Skip to main content
SpringerLink
Log in

Existence and Asymptotic Behavior of Positive Solutions for a Coupled System of Semilinear Fractional Differential Equations

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study a coupled system of semilinear Riemann–Liouville type fractional differential equations. Applying the Schäuder fixed point theorem, we prove the existence of positive solutions of the fractional differential system. Furthermore, the asymptotic behavior of the solutions is given by using Karamata regular variation theory.

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. Ahmad B., Nieto J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal R.P., Bǎleanu D., Hedayati V., Rezapour S.: Two fractional derivative inclusion problems via integral boundary condition. Appl. Math. Comput. 257, 205–212 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Bai Z., Fang J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Bai Z., Lü H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bǎleanu, D., Agarwal, R.P., Mohammadi, H., Rezapour, S.: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl. 2013, 8 pp. (2013)

  6. Bǎleanu D., Mustafa O.G., Agarwal R.P.: An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 23(9), 1129–1132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bǎleanu, D., Rezapour, S., Etemad, S., Alsaedi, A.: On a time-fractional integrodifferential equation via three-point boundary value conditions. Math. Probl. Eng. 2015, 12 pp. (2015)

  8. Cabada A., Wang G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389(1), 403–411 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chemmam R., Mâagli H., Masmoudi S., Zribi M.: Combined effects in nonlinear singular elliptic problems in a bounded domain. Adv. Nonlinear Anal. 1, 301–318 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Chen Y., An H.: Numerical solutions of coupled Burgers equations with time and space fractional derivatives. Appl. Math. Comput. 200, 87–95 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Diethelm K., Ford N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gafiychuk V., Datsko B., Meleshko V.: Mathematical modeling of time fractional reaction–diffusion systems. J. Comput. Appl. Math. 220, 215–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, H., Han, X.: Existence of positive solutions for fractional differential equation with nonlocal boundary condition. Int. J. Differ. Equ. 2011, 10 pp. (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam (2006)

    Google Scholar 

  16. Kilbas A.A., Trujillo J.J.: Differential equations of fractional order: methods, results and problems. I. Appl. Anal. 78, 153–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kilbas A.A., Trujillo J.J.: Differential equations of fractional order: methods, results and problems. II. Appl. Anal. 81, 435–493 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kosmatov N.: Countably many solutions of a fourth order boundary value problem. Electron. J. Differ. Equ. 2004, 1–15 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Lazarevi M.P.: Finite time stability analysis of PD α fractional control of robotic time-delay systems. Mech. Res. Commun. 33, 269–279 (2006)

    Article  Google Scholar 

  20. Li C.F., Luo X.N., Zhou Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Anal. 99, 1363–1375 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Liang S., Zhang J.: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 71(11), 5545–5550 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mâagli, H., Mhadhbi, N., Zeddini, N.: Existence and exact asymptotic behavior of positive solutions for a fractional boundary value problem. Abstr. Appl. Anal. 2013, 6 pp. (2013)

  23. Maric V.: Regular Variation and Equations. Lecture Notes in Mathematics, vol. 1726. Springer, Berlin (2000)

    Google Scholar 

  24. Metzler R., Klafter J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nyamoradi N.: Multiple positive solutions for fractional differential systems. Ann. Univ. Ferrara 2012(58), 359–369 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nyamoradi, N., Bǎleanu, D., Agarwal R.P.: Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions. Adv. Differ. Equ. 2013(266), 11 pp. (2013)

  27. Nyamoradi N., Bashiri T., Vaezpour S.M., Bǎleanu D.: Uniqueness and existence of positive solutions for singular fractional differential equations. Electron. J. Differ. Equ. 2014, 1–13 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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. 2012, 1–26 (2012)

    Article  MATH  Google Scholar 

  29. Oldham K.B., Spanier J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  30. Podlubny I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

    Google Scholar 

  31. Scher H., Montroll E.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B. 12, 2455–2477 (1975)

    Article  Google Scholar 

  32. Seneta R.: Regular Varying Functions. Lectures Notes in Mathematics, vol. 508. Springer, Berlin (1976)

    Book  Google Scholar 

  33. Su X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Timoshenko S.: Theory of Elastic Stability. McGraw-Hill, New York (1961)

    Google Scholar 

  35. Wang G., Agarwal R.P., Cabada A.: Existence results and monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25, 1019–1024 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, G., Ahmad, B., Zhang, L.: A Coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain. Abstr. Appl. Anal. 2012, 11 pp. (2012)

  37. Wang G., Liu W., Ren C.: Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions. Electron. J. Differ. Equ. 2012, 1–10 (2012)

    Article  MATH  Google Scholar 

  38. Wang, J., Xiang, H., Liu, Z.: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010, 12 pp. (2010)

  39. Yang, A., Ge, W.: Positive solutions for boundary value problems of N-Dimension nonlinear fractional differential system. Bound. Value Probl. 2008, 15 pp. (2008)

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang S.: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59(3), 1300–1309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Basic Sciences, College of Education, University of Dammam, Eastern Province, Saudi Arabia

    Hassan Yahya Alfifi

  2. Faculty of Sciences and Humanities of Khulais, Jeddah University, Jeddah, Saudi Arabia

    Imen Ben Saad

  3. Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, University of Dammam, P.O Box 1982, Dammam, 31441, Saudi Arabia

    Sameh Turki

  4. Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092, Tunis, Tunisia

    Zagharide Zine El Abidine

Authors
  1. Hassan Yahya Alfifi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Imen Ben Saad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sameh Turki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zagharide Zine El Abidine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zagharide Zine El Abidine.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alfifi, H.Y., Ben Saad, I., Turki, S. et al. Existence and Asymptotic Behavior of Positive Solutions for a Coupled System of Semilinear Fractional Differential Equations. Results Math 71, 705–730 (2017). https://doi.org/10.1007/s00025-016-0528-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-016-0528-9

Mathematics Subject Classification

Keywords

Search

Navigation