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; cone
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 , Karakostas and Tsamatos , Lomtatidze and Malaguti , 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., Podlubny  and Kilbas et al.. 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  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  Hao et al. consider the nth-order singular nonlocal BVP
In two recent papers  and , 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:
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
From the definition of the Riemann-Liouville derivative, we can obtain the statement.
Lemma 2.1 ()
Lemma 2.2 ()
In the following, we present Green’s function of the fractional differential equation boundary value problem.
Proof We may apply Lemma 2.2 to reduce (2) to an equivalent integral equation
Therefore, the unique solution of the problem (2) is
The proof is complete. □
From above, (a1), (a2), (a3), (a5) are complete. Clearly, (a4) is true. The proof is complete. □
Throughout this article, we adopt the following conditions.
We define an operator A as follows:
By (10) and Lemma 2.4, we have
which together with (H3) means that operator A defined by (9) is well defined.
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.
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 )
3 Main result
Proof Firstly, we show that the operator A has a fixed point in Q. Let
which together with (H3) implies that
In the following, we are in a position to show that
which together with (18), (19), (22), and (H3) implies that
By simple computation, we have that
which combined with (19) implies that
By (27) and (28), we have
By (7), (29), and (30), we obtain
Finally, we show that (16) holds. From (26) and Lemma 2.4, we know that
Consider the following singular semipositone fractional differential equations:
It follows from the left side of (33) that
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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