_{1}

In this paper, by means of constructing a special cone, we obtain a sufficient condition for the existence of positive solution to semipositone fractional differential equation.

The aim of this paper is to investigate the existence of positive solutions to the semipositone fractional differential equation

where

where

Fractional differential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modelling of many phenomena in engineering and sciences such as physics, mechanics, economics and biology. In recent years, there exist a great deal of researches on the existence and/or uniqueness of solutions (or positive solutions) to boundary value problems for fractional-order differential equations. Sun [

where

The purpose of the present paper is to apply the method of varying translation together with the fixed point theorems in cone to discuss (1) without nonnegativity imposed on the nonlinearity. Meanwhile, we also allow the nonlinearity to have many finite singularities on

In this section, we present several lemmas that are useful to the proof of our main results. For the forthcoming analysis, we need the following assumptions:

(H_{1})

such that

(H_{2})

In [

Lemma 2.1 [

is given by

where

Lemma 2.2 [

here

Remark 2.1. A simple computation shows that there exists a constant

Remark 2.2 [_{1}), then for any

and for any

Lemma 2.3 [

(i) Suppose that

(ii) Suppose that

Consider the Banach space

cone

to (2) for

where

Lemma 2.4. If the singular nonlinear boundary value problem (2) has a positive solution

Proof. In fact, if u is a positive solution to (6) such that

have

For any

Since for any fixed

_{1}), we have

Consequently, for any

Therefore, the operator T is well defined and

Lemma 2.5. Assume that (H_{1}), (H_{2}) hold. Then

Proof. For any

Whence, it follows from (8) that

Next we show that

Thus, we have

and

which implies

In what follows, we need to prove that

Let

Consequently

Therefore

Now we show that

Thus,

Lemma 2.6. Let

Proof. Assume that there exists

This contradiction shows that

Lemma 2.7. There exists a constant

Proof. Choose constants

From Remark (2.2), there exists

Let

fact, otherwise, there exists

So

Consequently,

contradiction shows that

Theorem 3.1. Suppose that (H_{1}), (H_{2}) hold. Then, the boundary value problems (1) has at least one positive solution

Proof of Theorem 3.1. Applying Lemma 2.6 and Lemma 2.7 and the definition of the fixed point index, we have

Let

We thank the Editor and the referee for their comments. This research was supported financially by the National Natural Science Foundation of China (11471187, 11571197), the Natural Science Foundation of Shandong Province of China (ZR2014AL004) and the Project of Shandong Province Higher Educational Science and Technology Program (J14LI08), the Project of Scientific and Technological of Qufu Normal University (XKJ201303).

Xinsheng Du, (2016) Existence of Positive Solutions to Semipositone Fractional Differential Equations. Applied Mathematics,07,1484-1489. doi: 10.4236/am.2016.714127