SciELO - Scientific Electronic Library Online

vol.34 número2On some I -convergent generalized difference sequence spaces associated with multiplier sequence defined by a sequence of modulliThe t-pebbling number of Jahangir graph J3,m índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados




Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google


Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.34 no.2 Antofagasta jun. 2015 

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

Cemil Tunc

Yuzuncu Yil University



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.

Subjclass : [2010]34K20.

Keywords : Boundedness, stability, non-autonomous, retarded.


[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.; Hanggi, 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 differential equations with deviating arguments. Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, (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]    Tung, C., Uniform ultimate boundedness of the solutions of third-order nonlinear differential equations. Kuwait J. Sci. Engrg. 32, No. 1, 39-48, (2005).

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

[24]    Tunc, 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]    Tung, 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]    Tung, 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]    Tunc, 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 third-order nonlinear differential equations. Math.Methods Appl. Sci. 36, No. 14, pp. 1845-1850, (2013).

Cemil Tunc

Department of Mathematics, Faculty of Sciences,

Yiizuncu Yil University, 65080, Van - Turkey Turkey

e-mail :

Received : October 2014. Accepted : April 2015

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons