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Journal of Applied Mathematics and Computing

, Volume 52, Issue 1–2, pp 477–488

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

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

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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