Stability, boundedness and existence of unique periodic solutions to a class of functional differential equations
DOI:
https://doi.org/10.22199/issn.0717-6279-2021-02-0017Keywords:
Fourth order, Non linear functional differential equation, Uniform stability, Uniform ultimate boundedness, Periodic solutionsAbstract
In this paper a novel class of fourth order functional differential equations is discussed. By reducing the fourth order functional differential equation to system of first order, a suitable complete Lyapunov functional is constructed and employed to obtain sufficient conditions that guarantee existence of a unique periodic solution, asymptotic and uniform asymptotic stability of the zero solutions, uniform boundedness and uniform ultimate boundedness of solutions. The obtained results are new and include many prominent results in literature. Finally, two examples are given to show the feasibility and reliability of the theoretical results.
Downloads
References
A. T. Ademola, “Periodicity, stability, and boundedness of solutions to certain fourth order delay differential equations”, International journal of nonlinear science, vol. 28, no. 1, pp. 20-39, 2019. [On line]. Available: https://bit.ly/2YwpD06
T. A. Ademola and M. O. Ogundiran, “On the existence and uniqueness of solutions of a generalized Lipschitz ordinary differential equations”, Ife journal of science, vol. 9, no. 2, pp. 241-246, 2007, doi: 10.4314/ijs.v9i2.32255.
O. A. Adesina and B. S. Ogundare, “Some new stability and boundedness results on a certain fourth order nonlinear differential equation”, Nonlinear studies, vol. 19, no. 3, pp. 359-369, 2012. [On line]. Available: https://bit.ly/3oETrlM
S. Balamuralitharan, “Periodic solutions of fourth order delay differential equation”, Bulletin of the Iranian Mathematical Society, vol. 41, no, 2, pp. 307-314, 2015. [On line]. Available: https://bit.ly/3oBAO1W
T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Orlando, FL: Academic Press, 1985.
T. A. Burton, Volterra integral and differential equations. New York, NY: Academic Press, 1983.
M. Cai and F. Meng, “Stability and boundedness of solutions for a certain fourth order delay differential equation”, Annals of applied mathematics, vol. 34, no. 4, pp. 345-357, 2018. [On line]. Available: https://bit.ly/3jfZ3C9
R. D. Driver, Ordinary and delay differential equations. New York, NY: Springer, 1977, doi: 10.1007/978-1-4684-9467-9
J. K. Hale, Theory of functional differential equations, New York, NY: Springer, 1977, doi: 10.1007/978-1-4612-9892-2
H. Kang and L. Si, “Stability of solutions to certain fourth order delay differential equations”, Annals of differential equations, no. 4, pp. 407-413, 2010. [On line]. Available: https://bit.ly/3aoNapl
E. Korkmaz and C. Tunç, “Boundedness and square integrability of solutions of nonlinear fourth order differential equations with bounded delay”, Electronic journal of differential equations, vol. 2017, Art ID: 47, pp. 1-13, 2017. [On line]. Available: https://bit.ly/3pzSexm
E. Korkmaz, “Stability and boundedness of solutions of nonlinear fourth order differential equations with bounded delay”, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 6, pp. 1317–1324, 2017, doi: 10.16984/saufenbilder.308097
E. Korkmaz, “Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay”, Journal of inequalities and applications, vol. 2017, pp. Art ID. 134, Jun. 2017, doi: 10.1186/s13660-017-1353-8
V. Lakshmikantham, L. Wen, and B. Zhang, Theory of differential equations with unbounded delay. Dordrecht: Kluwer, 1994, doi: 10.1007/978-1-4615-2606-3
E. O. Okoronkwo, “On stability and boundedness of solutions of a certain fourth order delay differential equation”, International journal of mathematics and mathematical sciences, vol. 12, Art ID. 607032, pp. 589-602, 1989, doi: 10.1155/S0161171289000724
M. Rahmane and M. Remili, “On stability and boundedness of solutions of certain non autonomous fourth order delay differential equations”, Acta Universitatis Matthiae Belii. Series mathematics (Online), vol. 23, pp. 101-114, 2015. [On line]. Available: https://bit.ly/2Mdxttb
A. I. Sadek, “On the stability of solutions of certain fourth order delay differential equation”, Applied mathematics and computation, vol. 148, no. 2, pp. 587-597, 2004, doi: 10.1016/S0096-3003(02)00925-6
A. S. C. Sinha, “On stability of solutions of some third and fourth order delay differential equations”, Information and control, vol. 23, no. 2, pp. 165-172, 1973, doi: 10.1016/S0019-9958(73)90651-7
H. O. Tejumola and B. Tchegnani, “Stability, boundedness and existence of periodic solutions of some third order and fourth-order nonlinear delay differential equations”, Journal of the Nigerian Mathematical Society, vol. 19, pp. 9-19, 2000.
C. Tunç, “A boundedness criterion for fourth order nonlinear ordinary differential equations with delay”, International journal of nonlinear science, vol. 6, no. 3, pp. 195-201, 2008. [On line]. Available; https://bit.ly/3ctxnYX
C. Tunç, “On stability of solutions of certain fourth order delay differential equations”, Applied mathematics and mechanics, vol. 27, pp. 1141-1148, 2006, doi: 10.1007/s10483-006-0815-y
C. Tunç, “On the uniform boundedness of solutions of some nonautonomous differential equations of the fourth order”, Applied mathematics and mechanics, vol. 20, pp. 622-628, 1999, doi: 10.1007/BF02464934
T. Yoshizawa, Stability theory by Liapunov’s second method. Tokyo: The Mathematical Society of Japan, 1966.
T. Yoshizawa, Stability theory and existence of periodic solutions and almost periodic solutions, New York, NY: Springer, 1975, doi: 10.1007/978-1-4612-6376-0
Downloads
Published
Issue
Section
License
Copyright (c) 2021 Adeleke Timothy Ademola
This work is licensed under a Creative Commons Attribution 4.0 International License.
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
