Asymptotic behavior of linear advanced dynamic equations on time scales.

Authors

Keywords:

Fixed points, Advanced dynamic equations, Asymptotic

Abstract

Let T be a time scale which is unbounded above and below and such that t0T. Let id + h, id + r: [t0,∞) ∩ TT  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.

Author Biographies

Malik Belaid, University of Annaba.

Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics.

Abdelouaheb Ardjouni, University of Souk Ahras.

Department of Mathematics and Informatics .

Ahcene Djoudi, University of Annaba.

Department of Mathematics, Applied Mathematics Lab, Faculty of Sciences.

References

M. Adıvar, Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations. Electronic Journal of Qualitative Theory of Differential Equations 2009, 1, pp. 1—20, (2009).

A. Ardjouni, I. Derrardjia and A. Djoudi, Stability in totally nonlinear neutral differential equations with variable delay, Acta Math. Univ. Comenianae, Vol. LXXXIII, 1, pp. 119-134, (2014).

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, pp. 5-19, (2013).

A. Ardjouni, A Djoudi, Stability in neutral nonlinear dynamic equations on time scale with unbounded delay, Stud. Univ. Babe¸ c-Bolyai Math. 57, No. 4, pp. 481-496, (2012).

A. Ardjouni, A Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Analysis 74, pp. 2062-2070, (2011).

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 Math. 33 (2), pp. 255-272, (2013).

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 Applications 16, pp. 561-570, (2007).

D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, London—New York, (1974).

Published

2019-02-26

How to Cite

[1]
M. Belaid, A. Ardjouni, and A. Djoudi, “Asymptotic behavior of linear advanced dynamic equations on time scales.”, Proyecciones (Antofagasta, On line), vol. 38, no. 1, pp. 97-110, Feb. 2019.

Issue

Section

Artículos

Most read articles by the same author(s)