Asymptotic properties of solutions to third order neutral differential equations with delay.

Authors

  • Linda D. Oudjedi University of Oran 1 Ahmed Ben Bella.
  • Belaib Lekhmissi University of Oran 1 Ahmed Ben Bella.
  • Moussadek Remili University of Oran 1 Ahmed Ben Bella. https://orcid.org/0000-0002-7920-1723

Keywords:

Uniform ultimate boundedness, Square integrability, Lyapunov functional, Neutral differential equation of third order

Abstract

This paper concerns the asymptotic properties of solutions of a class of third-order neutral differential equations with delay. We give sufficient conditions for every solution to be converges to zero, bounded and square integrable. An example is also given to illustrate the results.

Author Biographies

Linda D. Oudjedi, University of Oran 1 Ahmed Ben Bella.

Department of Mathematics.

Belaib Lekhmissi, University of Oran 1 Ahmed Ben Bella.

Department of Mathematics.

Moussadek Remili, University of Oran 1 Ahmed Ben Bella.

Department of Mathematics.

References

A. T. Ademola, and P.O.Arawomo, Uniform stability and boundedness of solutions of nonlinear delay differential equations of third order. Math. J. Okayama Univ. 55, pp. 157-166, (2013).

A. T. Ademola, and O. M. Arawomo, Ogunlaran and E. A. Oyekan, Uniform stability, boundedness and asymptotic behaviour of solutions of some third order nonlinear delay differential equations. Differential Equations and Control Processes, Issue 4, pp. 43-66, (2013).

O. Arino, M. L. Hbid, and E. Ait Dads, Delay differential equations and applications. NATO sciences series. Berlin, Springer, (2006).

B. Baculíková, J. Džurina, On the asymptotic behavior of a class of third order nonlinear neutral differential equations, Cent. Eur. J. Math., 8, pp. 1091-1103, (2010).

B. Mihalíková and E. Kostiková, Boundedness and oscillation of third order neutral differential equations. Tatra Mt. Math. Publ. 43, pp. 137—144, (2009).

P. Das and N. Misra, A necessary and sufficient condition for the solution of a functional differential equation to be oscillatory or tend to zero, J. Math. Anal. Appl. 204, pp. 78-87, (1997).

B. Dorociaková, Some nonoscillatory properties of third order differential equations of neutral type. Tatra Mt. Math. Publ. 38, pp. 71—76, (2007).

Z. Došla, P. Liška, Oscillation of third-order nonlinear neutral differential equations. Appl. Math. Lett., 56, pp. 42—48, (2016).

Z. Došla, P. Liška, Comparison theorems for third order neutral differential equations. Electronic Journal of Differential Equations, Vol. 38, pp. 1—13, (2016).

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

J. R. Graef, D. Beldjerd and M. Remili, On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay. PanAmerican Mathematical Journal 25, pp. 82-94, (2015).

J. R. Graef, L. D. Oudjedi and M. Remili Stability and square integrability of solutions of nonlinear third order differential equations. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 22, pp. 313—324, (2015).

M. R. S. Kulenovic, G. Ladas, A. Meimaridou, Stability of solutions of linear delay differential equations. Proc. Amer. Math. Soc., 100, pp. 433—441, (1987).

M. O. Omeike, New results on the stability of solution of some nonautonomous delay differential equations of the third order. Differential Equations and Control Processes 2010 1, pp. 18—29, (2010).

L. D. Oudjedi, D. Beldjerd, and M. Remili, On the Stability of Solutions for non- autonomous delay differential equations of third-order, Differential Equations and Control Processes 1, pp. 22—34, (2014).

M. Remili, D. Beldjerd, A boundedness and stability results for a kind of third order delay differential equations. Applications and Applied Mathematics, vol. 10, Issue 2, pp. 772-782, (2015).

M. Remili, and D. Beldjerd, On the asymptotic behavior of the solutions of third order delay differential equations. Rend. Circ. Mat. Palermo 63, pp. 447—455, (2014).

M. Remili, D. Beldjerd, On ultimate boundedness and existence of periodic solutions of kind of third order delay differential equations. Acta Universitatis Matthiae Belii, series Mathematics Issue, pp. 1-15, (2016).

M. Remili, D. Beldjerd, Stability and ultimate boundedness of solutions of some third order differential equations with delay. Journal of the Association of Arab Universities for Basic and Applied Sciences 23, pp. 90-95, (2017).

M. Remili, and L. D. Oudjedi, Boundedness and stability in third order nonlinear differential equations with bounded delay. Analele Universităţii Oradea Fasc. Matematica, Tom XXIII, Issue No. 1, pp. 135-143, (2016).

M. Remili, and L. D. Oudjedi, Boundedness and stability in third order nonlinear differential equations with multiple deviating arguments. Archivum Mathematicum, Vol. 52, No. 2, pp. 79-90, (2016).

M. Remili, and L. D. Oudjedi, Stability and boundedness of the solutions of non autonomous third order differential equations with delay. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 2, pp. 139—147, (2014).

M. Remili, and L. D. Oudjedi, Stability of the solutions of nonlinear third order differential equations with multiple deviating arguments. Acta Univ. Sapientiae, Mathematica, 8, 1, pp. 150—165, (2016).

M. Remili, and L. D. Oudjedi, On asymptotic stability of solutions to third order nonlinear delay differential equation. Filomat, 30, 12, pp. 3217—3226, (2016).

M. Remili, L. D. Oudjedi, and Beldjerd, D., On the qualitative behaviors of solutions to a kind of nonlinear third order differential equation with delay. Communications in Applied Analysis 20, pp. 53-64, (2016).

M. Remili, and L. D. Oudjedi, Uniform stability and boundedness of a kind of third order delay differential equations, Bull. Comput. Appl. Math., Vol. 2, No.1, pp. 25—35, (2014).

M. Remili, and L. D. Oudjedi, Uniform ultimate boundedness and asymptotic behaviour of third order nonlinear delay differential equation. Afrika Matematika, Vol. 27, Issue 7, pp. 1227—1237, (2016).

Y.-Z. Tian, Y.-L. Cai, Y.-L. Fu, T.-X. Li, Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments. Adv. Difference Equ., 267, 14 pages, (2015).

C. Tunç, 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).

C. Tunç, Some stability and boundedness conditions for nonautonomous differential equations with deviating arguments, E. J. Qualitative Theory of Diff. Equ., No. 1., pp. 1-12, (2010).

T.-X. Li, C.-H. Zhang, G.-J. Xing, Oscillation of third-order neutral delay differential equations. Abstr. Appl. Anal., 11 pages, (2012).

J. Yu, Z. Wang, C. Qian, Oscillation of neutral delay differential equation. Bull. Austral. Math. Soc. 45, pp. 195-200, (1992).

Yu Jianshe, Asymptotic stability for a class of nonautonomous neutral differential equations. Chin. Ann. of Math, 18 B (4), pp. 449-456, (1997).

Published

2019-02-26

How to Cite

[1]
L. D. Oudjedi, B. Lekhmissi, and M. Remili, “Asymptotic properties of solutions to third order neutral differential equations with delay.”, Proyecciones (Antofagasta, On line), vol. 38, no. 1, pp. 111-127, Feb. 2019.

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Artículos