SciELO - Scientific Electronic Library Online

 
vol.33 issue3Generalized b-closed sets in ideal bitopological spaces author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.33 no.3 Antofagasta Sept. 2014

http://dx.doi.org/10.4067/S0716-09172014000300007 

Proyecciones Journal of Mathematics Vol. 33, No 3, pp. 325-347, September 2014. Universidad Católica del Norte Antofagasta - Chile

 

On the qualitative properties of differential equations of third order with retarded argument

 

Cemil Tunc

Yüzüncü Yil University

Turkey

Sizar Abid Mohammed

University of Duhok

Iraq


ABSTRACT

By using the standard Liapunov-Krasovskii functional approach, in this paper, new stability, boundedness and ultimately boundedness criteria are established for a class of vector functional differential equations of third order with retarded argument.


REFERENCES

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

[2] A. U. Afuwape; J. E. Castellanos, Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: frequency domain method. Appl. Math. Comput. 216, No. 3, pp. 940950, (2010).

[3] A. U. Afuwape; P. Omari; F. Zanolin, Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems. J. Math. Anal. Appl. 143, No. 1, pp. 35-56, (1989).

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

[5] T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Academic Press, Orlando, (1985).

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

[7] E. N. Chukwu, On boundedness of solutions of third order differential equations. Ann. Math. Pura. Appl. 104, pp. 123-149, (1975).

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

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

[10] R. Eichhorn; S. J. Linz; P. HiAnggi, Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows. Phys Rev E 58, pp. 7151-7164, (1998).

[11] El'sgol'ts, L. E.; Norkin, S. B., Introduction to the theory and application of differential equations with deviating arguments. Translated from the Russian by John L. Casti. Mathematics in Science and Engineering, Vol. 105. Academic Press, New York-London, (1973).

[12] J. O. C. Ezeilo, On the stability of solutions of certain differential equations of the third order. Quart. J. Math. Oxford Ser. (2) 11, pp.64-69, (1960).

[13] Ezeilo, J. O. C., A generalization of a boundedness theorem for a certain third-order differential equation. Proc. Cambridge Philos. Soc. 63, pp. 735-742, (1967).

[14] K. O. Fridedrichs, On nonlinear vibrations of third order. Studies in Nonlinear Vibration Theory, pp. 65-103. Institute for Mathematics and Mechanics, New York University, (1946).

[15] J. Hale, Sufficient conditions for stability and instability of autonomous functional-differential equations. J. Differential Equations 1, pp. 452482, (1965).

[16] T. Hara, On the uniform ultimate boundedness of the solutions of certain third order differential equations. J. Math. Anal. Appl. 80, No.2, pp. 533-544, (1981).

[17] N. N. Krasovskii, Stability of Motion. Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Stanford, Calif.: Stanford University Press (1963).

[18] A. M. Liapunov, Stability of motion. With a contribution by V. A. Pliss and an introduction by V. P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni. Mathematics in Science and Engineering, Vol. 30 Academic Press, New York-London (1966).

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

[20] B. Mehri; D. Shadman, Boundedness of solutions of certain third order differential equation. Math. Inequal. Appl. 2, No.4, pp. 545-549, (1999).

[21] F. W. Meng, Ultimate boundedness results for a certain system of third order nonlinear differential equations. J. Math. Anal. Appl. 177, No. 2, pp. 496-509, (1993).

[22] M. O.Omeike; A. U. Afuwape, New result on the ultimate boundedness of solutions of certain thir d-order vector differential equations. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 49, No. 1, pp. 55-61, (2010).

[23] C. Qian, On global stability of third-order nonlinear differential equations. Nonlinear Anal. 42, No. 4, Ser. A: Theory Methods, pp. 651-661, (2000).

[24] L. L. Rauch, 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).

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

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

[27] K. E. Swick, Asymptotic behavior of the solutions of certain third order differential equations. SIAM J. Appl. Math. 19, pp. 96-102, (1970).

[28] H. O. Tejumola, A note on the boundedness and the stability of solutions of certain third-order differential equations. Ann. Mat. Pura Appl. (4) 92, pp. 65-75, (1972).

[29] C. Tunc, Global stability of solutions of certain third-order nonlinear differential equations. Panamer. Math. J. 14, No. 4, pp. 31-35, (2004).

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

[31] C. Tunc, Boundedness of solutions of a third-order nonlinear differen-tialequation.JIPAM.J.Inequal.PureAppl.Math.6,No.1,Article 3, 6 pp. —, (2005)

[32] C. Tunc, On the asymptotic behavior of solutions of certain third-order nonlinear differential equations. J. Appl. Math. Stoch. Anal., No. 1, pp. 29-35, (2005).

[33] C. Tunc, Some instability results on certain third order nonlinear vector differential equations. Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007), no. 1, 109-122.

[34] C. Tunc, 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).

[35] C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third order. Nonlinear Anal. 70, No. 6,pp. 2232-2236, (2009).

[36] C. Tunc, Stability and bounded of solutions to non-autonomous delay differential equations of third order. Nonlinear Dynam. 62 (2010), No.4, pp. 945-953, (2010).

[37] C. Tunc, On some qualitative behaviors of solutions to a kind of third order nonlinear delay differential equations. Electron. J. Qual. Theory Differ. Equ., No. 12, 19, pp. —, (2010).

[38] C. Tunc, 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).

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

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

[41] C. Tuncç; M. Ates, Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. Nonlinear Dynam. 45, No. 3-4, pp. 273-281, (2006).

[42] C. Tunc; H. Ergoren, Uniformly boundedness of a class of non-linear differential equations of third order with multiple deviating arguments. Cubo 14, No. 3, 63-69, (2012).

[43] E. T. Wall; M. L. Moe, An energy metric algorithm for the generation of Liapunov functions. IEEE Trans. Automat. Control. 13 (1), pp.121-122, (1968).

[44] L. Zhang; L. Yu, Global asymptotic stability of certain third-order nonlinear differential equations. Math. Methods Appl. Sci. Math. Methods Appl. Sci. 36, No. 14, pp. 1845-1850, (2013).


Cemil Tunç
Department of Mathematics
Faculty of Sciences
Yüzmicu Yil University
65080 Van
Turkey
e-mail : cemtunc@yahoo.com

Sizar Abid Mohammed
University of Duhok, Zakko Street 38
1006 AJ Duhok
Iraq
e-mail : sizar@uod.ac

Received : April 2014. Accepted : June 2014