Abstract
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.
Keywords:
fractional differential equations; integral boundary value problem; positive solution; semipositone; cone1 Introduction
In this article, we consider the existence of a positive solution for the following singular semipositone fractional differential equations:
where , , , , is the standard RiemannLiouville 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 [111] 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, thermoelasticity, and plasma physics. Moreover, boundary value problems with integral conditions constitute a very interesting and important class of problems. They include twopoint, threepoint, multipoint, 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 nthorder 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 nthorder 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 RiemannLiouville 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.
2 Preliminaries and several lemmas
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 [1417].
Definition 2.1 The RiemannLiouville fractional integral of order of a function is given by
provided the righthand side is pointwise defined on .
Definition 2.2 The RiemannLiouville fractional derivative of order of a continuous function is given by
where , denotes the integer part of the number α, provided that the righthand side is pointwise defined on .
From the definition of the RiemannLiouville 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.
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
The proof is complete. □
Lemma 2.4 The functiondefined by (3) satisfies
(a4) , andis not decreasing on;
From above, (a1), (a2), (a3), (a5) are complete. Clearly, (a4) is true. The proof is complete. □
Throughout this article, we adopt the following conditions.
(H_{1}) and there exist , , such that
(H_{2}) There exists such that uniformly for ;
(H_{3}) 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 (H_{1}). It follows from Lemma 2.4 and (H_{3}) that
For any , , . Consequently, by (6) and Lemma 2.4, we have
We define an operator A as follows:
Lemma 2.5Suppose that ()() hold. Thenis completely continuous.
Proof For any , it is clear that . By (H_{1}), we get
By (10) and Lemma 2.4, we have
which together with (H_{3}) means that operator A defined by (9) is well defined.
For any , by (H_{1}) 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 (H_{3}) 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 AscoliArzela 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), (H_{1}), (H_{3}), 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 wellknown 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:
3 Main result
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 (H_{3}). For any , by (10) and (12), we have that
Therefore,
which together with (H_{3}) implies that
For in (H_{2}), it is clear that
By (H_{3}), we know that there exists a natural number big enough such that
Choose
By (H_{2}), we know there exists such that
Take
In the following, we are in a position to show that
which together with (18), (19), (22), and (H_{3}) implies that
For , , it follows from (H_{1}), (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. □
4 Example
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
By (34) and (35) we know (H_{1}) holds. Obviously, (H_{2}) holds for .
Now, we check (H_{3}). By simple computation, we have , , , , . Take , then , . Thus, (H_{3}) is valid. It follows from Theorem 3.1 that BVP (32) has at least one positive solution.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
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 MiddleAged 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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