On uniform-ultimate boundedness and periodicity results of solutions to certain second order non-linear vector differential equations
Keywords:second order, ultimate boundedness, periodicity, Lyapunov function
In this paper, we employ the second method of Lyapunov to examine sufficient conditions for the uniform-ultimate boundedness of solutions and existence of at least one periodic solution to the following second order vector differential equation:
Ẍ+ F(X, Ẋ ) Ẋ + H(X) = P(t, X, Ẋ ),
when the non-linear term H(X) is: (i) differentiable, (ii) non-necessarily differentiable. The results contain in this paper are new and complement related ones in the literature.
A. M. A Abou-El-Ela and A. I. Sadek, “A stability result for the solutions of a certain system of fourth order differential equations”, Annals of Differential Equations, vol. 6, no. 2, pp. 131-141, 1990.
A.T. Ademola, “Boundedness and Stability of Solutions to Certain Second Order Differential Equations”, Differential Equations and Control Processes, vol. 3, pp. 38-50, 2015.
A. A. Adeyanju, “Existence of a limiting regime in the sense of demidovic for a certain class of second order non-linear vector differential equation”, Differential Equation and Control Processes, no. 4, pp. 63-79, 2018.
A.A. Adeyanju and D.O. Adams, “Some new results on the stability and boundedness of solutions of certain class of second order vector differential equations”, International Journal of Mathematical Analysis and Optimization: Theory and Applications, vol. 7, no. 1, pp. 108-115, 2021. https://doi.org/10.52968/28305999
A. A. Adeyanju, “Stability and Boundedness Criteria of Solutions of a Certain System of Second Order Differential Equations”, Annali Dell’Universita’ di Ferrara, vol. 69, no. 1, pp. 81-93, 2023.
A.U. Afuwape and M.O. Omeike, “Convergence of solutions of certain system of third order non-linear ordinary differential equations”, Annals of Differential Equations, vol. 21, pp. 533-540, 2005.
A.U. Afuwape, “Ultimate Boundedness Results for a Certain System of Third-Order Non-Linear Differential Equations”, Journal of Mathematical Analysis and Applications, vol. 97, pp. 140-150, 1983. https://doi.org/10.1016/0022-247X(83)90243-3
A.U. Afuwape and M.O. Omeike, Convergence of solutions of some system of second order non-linear ordinary differential equations. In Appreciating Mathematics in Contemporary World, Proceedings of the International Conference in Honour of Professor E.O. Oshobi and Dr. J. O. Amao, A. U. Afuwape and Shola Adeyemi (Eds.), 2005.
S. Ahmad and M. Rama Mohana Rao, Theory of Ordinary Differential Equations. With Applications in Biology and Engineering. New Delhi: East-West Press Pvt. Ltd., 1999.
J. G. Alaba and B.S. Ogundare, “On stability and boundedness properties of solutions of certain second order non-autonomous non-linear ordinary differential equation, “Kragujevac Journal of Mathematics, vol. 39, no. 2, pp. 255-266, 2015. https://doi.org/10.5937/KgJMath1502255A
J. Awrejcewicz, Ordinary Differential Equations and Mechanical Systems. Cham: Springer, 2014.
A. Baliki, M. Benchohra and J.R. Graef, “Global existence and stability for second order functional evolution equations with infinite delay”, Electronic Journal of Qualitative Theory of Differential Equations, vol. 2016, no. 23, pp. 10, 2016. https://doi.org/10.14232/ejqtde.2016.1.23
M. L. Cartwright and J. E. Littlewood, “On nonlinear differential equations of the second order“, Annals of Mathematics, vol. 48, pp. 472-494, 1947. https://doi.org/10.2307/1969181
C. Chicone, Ordinary Differential Equations with Applications. Texts in Applied Mathematics, vol. 34. New York: Springer, 1999.
J. O. C. Ezeilo, “On the existence of almost periodic solutions of some dissipative second order differential equations”, Annali di Matematica Pura ed Applicata, vol. 65, no. 4, pp. 389-406, 1964. https://doi.org/10.1007/BF02416464
J. O. C. Ezeilo, “On the convergence of solutions of certain system of second order differential equations”, Annali di Matematica Pura ed Applicata, vol. 72, no. 4, pp. 239-252, 1966. https://doi.org/10.1007/BF02414336
J. O. C. Ezeilo and H.O. Tejumola, “Boundedness and Periodicity of Solutions of a Certain Systems of Third Order Nonlinear Differential Equations”, Annali di Matematica Pura ed Applicata, vol. 74, pp. 283-316, 1966. https://doi.org/10.1007/BF02416460
J. O. C. Ezeilo, “n-Dimensional extensions of boundedness and stability theorem for some third order differential equation”, Journal of Mathematical Analysis and Applications, vol. 18, pp. 395-416, 1967. https://doi.org/10.1016/0022-247X(67)90035-2
G. A. Grigoryan, “Boundedness and stability criteria for linear ordinary differential equations of the second order”, Russian Mathematics, vol. 57, no. 12, pp. 8-15, 2013. https://doi.org/10.3103/S1066369X13120025
J. Hale, “Sufficient conditions for stability and instability of autonomous functional-differential equations”, Journal of Differential Equations, vol. 1, pp. 452-482, 1965. https://doi.org/10.1016/0022-0396(65)90005-7
D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: Problems and Solutions. A Sourcebook for Scientists and Engineers. Oxford: Oxford University Press, 2007.
