Functional Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Derivative and Integral Boundary Conditions

Main Article Content

Aida Irguedi, Khadidja Nisse, Samira Hamani


In this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. Our results are based on some fixed point theorems. Finally, we provide an example to illustrate the validity of our main results.

Article Details


  1. R.P. Agarwal, M. Benchohra, S. Hamani, A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions, Acta Appl. Math. 109 (2008), 973–1033.
  2. Y. Adjabi, F. Jarad, D. Baleanu, T. Abdeljawad, On Cauchy Problems With Caputo Hadamard Fractional Derivatives, J. Comput. Anal. Appl. 21 (2016), 661-681.
  3. P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Compositions of Hadamard-Type Fractional Integration Operators and the Semigroup Property, J. Math. Anal. Appl. 269 (2002), 387–400.
  4. P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Mellin Transform Analysis and Integration by Parts for Hadamard-Type Fractional Integrals, J. Math. Anal. Appl. 270 (2002), 1–15.
  5. P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional Calculus in the Mellin Setting and Hadamard-Type Fractional Integrals, J. Math. Anal. Appl. 269 (2002), 1–27.
  6. Z. Cui, P. Yu, Z. Mao, Existence of Solutions for Nonlocal Boundary Value Problems of Nonlinear Fractional Differential Equations, Adv. Dyn. Syst. Appl. 7 (2012), 31-40.
  7. Y.K. Chang, A. Anguraj, M. Mallika Arjunan, Existence Results for Impulsive Neutral Functional Differential Equations With Infinite Delay, Nonlinear Anal.: Hybrid Syst. 2 (2008), 209–218.
  8. J. Cao, Y. Luo, G. Liu, Some Results for Impulsive Fractional Differential Inclusions With Infinite Delay and Sectorial Operators in Banach Spaces, Appl. Math. Comput. 273 (2016), 237–257.
  9. J. Dabas, A. Chauhan, Existence and Uniqueness of Mild Solution for an Impulsive Neutral Fractional IntegroDifferential Equation With Infinite Delay, Math. Computer Model. 57 (2013), 754–763.
  10. J. Hadamard, Essai Sur l’Etude des Fonctions Donnees par Leur Developpement de Taylor, J. Math. Pure Appl. 8 (1892), 101-186.
  11. S. Hamani, A. Hammou, J. Henderson, Impulsive Fractional Differential Equations Involving The Hadamard Fractional Derivative, Commun. Appl. Nonlinear Anal. 24 (2017), 48-58.
  12. A. Hammou, S. Hamani, J. Henderson, Impulsive Hadamard Fractional Differential Equations in Banach Spaces, Commun. Appl. Nonlinear Anal. 28 (2018), 52-62.
  13. A. Hammou, S. Hamani, J. Henderson Initial Value Problems for Impulsive Caputo-Hadamard Fractional Differential Inclusions, Commun. Appl. Nonlinear Anal. 22 (2019), 17-35.
  14. S. Heidarkhani, A. Salari, G. Caristi, Infinitely Many Solutions for Impulsive Nonlinear Fractional Boundary Value Problems, Adv. Differ. Equ. 2016 (2016), 196.
  15. F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-Type Modification of the Hadamard Fractional Derivatives, Adv. Differ. Equ. 2012 (2012), 142.
  16. Y. Tian, Z. Bai, Impulsive Boundary Value Problem for Differential Equations with Fractional Order, Differ Equ. Dyn. Syst. 21 (2012), 253–260.
  17. P. Thiramanus, S.K. Ntouyas, J. Tariboon, Existence and Uniqueness Results for Hadamard-Type Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions, Abstr. Appl. Anal. 2014 (2014), 902054.
  18. A. Nain, R. Vats, A. Kumar, Caputo-Hadamard Fractional Differential Equation With Impulsive Boundary Conditions, J. Math. Model. 9 (2021), 93-106.
  19. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  20. S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018.
  21. S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012.