Title: Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale
Author(s): Manel Gouasmia, Abdelouaheb Ardjouni, Ahcene Djoudi
Pages: 162-177
Cite as:
Manel Gouasmia, Abdelouaheb Ardjouni, Ahcene Djoudi, Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale, Int. J. Anal. Appl., 16 (2) (2018), 162-177.

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].


Full Text: PDF

 

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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  7. L. Bi, M. Bohner and M. Fan, Periodic solutions of functional dynamic equations with infinite delay, Nonlinear Anal. 68 (2008), 1226–1245. Google Scholar

  8. M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, 2001. Google Scholar

  9. M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. Google Scholar

  10. T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006. Google Scholar

  11. S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten. PhD thesis, Universität Würzburg, 1988. Google Scholar

  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. Google Scholar

  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. Google Scholar

  14. V. Lakshmikantham, S. Sivasundaram, B. Kaymarkcalan, Dynamic Systems on Measure Chains, Kluwer Academic Pub- lishers, Dordrecht, 1996. Google Scholar

  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. Google Scholar

  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. Google Scholar

  17. D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, 1974. Google Scholar


COPYRIGHT INFORMATION

Copyright © 2020 IJAA, unless otherwise stated.