Skip to main content
SpringerLink
Log in

Qualitative properties of solution for hybrid nonlinear fractional differential equations

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this article we investigate some qualitative properties for a class of hybrid nonlinear fractional differential equations. The existence, uniqueness, monotonicity and positivity of the solution are studied by the method of upper and lower control functions and using Dhage fixed point theorem. Some examples are introduced to illustrate the applicability of 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. Abdeljawad, T., Alzabut, J., Jarad, F.: A generalized Lyaponuv-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. 2017, 321 (2017)

    Article  Google Scholar 

  2. 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. 2015, 220 (2015)

    Article  MathSciNet  Google Scholar 

  3. Ahmad, B., Matar, M.M., Ntouyas, S.K.: On general fractional differential inclusions with nonlocal integral boundary conditions. Differ. Equ. Dyn. Syst. (2016). https://doi.org/10.1007/s12591-016-0319-5

  4. Bai, Z.B., Qiu, T.T.: Existence of positive solution for singular fractional differential equation. Appl. Math. Comput. 215, 2761–2767 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Boulares, H., Ardjouni, A., Laskri, Y.: Positive solutions for nonlinear fractional differential equations. Positivity (2016). https://doi.org/10.1007/s11117-016-0461-x

    Article  MATH  Google Scholar 

  6. Dhage, B.C.: On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations. J. Math. Phys. Sci. 25, 603–611 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Dhage, B.C., Lakshmikantham, V.: Basic results on hybrid differential equations. Nonlinear Anal. Hybrid 4, 414–424 (2010)

    Article  MathSciNet  Google Scholar 

  8. Duprez, M., Perasso, A.: Criterion of positivity for semilinear problems with applications in biology. Positivity (2017). https://doi.org/10.1007/s11117-017-0474-0

    Article  MathSciNet  Google Scholar 

  9. Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 226, 3457–3471 (2017)

    Article  Google Scholar 

  10. Kaufmann, E., Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 3, 1–11 (2008)

    Article  MathSciNet  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  12. Li, N., Wang, C.: New existence results of positive solution for a class of nonlinear fractional differential equations. Acta Math. Sci. 33B, 847–854 (2013)

    Article  MathSciNet  Google Scholar 

  13. Matar, M.: Existence and uniqueness of solutions to fractional semilinear mixed Volterra–Fredholm integrodifferential equations with nonlocal conditions. Electron. J. Differ. Equ. 155, 1–7 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Matar, M.: On existence and uniqueness of the mild solution for fractional semilinear integro-differential equations. J. Integral Equ. Appl. 23, 457–466 (2011)

    Article  MathSciNet  Google Scholar 

  15. Matar, M.: On existence of positive solution for initial value problem of nonlinear fractional differential equations of arder \(1<\alpha \le 2\). Acta Math. Univ. Comenianae, vol. LXXXIV, 1, pp. 51–57 (2015)

  16. Matar, M., Trujillo, J.J.: Existence of local solutions for differential equations with arbitrary fractional order. Arab. J. Math. 5, 215 (2016). https://doi.org/10.1007/s40065-015-0139-4

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, C.: Existence and stability of periodic solutions for parabolic systems with time delays. J. Math. Anal. Appl. 339, 1354–1361 (2008)

    Article  MathSciNet  Google Scholar 

  18. Wang, C., Wang, R., Wang, S., Yang, C.: Positive solution of singular boundary value problem for a nonlinear fractional differential equation. Bound. Value Probl., Art ID 297026 (2011)

  19. Wang, C., Zhang, H., Wang, S.: Positive solution of a nonlinear fractional differential equation involving Caputo derivative. Discrete Dyn. Nat. Soc., Art ID 425408 (2012)

  20. Wang, C., Yang, Z.: Method of Upper and Lower Solutions for Reaction Diffusion Systems with Delay. Science Press, Beijing (2013) (in Chinese)

  21. Zhang, S.: The existence of a positive solution for a fractional differential equation. J. Math. Anal. Appl. 252, 804–812 (2000)

    Article  MathSciNet  Google Scholar 

  22. Zhao, Y., Sun, S., Han, S., Li, Q.: Theory of fractional hybrid differential equations. Comput. Math. Appl. 62(3), 1312–1324 (2011). https://doi.org/10.1016/j.camwa.2011.03.041

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhou, H., Alzabut, J., Yang, L.: On fractional Langevin differential equations with anti-periodic boundary conditions. Eur. Phys. J. Spec. Top. 226, 3577–3590 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Al-Azhar University-Gaza, Gaza, Palestine

    Mohammed M. Matar

Authors
  1. Mohammed M. Matar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed M. Matar.

Additional information

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

Matar, M.M. Qualitative properties of solution for hybrid nonlinear fractional differential equations. Afr. Mat. 30, 1169–1179 (2019). https://doi.org/10.1007/s13370-019-00710-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-019-00710-2

Keywords

Mathematics Subject Classification

Search

Navigation