Skip to main content
SpringerLink
Log in

On a Coupled Impulsive Fractional Integrodifferential System with Hadamard Derivatives

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

The main intention of the present research study is focused on the analysis of coupled impulsive fractional integrodifferential system having Hadamard derivatives. With the help of fixed point theorem attributed to Krasnoselskii’s, we investigate desired existence and uniqueness results. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability using the classical technique of functional analysis. Next, an example is designed to examine our findings based on the procedures applied in the theorems.

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

\footnotesize Figure 1
\footnotesize Figure 2

Similar content being viewed by others

Availability of Data and Materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Abbreviations

\(\mathcal {FDE}'s\) :

Fractional differential equations

\({\mathcal {R}}\)–\({\mathcal {L}}\) :

Riemann–Liouville

\(\mathcal {HUS}\) :

Hyers–Ulam stability

\(\mathcal {GHUS}\) :

Generalized Hyers–Ulam stability

\(\mathcal {HURS}\) :

Hyers–Ulam–Rassias stability

\(\mathcal {GHURS}\) :

Generalized Hyers–Ulam–Rassias stability

References

  1. Abdo, M.S., Abdeljawad, T., Shah, K., Jarad, F.: Study of impulsive problems under Mittag–Leffler power law. Heliyon 6, e05109 (2020)

  2. Ahmad, M., Zada, A., Alzabut, J.: Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type. Demonstr. Math. 52(1), 283–295 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alam, M., Shah, D.: Hyers–Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives. Chaos Solitons Fractals 150, 111122 (2021)

    Article  MathSciNet  Google Scholar 

  4. Alam, M., Zada, A., Popa, I.L., et al.: A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability. Bound. Value Probl. 2021, 73 (2021)

    Article  MathSciNet  Google Scholar 

  5. Ali, Z., Zada, A., Shah, K.: Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem. Bound. Value Probl. 2018(1), 1–16 (2018)

    Article  MathSciNet  Google Scholar 

  6. Altman, M.: A fixed point theorem for completely continuous operators in Banach spaces. Bull. Acad. Pol. Sci. 3, 409–413 (1955)

    MathSciNet  MATH  Google Scholar 

  7. Andronov, A., Witt, A., Haykin, S.: Oscilation Theory. Nauka, Moskow (1981)

    Google Scholar 

  8. Babitskii, V., Krupenin, V.: Vibration in Strongly Nonlinear Systems. Nauka, Moskow (1985)

    Google Scholar 

  9. Babusci, D., Dattoli, G., Sacchetti, D.: The Lamb–Bateman integral equation and the fractional derivatives. Fract. Calc. Appl. Anal. 14(2), 317–320 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bainov, D., Dimitrova, M., Dishliev, A.: Oscillation of the bounded solutions of impulsive differential–difference equations of second order. Appl. Math. Comput. 114(1), 61–68 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Benchohra, M., Lazreg, J.E.: Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative. Stud. Univ. Babes-Bolyai Math. 62(1), 27–38 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chernousko, F., Akulenko, L., Sokolov, B.: Control of Oscillations. Nauka, Moskow (1980)

    Google Scholar 

  13. Chua, L., Yang, L.: Cellular neural networks: applications. IEEE Trans Circuits Syst. CAS 1988(35), 1273–1290 (1988)

    Article  MathSciNet  Google Scholar 

  14. Garra, R., Polito, F.: On some operators involving Hadamard derivatives. Integr. Transf. Spec. Funct. 24(10), 773–782 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hadamard, J.: Essai sur l’étude des fonctions, données par leur développement de Taylor. Gauthier-Villars (1892)

  16. Hao, X., Sun, H., Liu, L., Wang, D.: Positive solutions for semipositone fractional integral boundary value problem on the half-line. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113(4), 3055–3067 (2019)

  17. Jiang, J., O’Regan, D., Xu, J., Fu, Z.: Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. J. Inequal. Appl. 2019(1), 1–18 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jung, S.M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 19, 854–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equation. North-Holland and Mathematics Studies, 204, Elsevier Science B. V, Amsterdam (2006)

    MATH  Google Scholar 

  20. Lakshmikanthan, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  21. Luo, D., Alam, M., Zada, A., Riaz, U., Luo, Z.: Existence and stability of implicit fractional differential equations with Stieltjes boundary conditions having Hadamard derivatives. Complexity 2021(3), 1–36 (2021)

