Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay

Erdal Korkmaz; Abdulhamit Özdemir

doi:10.33187/jmsm.1080204

Research Article

Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay

Year 2022, Volume: 5 Issue: 2, 40 - 47, 31.08.2022

Erdal Korkmaz Abdulhamit Özdemir

https://doi.org/10.33187/jmsm.1080204

Abstract

In this paper, we investigate the boundedness and uniformly asymptotically stability of the solutions to a certain third order non-autonomous differential equations with bounded delay. By constructing a Lyapunov functional, sufficient conditions for the stability and boundedness of solutions for equations considered are obtained. We used an example to demonstrate the feasibility of our results. The results essentially improve, include, and complement the results in the literature.

Keywords

Boundedness, Delay differential equations, Lyapunov functional, Stability, Third order

References

[1] V. B. Kolmanovskii, V. R. Nosov, Stability of functional-differential equations, Mathematics in Science and Engineering, 180, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986.
[2] V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
[3] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Mathematics in Science and Engineering, Vol. 178, Academic Press, Orlando, 1985.
[4] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Heidelberg, 1977.
[5] R. Reissig, G. Sansone, R. Conti, Non-Linear Differential Equations of Higher Order, Noordhoff International Publishing, Leyden, 1974.
[6] L. E. Elsgolts, Introduction to the Theory of Differential Equations with Deviating Arguments, Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966.
[7] L. E. Elsgolts, S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Translated from the Russian by John L. Casti. Mathematics in Science and Engineering, Vol. 105. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, 1973.
[8] N. N. Krasovskii, Stability of Motion, Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Translated by J. L. Brenner Stanford University Press, Stanford, Calif. 1963.
[9] A. M. Lyapunov, The General Problem of the Stability of Motion, Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Taylor & Francis, Ltd., London, 1992.
[10] J. O. C. Ezeilo, On the stability of solutions of certain differential equations of the third order, Quart. J. Math. Oxford Ser., 11(2) (1960), 64-69.
[11] K. Swick, On the boundedness and the stability of solutions of some nonautonomous differential equations of the third order, J. London Math. Soc., 44 (1969), 347-359.
[12] K. E. Swick, Asymptotic behavior of the solutions of certain third order differential equations, SIAM J. Appl. Math. 19 (1970), 96-102.
[13] T. Hara, On the asymptotic behavior of solutions of certain of certain third order ordinary differential equations, Proc. Japan Acad., 47 (1971), 903-908.
[14] H. O. Tejumola, A note on the boundedness and the stability of solutions of certain third-order differential equations, Ann. Mat. Pura Appl., 92(4) (1972), 65-75.
[15] T. Hara, On the asymptotic behavior of the solutions of some third and fourth order non-autonomous differential equations, Publ. Res. Inst. Math. Sci., 9(74) (1973), 649-673.
[16] T. Hara, On the asymptotic behavior of solutions of certain non-autonomous differential equations, Osaka J. Math., 12 (1975), 267-282.
[17] T. Hara, On the uniform ultimate boundedness of the solutions of certain third order differential equations, J. Math. Anal. Appl., 80 (1981), 533-544.
[18] Y. F. Zhu, On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system, Ann. Differential Equations, 8(2) (1992), 249-259.
[19] C. Qian, On global stability of third-order nonlinear differential equations, Nonlinear Anal. 42 (2000), 651-661.
[20] M. O. Omeike, Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (NS), 55(1) (2009), 49-58.
[21] M. Remili, L. D. Oudjedi, Stability and boundedness of the solutions of nonautonomous third order differential equations with delay, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 53 (2014), 139-147.
[22] C. Tunç, On the asymptotic behavior of solutions of certain third-order nonlinear differential equations, J. Appl. Math. Stoch. Anal., 1 (2005), 29-35.
[23] C. Tunç, Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait J. Sci. Engrg., 32 (2005), 39-48.
[24] C. Tunç, Boundedness of solutions of a third-order nonlinear differential equation, J. Inequal. Pure Appl. Math., 6(1) (2005), Article 3, 1-6.
[25] B. S. Ogundare, G. E. Okecha, On the boundedness and the stability of solution to third order non-linear differential equations, Ann. Differential Equations, 24 (2008), 1-8.
[26] C. Tunç, The boundedness of solutions to nonlinear third order differential equations, Nonlinear Dyn. Syst. Theory, 10 (2010), 97-102.
[27] A. T. Ademola, P. O. Arawomo, Asymptotic behaviour of solutions of third order nonlinear differential equations, Acta Univ. Sapientiae Math., 3 (2011), 197-211.
[28] L. Zhang, L. Yu, Global asymptotic stability of certain third-order nonlinear differential equations, Math. Methods Appl. Sci., 36 (2013), 1845-1850.
[29] M. Remili, D. Beldjerd, On the asymptotic behavior of the solutions of third order delay differential equations, Rend. Circ. Mat. Palermo, 63(2) (2014), 447-455.
[30] L. Oudjedi, D. Beldjerd, M. Remili, On the stability of solutions for nonautonomous delay differential equations of third-order, Differential Equations and Control Processes, 2014 (2014), 22-34.
[31] E. I. Verriest, A. Woihida, Stability of nonlinear differential delay systems, Math. Comput. Simul., 45(3-4), (1998), 257-267.
[32] J. R. Graef, D. L. Oudjedi, M. Remili, Stability and square integrability of solutions of nonlinear third order differential equations, Dyn. Continuous Discrete Impulsive Syst. Ser. A: Math. Anal., 22 (2015), 313-324.
[33] J. R. Graef, D. Beldjerd, M. Remili, On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay, Panam. Math. J., 25 (2015), 82-94.

