Advertisement

Qualitative properties of solution for hybrid nonlinear fractional differential equations

  • Mohammed M. MatarEmail author
Article
  • 9 Downloads

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.

Keywords

Fractional differential equations Positive solution Control functions Existence Uniqueness Monotonicity Dhage 

Mathematics Subject Classification

26A33 34A12 34G20 

Notes

References

  1. 1.
    Abdeljawad, T., Alzabut, J., Jarad, F.: A generalized Lyaponuv-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. 2017, 321 (2017)CrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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. 4.
    Bai, Z.B., Qiu, T.T.: Existence of positive solution for singular fractional differential equation. Appl. Math. Comput. 215, 2761–2767 (2009)MathSciNetzbMATHGoogle Scholar
  5. 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 zbMATHGoogle Scholar
  6. 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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dhage, B.C., Lakshmikantham, V.: Basic results on hybrid differential equations. Nonlinear Anal. Hybrid 4, 414–424 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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
  9. 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)CrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)Google Scholar
  16. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, C.: Existence and stability of periodic solutions for parabolic systems with time delays. J. Math. Anal. Appl. 339, 1354–1361 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)Google Scholar
  19. 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)Google Scholar
  20. 20.
    Wang, C., Yang, Z.: Method of Upper and Lower Solutions for Reaction Diffusion Systems with Delay. Science Press, Beijing (2013) (in Chinese) Google Scholar
  21. 21.
    Zhang, S.: The existence of a positive solution for a fractional differential equation. J. Math. Anal. Appl. 252, 804–812 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)CrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsAl-Azhar University-GazaGazaPalestine

Personalised recommendations