In this article, by employing a fixed point theorem in cones, we investigate the existence of a positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions. We also obtain some relations between the solution and Green’s function.

MSC: 26A33, 34B15, 34B16, 34G20.

In this article, we consider the existence of a positive solution for the following singular semipositone fractional differential equations:

where , , , , is the standard Riemann-Liouville derivative, may be singular at and/or . Since the nonlinearity may change sign, the problem studied in this paper is called the semipositone problem in the literature which arises naturally in chemical reactor theory. Up to now, much attention has been attached to the existence of positive solutions for semipositone differential equations and the system of differential equations; see [1-11] and references therein to name a few.

Boundary value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics. Moreover, boundary value problems with integral conditions constitute a very interesting and important class of problems. They include two-point, three-point, multi-point, and nonlocal boundary value problems as special cases, which have received much attention from many authors. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [12], Karakostas and Tsamatos [13], Lomtatidze and Malaguti [14], and the references therein.

On the other hand, fractional differential equations have been of great interest for many researchers recently. This is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of science and engineering such as control, porous media, electromagnetic, and other fields. For an extensive collection of such results, we refer the readers to the monographs by Samko et al.[15], Podlubny [16] and Kilbas et al.[17]. For the case where α is an integer, a lot of work has been done dealing with local and nonlocal boundary value problems. For example, in [18] Webb studied the nth-order nonlocal BVP

where can have singularities, and the nonlinearity f satisfies Carathéodory conditions. Under weak assumptions, Webb obtained sharp results on the existence of positive solutions under a suitable condition on f. In [19] Hao et al. consider the nth-order singular nonlocal BVP

where is a parameter, a may be singular at and/or , may also have singularity at .

In two recent papers [20] and [21], by means of the fixed point theory and fixed point index theory, the authors investigated the existence and multiplicity of positive solutions for the following two kinds of fractional differential equations with integral boundary value problems:

and

where , and are the standard Riemann-Liouville derivative and the Caputo fractional derivative, respectively.

To the author’s knowledge, there are few papers in the literature to consider fractional differential equations with integral boundary value conditions. Motivated by above papers, the purpose of this article is to investigate the existence of positive solutions for the more general fractional differential equations BVP (1). Firstly, we derive corresponding Green’s function known as fractional Green’s function and argue its positivity. Then a fixed point theorem is used to obtain the existence of positive solutions for BVP (1). We also obtain some relations between the solution and Green’s function. From the example given in Section 4, we know that λ in this article may be greater than 2 and η may take the value 1. Therefore, compared with that in [21], BVP (1) considered in this article has a more general form.

The rest of this article is organized as follows. In Section 2, we give some preliminaries and lemmas. The main result is formulated in Section 3, and an example is worked out in Section 4 to illustrate how to use our main result.

Let , , then is a Banach space. Denote , , .

For the reader’s convenience, we present some necessary definitions from fractional calculus theory and lemmas. They can be found in the recent literature; see [14-17].

Definition 2.1 The Riemann-Liouville fractional integral of order of a function is given by

provided the right-hand side is pointwise defined on .

Definition 2.2 The Riemann-Liouville fractional derivative of order of a continuous function is given by

where , denotes the integer part of the number α, provided that the right-hand side is pointwise defined on .

From the definition of the Riemann-Liouville derivative, we can obtain the statement.

Lemma 2.1 ([17])

Let. If we assume, then the fractional differential equation

has, , , as unique solutions, whereNis the smallest integer greater than or equal toα.

Lemma 2.2 ([17])

Assume thatwith a fractional derivative of orderthat belongs to.

Then

for some, , whereNis the smallest integer greater than or equal toα.

In the following, we present Green’s function of the fractional differential equation boundary value problem.

Lemma 2.3 Given, the problem

where, , , , is equivalent to

where

Here, , is called the Green function of BVP (2). Obviously, is continuous on.

Proof We may apply Lemma 2.2 to reduce (2) to an equivalent integral equation

for some . Consequently, the general solution of (2) is

By , one gets that . On the other hand, combining with

yields

Therefore, the unique solution of the problem (2) is

For , one has

For , one has

The proof is complete. □

Lemma 2.4 The functiondefined by (3) satisfies

(a1) , ;

(a2) , ;

(a3) , ;

(a4) , andis not decreasing on;

(a5) , ,

where, , .

