On the Novel Ulam–Hyers Stability for a Class of Nonlinear \(\psi \)-Hilfer Fractional Differential Equation with Time-Varying Delays

  • Danfeng Luo
  • Kamal Shah
  • Zhiguo LuoEmail author


In this paper, we present some alternative results concerning the uniqueness and Ulam–Hyers stability of solutions for a kind of \(\psi \)-Hilfer fractional differential equations with time-varying delays. Under some updated criteria along with the generalized Gronwall inequality, the new constructive results have been established in the literature. The derived analysis has the ability to generalize and improve some other results from the literature. As an application, two typical examples are delineated to demonstrate the effectiveness of our theoretical results.


Ulam–Hyers–Rassias stability Ulam–Hyers stability Uniqueness \(\psi \)-Hilfer fractional derivative Time-varying delays 

Mathematics Subject Classification

26A33 34D20 



The authors thank the referees for the helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11471109).


  1. 1.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, vol. 207. Elsevier, Amsterdam (2006)Google Scholar
  2. 2.
    Miller, K., Rose, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993). (ISBN: 0-471-58884-9)Google Scholar
  3. 3.
    Goodrich, C., Peterson, A.: Discrete Fractional Calculus. Springer, Berlin (2016). (ISBN: 3319255606; 9783319255606)zbMATHGoogle Scholar
  4. 4.
    Podlubny, I., Thimann, K.: Fractional Differential Equation: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999). (ISBN: 0125588402)Google Scholar
  5. 5.
    Chen, F., Liu, Z.: Asymptotic stability results for nonlinear fractional difference eqautions. J. Appl. Math. 2012, 155–172 (2012)Google Scholar
  6. 6.
    Abu-Saris, R., Al-Mdallal, Q.: On the asymptotic stability of linear system of fractional-order difference equations. Fract. Calc. Appl. Anal. 16(3), 613–629 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68–73 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Khudair, A., Haddad, S., Khalaf, S.: Restricted fractional differential transform for solving irrational order fractional differential equations. Chaos Solitons Fractals 101, 81–85 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu, K., Jiang, W.: Stability of nonlinear Caputo fractional differential equations. Appl. Math. Model. 40(5–6), 3919–3924 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Baleanu, D., Wu, G., Bai, Y., Chen, F.: Stability analysis of Caputo-like discrete fractional systems. Commun. Nonlinear Sci. Numer. Simul. 48, 520–530 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sousa, J., Oliveira, E.: Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81, 50–56 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sousa, J., Oliveira, E.: On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the \(\psi \)-Hilfer operator. J. Fixed Point Theory Appl. 20, 96 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mozyrska, D., Wyrwas, M.: Stability of discrete fractional linear systems with positive orders. Conf. Pap. Arch. 51–1, 8115–8120 (2017)zbMATHGoogle Scholar
  14. 14.
    Wu, G., Baleanu, G., Luo, W.: Lyapunov functions for Riemann–Liouville-like fractional difference equations. Appl. Math. Comput. 314, 228–236 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lizama, C., Murilla-Arcila, M.: Maximal regularity in \(l_p\) spaces for discrete time fractional shifted equations. J. Differ. Equ. 263(6), 3175–3196 (2017)CrossRefGoogle Scholar
  16. 16.
    Wang, J., Zhou, Y.: Mittag–Leffler–Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 25(4), 723–728 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ali, Z., Zada, A., Shah, K.: On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations. Bull. Malays. Math. Sci. Soc. 2, 4 (2019). MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, Y.: On piecewise continuous solutions of higher order impulsive fractional differential equations and applications. Appl. Math. Comput. 287–288, 38–49 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sousa, J., Oliveira, E.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Abdeljawad, T., TORRES, F.: Symmetric duality for left and right Riemann–Liouville and Caputo fractional differences. Arab J. Math. Sci 23, 157–172 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Dassios, I., Baleanu, D.: Duality of singular linear systems of fractional nabla difference equations. Appl. Math. Model. 39, 4180–4195 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Chen, F., Zhou, Y.: Existence and Ulam stability of solutions for discrete fractional boundary value problem. Discret. Dyn. Nat. Soc. 2013, 7 (2013). (Article ID.459161)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sousa, J., Oliveira, D., Oliveira, E.: On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation. Math. Methods Appl. Sci. 42, 1249–1261 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, J., Li, X.: Ulam–Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258, 72–83 (2015)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Li, M., Wang, J.: Exploring delayed Mittag–Leffler type matrix functions to study finite time stability of fractional delay differential equations. Appl. Math. Comput. 324, 254–265 (2018)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hei, X., Wu, R.: Finite-time stability of impulsive fractional-order systems with time-delay. Appl. Math. Model. 40, 4285–4290 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, Q., Lu, D., Fang, Y.: Stability analysis of impulsive fractional differential systems with delay. Appl. Math. Lett. 40, 1–6 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Liu, K., Wang, J., O’Regan, D.: Ulam–Hyers–Mittag–Leffler stability for \(\psi \)-Hilfer fractional-order delay differential equations. Adv. Differ. Equ. 2019, 50 (2019). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sousa, J., Oliveira, E.: A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator. Differ. Equ. Appl. 11(n.1), 87–106 (2019)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of MalakandLower DirPakistan
  3. 3.Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China

Personalised recommendations