Stability in Totally Nonlinear Neutral Dynamic Equations on Time Scales

Belaid, Malik; Ardjouni, Abdelouaheb; Djoudi, Ahcene

Research Article

Stability in Totally Nonlinear Neutral Dynamic Equations on Time Scales

Malik Belaid
Abdelouaheb Ardjouni*
Ahcene Djoudi

* Corresponding author

Abstract

Let T be a time scale which is unbounded above and below and such that 0∈T. Let id-Ï„:[0,∞)∩T→T be such that (id-Ï„)([0,∞)∩T) is a time scale. We use the Krasnoselskii-Burton's fixed point theorem to obtain stability results about the zero solution for the following totally nonlinear neutral dynamic equation with variable delay

x^{â–³}(t)=-a(t)h(x^{σ}(t))+c(t)x^{â–³}(t-Ï„(t))+b(t)G(x(t),x(t-Ï„(t))), t∈[0,∞)∩T,

where f^{â–³} is the â–³-derivative on T and f^{â–³} is the â–³-derivative on (id-Ï„)(T). The results obtained here extend the work of Ardjouni, Derrardjia and Djoudi [2].

How to Cite

M. Belaid, A. Ardjouni, A. Djoudi, Stability in Totally Nonlinear Neutral Dynamic Equations on Time Scales, Int. J. Anal. Appl. 11 (2016), 110–123.

BibTeX

@article{Belaid2016,
  author  = {Belaid, Malik and Ardjouni, Abdelouaheb and Djoudi, Ahcene},
  title   = {Stability in Totally Nonlinear Neutral Dynamic Equations on Time Scales},
  journal = {Int. J. Anal. Appl.},
  volume  = {11},
  year    = {2016},
  pages   = {110--123},
  doi     = {}
}

References

[1] M. Adivar, Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations. Electron. J. Qual. Theory Differ. Equ., Spec. Ed. 1 (2009), 1-20.
[2] A. Ardjouni, I. Derrardjia and A. Djoudi, Stability in totally nonlinear neutral differential equations with variable delay, Acta Math. Univ. Comenianae, 83 (2014), 119-134.
[3] A. Ardjouni, A Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale, Acta Univ. Palacki. Olomnc., Fac. rer. nat., Mathematica 52, 1 (2013) 5-19.
[4] A. Ardjouni, A Djoudi, Stability in neutral nonlinear dynamic equations on time scale with unbounded delay, Stud. Univ. Babe ¸c-Bolyai Math. 57(2012), No. 4, 481-496.
[5] A. Ardjouni, A Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis 74 (2011), 2062-2070.
[6] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, 2001.
[7] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
[8] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2001), 181-190.
[9] T. A. Burton, Stability by fixed point theory or Liapunov theory: A Comparaison, Fixed Point Theory, 4(2003), 15-32.
[10] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
[11] I. Derrardjia, A. Ardjouni and A. Djoudi, Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations, Opuscula Math. 33(2) (2013), 255-272.
[12] S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, Universität Würzburg, Würzburg, 1988.
[13] E. R. Kaufmann, Y. N. Raffoul, Stability in neutral nonlinear dynamic equations on a time scale with functional delay, Dynamic Systems and Applications 16 (2007) 561-570.
[14] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, London-New York, 1974.