Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

  7. Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, Berlin (2012)

    Book  Google Scholar 

  8. Burger, M., Ozawa, N., Thom, A.: On Ulam stability. Isr. J. Math. 193, 109–129 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Gupta, V., Dabas, J.: Nonlinear fractional boundary value problem with not instantaneous impulse. AIMS Math. 2, 365–376 (2017)

    Article  Google Scholar 

  11. Hiffer, R.: Applications of Fractional Calculus in Physics. Word Scientific, Singapore (2000)

    Google Scholar 

  12. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  13. Hyers, D.H., Isac, G., Rassias, T.: Stability of Functional Equations in Several Variables. Birkhauser, Boston (1998)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  16. Jung, C.J.: On Generalized Complete Metric Spaces. Kansas State University, Manhattan (1968)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  19. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  22. Li, T., Zada, A., Faisal, S.: Hyers–Ulam stability of nth order linear differential equations. J. Nonlinear Sci. Appl. 9, 2070–2075 (2016)

    Article  MathSciNet  Google Scholar 

  23. Li, T., Viglialoro, G.: Analysis and explicit solvability of degenerate tensorial problems. Bound. Value Probl. 2018, 2 (2018)

    Article  MathSciNet  Google Scholar 

  24. Li, T., Rogovchenko, YuV: Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations. Monatsh. Math. 184, 489–500 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  26. Obloza, M.: Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Prace. Mat. 13, 259–270 (1993)

    MathSciNet  MATH  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Math. Sci. Eng. 198, 1–340 (1999)

    Article  MathSciNet  Google Scholar 

  28. Popa, D., Rasa, I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  30. Rassias, T.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MathSciNet  Google Scholar 

  31. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  32. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  33. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s12346-019-00315-x

  38. Tarasov, V.E.: Fractional Dynamics. Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2011)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  40. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  42. Wang, X., Arif, M., Zada, A.: \(\beta \)–Hyers–Ulam–Rassias stability of semilinear nonautonomous impulsive system. Symmetry 11(2), 231 (2019)

    Article  Google Scholar 

  43. Wang, J., Feckan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MathSciNet  Google Scholar 

  44. Wang, P., Liu, X.: Rapid convergence for telegraph systems with periodic boundary conditions. J. Funct. Spaces vol. 2017, 10 (2017)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  47. Wang, J., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649–657 (2014)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  53. Zada, A., Riaz, U., Khan, F.U.: Hyers–Ulam stability of impulsive integral equations. Boll. Unione Mat. Ital. (2018). https://doi.org/10.1007/s40574-018-0180-2

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  58. Zhang, S.: Positive solutions for boundary value problem of nonlinear fractional differential equations. Electron. J. Differ. Equ. 36, 1–12 (2016)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan

    Akbar Zada & Sartaj Ali

Authors
  1. Akbar Zada
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sartaj Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akbar Zada.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zada, A., Ali, S. Stability of Integral Caputo-Type Boundary Value Problem with Noninstantaneous Impulses. Int. J. Appl. Comput. Math 5, 55 (2019). https://doi.org/10.1007/s40819-019-0640-0

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-019-0640-0

Keywords

Mathematics Subject Classification

Search

Navigation