Stability of solutions to fractional differential equations with time-delays


  • Fatima Fenizri ENSET Skikda.
  • Assia Guezane Lakoud Badji Mokhtar Annaba University.
  • Rabah Khaldi Badji Mokhtar-Annaba University.



fractional derivative, existence of solution, stability of solution, boundary value problem


This paper deals with a fractional boundary value problem involving variable delays. Sufficient conditions for the existence of a unique solution are investigated. Moreover the stability of the unique solution is discussed. A numerical example that emphasizes the importance of the results obtained in this article is also included.

Author Biographies

Assia Guezane Lakoud, Badji Mokhtar Annaba University.

Laboratory of Advanced Materials, Faculty of Sciences.

Rabah Khaldi, Badji Mokhtar-Annaba University.

Laboratory of Advanced Materials, Faculty of Sciences.


S. Abbas, “Existence of solutions to fractional order ordinary and delay differential equations and applications”, Electronic Journal of Differential Equations, vol. 2011, no. 09, pp. 1-11, 2011.

E. Ahmed, A. Hashish, F. A. Rihan, “On fractional order cancer model”, Journal of Fractional Calculus and Applications, vol. 3, no. 2, pp. 1-6, 2012.

D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods. Singapore: World Scientific, 2012.

M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahaba, “Existence results for fractional order functional differential equations with infinite delay”, Journal of Mathematical Analysis and Applications, vol. 338, pp. 1340-13, 2008.

L. Bingwen, “Existence and uniqueness of periodic solutions for a class of nonlinear n-th order differential equations with delays”, Mathematische Nachrichten, vol. 282, no. 4, pp. 581-590, 2009.

N. D. Cong and H. T. Tuan, “Existence, uniqueness and exponential bound-edness of global solutions to delay fractional differential equations”, Mediterranean Journal of Mathematics, 2017.

K. S. Cole, “Electric conductance of biological systems”, Cold Spring Harbor Symposia on Quantitative Biology, vol. 1, pp. 107-116, 1933.

W. C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system”, Chaos Solitons & Fractals, vol. 36, no. 5, pp. 1305-1314, 2008.

L. Debnath, “Recent applications of fractional calculus to science and engineering”, International Journal of Mathematics and Mathematical Sciences, vol. 54, pp. 3413-3442, 2003.

F. A. Rihan, ”Computational methods for delay parabolic and time fractional partial differential equations”, Numerical Methods for Partial Differential Equations, vol. 26, no. 6, pp. 1556-1571, 2010.

F. Ge and C. Kou, “Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations”, Applied Mathematics and Computation, vol. 257, pp. 308-316, 2015.

A. Guezane-Lakoud and A. Kiliçman, “Unbounded solution for a fractional boundary value problema”, Advances in Difference Equations, vol. 2014, 2014.

A. Guezane-Lakoud and R. Rodríguez-López, “On a fractional boundary value problem in a weighted space”, SeMA Journal, vol. 75, no. 3, pp. 435-443, 2018.

A. Guezane-Lakoud, R. Khaldi and Delfim F. M. Torres, “On a fractional oscillator equation with natural boundary conditions”, Progress in Fractional Differentiation and Applications, vol. 3, No. 3, pp. 1-7, 2017.

N. M. Grahovac and M. Zigic, “Modelling of the hamstring muscle group by use of fractional derivatives”, Computers & Mathematics with Applications, vol. 59, pp. 1695-1700, 2010.

K. Hadi, A. Babakhani and D. Baleanu, “Existence results for a class of fractional differential equations with periodic boundary value conditions and with delay”, Abstract and Applied Analysis, vol. 2013, Art. ID 176180, 2013.

L. Kexue and J. Junxiong, “Existence and uniqueness of mild solutions for abstract delay fractional differential equations”, Computers and Mathematics with Applications, vol. 62, pp. 1398-1404, 2011.

A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, Amsterdam, 2006.

R. Khaldi and A. Guezane-Lakoud, “Upper and Lower Solutions, method for fractional oscillation equations”, Proceedings of the Institute of Mathematics and Mechanics, vol. 43, no. 2, pp. 214-220, 2017.

C. Li and F. Zhang, “A survey on the stability of fractional differential equations”, The European Physical Journal Special Topics, vol. 193, pp. 27-47, 2011.

N. Laskin and G. M. Zaslavsky, “Nonlinear fractional dynamics on a lattice with long-range interactions”, Physica A: Statistical Mechanics and its Applications, vol. 368, pp. 38-54. 2006.

W. Lin, “Global existence theory and chaos control of fractional differential equations”, Journal of Mathematical Analysis and Applications, vol. 332, pp. 709-726, 2007.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

A. El-Sayed and F. Gaafar, “Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions”, Advances in Difference Equations, vol. 2011, no. 1, pp.47, 2011.

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives. Theory and Applications. Gordon and Breach, Yverdon, 1993.

X. Zhang, “Some results of linear fractional order time-delay system”, Applied Mathematics and Computation, vol. 197, pp. 407-411, 2008.

G. Zhenghui, Y. Liu and L. Zhenguo, “Stability of the solutions for nonlinear fractional differential equations with delays and integral boundary conditions”, Advances in Difference Equations, vol. 2013, no. 1, pp. 43, 2013.

Y. Zhou, “Existence and uniqueness of fractional differential equations with unbounded delay”, International Journal of Dynamical Systems and Differential Equations, vol. 1, no. 4, pp. 239-244, 2008.

G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport”, Physics Reports, vol. 371, pp. 461-580, 2002.



How to Cite

F. Fenizri, A. Guezane Lakoud, and R. Khaldi, “Stability of solutions to fractional differential equations with time-delays”, Proyecciones (Antofagasta, On line), vol. 42, no. 2, pp. 261-272, Mar. 2023.