W. S. Loud, “Boundedness and convergence of solutions of x ̈ + cx ̇ + g(x) = e(t)”, Duke Mathematical Journal, vol. 24, pp. 63-72, 1957. https://doi.org/10.1215/S0012-7094-57-02412-2
F. W. Meng, “Ultimate boundedness results for a certain system of third order non-linear differential equations”, Journal of Mathematical Analysis and Applications, vol. 177, pp. 496-509, 1993.
B. S. Ogundare, A.T. Ademola, M. O. Ogundiran and O.A. Adesina, “On the qualitative behaviour of solutions to certain second order nonlinear differential equation with delay”, Annali dell’Universita’ di Ferrara, 2016. https://doi.org/10.1007/s11565-016-0262-y
M. O. Omeike, O.O. Oyetune and A.L. Olutimo, “Boundedness of solutions of a certain system of second-order ordinary differential equations”, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, vol. 53, pp. 107-115, 2014.
M. O. Omeike, A.A. Adeyanju and D.O. Adams, “Stability and Boundedness of Solutions of Certain Vector Delay Differential Equations”, Journal of the Nigerian Mathematical Society, vol. 37, no. 2, pp. 77-87, 2018.
M. O. Omeike, A.A. Adeyanju, D.O. Adams and A.L. Olutimo, “Boundedness of Certain System of Second Order Differential Equations”, Kragujevac Journal of Mathematics, vol. 45, no. 5, pp. 787-796, 2021.
R. Reissig, G. Sansone and R. Conti, Non-linear Differential Equations of Higher Order. Leyden: Noordhoff, 1974. Translated from the German
A. I. Sadek, “On the stability of a nonhomogeneous vector differential equations of the fourth-order”, Applied Mathematics and Computation, vol. 150, pp. 279-289, 2004. https://doi.org/10.1016/S0096-3003(03)00227-3
H.O. Tejumola, “On the Boundedness and periodicity of solutions of certain third-order nonlinear differential equation”, Annali di Matematica Pura ed Applicata, vol. 83, no. 4, pp. 195-212, 1969. https://doi.org/10.1007/BF02411167
H.O. Tejumola, “Boundedness Criteria for solutions of some second-order differential equations”, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, vol. 50, no. 8, pp. 432-437, 1971.
H.O. Tejumola, “On a Lienard type matrix differential equations”, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, vol. 60, no. 8, pp. 100-107, 1976.
C. Tunç., “On the stability and boundedness of solutions of nonlinear vector differential equations of third order”, Nonlinear Analysis, vol. 70, no. 6, pp. 2232-2236, 2009. https://doi.org/10.1016/j.na.2008.03.002
C. Tunç and O. Tunç, “A note on the stability and boundedness of solutions to non-linear differential systems of second order”, Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 24, pp. 169-175, 2017. https://doi.org/10.1016/j.jaubas.2016.12.004
C. Tunc and S.A. Mohammed, “On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order”, Advances in Difference Equations , vol. 2019, no. 461, 2019. https://doi.org/10.1186/s13662-019-2384-x
T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. New York: Springer-Verlag, 1978.
T. Yoshizawa, Stability theory by Liapunov’s second method. Japan: Publications of the Mathematical Society of Japan, 1966.
T. Yoshizawa, “Liapunov’s function and boundedness of solutions”, Funkcialaj Ekvacioj, vol. 2, pp. 71-103, 1958.
T. Yoshizawa, Stability theory by Liapunov’s second method. The Mathematical Society of Japan, 1966.
M. J. Zainab, “Bounded Solution of the Second Order Differential Equation ẍ+ f(x) ẋ + g(x) = u(t), Baghdad Science Journal, vol. 12, no. 4, pp. 822-825, 2015. https://doi.org/10.21123/bsj.12.4.822-825
How to Cite
Copyright (c) 2023 Adetunji Adeyanju
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.