Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms
Abstract
:1. Introduction
- and , where ;
- , are positive constants with ;
- )
- , , , where and .
2. Main Results
- , , ,
- , , , , for is large enough.
- , , , , for is large enough.
- , , , , for is large enough.
3. Examples
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Fowler, R.H. Further studies of Emden’s and similar differential equations. Q. J. Math. 1931, 2, 259–288. [Google Scholar] [CrossRef]
- Berkovich, L.M. The generalized Emden–Fowler equation. Symmetry Nonlinear Math. Phys. 1997, 1, 155–163. [Google Scholar]
- Wong, J.S.W. On the generalized Emden–Fowler equation. Siam Rev. 1975, 17, 339–360. [Google Scholar] [CrossRef]
- Hale, J.K. Theory of Functional Differential Equations; Springer: New York, NY, USA, 1953. [Google Scholar]
- Agarwal, R.P.; Bohner, M.; Li, W.T. Nonoscillation and Oscillation: Theory for Functional Differential Equations; In Pure and Applied Mathematics; Marcel Dekker: New York, NY, USA, 2004; Volume 267. [Google Scholar]
- Agarwal, R.P.; Bohner, L.T.; Zhang, C. Oscillation of second order Emden-Fowler neutral delay differential equations. Ann. Mat. 2014, 193, 1861–1875. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Grace, S.R.; O’Regan, D. Oscillation Theory for Difference and Functional Differential Equations; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000. [Google Scholar]
- Alharbi, A.R. Almatrafi, M.B. New exact and numerical solutions with their stability for Ito integro-differential equation via Riccati–Bernoulli sub-ODE method. J. Taibah Univ. Sci. 2020, 14, 1447–1456. [Google Scholar] [CrossRef]
- Bellman, R. Stability Theory of Differential Equations; MaGraw-Hill: New York, NY, USA, 1953. [Google Scholar]
- Domoshnitsky, A.; Koplatadze, R. On asymptotic behavior of solutions of generalized Emden-Fowler differential equations with delay argument. Abstr. Appl. Anal. 2014, 2014, 168425. [Google Scholar] [CrossRef] [Green Version]
- Dosla, Z.; Marini, M. On super-linear Emden-Fowler type differential equations. J. Math. Anal. Appl. 2014, 416, 497–510. [Google Scholar] [CrossRef]
- Erbe, L.H.; Kong, Q.; Zhang, B.G. Oscillation Theory for Functional-Differential Equations; Monographs and Textbooks in Pure and Applied Mathematics; Marcel Dekker: New York, NY, USA, 1995; Volume 190. [Google Scholar]
- Almarri, B.; Ali, A.H.; Al-Ghafri, K.S.; Almutairi, A.; Bazighifan, O.; Awrejcewicz, J. Symmetric and Non-Oscillatory Characteristics of the Neutral Differential Equations Solutions Related to p-Laplacian Operators. Symmetry 2022, 14, 566. [Google Scholar] [CrossRef]
- Almarri, B.; Janaki, S.; Ganesan, V.; Ali, A.H.; Nonlaopon, K.; Bazighifan, O. Novel Oscillation Theorems and Symmetric Properties of Nonlinear Delay Differential Equations of Fourth-Order with a Middle Term. Symmetry 2022, 14, 585. [Google Scholar] [CrossRef]
- Ali, A.H.; Meften, G.; Bazighifan, O.; Iqbal, M.; Elaskar, S.; Awrejcewicz, J. A Study of Continuous Dependence and Symmetric Properties of Double Diffusive Convection: Forchheimer Model. Symmetry 2022, 14, 682. [Google Scholar] [CrossRef]
- Bazighifan, O.; Kumam, P. Oscillation Theorems for Advanced Differential Equations with P-Laplacian Like Operators. Mathematics 2020, 8, 821. [Google Scholar] [CrossRef]
- Moaaz, O.; El-Nabulsi, R.A.; Bazighifan, O. Oscillatory Behavior of Fourth-Order Differential Equations with Neutral Delay. Symmetry 2020, 12, 371. [Google Scholar] [CrossRef] [Green Version]
- Györi, I.; Ladas, G. Oscillation Theory of Delay Differential Equations; Clarendon Press: Oxford, UK, 1991. [Google Scholar]