    Google Scholar 

  22. Obloza, M.: Hyers stability of the linear differential equation. Rocznik NaukDydakt Prace Mat. 13, 259–270 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Oldham, K.B.: Fractional differential equations in electrochemistry. Adv. Eng. Softw. 41, 9–12 (2010)

    Article  MATH  Google Scholar 

  24. Popov, E.: The Dynamics of Automatic Control Systems. Gostehizdat, Moskow (1964)

    Google Scholar 

  25. Riaz, U., Zada, A., Ali, Z., et al.: Analysis of nonlinear coupled systems of impulsive fractional differential equations with Hadamard derivatives. Math. Probl. Eng. 2019, 20 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Riaz, U., Zada, A., Ali, Z., et al.: On a Riemann–Liouville type implicit coupled system via generalized boundary conditions. Mathematics 2021(9), 1205 (2021)

    Article  Google Scholar 

  27. Riaz, U., Zada, A., Ali, Z., Cui, Y., Xu, J.: Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives. Adv. Differ. Equ. 2019(226), 1–27 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Rihan, F.A.: Numerical modeling of fractional order biological systems. Abs. Appl. Anal. 2013, 816803 (2013). https://doi.org/10.1155/2013/816803

  29. Rizwan, R., Zada, A., Wang, X.: Stability analysis of non linear implicit fractional Langevin equation with noninstantaneous impulses. Adv. Differ. Equ. 2019(1), 1–31 (2019)

    Article  MATH  Google Scholar 

  30. Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpath. J. Math. 26, 103–107 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus. Springer, Dordrecht (2007)

    Book  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. Shah, K., Ali, A., Bushnaq, S.: Hyers–Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions. Math. Methods Appl. Sci. 41(17), 8329–8343 (2018)

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

  35. Subramanian, M., Alzabut, J., Baleanu, D., et al.: Existence, uniqueness and stability analysis of a coupled fractional-order differential systems involving Hadamard derivatives and associated with multi-point boundary conditions. Adv. Differ. Equ. 2021, 267 (2021)

    Article  MathSciNet  Google Scholar 

  36. Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  37. Ulam, S.M.: A Collection of the Mathematical Problems. Interscience, New York (1960)

    MATH  Google Scholar 

  38. Wang, X., Alam, M., Zada, A.: On coupled impulsive fractional integro-differential equations with Riemann–Liouville derivatives. AIMS Math. 6(2), 1561–1595 (2020)

    Article  MathSciNet  Google Scholar 

  39. Wang, G., Pei, K., Chen, Y.: Stability analysis of nonlinear Hadamard fractional differential system. J. Frankl. Inst. 356(12), 6538–6546 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, J., Shah, K., Ali, A.: Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations. Math. Methods Appl. Sci. 41(6), 2392–2402 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Xu, J., Goodrich, C.S., Cui, Y.: Positive solutions for a system of first-order discrete fractional boundary value problems with semipositone nonlinearities. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113(2), 1343–1358 (2019)

  42. Zada, A., Alam, M., Riaz, U.: Analysis of q-fractional implicit boundary value problem having Stieltjes integral conditions. Math. Methods Appl. Sci. 44, 4381–4413 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zada, A., Ali, W., Park, C.: Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall–Bellman–Bihari’s type. Appl. Math. Comput. 350, 60–65 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Zada, A., Alzabut, J., Waheed, H., Popa, I.L.: Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions. Adv. Differ. Equ. 2020, 64 (2020)

    Article  MathSciNet  Google Scholar 

  45. Zada, A., Riaz, U., Khan, F.: Hyers–Ulam stability of impulsive integral equations. Boll. Unione Mat. Ital. 12(3), 453–467 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zavalishchin, S., Sesekin, A.: Impulsive Processes: Models and Applications. Nauka, Moskow (1991)

    MATH  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Mehboob Alam, Akbar Zada & Usman Riaz

Authors
  1. Mehboob Alam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akbar Zada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Usman Riaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Ethics declarations

Conflict of interest

The authors declared that they don’t have any conflict of interest regarding this manuscript.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alam, M., Zada, A. & Riaz, U. On a Coupled Impulsive Fractional Integrodifferential System with Hadamard Derivatives. Qual. Theory Dyn. Syst. 21, 8 (2022). https://doi.org/10.1007/s12346-021-00535-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00535-0

Keywords

Mathematics Subject Classification

Search

Navigation