# On the Global Existence and Boundedness of Solutions of a Certain Integro-Vector Differential Equation of Second Order

## DOI:

https://doi.org/10.5614/j.math.fund.sci.2018.50.1.1## Keywords:

Lyapunov functional, second order, integro-differential equation, global existence, boundedness## Abstract

In this paper, we consider a nonlinear integro-vector differential equation of the second order. We establish sufficient conditions that guarantee the global existence and boundedness of solutions of the equation considered. The method of proof involves constructing a suitable Lyapunov functional that gives meaningful results for the problem to be investigated. The result obtained is new and complements that found in the literature. We give an example to verify the result obtained and for illustration purposes. Using MATLAB-Simulink, the behaviors of the orbits of the equation considered are clearly shown.## References

Ahmad, S. & Rao, M.R.M., Theory of Ordinary Differential Equations: with Applications in Biology and Engineering, Affiliated East-West Press Pvt. Ltd., New Delhi, 1999.

Bellman, R., Stability Theory of Differential Equations, New York-Toronto-London, McGraw-Hill Book Company, Inc., 1953.

Burton, T.A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, 1985.

Graef, J.R. & Tun, C., Continuability and Boundedness of Multi-Delay Functional Integro-Differential Equations of the Second Order, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 109(1), pp. 169-173, 2015.

Hale, J., Sufficient Conditions for Stability and Instability of Autonomous Functional-Differential Equations, J. Differential Equations, 1(4), pp. 452-482, 1965.

Hara, T. & Yoneyama, T., On the Global Center of Generalized Li'enard Equation and Its Application to Stability Problems, Funkcial. Ekvac., 31(2), pp. 221-225, 1988.

Hsu, S-B., Ordinary Differential Equations with Applications, World Scientific Publishing Co. Pte. Ltd., 244, 2006.

Huang, L.H. & Yu, J.S., On Boundedness of Solutions of Generalized Lienard's System and Its Application, Ann. Differential Equations, 9(3), pp. 311-318, 1993.

Kato, J., On a Boundedness Condition for Solutions of a Generalized Lienard Equation, J. Differential Equations, 65(2), pp. 269-286, 1986.

Kolmanovskii V. & Myshkis, A., Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.

Krasovskii, N.N., Stability of Motion, Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Stanford University Press, Stanford, California, 1963.

Lyapunov, A.M., Stability of Motion, Academic Press, London, 1966.

Mustafa, G.O. & Rogovchenko, V.Y., Global Existence of Solutions with Prescribed Asymptotic Behavior for Second-Order Nonlinear Differential Equations, Nonlinear Anal., 51, pp. 339-368, 2002.

Valdes, J.E.N., A Note on the Boundedness of an Integro-Differential Equation, Quaest. Math., 24(2), pp. 213-216, 2001.

Qian, C.X., Boundedness and Asymptotic Behaviour of Solutions of a Second-Order Nonlinear System, Bull. London Math. Soc., 24(3), pp. 281-288, 1992.

Sugie, J. & Amano, Y., Global Asymptotic Stability of Non-Autonomous Systems of Lienard Type, J. Math. Anal. Appl., 289(2), pp. 673-690, 2004.

Tun, C., Some New Stability and Boundedness Results on the Solutions of the Nonlinear Vector Differential Equations of Second Order, Iranian Journal of Science & Technology, Transaction A, 30(A2), pp. 213-221, 2006.

Tun, C., Stability and Boundedness of Solutions of Non-Autonomous Differential Equations of Second Order, J. Comput. Anal. Appl., 13(6), pp. 1067-1074, 2011.

Tun, C., Properties of Solutions to Volterra Integro-Differential Equations with Delay, Appl. Math. Inf. Sci., 10(5), pp. 1775-1780, 2016.

Tun, C., Instability of Solutions of Vector Lienard Equation with Constant Delay, Bull. Math. Soc. Sci. Math. Roumanie, 59(107), No. 2, pp. 197-204, 2016.

Tun, C., Qualitative Properties in Nonlinear Volterra Integro-Differential Equations with Delay, Journal of Taibah University for Science, 11(2), pp. 309-314, 2017.

Tun, C. & Sevli, H., Stability and Boundedness Properties of Certain Second-Order Differential Equations, J. Franklin Inst., 344(5), pp. 399-405, 2007.

Tun, C. & Din, Y., Qualitative Properties of Certain Non-linear Differential Systems of Second Order, Journal of Taibah University for Science, 11(2), pp.359-366, 2017.

Tun, C. & Tun, O., A Note on Certain Qualitative Properties of A Second Order Linear Differential System, Appl. Math. Inf. Sci., 9(2), pp. 953-956, 2015.

Tun, C. & Tun, O., On the Boundedness and Integration of Non-Oscillatory Solutions of Certain Linear Differential Equations of Second Order, Journal of Advanced Research, 7(1), pp. 165-168, 2016.

Yang, X.S., A Boundedness Theorem on Higher-Dimensional Hill Equations, Math. Inequal. Appl., 2(3), pp. 363-365, 1999.

Yoshizawa, T., Stability Theory by Liapunov's Second Method. Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966.

Zhou, J. & Liu, Z.R., The Global Asymptotic Behavior of Solutions for a Non-autonomous Generalized Lienard System, J. Mat. Res. Exposition, 21(3), pp. 410-414, 2001.

Wiandt, T., On the Boundedness of Solutions of the Vector Lienard Equation, Dynam. Systems Appl., 7(1), pp. 141-143, 1998.

Mirsky, L., An Introduction to Linear Algebra, Dover Publications, Inc., New York, 1990.