SpringerLink
Log in

Existence of mild solutions for multi-term time fractional measure differential equations

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

This paper focuses on a class of multi-term time-fractional measure differential equations involving nonlocal conditions. A set of sufficient conditions for the existence of mild solutions is provided. Utilizing \((\beta ,\gamma _{k})\)-resolvent family in the sense of Pardo and Lizama, measure functional (Henstock–Lebesgue–Stieltjes integral) and Schauder’s fixed point theorem, we prove the existence of mild solutions. Finally, an illustrated example is presented.

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

Similar content being viewed by others

References

  1. Podlubny, I. 1999. Fractional differential equations. New York: Academic Press.

    MATH  Google Scholar 

  2. Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo. 2006. Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204. Amsterdam: Elsevier Science B. V.

    Google Scholar 

  3. Keyantuo, V., C. Lizama, and M. Warma. 2013. Asymptotic behavior of fractional order semilinear evolution equations. Differential and Integral Equations 26 (7/8): 757–780.

    MathSciNet  MATH  Google Scholar 

  4. Trong, L.V. 2016. Decay mild solutions for two-term time fractional differential equations in Banach spaces. Journal Fixed Point Theory and Applications 18 (2016): 417–432.

    MathSciNet  MATH  Google Scholar 

  5. Singh, V., and D.N. Pandey. 2020. Controllability of multi-term time-fractional differential systems. Journal of Control and Decision 7 (2): 109–125.

    Article  MathSciNet  Google Scholar 

  6. Pardo, E.A., and C. Lizama. 2014. Mild solutions for multi-term time-fractional differential equations with nonlocal initial conditions. Electronic Journal of Differential Equations 39: 1–10.

    MathSciNet  MATH  Google Scholar 

  7. Brogliato, B. 1996. Nonsmooth Mechanics: Models, Dynamics, and Control. Berlin: Springer.

    MATH  Google Scholar 

  8. Lygeros, J., Tomlin, C., and S. Sastry, 2008. Hybrid Systems: Modeling, Analysis and Control.

  9. Cao, Y., and J. Sun. 2015. Existence of solutions for semilinear measure driven equations. Journal of Mathematical Analysis and Applications 425 (2): 621–631.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, Y., and J. Sun. 2016. Measures of noncompactness in spaces of regulated functions with application to semilinear measure driven equations. Boundary Value Problems 2016 (1): 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  11. Gou, H., and Li, Y. 2021 . Existence and Approximate Controllability of Semilinear Measure Driven Systems with Nonlocal Conditions. Bulletin of the Iranian Mathematical Society, 1–21.

  12. Gu, H., and Y. Sun. 2020. Nonlocal controllability of fractional measure evolution equation. Journal of Inequalities and Applications 1: 1–18.

    MathSciNet  MATH  Google Scholar 

  13. Kumar, S., and S. Mohammad Abdal. 2021. Approximate controllability of non-instantaneous impulsive semilinear measure driven control system with infinite delay via fundamental solution. IMA Journal of Mathematical Control and Information 38 (2): 552–575.

    Article  MathSciNet  MATH  Google Scholar 

  14. Surendra, K., and P.A. Ravi. 2020. Existence of solution non-autonomous semilinear measure driven equations. Differential Equation & Application 12 (3): 313–322. https://doi.org/10.7153/dea-2020-12-20.

    Article  MathSciNet  MATH  Google Scholar 

  15. Cao, Y., and J. Sun. 2018. Approximate controllability of semilinear measure driven systems. Mathematische Nachrichten 291 (13): 1979–1988.

    Article  MathSciNet  MATH  Google Scholar 

  16. Zavalishchin, S.T., and A.N. Sesekin. 1997. Dynamic impulse systems: Theory and applications. Dordrecht: Kluwer Academic Publishers.

    Book  MATH  Google Scholar 

  17. Satco, B. 2014. Regulated solutions for nonlinear measure driven equations. Nonlinear Analysis: Hybrid Systems 13: 22–31.

    MathSciNet  MATH  Google Scholar 

  18. Mesquita, J.G. 2012. Measure Functional Differential Equations and Impulsive Functional Dynamic Equations on Time Scales, Universidade de Sao Paulo, Brazil, Ph.D. thesis.

  19. Cao, Y., and J. Sun. 2016. On existence of nonlinear measure driven equations involving non-absolutely convergent integrals. Nonlinear Analysis: Hybrid Systems 20: 72–81.

    MathSciNet  MATH  Google Scholar 

  20. Hernandez, E., and D. O’Regan. 2013. On a new class of abstract impulsive differential equations. Proceedings of the American Mathematical Society 141: 1641–1649.

    Article  MathSciNet  MATH  Google Scholar 

  21. Diop, Amadou, Mamadou Abdoul Diop, Khalil Ezzinbi, and Paul dit Akouni Guindo. 2022. Optimal controls problems for some impulsive stochastic integro-differential equations with state-dependent delay. Stochastics. https://doi.org/10.1080/17442508.2022.2029446.

    Article  Google Scholar 

  22. Gordon, R.A. 1994. The Integrals of Lebesgue, Denjoy, Perron and Henstock. In Grad. Stud. Math., vol. 4. Providence: AMS.

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Analyse Numérique et Informatique, Département de Mathématiques, Université Gaston Berger de Saint-Louis,  UFR SAT, B. P. 234, Saint-Louis, Senegal

    Amadou Diop

Authors
  1. Amadou Diop
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amadou Diop.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Human participants

This article does not contain any studies with human participants contain any studies with human participants or animals performed by the author.

Additional information

Communicated by Samy Ponnusamy.

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

Diop, A. Existence of mild solutions for multi-term time fractional measure differential equations. J Anal 30, 1609–1623 (2022). https://doi.org/10.1007/s41478-022-00420-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-022-00420-2

Keywords

Mathematics Subject Classification

Search

Navigation