Journal of Applied Mathematics and Computing

, Volume 52, Issue 1–2, pp 477–488 | Cite as

Positive solutions for singular fractional differential equations with three-point boundary conditions

  • Bingxian Li
  • Shurong Sun
  • Zhenlai HanEmail author
Original Research


In this paper, we consider the nonlinear three-point boundary value problem of singular fractional differential equations
$$\begin{aligned} D^{\alpha }_{0^+}u(t)+a(t)f(t,u(t))=0,\quad 0<t<1,\;2<\alpha \le 3 \end{aligned}$$
with boundary conditions
$$\begin{aligned}u(0)=0,\quad D^{\beta }_{0^+}u(0)=0,\quad D^{\beta }_{0^{+}}u(1)=bD^{\beta }_{0^{+}}u(\xi ),\quad 1\le \beta \le 2 \end{aligned}$$
involving Riemann–Liouville fractional derivatives \(D^{\alpha }_{0^{+}}\) and \(D^{\beta }_{0^{+}}\). The nonlinear term f permits singularities with respect to both the time and space variables. We obtain several local existence and multiplicity of positive solutions theorems by introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions. An example is given to show the applicability of our main results.


Fractional differential equations Existence and multiplicity  Singular boundary value problem Positive solution 

Mathematics Subject Classification

34A08 34B16 34B18 



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 the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003).


  1. 1.
    Oldham, K., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)zbMATHGoogle Scholar
  2. 2.
    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993)zbMATHGoogle Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  4. 4.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. The Netherlands, Amsterdam (2006)zbMATHGoogle Scholar
  5. 5.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integral and Derivative. Theory and Applications. Gordon and Breach, Switzerland (1993)zbMATHGoogle Scholar
  6. 6.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Berlin. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  7. 7.
    Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)CrossRefzbMATHGoogle Scholar
  8. 8.
    Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Elsevier, Amsterdam (2015)Google Scholar
  9. 9.
    Zhai, C., Yan, W., Yang, C.: A sum operator method for the existence and uniqueness of positive solutions to Riemann–Liouville fractional differential equation boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 18, 858–866 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, C., Luo, X., Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363–1375 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, B., Sun, S., Li, Y.: Multi-point boundary value problems for a class of Riemann–Liouville fractional differential equations. Adv. Differ. Equ. 151, 1–11 (2014)MathSciNetGoogle Scholar
  12. 12.
    Feng, W., Sun, S., Li, X., Xu, M.: Positive solutions to fractional boundary value problems with nonlinear boundary conditions. Bound. Value Probl. 225, 1–15 (2014)MathSciNetGoogle Scholar
  13. 13.
    Zhang, X.: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22–27 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Feng, W., Sun, S., Han, Z., Zhao, Y.: Existence of solutions for a singular system of nonlinear fractional diffrential equations. Comput. Math. Appl. 62, 1370–1378 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sun, S., Zhao, Y., Han, Z., Xu, M.: Uniqueness of positive solutions for boundary value problems of singular fractional diffrential equations. Inverse Probl. Sci. Eng. 20, 299–309 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Staněk, S.: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 62, 1379–1388 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)zbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2015

Authors and Affiliations

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

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