Stability of Integral Caputo-Type Boundary Value Problem with Noninstantaneous Impulses

  • Akbar ZadaEmail author
  • Sartaj Ali
Original Paper


The modeling of a natural phenomena give soar to impulsive (instantaneous and noninstantaneous) fractional Caputo differential equations with boundary conditions. The behavior of the natural real world phenomena can be observed from the solutions of corresponding impulsive fractional Caputo differential equations with boundary conditions. Therefore, the existence, uniqueness and Ulam’s stability of the solutions of impulsive fractional Caputo differential equations are the most important concepts in fractional calculus. In this article, we take a noninstantaneous impulsive fractional Caputo differential equations with integral boundary conditions. The main objective of this article is, to study the existence, uniqueness and different types of Ulam’s stability for the solutions of fractional Caputo differential equations with noninstantaneous impulses and integral boundary conditions. At last, few examples are given to illustrate the new work.


Caputo fractional derivative Riemann–Liouville fractional integral Impulses Ulam–Hyers–Rassias stability Fixed point theorem 

Mathematics Subject Classification

34A08 34B27 



The authors express their sincere gratitude to the Editor and referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Abbas, M.I.: Existence and uniqueness results for fractional differential equations with Riemann–Liouville fractional integral boundary conditions. Abstr. Appl. Anal. 2015, 6 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ahmad, B., Nieto, J.J.: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009, 708576 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ali, A., Rabiei, F., Shah, K.: On Ulams type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions. J. Nonlinear Sci. Appl. 10, 4760–4775 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional calculus models and numerical methods. In: Series on Complexity, Nonlinearity and Chaos, vol. 3, World Scientific, Singapore (2012)Google Scholar
  6. 6.
    Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with non-linear fractional differential equations. Appl. Anal. 87, 851–863 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, Berlin (2012)CrossRefGoogle Scholar
  8. 8.
    Burger, M., Ozawa, N., Thom, A.: On Ulam stability. Isr. J. Math. 193, 109–129 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diaz, J.B., Margolis, B.: A fixed point theorem of alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gupta, V., Dabas, J.: Nonlinear fractional boundary value problem with not instantaneous impulse. AIMS Math. 2, 365–376 (2017)CrossRefGoogle Scholar
  11. 11.
    Hiffer, R.: Applications of Fractional Calculus in Physics. Word Scientific, Singapore (2000)Google Scholar
  12. 12.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hyers, D.H., Isac, G., Rassias, T.: Stability of Functional Equations in Several Variables. Birkhauser, Boston (1998)CrossRefGoogle Scholar
  14. 14.
    Huang, J., Jung, S.M., Li, Y.: On the Hyers–Ulam stability of non-linear differetial equations. Bull. Korean Math. Soc. 52, 685–697 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Haq, F., Shah, K., Ur Rahman, G., Shahzad, M.: Hyers–Ulam stability to a class of fractional differential equations with boundary conditions. Int. J. Appl. Comput. Math. 3, 1135–1147 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jung, C.J.: On Generalized Complete Metric Spaces. Kansas State University, Manhattan (1968)Google Scholar
  17. 17.
    Jiang, C., Zhang, F., Li, T.: Synchronization and antisynchronization of N-coupled fractional order complex chaotic systems with ring connection. Math. Methods Appl. Sci. 41, 2625–2638 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  19. 19.
    Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)zbMATHGoogle Scholar
  20. 20.
    Liu, Y., Yang, X.: New boundary value problems for higher order impulsive fractional differential equations and their solvability. Fract. Differ. Calc. 7, 1–121 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Li, T., Zada, A.: Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv. Differ. Equ. 2016, 153 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, T., Zada, A., Faisal, S.: Hyers–Ulam stability of nth order linear differential equations. J. Nonlinear Sci. Appl. 9, 2070–2075 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Li, T., Viglialoro, G.: Analysis and explicit solvability of degenerate tensorial problems. Bound. Value Probl. 2018, 2 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, T., Rogovchenko, YuV: Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations. Monatsh. Math. 184, 489–500 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mardanov, M.J., Mahmudov, N.I., Sharifov, Y.A.: Existence and uniqueness theorems for impulsive fractional differential equations with two-point and integral boundary conditions. Sci. World J. 2014, 1–8 (2014)CrossRefGoogle Scholar
  26. 26.
