Research Article
Stability and boundedness of solutions of nonlinear fourth order differential equations with bounded delay
Abstract
In this paper, we determine sufficient conditions for the
boundedness, uniformly asymtotically stability of the solutions to a certain
fourth-order non-autonomous differential equations with bounded delay by
considering second method of Lyapunov. The results obtain essentially improve,
include and complement the consequences in the current literature.
References
- [1] A. M. A. Abou-Ela-Ela and A. I. Sadek, "A stability result for certain fourth order differential equations," Ann. Differential Equations, vol. 1, no. 6, pp. 1-9, 1990. [2] O. A. Adesina and B. S. Ogundare, "Some new stability and boundedness results on a certainfourth order nonlinear differential equation.," Nonlinear Studies, vol. 3, no. 19, pp. 359-369, 2012. [3] H. Bereketoğlu, "Asymptotic stability in a fourth order delay differential equation.," Dynam. Systems Appl, vol. 1, no. 7, pp. 105-115, 1998. [4] T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Mathematics in science and engineering, 1985. [5] L. 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. [6] J. O. C. Ezeilo and H. O. Tejumola, "On the boundedness and the stability properties of solutions of certain fourth order differential equations," Ann. Mat. Pura Appl., vol. 4, no. 95, pp. 131-145, 1973. [7] J. O. C. Ezeilo, "Stability results for the solutions of some third and fourth order differential equations," Ann. Mat. Pura Appl., vol. 4, no. 66, pp. 233-249, 1964. [8] T. Hara, "On the asymptotic behavior of the solutions of some third and fourth order nonautonomous differential equations," Publ. RIMS, Kyoto Univ., no. 9, pp. 649-673, 1974. [9] M. Harrow, "Further results on the boundedness and the stability of solutions of some differential equations of the fourth order," SIAM J. Math. Anal., no. 1, pp. 189-194, 1970. [10] M. Harrow, "A Stability result for Solutions of a Certain Fourth Order Homogeneous Differential Equations," J. London Math. Soc., no. 42, pp. 51-56, 1967. [11] E. Korkmaz and C. Tunç, "On some qualitative behaviors of certain differential equations of fourth order with multiple retardations," Journal of Applied Analysis and Computation, vol. 2, no. 6, pp. 336-349, 2016. [12] N. N. Krasovskii, "On the stability in the large of the solution of a nonlinear system of differential equations (Russian)," Prikl. Mat. Meh., no. 18, pp. 735-737, 1954. [13] 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., London: Taylor & Francis, Ltd., 1992. [14] P. S. M. Omeike, "Boundedness Of Solutions To Fourth-Order Differential Equation With Oscillatory Restoring And Forcing Terms," Electronic Journal of Differential Equations, no. 104, pp. 1-8, 2007. [15] 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 Issue, no. 17-30, 2015. [16] R. Reissig, G. Sansone and R. Conti, Non-linear Differential Equations of Higher Order. Translated from the German., Leyden: Noordhoff International Publishing, 1974. [17] M. Remili and M. Rahmane, "Boundedness and square integrability of solutions of nonlinear fourth-order differential equations," Nonlinear Dynamics and Systems Theory, vol. 2, no. 16, pp. 192-205, 2016. [18] M. Remili and M. Rahmane, "Sufficient conditions for the boundedness and square integrability of solutions of fourth-order differential equations," Proyecciones Journal of Mathematics., vol. 1, no. 35, pp. 41-61, 2016. [19] A. Shair, "Asymptotic properties of linear fourth order differential equations," American mathematical society, vol. 1, no. 59, pp. 45-51, 1976. [20] A. S. C. Sinha, "On stability of solutions of some third and fourth order delay-differential equations," Information and Control, no. 23, pp. 165-172, 1973. [21] H. O. Tejumola, "Further results on the boundedness and the stability of certain fourth order differential equations," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 8, no. 52, pp. 16-23, 1972. [22] C. Tunç, "Some remarks on the stability and boundedness of solutions of certain differential equations of fourth-order," Computational and Applied Mathematics, vol. 1, no. 26, pp. 1-17, 2007. [23] C. Tunç, "Some stability and boundedness results for the solutions of certain fourth order differential equations," Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., no. 44, pp. 161-171, 2005. [24] C. Tunç, "Some stability results for the solutions of certain fourth order delay differential equations," Differential Equations and Applications, no. 4, pp. 165-174, 2004. [25] C. Tunç, "Stability and boundedness of solutions to certain fourth order differential equations," Electron. J. Differential Equations, no. 35, p. 10pp., 2006. [26] C. Tunç, "Stability and boundedness of solutions to certain fourth order differential equations," Electronic Journal of Differential Equations, no. 35, pp. 1-10, 2006. [27] X. Wu and K. Xiong, "Remarks on stability results for the solutions of certain fourth-order autonomous differential equations," Internat. J. Control, vol. 2, no. 69, p. 353, 1998. [28] T. Yoshizawa, Stability Theory by Liapunovs Second Method, Tokyo,: The Mathematical Society of Japan, 1966.
Dördüncü mertebeden sınırlı gecikmeli nonlineer diferansiyel denklemlerin çözümlerinin kararlılığı ve sınırlılığı
Abstract
Bu makalede Lyapunov’un ikinci metodu kullanılarak dördüncü
mertebeden otonom olmayan değişken gecikmeli diferansiyel denklemlerin
çözümlerinin düzgün asimptotik kararlılığı ve sınırlılığı için yeterli şartları
veririz. Elde edilen sonuçlar literatürdeki sonuçları tamamlar, kapsar ve
geliştirir.
