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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.1 Antofagasta mar. 2019 


Asymptotic behavior of linear advanced dynamic equations on time scales

Malik Belaid1 

Abdelouaheb Ardjouni2 

Ahcene Djoudi3 

1University of Annaba, Department of Mathematics, Faculty of Sciences, Applied Mathematics Lab, P. O. Box 12, Annaba 23000, Algeria, e-mail:

2University of Souk Ahras, Department of Mathematics and Informatics, P. O. Box 1553, Souk Ahras 41000, Algeria, e-mail: abd

3University of Annaba, Faculty of Sciences, Applied Mathematics Lab, Department of Mathematics, P. O. Box 12, Annaba 23000, Algeria, e-mail:


Let T be a time scale which is unbounded above and below and such that t0 T. Let id be such that are time scales. We use the contraction mapping theorem to obtain convergence to zero about the solution for the following linear advanced dynamic equation

where 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.

Keywords : Fixed points; advanced dynamic equations; Asymptotic

Texto completo disponible sólo en PDF.

Full text available only in PDF format.


The authors would like to thank the anonymous referee for his valuable comments.


[1] 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). [ Links ]

[2] 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). [ Links ]

[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, pp. 5-19, (2013). [ Links ]

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

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

[6] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, (2001). [ Links ]

[7] M. Bohner , A. Peterson , Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, (2003). [ Links ]

[8] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9, pp. 181-190, (2001). [ Links ]

[9] T. A. Burton, Stability by fixed point theory or Liapunov theory: A Comparaison, Fixed Point Theory, 4, pp. 15-32, (2003). [ Links ]

[10] T. A. Burton , Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, (2006). [ Links ]

[11] N. T. Dung, Asymptotic behavior of linear advanced differential equations, Acta Mathematica Scientia, 35B (3): pp. 610-618, (2015). [ Links ]

[12] I. Derrardjia, A. Ardjouni andA. Djoudi , Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential equations, Opuscula Math. 33 (2), pp. 255-272, (2013). [ Links ]

[13] S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, Universität Würzburg, Würzburg, (1988). [ Links ]

[14] 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). [ Links ]

[15] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, London-New York, (1974). [ Links ]

Received: January 2018; Accepted: October 2018

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