On the stability and boundedness of certain third order non-autonomous differential equations of retarded type


  • Cemil Tunc Yuzuncu Yil University.




Boundedness, Stability, Non-autonomous, Retarded.


In this paper, based on the Lyapunov-Krasovskii functional approach, we obtain sufficient conditions which guarantee stability, uniformly stability, boundedness and uniformly boundedness of solutions of certain third order non- autonomous differential equations of retarded type. Our results complement and improve some recent ones.

Author Biography

Cemil Tunc, Yuzuncu Yil University.

Department of Mathematics, Faculty of Sciences.


[1] Adams, D. O.; Omeike, M. O.; Mewomo, O. T.; Olusola, I. O., Boundedness of solutions of some third order non-autonomous ordinary differential equations. J. Nigerian Math. Soc. 32, pp. 229-240, (2013).

[2] Ademola, T. A.; Arawomo,P. O., Asymptotic behaviour of solutions of third order nonlinear differential equations. Acta Univ. Sapientiae Math. 3, No. 2, pp. 197-211, (2011).

[3] Afuwape, A. U.; Adesina, O. A., On the bounds for mean-values of solutions to certain third-order non-linear differential equations. Fasc. Math. No. 36, pp. 5-14, (2005).

[4] Ahmad, S.; Rama Mohana Rao, M., Theory of ordinary differential equations. With applications in biology and engineering. Affiliated East-West Press Pvt. Ltd., New Delhi, (1999).

[5] Bai, Yuzhen; Guo, C., New results on stability and boundedness of third order nonlinear delay differential equations. Dynam.Systems Appl. 22, No. 1, pp. 95-104, (2013).

[6] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Orlando, (1985).

[7] Chlouverakis, K. E.; Sprott, J. C., Chaotic hyperjerk systems. Chaos Solitons Fractals 28, No. 3, pp. 739-746, (2006).

[8] Cronin-Scanlon, J., Some mathematics of biological oscillations. SIAM Rev. 19, No. 1, pp. 100-138, (1977).

[9] Eichhorn, R.; Linz, S. J.; Hänggi, P., Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows. Phys Rev E58, pp. 7151-7164, (1998).

[10] Elhadj, Z.; Sprott, J. C., Boundedness of certain forms of jerky dynamics. Qual. Theory Dyn.Syst. 11, No. 2, pp. 199-213, (2012).

[11] El-Nahhas, A., Stability of a third-order differential equation of
retarded type. Appl. Math. Comput. 60, No. 2-3, pp. 147-152, (1994).

[12] El’sgol’ts, L. E., Introduction to the theory of dìfferential equations with deviating arguments. Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-LondonAmsterdam, (1966).

[13] Krasovskii, N. N., Stability of motion. Applications of Liapunov’s second method to differential systems and equations with delay. Stanford, Calif.: Stanford University Press, (1963).

[14] Linz, S. J., Onhyperjerky systems. Chaos Solitons Fractals 37, No. 3, pp. 741-747, (2008).

[15] Ogundare, B. S.; Okecha, G.E., On the boundedness and the stability of solution to third order non-linear differential equations. Ann. Differential Equations 24, No. 1, 1-8, (2008).

[16] Rauch, L. L., Oscillation of a third order nonlinear autonomous system. Contributions to the Theory of Nonlinear Oscillations, pp. 39-88. Annals of Mathematics Studies, No. 20. Princeton University Press, Princeton, N. J., (1950).

[17] Reissig, R.; Sansone, G.; Conti, R., Non-linear differential equations of higher order. Translated from the German. Noordhoff International Publishing, Leyden, (1974).

[18] Sadek, A. I., On the stability of solutions of some non-autonomous delay differential equations of the third order. Asymptot. Anal. 43,no. 1-2, pp. 1-7, (2005).

[19] Smith, H., An introduction to delay differential equations with applications to the life sciences. Texts in Applied Mathematics, 57. Springer, New York, (2011).

[20] Tun¸c, C., Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations. Kuwait J. Sci. Engrg. 32, No. 1, 39-48, (2005).

[21] Tun¸c, C., On the stability of solutions to a certain fourth-order delay differential equation. Nonlinear Dynam. 51(2008), No. 1-2, pp. 71-81, (2008).

[22] Tun¸c, C., On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dynam. 57, No. 1-2, pp. 97-106, (2009).

[23] Tun¸c, C., Bound of solutions to third-order nonlinear differential equations with bounded delay. J. Franklin Inst. 347, No. 2, pp. 415-425, (2010).

[24] Tun¸c, C., On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24, No. 3, pp. 1-10, (2010).

[25] Tun¸c, C., On the qualitative behaviors of solutions of some differential equations of higher order with multiple deviating arguments. J. Franklin Inst. 351, No. 2, 643-655, (2014).

[26] Tun¸c, C., Stability and boundedness of the nonlinear differential equations of third order with multiple deviating arguments. Afr. Mat. 24, No. 3, pp. 381-390, (2013).

[27] Tun¸c, C., Stability and boundedness for a kind of non-autonomous differential equations with constant delay. Appl. Math. Inf. Sci. 7, No. 1, pp. 355-361, (2013).

[28] 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).

[29] Zhang, Lijuan; Yu, Lixin, Global asymptotic stability of certain thirdorder nonlinear differential equations. Math.Methods Appl. Sci. 36, No. 14, pp. 1845-1850, (2013).

How to Cite

C. Tunc, “On the stability and boundedness of certain third order non-autonomous differential equations of retarded type”, Proyecciones (Antofagasta, On line), vol. 34, no. 2, pp. 147-159, 1.




Most read articles by the same author(s)