Skip to main content
SpringerLink
Log in

On General Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper investigates the existence of solutions for fractional differential inclusions involving Caputo fractional derivative of any order together with nonlocal integral boundary conditions. Our study includes the cases when the multivalued map involved in the problem has convex as well as non-convex values. Some standard fixed point theorems for multivalued maps are applied to establish the main results, which are well illustrated with the aid of examples.

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.

Similar content being viewed by others

References

  1. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. Vol. 3 of Series on complexity, nonlinearity and chaos. World Scientific Publishing Co. Pte. Ltd., Singapore (2012)

  2. Gafiychuk, V., Datsko, B.: Mathematical modelling of different types ofinstabilities in time fractional reaction-diffusion system. Comput. Math. Appl. 59, 1101–1107 (2010)

    Article  MathSciNet  Google Scholar 

  3. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers Inc., USA (2006)

    Google Scholar 

  4. Kisielewicz, M.: Differential inclusions and optimal control. Mathematics and its Applications (East European Series), vol. 44. Kluwer, Dordrecht (1991)

  5. Agarwal, R.P., Belmekki, M. Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ., 47 (2009) (Art. ID: 981728)

  6. Agarwal, R.P., Ahmad, B.: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1200–1214 (2011)

    Article  MathSciNet  Google Scholar 

  7. Ahmad, B., Ntouyas, S.K.: A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order. Electron. J. Qual. Theory Differ. Equ. 22, 1–15 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Nyamoradi, N., Javidi, M.: Existence of multiple positive solutions for fractional differential inclusions with \(m\)-point boundary conditions and two fractional orders. Electron. J. Differ. Equ. 187, 26 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Nyamoradi, N., Baleanu, D., Agarwal, R.P.: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys., 9 (2013) (Art. ID: 823961)

  10. Ahmad, B., Ntouyas, S.K.: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 20, 1–19 (2013)

    Article  MathSciNet  Google Scholar 

  11. Ahmad, B., Ntouyas, S.K., Alsaedi, A.: On fractional differential inclusions with anti-periodic type integral boundary conditions. Bound. Value Probl. 2013(82), 1–15 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Ahmad, B., Matar, M.M., Agarwal, R.P.: Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions. Bound. Value Probl. 220, 13 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Matar, M.M.: On Existence of positive solution for initial value problem of nonlinear fractional differential equations of order \(1< \alpha \le 2\). Acta Math. Univ. Comen. LXXXIV, 51–57 (2015)

  14. Matar, M.M., El-Bohisie, F.A.: On Existence of solution for higher-order fractional differential inclusions with anti-periodic type boundary conditions. Br. J. Math. Comput. Sci. 7, 328–340 (2015)

    Article  Google Scholar 

  15. Abbas, S., Benchohra, M., N’Guerekata, G.M.: Topics in fractional differential equations. Springer, New York (2012)

    Book  Google Scholar 

  16. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  17. Hu, Sh., Papageorgiou, N.: Handbook of multivalued analysis, vol. I: Theory. Kluwer, Dordrecht (1997)

  18. Deimling, K.: Multivalued differential equations. Walter De Gruyter, New York (1992)

    Book  Google Scholar 

  19. Smirnov, G.V.: Introduction to the theory of differential inclusions. Graduate Studies in Mathematics, vol. 41. American Mathematical Society, Providence (2002)

  20. Tolstonogov, A.A.: Differential inclusions in a Banach Space. Kluwer, Dordrecht (2000)

    Book  Google Scholar 

  21. Bressan, A., Colombo, G.: Extensions and selections of maps with decomposable values. Stud. Math. 90, 69–86 (1988)

    Article  MathSciNet  Google Scholar 

  22. Granas, A., Dugundji, J.: Fixed point theory. Springer, New York (2005)

    MATH  Google Scholar 

  23. Covitz, H., Nadler Jr., S.B.: Multivalued contraction mappings in generalized metric spaces. Isr. J. Math. 8, 5–11 (1970)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Bashir Ahmad & Sotiris K. Ntouyas

  2. Mathematics Department, Al-Azhar University-Gaza, Gaza, Palestine

    Mohammed M. Matar

  3. Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece

    Sotiris K. Ntouyas

Authors
  1. Bashir Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammed M. Matar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sotiris K. Ntouyas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bashir Ahmad.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmad, B., Matar, M.M. & Ntouyas, S.K. On General Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions. Differ Equ Dyn Syst 28, 241–254 (2020). https://doi.org/10.1007/s12591-016-0319-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-016-0319-5

Keywords

Mathematics Subject Classification

Search

Navigation