Stability and boundedness of solutions of certain Lienard-type non-autonomous differential equations

Authors

  • Abdullai Abdurasid Lagos State University of Science and Technology.
  • A. L. Olutimo Lagos State University of Science and Technology.
  • M. O. Omeike Federal University of Agriculture.

DOI:

https://doi.org/10.22199/issn.0717-6279-6155

Keywords:

Stability of solutions, Boundedness of solutions, Non-autonomous, Li´enard-type differential equations, Lyapunov’s direct method

Abstract

In this paper, the Liénard-type non-autonomous nonlinear differential equation

ẍ+ p(t)f(x, ẋ) ẋ + q(t)g(x) = ϕ(t, x, ẋ)

is investigated for uniform exponential asymptotic stability of solutions when ϕ(t, x, ẋ) ≡ 0, and uniform ultimate boundedness of solutions when ϕ(t, x, ẋ) ≠ 0, using the Lyapunov’s direct method

References

A. Y. Aleksandrov, "On the stability of the Liénard equation with unsteady perturbations," Differ. Uravn., vol. 32, no. 5, pp. 702 - 703, 1996.

T. A. Burton, "The generalized Liénard equation," J. Soc. Indust. Appl., vol. 3, pp. 223 - 230, 1965. https://doi.org/10.1137/0303018

T. A. Burton, "On the equation x + f(x)h(x )x+g(x) = e(t)," Ann. Mat. Pura Appl., vol. 85, pp. 277 - 285, 1970. https://doi.org/10.1007/BF02413538

T. A. Burton and B. Zhang, "Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation," Ann. Mat. Pura Appl., vol. 165, no. 4, pp. 351 - 368, 1993. https://doi.org/10.1007/BF01765856

G. Cantarelli, "On the stability of the origin of a nonautonomous Liénard equation," Boll. Un. Mat. Ital., vol. 10, no. 7, pp. 563 - 573, 1966.

L. A. Cerkas, "The degree of structural instability of the focus in the Liénard equation," Dokl. Akad. Nauk BSSR, vol. 23, no. 8, pp. 681 - 683, 1979.

L. Hatvani, "On the stability of the zero solution of nonlinear second order differential equations," Acta Sci. Math., vol. 57, pp. 367 - 371, 1993.

J. W. Heidel, "Global asymptotic stability of a generalized Liénard equation," SIAM J. App. Math., vol. 19, no. 3, pp. 629 - 636, 1970. https://doi.org/10.1137/0119061

J. W. Heidel, "A liapunov function for a generalized Liénard equation," J.Math. Anal. Appl., vol. 39, pp. 192 - 197, 1972.

https://doi.org/10.1016/0022-247X(72)90234-X

L. Jia, "Another elementary proof of the stability criterion of the Liénard and chipart," Quart. J. Math., vol. 14, no. 3, pp. 76 - 79, 1999.

J. Jifa, "The global stability of a class of second order differential equations," Nonlinear Anal., vol. 28, pp. 855 - 870, 1997. https://doi.org/10.1016/0362-546X(95)00187-Z

J. Sugie and Y. Amano, "Global asymptotic stability of non-autonomous systems of Liénard type," J. Math. Anal. Appl., vol. 289, no. 2, pp. 675 - 690, 2004. https://doi.org/10.1016/j.jmaa.2003.09.023

J. Kato, "A simple boundedness theorem for a Liénard equation with damp ing," Ann. Polon. Math., vol. 51, pp. 183 - 188, 1990. https://doi.org/10.4064/ap-51-1-183-188

E. Korkmaz and C. Tunc "Convergence to non-autonomous differential equa tions of second order," J. Egypt. Math. Soc., vol. 23, no. 1, pp. 27 - 30, 2015. https://doi.org/10.1016/j.joems.2014.02.009

J. L. Salle and S. Lafschetz, Stability by Liapunov's direct method with ap plications. New York: Academic Press, 1961.

F. Lin, "Stability and existence of periodic solutions and almost periodic solutions on Liénard systems," Ann. Differential Equations, vol. 13, no. 3, pp. 248 - 253, 1997.

