; Chanturia, T.A. Asymptotic Properties of Solutions of Nonautonomous Ordinary Diferential Equations; Translated from the 1985 Russian Original; Kluwer Academic: Dordrecht, The Netherlands, 1993. [Google Scholar]
- Baculikova, B.; Dzurina, J. Oscillation of third-order neutral differential equations. Math. Comput. Model. 2010, 52, 215–226. [Google Scholar] [CrossRef]
- Philos, H.G. Oscillation theroms for linear differential equations of second order. Arch. Math. 1989, 53, 482–492. [Google Scholar] [CrossRef]
- Philos, C.G. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. 1981, 36, 168–178. [Google Scholar] [CrossRef]
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Alsharidi, A.K.; Muhib, A. Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators. Mathematics 2024, 12, 2488. https://doi.org/10.3390/math12162488
Alsharidi AK, Muhib A. Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators. Mathematics. 2024; 12(16):2488. https://doi.org/10.3390/math12162488
Chicago/Turabian StyleAlsharidi, Abdulaziz Khalid, and Ali Muhib. 2024. "Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators" Mathematics 12, no. 16: 2488. https://doi.org/10.3390/math12162488