Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale

Gouasmia, Manel; Ardjouni, Abdelouaheb; Djoudi, Ahcene

doi:10.28924/2291-8639-16-2018-162

Research Article

Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale

Manel Gouasmia
Abdelouaheb Ardjouni*
Ahcene Djoudi

* Corresponding author

Abstract

Let T be a periodic time scale. We use Krasnoselskii--Burton's fixed point theorem to show new results on the existence of periodic and nonnegative periodic solutions of nonlinear neutral dynamic equation with variable delay of the form

$x^{\Delta }(t)=-a(t)h(x^{\sigma }(t))+Q(t,x(t-\tau (t)))^{\Delta}+G(t,x(t),x(t-\tau (t))),\text{ }t\in \mathbb{T}.$

We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a completely continuous map. The Caratheodory condition is used for the functions $Q$ and $G$. The results obtained here extend the work of Mesmouli, Ardjouni and Djoudi [16].

How to Cite

M. Gouasmia, A. Ardjouni, A. Djoudi, Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale, Int. J. Anal. Appl. 16 (2018), 162–177. https://doi.org/10.28924/2291-8639-16-2018-162

BibTeX

@article{Gouasmia2018,
  author  = {Gouasmia, Manel and Ardjouni, Abdelouaheb and Djoudi, Ahcene},
  title   = {Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale},
  journal = {Int. J. Anal. Appl.},
  volume  = {16},
  year    = {2018},
  pages   = {162--177},
  doi     = {10.28924/2291-8639-16-2018-162}
}

References

[1] M. Adivar and Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations, Electronic Journal of Qualitative Theory of Differential Equations, 2009, No. 1, 1-20.
[2] A. Ardjouni and A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 3061-3069.
[3] A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale, Malaya J. Mat. 2(1) (2013) 60-67.
[4] A. Ardjouni and 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.
[5] A. Ardjouni and A. Djoudi, A. Existence, uniqueness and positivity of solutions for a neutral nonlinear periodic dynamic equation on a time scale, J. Nonlinear Anal. Optim. 6 (2) (2015), 19-29.
[6] M. Belaid, A. Ardjouni and A.Djoudi, Stability in totally nonlinear neutral dynamic equations on time scales, Int. J. Anal. Appl. 11 (2) (2016), 110-123.
[7] L. Bi, M. Bohner and M. Fan, Periodic solutions of functional dynamic equations with infinite delay, Nonlinear Anal. 68 (2008), 1226-1245.
[8] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, 2001.
[9] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
[10] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
[11] S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten. PhD thesis, Universität Würzburg, 1988.
[12] E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl. 319 (2006), no. 1, 315-325.
[13] E. R. Kaufmann and Y. N. Raffoul, Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale, Electron. J. Differential Equations, 2007 (2007), No. 27, 1-12.
[14] V. Lakshmikantham, S. Sivasundaram, B. Kaymarkcalan, Dynamic Systems on Measure Chains, Kluwer Academic Pub- lishers, Dordrecht, 1996.
[15] M. B. Mesmouli, A. Ardjouni, A. Djoudi, Existence and stability of periodic solutions for nonlinear neutral differential equations with variable delay using fixed point technique, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 54 (1) (2015), 95-108.
[16] M. B. Mesmouli, A. Ardjouni and A. Djoudi, Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points, Acta Univ. Sapientiae, Mathematica, 8 (2) (2016), 255-270.
[17] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, 1974.