W. S. Luk, "Some results concerning the boundedness of solutions of Liénard equations with delay," SIAM J. Appl. Math., vol. 30, no. 4, pp. 768 - 774, 1976. https://doi.org/10.1137/0130069

M. Muresan, "Boundedness of solutions for Liénard type equations," Mathe matica, vol. 62, no. 2, pp. 243 - 257, 1998.

F. Nakajima, "Ultimate boundedness of solutions for a generalized Liénard equation with forcing term," Tohoku Math. J., vol. 46, no. 3, pp. 295 - 310, 1994. https://doi.org/10.2748/tmj/1178225713

J. E. V. Napoles, "Ultimate boundedness of solutions for a generalized Liénard equation with forcing term," Rev. Un. Mat. Argentina, vol. 41, no. 4, pp. 47 - 59, 2001.

M. O. Omeike, K. D. Aduloju, and A. A. Abdurasid, "Convergence of so lutions of some second-order nonlinear differential equations," J. Nig. Math. Soc., vol. 41, no. 2, pp. 143 - 150, 2022.

J. Sugie, "On the boundedness of solutions of generalized Liénard equation without the signum condition," Nonlinear Anal., vol. 11, no. 12, pp. 1391 -1397, 1987. https://doi.org/10.1016/0362-546X(87)90091-5

J. Sugie, D. L. Chen, and H. Matsunaga, "On global asymptotic stability of systems of Liénard type," J. Math. Anal. Appl., vol. 219, no. 1, pp. 140 - 164, 1998. https://doi.org/10.1006/jmaa.1997.5773

J. Sugie and Y. Amano, "Global asymptotic stability of nonautonomous sys tems of Liénard type," J. Math. Anal. Appl., vol. 289, no. 2, pp. 673 - 690, 2004. https://doi.org/10.1016/j.jmaa.2003.09.023

C. Tunc and E. Tunc, "On the asymptotic behavior of solutions of cer tain second-order differential equations," Journal of the Franklin Institute, vol. 344, pp. 391 - 398, 2007. https://doi.org/10.1016/j.jfranklin.2006.02.011

O. Tunc, "Stability tests and solution estimates for non-linear differential equations," An International Journal of Optimization and Control: Theory & Applications, vol. 13, no. 1, pp. 92 - 103, 2023. https://doi.org/10.11121/ijocta.2023.1251

T. Wiandt, "On the boundedness of solutions of the vector Liénard equation," Dynam. Systems Appl., vol. 7, no. 1, pp. 141 - 143, 1998.

Q. G. Yang, "Boundedness and global asymptotic behavior of solutions to the Liénard equation," J. Systems Sci. Math. Sci., vol. 19, no. 2, pp. 673 - 690, 2004.

T. Yoshizawa, Stability theory by Liapunov's second method. Publications of the Mathematical Society of Japan, 1966.

B. Zhang, "Boundedness and stability of solutions of the retarded Liénard equation with negative damping," Nonlinear Anal., vol. 20, no. 3, pp. 303 -313, 2004. https://doi.org/10.1016/0362-546X(93)90165-O

J. Zhou and Z. R. Liu, "The global asymptotic behavior of solutions for a nonautonomous generalized Liénard system," J. Math. Res. Exposition, vol. 21, no. 3, pp. 410 - 414, 2001.

J. Zhou and L. Xiang, "On the stability and boundedness of solutions for the retarded Liénard-type equation," Ann. Differential Equations, vol. 15, no. 4, pp. 460 - 465, 1999.

Published

2024-06-17

How to Cite

[1]
A. Abdurasid, A. L. Olutimo, and M. O. Omeike, “Stability and boundedness of solutions of certain Lienard-type non-autonomous differential equations”, Proyecciones (Antofagasta, On line), vol. 43, no. 4, pp. 873-882, Jun. 2024.

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Artículos