Proof For , ,

For ,

For ,

For , ,

From above, (a1), (a2), (a3), (a5) are complete. Clearly, (a4) is true. The proof is complete. □

Throughout this article, we adopt the following conditions.

(H₁) and there exist , , such that

(H₂) There exists such that uniformly for ;

(H₃) There exists such that

Let

Obviously, Q is a cone in a Banach space E and is an ordering Banach space.

Let

where is defined as that in (H₁). It follows from Lemma 2.4 and (H₃) that

So, and it satisfies

For any , , . Consequently, by (6) and Lemma 2.4, we have

For any , denote

We define an operator A as follows:

Lemma 2.5Suppose that ()-() hold. Thenis completely continuous.

Proof For any , it is clear that . By (H₁), we get

By (10) and Lemma 2.4, we have

which together with (H₃) means that operator A defined by (9) is well defined.

Now, we show that .

For any , by (H₁) we have by (9) and Lemma 2.4 that

which means that

It follows from (12) and Lemma 2.4 that

Thus, A maps Q into Q.

Finally, we prove that A maps Q into Q is completely continuous.

Let be any bounded set. Then there exists a constant such that for any . Notice that , for any , , by (H₃) and (11), we have

Therefore, is uniformly bounded.

On the other hand, since is continuous on , it is uniformly continuous on as well. Thus, for fixed and for any , there exists a constant such that for any and ,

Therefore, for any , we get by (10) and (13)

which implies that the operator A is equicontinuous. Thus, the Ascoli-Arzela theorem guarantees that is a relatively compact set.

Let , (). Then is bounded. Let , by (10), we get

By (9), we have

It follows from (14), (15), (H₁), (H₃), and the Lebesgue dominated convergence theorem that A is continuous. Thus, we have proved the continuity of the operator A. This completes the complete continuity of A. □

To prove the main result, we need the following well-known fixed point theorem.

Lemma 2.6 (Fixed point theorem of cone expansion and compression of norm type [22])

Letandbe two bounded open sets in a Banach spaceEsuch thatandbe a completely continuous operator, whereθdenotes the zero element ofEandPa cone ofE. Suppose that one of the two conditions holds:

(i) , ; , ;

(ii) , ; , .

ThenAhas a fixed point in.

Theorem 3.1Assume that conditions ()-() are satisfied. Then the singular semipositone BVP (1) has at least one positive solution. Furthermore, there exist two constantssuch that

Proof Firstly, we show that the operator A has a fixed point in Q. Let

where r is the same as that defined in (H₃). For any , by (10) and (12), we have that

Therefore,

which together with (H₃) implies that

For in (H₂), it is clear that

By (H₃), we know that there exists a natural number big enough such that

Choose

By (H₂), we know there exists such that

Take

In the following, we are in a position to show that

For any , by (8) we get

which together with (18), (19), (22), and (H₃) implies that

For , , it follows from (H₁), (20), (21), (22), and (24) that

By (25), we know that (23) holds. So, (17), (23), and Lemma 2.6 guarantee that A has at least one fixed point in and . Furthermore,

By simple computation, we have that

Secondly, we show BVP (1) has a positive solution. It follows from (8) and the fact that

which combined with (19) implies that

By (27) and (28), we have

Let , . It follows from (28) and that

By (7), (29), and (30), we obtain

Thus, we have proved that is a positive solution for BVP (1).

Finally, we show that (16) holds. From (26) and Lemma 2.4, we know that

Since , (30), and (31) mean that (16) holds for and holds. This completes the proof of Theorem 3.1. □

Consider the following singular semipositone fractional differential equations:

where . It is clear (32) has the form of (1), where , , . By simple computation, we know that , . Let

Notice that

we have

It follows from the left side of (33) that

Considering , we get

By (34) and (35) we know (H₁) holds. Obviously, (H₂) holds for .

Now, we check (H₃). By simple computation, we have , , , , . Take , then , . Thus, (H₃) is valid. It follows from Theorem 3.1 that BVP (32) has at least one positive solution.

The author declares that he has no competing interests.

The author thanks the referee for his/her careful reading of the manuscript and useful suggestions. The project is supported financially by the Foundation for Outstanding Middle-Aged and Young Scientists of Shandong Province (Grant No. BS2010SF004), a Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J10LA53, No. J11LA02), the China Postdoctoral Science Foundation (Grant No. 20110491154) and the National Natural Science Foundation of China (Grant No. 10971179).

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