- Grace, S.R.; Abbas, S.; Sajid, M. Oscillation of nonlinear even order differential equations with mixed neutral terms. Math. Methods Appl. Sci. 2022, 45, 1063–1071. [Google Scholar] [CrossRef]
- Kusano, T.; Manojlovič, J. Asymptotic behavior of positive solutions of sublinear differential equations of Emden- Fowler type. Comput. Math. Appl. 2011, 62, 551–565. [Google Scholar] [CrossRef] [Green Version]
- Li, T.; Han, Z.; Zhang, C.; Sun, S. On the oscillation of second order Emden-Fowler neutral delay differential equations. J. Appl. Math. 2011, 37, 601–610. [Google Scholar]
- Sathish Kumar, M.; Janaki, S.; Ganesan, V. Some new oscillatory behavior of certain third-order nonlinear neutral differential equations of mixed type. Int. J. Appl. Comput. Math. 2018, 78, 1–14. [Google Scholar] [CrossRef]
- Takasi, K.; Manojlovic, J. Precise asymptotic behavior of solutions of the sublinear Emden-Fowler differential equation. Appl. Math. Comput. 2011, 217, 4382–4396. [Google Scholar] [CrossRef]
- Wu, Y.; Yu, Y.; Zhang, J.; Xiao, J. Oscillation criteria for second order Emden-Fowler functional differential equations of neutral type. J. Inequalities Appl. 2016, 328, 1–11. [Google Scholar] [CrossRef] [Green Version]
- Xu, Z. Oscillation theorems related to technique for second order Emden-Fowler type neutral differential equations. Rocky Mt. J. Math. 2008, 38, 649–667. [Google Scholar] [CrossRef]
- Almarri, B.; Ali, A.H.; Lopes, A.M.; Bazighifan, O. Nonlinear Differential Equations with Distributed Delay: Some New Oscillatory Solutions. Mathematics 2022, 10, 995. [Google Scholar] [CrossRef]
- Bazighifan, O.; Ali, A.H.; Mofarreh, F.; Raffoul, Y.N. Extended Approach to the Asymptotic Behavior and Symmetric Solutions of Advanced Differential Equations. Symmetry 2022, 14, 686. [Google Scholar] [CrossRef]
- Qaraad, B.; Bazighifan, O.; Nofal, T.A.; Ali, A.H. Neutral differential equations with distribution deviating arguments: Oscillation conditions. J. Ocean Eng. Sci. 2022, in press. [Google Scholar] [CrossRef]
- Qaraad, B.; Bazighifan, O.; Ali, A.H.; Al-Moneef, A.A.; Alqarni, A.J.; Nonlaopon, K. Oscillation Results of Third-Order Differential Equations with Symmetrical Distributed Arguments. Symmetry 2022, 14, 2038. [Google Scholar] [CrossRef]
- Xu, Z.; Liu, X. Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations. J. Comput. Appl. Math. 2007, 206, 1116–1126. [Google Scholar] [CrossRef] [Green Version]
- Sathish Kumar, M.; Ganesan, V. Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay. Aims Math. 2020, 5, 3851–3874. [Google Scholar] [CrossRef]
- Xu, R.; Meng, F. Some new oscillation for second order quasi-linear neutral delay differential equations. Appl. Math. Comput. 2006, 182, 797–803. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Marappan, S.K.; Almutairi, A.; Iambor, L.F.; Bazighifan, O. Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms. Symmetry 2023, 15, 553. https://doi.org/10.3390/sym15020553
Marappan SK, Almutairi A, Iambor LF, Bazighifan O. Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms. Symmetry. 2023; 15(2):553. https://doi.org/10.3390/sym15020553
Chicago/Turabian StyleMarappan, Sathish Kumar, Alanoud Almutairi, Loredana Florentina Iambor, and Omar Bazighifan. 2023. "Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms" Symmetry 15, no. 2: 553. https://doi.org/10.3390/sym15020553
APA StyleMarappan, S. K., Almutairi, A., Iambor, L. F., & Bazighifan, O. (2023). Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms. Symmetry, 15(2), 553. https://doi.org/10.3390/sym15020553