Skip to main content
SpringerLink
Log in

Existence and Stability of Integro Differential Equation with Generalized Proportional Fractional Derivative

  • Published:
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) Aims and scope Submit manuscript

Abstract

In this study, integro-differential equations of arbitrary order are studied. The fractional order is expressed in terms of the \(\psi\)-Hilfer type proportional fractional operator. This research exposes the dynamical behavior of integro-differential equations with fractional order, such as existence, uniqueness, and stability solutions. The initial value problem and nonlocal conditions are used to prove the results.

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. M. Abbas and M. Ragusa, ‘‘On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function,’’ Symmetry 13, 264 (2021). https://doi.org/10.3390/sym13020264

    Article  Google Scholar 

  2. M. I. Abbas, ‘‘Controllability and Hyers–Ulam stability results of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivative,’’ Miskolc Math. Notes 22, 491 (2021). https://doi.org/10.18514/mmn.2021.3470

    Article  MathSciNet  MATH  Google Scholar 

  3. M. I. Abbas, ‘‘Existence results and the Ulam stability for fractional differential equations with hybrid proportional-Caputo derivatives,’’ J. Nonlinear Funct. Anal. 2020, 48 (2020). https://doi.org/10.23952/jnfa.2020.48

    Article  Google Scholar 

  4. B. Ahmad and J. J. Nieto, ‘‘Riemann–Liouville fractional differential equations with fractional boundary conditions,’’ Fixed Point Theory 13, 329–336 (2013).

    MathSciNet  MATH  Google Scholar 

  5. R. Almeida, ‘‘A Caputo fractional derivative of a function with respect to another function,’’ Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017). https://doi.org/10.1016/j.cnsns.2016.09.006

    Article  MathSciNet  MATH  Google Scholar 

  6. I. Ahmed, P. Kumam, F. Jarad, P. Borisut, and W. Jirakitpuwapat, ‘‘On Hilfer generalized proportional fractional derivative,’’ Adv. Difference Equations 2020, 329 (2020). https://doi.org/10.1186/s13662-020-02792-w

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Bashir and S. Sivasundaram, ‘‘Some existence results for fractional integro-differential equations with nonlocal conditions,’’ Commun. Appl. Anal. 12, 107–112 (2008).

    MathSciNet  MATH  Google Scholar 

  8. A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics (Springer, New York, 2003). https://doi.org/10.1007/978-0-387-21593-8

  9. Application of Fractional Calculus in Physics, Ed. by R. Hilfer (World Scientific, Singapore, 1999).

    Google Scholar 

  10. S. Harikrishnan, K. Shah, D. Baleanu, and K. Kanagarajan, ‘‘Note on the solution of random differential equations via \(\psi\)-Hilfer fractional derivative,’’ Adv. Difference Equations 2018, 224 (2018). https://doi.org/10.1186/s13662-018-1678-8

    Article  MATH  Google Scholar 

  11. Theory and Applications of Fractional Differential Equations, Ed. by A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, North-Holland Mathematics Studies, Vol. 204 (Elsevier, 2006). https://doi.org/10.1016/s0304-0208(06)80001-0

  12. V. Lupulescu and S. K. Ntouyas, ‘‘Random fractional differential equations,’’ Int. J. Pure Appl. Math. 4, 119–136 (2012).

    MathSciNet  MATH  Google Scholar 

  13. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Ed. by I. Podlubny, Mathematics in Science and Engineering, Vol. 198 (Elsevier, 1999). https://doi.org/10.1016/S0076-5392(13)60008-9

  14. Random Differential Equations in Science and Engineering, Ed. by T. T. Soong, Mathematics in Science and Engineering, Vol. 103 (Elsevier, New York, 1973). https://doi.org/10.1016/S0076-5392(08)60152-6

  15. J. V. da C. Sousa and E. Capelas de Oliveira, ‘‘On the \(\Psi\)-Hilfer fractional derivative,’’ Commun. Nonlinear Sci. Numer. Simul 60, 72–91 (2018).

    MathSciNet  Google Scholar 

  16. J. V. da C. Sousa and E. Capelas de Oliveira, ‘‘Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation,’’ Appl. Math. Lett. 81, 50–56 (2018). https://doi.org/10.1016/j.aml.2018.01.016

    Article  MathSciNet  Google Scholar 

  17. J. V. Da C. Sousa and E. Capelas de Oliveira, ‘‘On a new operator in fractional calculus and applications,’’ J. Fixed Point Theory Appl. 20, 1–21 (2018).

    MathSciNet  Google Scholar 

  18. J. V. Da C. Sousa and E. Capelas de Oliveira, ‘‘On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the \(\psi\)-Hilfer operator,’’ J. Fixed Point Theory Appl. 20, 96 (2018). https://doi.org/10.1007/s11784-018-0587-5

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Vu, ‘‘Random fractional functional differential equations,’’ Int. J. Nonlinear Anal. Appl. 7, 253–267 (2016). https://doi.org/10.22075/ijnaa.2017.980.1185

    Article  MATH  Google Scholar 

  20. D. Vivek, K. Kanagarajan, and E. M. Elsayed, ‘‘Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions,’’ Mediterr. J. Math. 15, 15 (2018). https://doi.org/10.1007/s00009-017-1061-0

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Wang, L. Lv, and Yo. Zhou, ‘‘Ulam stability and data dependence for fractional differential equations with Caputo derivative,’’ Electron. J. Qualitative Theory Differ. Equations 63, 1–10 (2011). https://doi.org/10.14232/ejqtde.2011.1.63

    Article  MathSciNet  MATH  Google Scholar 

Download references

ACKNOWLEDGEMENT

The authors are thankful to the anonymous reviewers and the handling editor for the fruitful comments that made the presentation of the work more interesting.

Author information

Authors and Affiliations

  1. Department of Mathematics, TIPS College of Arts and Science, Coimbatore, India

    S. Harikrishnan

  2. Department of Mathematics, PSG College of Arts and Science, Coimbatore, India

    D. Vivek

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    E. M. Elsayed

  4. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

    E. M. Elsayed

Authors
  1. S. Harikrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Vivek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. M. Elsayed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. Harikrishnan, D. Vivek or E. M. Elsayed.

Ethics declarations

The authors declare that they have no conflicts of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harikrishnan, S., Vivek, D. & Elsayed, E.M. Existence and Stability of Integro Differential Equation with Generalized Proportional Fractional Derivative. J. Contemp. Mathemat. Anal. 58, 253–263 (2023). https://doi.org/10.3103/S1068362323040040

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362323040040

Keywords:

Search

Navigation