SpringerLink
Log in

Existence of a system of fractional order differential equations via generalized contraction map** in partially ordered Banach space

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

H. K. Nashine et al. introduced \(\chi \)-set contraction condition in partially ordered Banach space (Nashine in Adv Differ Equ 2020:1–13, 2020). Motivated by the work of H. K. Nashine et al., we present generalizations of new Darbo-type fixed point and coupled fixed point theorems having some relaxed conditions on operator in partially ordered Banach space. The fixed point theorem presented in this article is a generalization of an analogues work of V. Parvaneh et al. (Adv Differ Equ 2020:243, 2020) and some other results defined for the abstract Banach space. As an application, we illustrate the existence of a solution for a system of fractional integral equations with some local conditions. Finally, the realization of our findings is shown by a real example.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data availability

There are no data associated with this paper.

References

  1. Guo D, Lakshmikantham V, Liu X (2013) Nonlinear integral equations in abstract spaces, vol 373. Springer, New York

    Google Scholar 

  2. Banaś J (1980) On measures of noncompactness in Banach spaces. Commentationes Mathematicae Universitatis Carolinae 21(1):131–143

    MathSciNet  Google Scholar 

  3. Akhmerov RR, Kamenskii MI, Potapov AS, Rodkina A, Sadovskii BN (1992) Measures of noncompactness and condensing operators, vol 55. Springer, New York

    Book  Google Scholar 

  4. Kiryakova V (2010) The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput Math Appl 59(5):1885–1895

  5. Machado JT, Lopes AM (2017) A fractional perspective on the trajectory control of redundant and hyper-redundant robot manipulators. Appl Math Modell 46:716–726

    Article  MathSciNet  Google Scholar 

  6. Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59(5):1586–1593

    Article  MathSciNet  Google Scholar 

  7. Nashine HK, Ibrahim RW, Agarwal RP, Can N (2020) Existence of local fractional integral equation via a measure of non-compactness with monotone property on Banach spaces. Adv Differ Equ 2020(1):1–13

    Article  MathSciNet  Google Scholar 

  8. Parvaneh V, Khorshidi M, De La Sen M, Işık H, Mursaleen M (2020) Measure of noncompactness and a generalized darbo’s fixed point theorem and its applications to a system of integral equations. Adv Differ Equ 2020(1):243

    Article  MathSciNet  Google Scholar 

  9. Winkel B (2017) 2017-Gustafson, G.B.-differential equations course materials

  10. Altun I, Turkoglu D (2007) A fixed point theorem for map**s satisfying a general contractive condition of operator type. J Comput Anal Appl 9(1):9

    MathSciNet  Google Scholar 

  11. Garcia-Falset J, Latrach K (2015) On Darbo-Sadovskii’s fixed point theorems type for abstract measures of (weak) noncompactness. Bull Belgian Math Soc Simon Stevin 22(5):797–812

    Article  MathSciNet  Google Scholar 

  12. Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):73–85

    Google Scholar 

  13. Yépez-Martínez H, Gómez-Aguilar JF (2019) A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (mham). J Comput Appl Math 346:247–260

    Article  MathSciNet  Google Scholar 

  14. Yameni Noupoue YY, Tandoğdu Y, Awadalla M (2019) On numerical techniques for solving the fractional logistic differential equation. Adv Differ Equ 2019(1):108

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author expresses his gratitude to the referees for constructive and useful remarks and suggestions.

Funding

This research work was carried out without specific grant from any funding agency.

Author information

Author notes

  1. Ajay Kumar Shukla and Dhananjay Gopal have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics and Humanities, S. V. National Institute of Technology, Ichhanath, Surat, Gujrat, 395007, India

    Vishal Nikam, Ajay Kumar Shukla & Dhananjay Gopal

  2. Department of Mathematics, Guru Ghasidas Vishwavidyalaya, Koni, Bilaspur, Chattisgarh, 495009, India

    Dhananjay Gopal

Authors
  1. Vishal Nikam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ajay Kumar Shukla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dhananjay Gopal
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All three authors contributed equally to the completion of this research work.

Corresponding author

Correspondence to Vishal Nikam.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nikam, V., Shukla, A.K. & Gopal, D. Existence of a system of fractional order differential equations via generalized contraction map** in partially ordered Banach space. Int. J. Dynam. Control 12, 125–135 (2024). https://doi.org/10.1007/s40435-023-01245-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-023-01245-y

Keywords

Search

Navigation