Positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions
© Zhang; licensee Springer 2012
Received: 20 July 2012
Accepted: 10 October 2012
Published: 24 October 2012
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.
Keywordsfractional differential equations integral boundary value problem positive solution semipositone cone
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 , Karakostas and Tsamatos , Lomtatidze and Malaguti , and the references therein.
where is a parameter, a may be singular at and/or , may also have singularity at .
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 , 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 , , .
provided the right-hand side is pointwise defined on .
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 ()
has, , , as unique solutions, where N is the smallest integer greater than or equal to α.
Lemma 2.2 ()
Assume thatwith a fractional derivative of orderthat belongs to.
for some, , where N is the smallest integer greater than or equal to α.
In the following, we present Green’s function of the fractional differential equation boundary value problem.
Here, , is called the Green function of BVP (2). Obviously, is continuous on.
The proof is complete. □
Lemma 2.4 The functiondefined by (3) satisfies
(a1) , ;
(a2) , ;
(a3) , ;
(a4) , andis not decreasing on;
(a5) , ,
where, , .
From above, (a1), (a2), (a3), (a5) are complete. Clearly, (a4) is true. The proof is complete. □
Throughout this article, we adopt the following conditions.
(H2) There exists such that uniformly for ;
Obviously, Q is a cone in a Banach space E and is an ordering Banach space.
Lemma 2.5 Suppose that ()-() hold. Thenis completely continuous.
which together with (H3) means that operator A defined by (9) is well defined.
Now, we show that .
Thus, A maps Q into Q.
Finally, we prove that A maps Q into Q is completely continuous.
Therefore, is uniformly bounded.
which implies that the operator A is equicontinuous. Thus, the Ascoli-Arzela theorem guarantees that is a relatively compact set.
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 )
, ; , ;
, ; , .
Then A has a fixed point in.
3 Main result
Thus, we have proved that is a positive solution for BVP (1).
Since , (30), and (31) mean that (16) holds for and holds. This completes the proof of Theorem 3.1. □
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 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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