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We discuss the existence of positive solutions of a boundary value problem of nonlinear fractional differential equation with changing sign nonlinearity. We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.

Recently, much attention has been paid to the existence of solutions for fractional differential equations due to its wide range of applications in engineering, economics, and many other fields, and for more details see, for instance, [

To the best of our knowledge, there are only very few papers dealing with the existence of positive solutions of semipositone fractional boundary value problems due to the difficulties in finding and analyzing the corresponding Green function. The purpose of this paper is to establish the existence of positive solutions to the following nonlinear fractional differential equation boundary value problem:

The rest of the paper is organized as follows. In Section

In this section, we present some preliminaries and lemmas that are useful to the proof of our main results. For the convenience of the reader, we also present here some necessary definitions from fractional calculus theory. These definitions can be found in the recent literature.

The Riemann-Liouville fractional integral of order

The Caputo’s fractional derivative of order

Given

The function

It is obvious that (1) holds. In the following, we will prove (2) and (3).

(i) When

On the other hand, since

(ii) When

On the other hand, as

By Lemma

In [

For any

If

For the convenience of presentation, we list here the hypotheses to be used later.

Assume that

By Lemma

Let

Next we consider the following boundary value problem:

Let

For any

According to the Ascoli-Arzela theorem, we can easily get that

Let

Suppose that

Choose

For any

Now choose a real number

Suppose that

By the first limit of

Let

On the other hand, as

Select

The conclusion of Theorem

Consider the following problem

Consider the following problem

The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11101237) and the Natural Science Foundation of Shandong Province of China (ZR2011AQ008