Keywords
Kararlılık Sınırlılık Lyapunov fonksiyonu Gecikmeli diferansiyel denklemler Dördüncü mertebe
References
- [1] A. M. A. Abou-Ela-Ela and A. I. Sadek, "A stability result for certain fourth order differential equations," Ann. Differential Equations, vol. 1, no. 6, pp. 1-9, 1990. [2] O. A. Adesina and B. S. Ogundare, "Some new stability and boundedness results on a certainfourth order nonlinear differential equation.," Nonlinear Studies, vol. 3, no. 19, pp. 359-369, 2012. [3] H. Bereketoğlu, "Asymptotic stability in a fourth order delay differential equation.," Dynam. Systems Appl, vol. 1, no. 7, pp. 105-115, 1998. [4] T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Mathematics in science and engineering, 1985. [5] L. 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. [6] J. O. C. Ezeilo and H. O. Tejumola, "On the boundedness and the stability properties of solutions of certain fourth order differential equations," Ann. Mat. Pura Appl., vol. 4, no. 95, pp. 131-145, 1973. [7] J. O. C. Ezeilo, "Stability results for the solutions of some third and fourth order differential equations," Ann. Mat. Pura Appl., vol. 4, no. 66, pp. 233-249, 1964. [8] T. Hara, "On the asymptotic behavior of the solutions of some third and fourth order nonautonomous differential equations," Publ. RIMS, Kyoto Univ., no. 9, pp. 649-673, 1974. [9] M. Harrow, "Further results on the boundedness and the stability of solutions of some differential equations of the fourth order," SIAM J. Math. Anal., no. 1, pp. 189-194, 1970. [10] M. Harrow, "A Stability result for Solutions of a Certain Fourth Order Homogeneous Differential Equations," J. London Math. Soc., no. 42, pp. 51-56, 1967. [11] E. Korkmaz and C. Tunç, "On some qualitative behaviors of certain differential equations of fourth order with multiple retardations," Journal of Applied Analysis and Computation, vol. 2, no. 6, pp. 336-349, 2016. [12] N. N. Krasovskii, "On the stability in the large of the solution of a nonlinear system of differential equations (Russian)," Prikl. Mat. Meh., no. 18, pp. 735-737, 1954. [13] 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., London: Taylor & Francis, Ltd., 1992. [14] P. S. M. Omeike, "Boundedness Of Solutions To Fourth-Order Differential Equation With Oscillatory Restoring And Forcing Terms," Electronic Journal of Differential Equations, no. 104, pp. 1-8, 2007. [15] 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 Issue, no. 17-30, 2015. [16] R. Reissig, G. Sansone and R. Conti, Non-linear Differential Equations of Higher Order. Translated from the German., Leyden: Noordhoff International Publishing, 1974. [17] M. Remili and M. Rahmane, "Boundedness and square integrability of solutions of nonlinear fourth-order differential equations," Nonlinear Dynamics and Systems Theory, vol. 2, no. 16, pp. 192-205, 2016. [18] M. Remili and M. Rahmane, "Sufficient conditions for the boundedness and square integrability of solutions of fourth-order differential equations," Proyecciones Journal of Mathematics., vol. 1, no. 35, pp. 41-61, 2016. [19] A. Shair, "Asymptotic properties of linear fourth order differential equations," American mathematical society, vol. 1, no. 59, pp. 45-51, 1976. [20] A. S. C. Sinha, "On stability of solutions of some third and fourth order delay-differential equations," Information and Control, no. 23, pp. 165-172, 1973. [21] H. O. Tejumola, "Further results on the boundedness and the stability of certain fourth order differential equations," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., vol. 8, no. 52, pp. 16-23, 1972. [22] C. Tunç, "Some remarks on the stability and boundedness of solutions of certain differential equations of fourth-order," Computational and Applied Mathematics, vol. 1, no. 26, pp. 1-17, 2007. [23] C. Tunç, "Some stability and boundedness results for the solutions of certain fourth order differential equations," Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., no. 44, pp. 161-171, 2005. [24] C. Tunç, "Some stability results for the solutions of certain fourth order delay differential equations," Differential Equations and Applications, no. 4, pp. 165-174, 2004. [25] C. Tunç, "Stability and boundedness of solutions to certain fourth order differential equations," Electron. J. Differential Equations, no. 35, p. 10pp., 2006. [26] C. Tunç, "Stability and boundedness of solutions to certain fourth order differential equations," Electronic Journal of Differential Equations, no. 35, pp. 1-10, 2006. [27] X. Wu and K. Xiong, "Remarks on stability results for the solutions of certain fourth-order autonomous differential equations," Internat. J. Control, vol. 2, no. 69, p. 353, 1998. [28] T. Yoshizawa, Stability Theory by Liapunovs Second Method, Tokyo,: The Mathematical Society of Japan, 1966.
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Details
| Subjects | Mathematical Sciences |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | December 1, 2017 |
| Submission Date | April 24, 2017 |
| Acceptance Date | August 1, 2017 |
| Published in Issue | Year 2017 Volume: 21 Issue: 6 |
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