Asymptotic behavior of linear advanced dynamic equations on time scales.

• Belaid, Malik ^[1] ; Ardjouni, Abdelouaheb ^[2] ; Djoudi, Ahcene ^[1]
  1. [1] University of Annaba.
  2. [2] University of Souk Ahras.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 38, Nº. 1, 2019, págs. 97-110
• Idioma: inglés
• DOI: 10.4067/s0716-09172019000100097
• Enlaces
  • Texto completo
• Resumen

  • Let T be a time scale which is unbounded above and below and such that t0 ∈ T. Let id + h, id + r: [t0,∞) ∩ T → T  be such that (id + h)([t0,∞) ∩ T) and (id + r)([t0,∞) ∩ T) are time scales. We use the contraction mapping theorem to obtain convergence to zero about the solution for the following linear advanced dynamic equation  x△ (t) + a (t) xσ (t + h (t)) + b (t) xσ (t + r (t)) = 0, t ∈ [t0, ∞) ∩ T   where f△ is the △-derivative on T. A convergence theorem with a necessary and sufficient condition is proved. The results obtained here extend the work of Dung [11]. In addition, the case of the equation with several terms is studied.

• Referencias bibliográficas
  • M. Adıvar, Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations. Electronic Journal of Qualitative...
  • A. Ardjouni, I. Derrardjia and A. Djoudi, Stability in totally nonlinear neutral differential equations with variable delay, Acta Math. Univ....
  • A. Ardjouni, A Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale, Acta...
  • A. Ardjouni, A Djoudi, Stability in neutral nonlinear dynamic equations on time scale with unbounded delay, Stud. Univ. Babe¸ c-Bolyai Math....
  • A. Ardjouni, A Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis 74, pp....
  • M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, (2001).
  • M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, (2003).
  • T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9, pp. 181—190, (2001).
  • T. A. Burton, Stability by fixed point theory or Liapunov theory: A Comparaison, Fixed Point Theory, 4, pp. 15-32, (2003).
  • T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, (2006).
  • N. T. Dung, Asymptotic behavior of linear advanced differential equations, Acta Mathematica Scientia, 35B (3): pp. 610-618, (2015).
  • I. Derrardjia, A. Ardjouni and A. Djoudi, Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential equations, Opuscula...
  • S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, Universität Würzburg, Würzburg, (1988).
  • E. R. Kaufmann, Y. N. Raffoul, Stability in neutral nonlinear dynamic equations on a time scale with functional delay, Dynamic Systems and...
  • D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, London—New York, (1974).