Year 2022, Volume: 5 Issue: 2, 40 - 47, 31.08.2022

Erdal Korkmaz Abdulhamit Özdemir

https://doi.org/10.33187/jmsm.1080204

Abstract

References

[1] V. B. Kolmanovskii, V. R. Nosov, Stability of functional-differential equations, Mathematics in Science and Engineering, 180, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986.
[2] V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.
[3] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Mathematics in Science and Engineering, Vol. 178, Academic Press, Orlando, 1985.
[4] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Heidelberg, 1977.
[5] R. Reissig, G. Sansone, R. Conti, Non-Linear Differential Equations of Higher Order, Noordhoff International Publishing, Leyden, 1974.
[6] L. E. Elsgolts, Introduction to the Theory of Differential Equations with Deviating Arguments, Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966.
[7] L. E. Elsgolts, S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Translated from the Russian by John L. Casti. Mathematics in Science and Engineering, Vol. 105. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, 1973.
[8] N. N. Krasovskii, Stability of Motion, Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Translated by J. L. Brenner Stanford University Press, Stanford, Calif. 1963.
[9] A. M. Lyapunov, The General Problem of the Stability of Motion, Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Taylor & Francis, Ltd., London, 1992.
[10] J. O. C. Ezeilo, On the stability of solutions of certain differential equations of the third order, Quart. J. Math. Oxford Ser., 11(2) (1960), 64-69.
[11] K. Swick, On the boundedness and the stability of solutions of some nonautonomous differential equations of the third order, J. London Math. Soc., 44 (1969), 347-359.
[12] K. E. Swick, Asymptotic behavior of the solutions of certain third order differential equations, SIAM J. Appl. Math. 19 (1970), 96-102.
[13] T. Hara, On the asymptotic behavior of solutions of certain of certain third order ordinary differential equations, Proc. Japan Acad., 47 (1971), 903-908.
[14] H. O. Tejumola, A note on the boundedness and the stability of solutions of certain third-order differential equations, Ann. Mat. Pura Appl., 92(4) (1972), 65-75.
[15] T. Hara, On the asymptotic behavior of the solutions of some third and fourth order non-autonomous differential equations, Publ. Res. Inst. Math. Sci., 9(74) (1973), 649-673.
[16] T. Hara, On the asymptotic behavior of solutions of certain non-autonomous differential equations, Osaka J. Math., 12 (1975), 267-282.
[17] T. Hara, On the uniform ultimate boundedness of the solutions of certain third order differential equations, J. Math. Anal. Appl., 80 (1981), 533-544.
[18] Y. F. Zhu, On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system, Ann. Differential Equations, 8(2) (1992), 249-259.
[19] C. Qian, On global stability of third-order nonlinear differential equations, Nonlinear Anal. 42 (2000), 651-661.
[20] M. O. Omeike, Stability and boundedness of solutions of some non-autonomous delay differential equation of the third order, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (NS), 55(1) (2009), 49-58.
[21] M. Remili, L. D. Oudjedi, Stability and boundedness of the solutions of nonautonomous third order differential equations with delay, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 53 (2014), 139-147.
[22] C. Tunç, On the asymptotic behavior of solutions of certain third-order nonlinear differential equations, J. Appl. Math. Stoch. Anal., 1 (2005), 29-35.
[23] C. Tunç, Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations, Kuwait J. Sci. Engrg., 32 (2005), 39-48.
[24] C. Tunç, Boundedness of solutions of a third-order nonlinear differential equation, J. Inequal. Pure Appl. Math., 6(1) (2005), Article 3, 1-6.
[25] B. S. Ogundare, G. E. Okecha, On the boundedness and the stability of solution to third order non-linear differential equations, Ann. Differential Equations, 24 (2008), 1-8.
[26] C. Tunç, The boundedness of solutions to nonlinear third order differential equations, Nonlinear Dyn. Syst. Theory, 10 (2010), 97-102.
[27] A. T. Ademola, P. O. Arawomo, Asymptotic behaviour of solutions of third order nonlinear differential equations, Acta Univ. Sapientiae Math., 3 (2011), 197-211.
[28] L. Zhang, L. Yu, Global asymptotic stability of certain third-order nonlinear differential equations, Math. Methods Appl. Sci., 36 (2013), 1845-1850.
[29] M. Remili, D. Beldjerd, On the asymptotic behavior of the solutions of third order delay differential equations, Rend. Circ. Mat. Palermo, 63(2) (2014), 447-455.
[30] L. Oudjedi, D. Beldjerd, M. Remili, On the stability of solutions for nonautonomous delay differential equations of third-order, Differential Equations and Control Processes, 2014 (2014), 22-34.
[31] E. I. Verriest, A. Woihida, Stability of nonlinear differential delay systems, Math. Comput. Simul., 45(3-4), (1998), 257-267.
[32] J. R. Graef, D. L. Oudjedi, M. Remili, Stability and square integrability of solutions of nonlinear third order differential equations, Dyn. Continuous Discrete Impulsive Syst. Ser. A: Math. Anal., 22 (2015), 313-324.
[33] J. R. Graef, D. Beldjerd, M. Remili, On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay, Panam. Math. J., 25 (2015), 82-94.

