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.
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).
- Anuradha V, Hai DD, Shivaji R: Existence results for superlinear semipositone BVP’s. Proc. Am. Math. Soc. 1996, 124: 747-763. doi:10.1090/S0002-9939-96-03256-XMathSciNetView ArticleGoogle Scholar
- Agarwal RP, O’Regan D: A note on existence of nonnegative solutions to singular semi-positone problems. Nonlinear Anal. 1999, 36: 615-622. doi:10.1016/S0362-546X(98)00181-3 10.1016/S0362-546X(98)00181-3MathSciNetView ArticleGoogle Scholar
- Xu X: Positive solutions for singular semi-positone boundary value problems. J. Math. Anal. Appl. 2002, 273: 480-491. doi:10.1016/S0022-247X(02)00259-7 10.1016/S0022-247X(02)00259-7MathSciNetView ArticleGoogle Scholar
- Zhao Z: Existence of positive solutions for 2 n th-order singular semipositone differential equations with Sturm-Liouville boundary conditions. Nonlinear Anal. 2010, 72: 1348-1357. doi:10.1016/j.na.2009.08.013 10.1016/j.na.2009.08.013MathSciNetView ArticleGoogle Scholar
- Ma R: Positive solutions for semipositone conjugate boundary value problems. J. Math. Anal. Appl. 2000, 252: 220-229. doi:10.1006/jmaa.2000.6987 10.1006/jmaa.2000.6987MathSciNetView ArticleGoogle Scholar
- Ma R, Ma Q: Positive solutions for semipositone m -point boundary-value problems. Acta Math. Sin. 2004, 20(2):273-282. doi:10.1007/s10114-003-0251-9 10.1007/s10114-003-0251-9MathSciNetView ArticleGoogle Scholar
- Su H, Liu L, Wu Y: Positive solutions for a nonlinear second-order semipositone boundary value system. Nonlinear Anal. 2009, 71: 3240-3248. doi:10.1016/j.na.2009.01.201 10.1016/j.na.2009.01.201MathSciNetView ArticleGoogle Scholar
- Liu Y: Twin solutions to singular semipositone problems. J. Math. Anal. Appl. 2003, 286: 248-260. doi:10.1016/S0022-247X(03)00478-5 10.1016/S0022-247X(03)00478-5MathSciNetView ArticleGoogle Scholar
- Zhang X, Liu L: Positive solutions of superlinear semipositone singular Dirichlet boundary value problems. J. Math. Anal. Appl. 2006, 316: 535-537. doi:10.1016/j.jmaa.2005.04.081Google Scholar
- Zhang X, Liu L: Existence of positive solutions for a singular semipositone differential system. Math. Comput. Model. 2008, 47: 115-126. doi:10.1016/j.mcm.2007.02.008 10.1016/j.mcm.2007.02.008View ArticleGoogle Scholar
- Zhang X, Liu L, Wu Y: On existence of positive solutions of a two-point boundary value problem for a nonlinear singular semipositone system. Appl. Math. Comput. 2007, 192: 223-232. doi:10.1016/j.amc.2007.03.002 10.1016/j.amc.2007.03.002MathSciNetView ArticleGoogle Scholar
- Gallardo JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 2000, 30: 1265-1292. doi:10.1216/rmjm/1021477351 10.1216/rmjm/1021477351MathSciNetView ArticleGoogle Scholar
- Karakostas GL, Tsamatos PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002., 2002: Article ID 30Google Scholar
- Lomtatidze A, Malaguti L: On a nonlocal boundary-value problems for second order nonlinear singular differential equations. Georgian Math. J. 2000, 7: 133-154. doi:10.1515/GMJ.2000.133MathSciNetGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivative: Theory and Applications. Gordon & Breach, Switzerland; 1993.Google Scholar
- Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Webb JRL: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 2009, 71: 1933-1940. doi:10.1016/j.na.2009.01.033 10.1016/j.na.2009.01.033MathSciNetView ArticleGoogle Scholar
- Hao X, Liu L, Wu Y, Sun Q: Positive solutions for nonlinear n th-order singular eigenvalue problem with nonlocal conditions. Nonlinear Anal. 2010, 73: 1653-1662. doi:10.1016/j.na.2010.04.074 10.1016/j.na.2010.04.074MathSciNetView ArticleGoogle Scholar
- Wang Y, Liu L, Wu Y: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 2011, 74: 3599-3605. doi:10.1016/j.na.2011.02.043 10.1016/j.na.2011.02.043MathSciNetView ArticleGoogle Scholar
- Cabada A, Wang G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 2012, 389: 403-411. doi:10.1016/j.jmaa.2011.11.065 10.1016/j.jmaa.2011.11.065MathSciNetView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.Google Scholar
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