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Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 271–283 | Cite as

Positivity for integral boundary value problems of fractional differential equations with two nonlinear terms

  • Mengrui Xu
  • Shurong SunEmail author
Original Research
  • 225 Downloads

Abstract

We investigate a class of integral boundary value problems of fractional differential equations with two nonlinear terms, one is non-monotone and the other contain fractional derivative. Existence of positive solutions is obtained by the method of upper and lower solutions and Schauder fixed point theorem and unique result is presented from Banach contraction mapping principle. Several examples are given to show the applicability of our main results.

Keywords

Fractional differential equations Boundary value problems Positive solutions Existence 

Mathematics Subject Classification

34A08 34B18 34A12 

Notes

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by Shandong Provincial Natural Science Foundation (ZR2016AM17).

References

  1. 1.
    Podlubny, I.: Fractional Differential Equation. Academic Press, New York (1999)zbMATHGoogle Scholar
  2. 2.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, The Netherlands (2006)zbMATHGoogle Scholar
  3. 3.
    Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)CrossRefzbMATHGoogle Scholar
  4. 4.
    Sun, S., Zhao, Y., Han, Z., Li, Y.: The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun Nonlinear Sci. Numer. Simul. 17, 4961–4967 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jia, M., Zhang, H., Chen, Q.: Existence of positive solutions for fractional differential equation with integral boundary conditions on the half-line. Bound. Value Probl. 104, 1–16 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Zhao, Y., Sun, S., Han, Z., Li, Q.: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 2086–2097 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhao, Y., Sun, S., Han, Z., Zhang, M.: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 217, 6950–6958 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Liu, X., Jia, M.: Multiple solutions for fractional differential equations with nonlinear boundary conditions. Comput. Math. Appl. 59, 2880–2886 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Staněk, S.: Periodic problem for two-term fractional differential equations. Fract. Calc. Appl. Anal. 20, 662–678 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Thiramanus, P., Ntouyas, S., Tariboon, J.: Existence of solutions for Riemann–Liouville fractional differential equations with nonlocal Erdélyi–Kober integral boundary conditions on the half-line. Bound. Value Probl. 196, 1–15 (2015)zbMATHGoogle Scholar
  13. 13.
    Liu, X., Jia, M., Ge, W.: The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator. Appl. Math. Lett. 65, 56–62 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jia, M., Liu, X.: Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions. Appl. Math. Comput. 232, 313–323 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bǎleanu, D., Agarwal, R.P., Khan, H., Khan, R.A., Jafari, H.: On the existence of solution for fractional differential equations of order \(3 < \delta _1 \le 4\). Adv. Differ. Equ. 2015, 362 (2015)CrossRefzbMATHGoogle Scholar
  16. 16.
    Agarwal, R.P., Bǎleanu, D., Rezapour, S., Salehi, S.: The existence of solutions for some fractional finite difference equations via sum boundary conditions. Adv. Differ. Equ. 2014, 282 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bǎleanu, D., Mustafa, O.G., Agarwal, R.P.: On \(L^p\)-solutions for a class of sequential fractional differential equations. Appl. Math. Comput. 218, 2074–2081 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Boulares, H., Ardjouni, A., Laskri, Y.: Positive solutions for nonlinear fractional differential equations. Positivity 21, 1201–1212 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of JinanJinanPeople’s Republic of China

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