There are 33 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Erdal Korkmaz 0000-0002-6647-9312 Abdulhamit Özdemir 0000-0002-5310-6285
Publication Date	August 31, 2022
Submission Date	February 28, 2022
Acceptance Date	April 4, 2022
Published in Issue	Year 2022 Volume: 5 Issue: 2

Cite

APA	Korkmaz, E., & Özdemir, A. (2022). Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay. Journal of Mathematical Sciences and Modelling, 5(2), 40-47. https://doi.org/10.33187/jmsm.1080204
AMA	Korkmaz E, Özdemir A. Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay. Journal of Mathematical Sciences and Modelling. August 2022;5(2):40-47. doi:10.33187/jmsm.1080204
Chicago	Korkmaz, Erdal, and Abdulhamit Özdemir. “Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations With Bounded Delay”. Journal of Mathematical Sciences and Modelling 5, no. 2 (August 2022): 40-47. https://doi.org/10.33187/jmsm.1080204.
EndNote	Korkmaz E, Özdemir A (August 1, 2022) Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay. Journal of Mathematical Sciences and Modelling 5 2 40–47.
IEEE	E. Korkmaz and A. Özdemir, “Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay”, Journal of Mathematical Sciences and Modelling, vol. 5, no. 2, pp. 40–47, 2022, doi: 10.33187/jmsm.1080204.
ISNAD	Korkmaz, Erdal - Özdemir, Abdulhamit. “Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations With Bounded Delay”. Journal of Mathematical Sciences and Modelling 5/2 (August 2022), 40-47. https://doi.org/10.33187/jmsm.1080204.
JAMA	Korkmaz E, Özdemir A. Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay. Journal of Mathematical Sciences and Modelling. 2022;5:40–47.
MLA	Korkmaz, Erdal and Abdulhamit Özdemir. “Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations With Bounded Delay”. Journal of Mathematical Sciences and Modelling, vol. 5, no. 2, 2022, pp. 40-47, doi:10.33187/jmsm.1080204.
Vancouver	Korkmaz E, Özdemir A. Stability and Boundedness of Solutions of Nonlinear Third Order Differential Equations with Bounded Delay. Journal of Mathematical Sciences and Modelling. 2022;5(2):40-7.