    Obloza, M.: Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Prace. Mat. 13, 259–270 (1993)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Podlubny, I.: Fractional Differential Equations. Math. Sci. Eng. 198, 1–340 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Popa, D., Rasa, I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Qin, H., Gu, Z., Fu, Y., Li, T.: Existence of mild solutions and controllability of fractional impulsive integrodifferential systems with nonlocal conditions. J. Funct. Space 2017, 1–11 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Rassias, T.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)zbMATHGoogle Scholar
  32. 32.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  33. 33.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)CrossRefGoogle Scholar
  34. 34.
    Shah, K., Khalil, H., Khan, R.A.: Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos, Solitons Fractals 77, 240–246 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Shah, K., Wang, J., Khalil, H., Khan, R.A.: Existence and numerical solutions of a coupled system of integral BVP for fractional differential equations. Adv. Differ. Equ 2018, 149 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Shah, R., Zada, A.: A fixed point approach to the stability of a nonlinear volterra integrodiferential equation with delay. Hacet. J. Math. Stat. 47(3), 615–623 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Shah, S.O., Zada, A., Hamza, A.E.: Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales. Qual. Theory Dyn. Syst. (2019).
  38. 38.
    Tarasov, V.E.: Fractional Dynamics. Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2011)Google Scholar
  39. 39.
    Tang, S., Zada, A., Faisal, S., EL–Sheikh, M.M.A., Li, T.: Stability of higher-order nonlinear impulsive differential equations. J. Nonlinear Sci. Appl. 9, 4713–4721 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968)zbMATHGoogle Scholar
  41. 41.
    Wang, G., Ahmad, B., Zhang, L.: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput. Math. Appl. 62, 1389–1397 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wang, X., Arif, M., Zada, A.: \(\beta \)–Hyers–Ulam–Rassias stability of semilinear nonautonomous impulsive system. Symmetry 11(2), 231 (2019)Google Scholar
  43. 43.
    Wang, J., Feckan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wang, P., Liu, X.: Rapid convergence for telegraph systems with periodic boundary conditions. J. Funct. Spaces vol. 2017, 10 (2017)zbMATHGoogle Scholar
  45. 45.
    Wang, J., Shah, K., Ali, A.: Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations. Math. Methods Appl. Sci. 41, 2392–2402 (2018)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Wang, J., Zada, A., Ali, W.: Ulam’s-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces. Int. J. Nonlinear Sci. Num. 19(5), 553–560 (2018)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Wang, J., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649–657 (2014)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Zada, A., Ali, S.: Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul. 19(7), 763–774 (2018)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Zada, A., Ali, S., Li, Y.: Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition. Adv. Differ. Equ. 2017, 317 (2017)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zada, A., Ali, W., Farina, S.: Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods App. Sci. 40(15), 5502–5514 (2017)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zada, A., Ali, W., Park, C.: Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Gr\(\ddot{o}\)nwall–Bellman–Bihari’s type. Appl. Math. Comput. 350, 60–65 (2019)Google Scholar
  52. 52.
    Zada, A., Wang, P., Lassoued, D., Li, T.: Connections between Hyers–Ulam stability and uniform exponential stability of \(2\)-periodic linear nonautonomous systems. Adv. Differ. Equ. 2017, 192 (2017)Google Scholar
  53. 53.
    Zada, A., Riaz, U., Khan, F.U.: Hyers–Ulam stability of impulsive integral equations. Boll. Unione Mat. Ital. (2018).
  54. 54.
    Zada, A., Shah, S.O.: Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses. Hacet. J. Math. Stat. 47(5), 1196–1205 (2018)Google Scholar
  55. 55.
    Zada, A., Shah, O., Shah, R.: Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271, 512–518 (2015)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Zada, A., Shaleena, S., Li, T.: Stability analysis of higher order nonlinear differential equations in \(\beta \)-normed spaces. Math. Methods Appl. Sci. 42(4), 1151–1166 (2019)Google Scholar
  57. 57.
    Zada, A., Yar, M., Li, T.: Existence and stability analysis of nonlinearsequential coupled system of Caputo fractionaldifferential equations with integral boundaryconditions. Ann. Univ. Paedagog. Crac. Stud. Math. 17, 103–125 (2018)MathSciNetGoogle Scholar
  58. 58.
    Zhang, S.: Positive solutions for boundary value problem of nonlinear fractional differential equations. Electron. J. Differ. Equ. 36, 1–12 (2016)MathSciNetGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PeshawarPeshawarPakistan

Personalised recommendations