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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

Abstract

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.

Keywords

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

Mathematics Subject Classification

34A08 34B16 34B18 

Notes

Acknowledgments

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).

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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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