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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 740760, 14 pages
Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations
1School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, Sichuan, China
2School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, China
Received 22 October 2012; Accepted 19 November 2012
Academic Editor: Dragoş-Pătru Covei
Copyright © 2012 Jing Wu and Xinguang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation = = where , , , , satisfying , is the standard Riemann-Liouville derivative, , and is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.
In this paper, we consider the eigenvalue interval for existence of positive solutions to the following semipositone higher order fractional differential equation: where , , , , and , , satisfying , is the standard Riemann-Liouville derivative, is continuous.
Recently, one has found that fractional models can sufficiently describe the operation of variety of computational, economic mathematics, physical, and biological processes and systems, see [1–9]. Accordingly, considerable attention has been paid to the solution of fractional differential equations, integral equations, and fractional partial differential equations of physical phenomena [10–24]. One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. It possesses advantages of fast convergence, higher stability and higher accuracy to derive the solution of different types of fractional equations.
In this work, we will deal with the eigenvalue interval for existence of positive solutions to the higher order fractional differential equation when may be negative. This type of differential equation is called semipositone problem which arises in many interesting applications as pointed out by Lions in . For example, the semipositone differential equation which can be derived from chemical reactor theory, design of suspension bridges, combustion, and management of natural resources, see [26–28]. To our knowledge, few results were established, especially for higher order multipoint boundary value problems with the fractional derivatives.
2. Preliminaries and Lemmas
We use the following assumptions in this paper: is continuous, and there exist functions and continuous function such that
Now we begin this section with some preliminaries of fractional calculus. Let 0 and , where is the smallest integer greater than or equal to . For a function , we define the fractional integral of order of as provided the integral exists. The fractional derivative of order of a continuous function is defined by provided the right side is pointwise defined on . We recall the following properties [8, 9] which are useful for the sequel.
Lemma 2.2 (see ). Assume that and . Then where , is the smallest integer greater than or equal to .
Let , and consider the following modified integro-differential equation:
Lemma 2.3. The higher order multipoint boundary value problem (1.1) has a positive solution if and only if nonlinear integro-differential equation (2.8) has a positive solution. Moreover, if is a positive solution of (2.8), then is positive solution of the higher order multipoint boundary value problem (1.1).
Lemma 2.4. If and , then the boundary value problem has the unique solution where is the Green function of the boundary value problem (2.9), and
Lemma 2.5. The Green function of the boundary value problem (2.9) satisfies
Define a modified function for any by and consider the following boundary value problem
Lemma 2.6. Suppose , is a solution of the problem (2.16), then is a positive solution of the problem (2.8), consequently, is also a positive solution of the semipositone higher differential equation (1.1).
Proof. Since is a solution of the BVP (2.16) and for any , then we have Let , then we have and , which implies that Substituting the above into (2.17), then solves the (2.8), that is, is a positive solution of the semipositone differential equation (2.8). By Lemma 2.3, is a positive solution of the singular semipositone differential equation (1.1). This completes the proof of Lemma 2.5.
It is well known that the BVP (2.17) is equivalent to the fixed points for the mapping by
The basic space used in this paper is , where is the set of real numbers. Obviously, the space is a Banach space if it is endowed with the norm as follows: for any . Let then is a cone of .
Lemma 2.8. Assume that holds. Then is a completely continuous operator.
Proof. By using similar method to  and standard arguments, according to the Ascoli-Arzela Theorem, one can show that is a completely continuous operator.
Lemma 2.9 (see ). Let be a real Banach space, be a cone. Assume are two bounded open subsets of with , , and let be a completely continuous operator such that either(1) and , or (2) and .Then has a fixed point in ).
3. Main Results
Theorem 3.1. Suppose that holds, and Then there exists some constant such that the higher order multipoint boundary value problem (1.1) has at least one positive solution for any .
Proof. By Lemma 2.8, we know is a completely continuous operator. Take
where is defined by Lemma 2.5. Let . Then, for any , , we have
and is defined by Lemma 2.7. Thus, for any , , and , by (3.3), we have
On the other hand, choose a real number such that By (3.1), for any , there exists a constant such that Take let and . Then for any , , by Lemma 2.7, we have And then, for any , , one gets It follows from (3.12) that, for any , So, we have By Lemma 2.9, has at least a fixed point such that .
It follows from and that
Let , then By Lemma 2.6, we know that the differential equation (1.1) has at least a positive solutions .
Theorem 3.2. Suppose holds and Then there exists such that the higher order multipoint boundary value problem (1.1) has at least one positive solution for any .
Proof. By (3.17), there exists such that for any we have
In the following of the proof, we suppose . Take
and let and . Then for any , , by Lemma 2.7, we have
So for any , , one gets Thus, by (3.22), for any , we have So, we have
Next, take Let us choose such that Then for the above , by (3.17), there exists such that, for any , Thus, by (3.3) and (3.27), if we have
Take where Then .
Now let and . Then, for any , we have which implies that By Lemma 2.9, has at least a fixed point such that .
It follows from that
Let , then By Lemma 2.6, we know that the differential equation (1.1) has at least a positive solutions .
Example 3.3. Consider the existence of positive solutions for the nonlinear higher order fractional differential equation with four-point boundary condition Then there exists such that the higher order four-point boundary value problem (1.1) has at least one positive solution for any .
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