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- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 214042, 17 pages
Positive Solutions for a Fractional Boundary Value Problem with Changing Sign Nonlinearity
1School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
2Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
Received 14 April 2012; Accepted 25 June 2012
Academic Editor: Bashir Ahmad
Copyright © 2012 Yongqing Wang et al. 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 discuss the existence of positive solutions to the following fractional m-point boundary value problem with changing sign nonlinearity , where is a positive parameter, , , with , is the standard Riemann-Liouville derivative, and may be singular at and/or and also may change sign. The work improves and generalizes some previous results.
In this paper, we consider the following fractional differential equation with -point boundary conditions: where , is a parameter, , with , is the standard Riemann-Liouville derivative, and may be singular at and/or and also may change sign. In this paper, by a positive solution to (1.1), we mean a function which is positive on and satisfies (1.1).
In recent years, great efforts have been made worldwide to study the existence of solutions for nonlinear fractional differential equations by using nonlinear analysis methods [1–24]. Fractional-order multipoint boundary value problems (BVP) have particularly attracted a great deal of attention (see, e.g., [13–19]). In , the authors discussed some properties of the Green function for the Direchlet-type BVP of nonlinear fractional differential equations where , is the standard Riemann-Liouville derivative and is continuous. By using the Krasnosel'skii fixed-point theorem, the existence of positive solutions was obtained under some suitable conditions on .
In , the authors investigated the existence and multiplicity of positive solutions by using some fixed-point theorems for the fractional differential equation given by where , , , with , , is nonnegative.
It should be noted that in most of the works in literature the nonlinearity needs to be nonnegative in order to establish positive solutions. As far as we know, semipositone fractional nonlocal boundary value problems with have been seldom studied due to the difficulties in finding and analyzing the corresponding Green function.
In , the authors investigated the following fractional differential equation with three-point boundary conditions: where , , , ,, , and satisfies the Caratheodory conditions. The authors obtained the properties of the Green function for (1.4) as follows: By using the Schauder fixed-point theorem, the authors obtained the existence of positive solution of (1.4) with the following assumptions: for each , there exists a function such that for a.e. , for all ; there exist , and , such that here is continuous and nonincreasing, is continuous, and is nondecreasing; There exist two positive constants such that Here The assumptions on nonlinearity are not suitable for frequently used conditions, such as superlinear or some sublinear. For instance, , , obviously, does not satisfy .
Inspired by the previous work, the aim of this paper is to establish conditions for the existence of positive solutions of the more general BVP (1.1). Our work presented in this paper has the following new features. Firstly, we consider few cases of which has been studied before, and in dealing with the difficulties related to the Green function for this case, some new properties of the Green function have been discovered. Secondly, the BVP (1.1) possesses singularity; that is, may be singular at and/or . Thirdly, the nonlinearity may change sign and may be unbounded from below. Finally, we impose weaker positivity conditions on the nonlocal boundary term; that is, some of the coefficients may be negative.
The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that are to be used to prove our main results. We also discover some new positive properties of the corresponding Green function. In Section 3, we discuss the existence of positive solutions of the semipositone BVP (1.1). In Section 4, we give an example to demonstrate the application of our theoretical results.
2. Basic Definitions and Preliminaries
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can also be found in the recent literature.
Definition 2.1. The fractional integral of order of a function is given by provided that the right-hand side is pointwisely defined on .
Definition 2.2. The Riemann-Liouville fractional derivative of order of a function is given by where and denotes the integer part of the number , provided that the right-hand side is pointwisely defined on .
Lemma 2.3 (see ). Let . Then the following equality holds for , ; where , , .
Set where For convenience in presentation, we here list the assumption to be used throughout the paper., on .
Remark 2.4. If , we have and . If and , we have on .
Lemma 2.5 (see ). Assume that and . Then
Lemma 2.6. Assume holds, and . Then the unique solution of the problem is where is the Green function of the boundary value problem (2.9).
Lemma 2.7. The function has the following properties:(1), for ;(2), for ;(3), for ,where
Proof. (1) When , it is clear that
When , we have
(2) By (2.4), for any , we have
In the following, we will prove (3).(i) When , noticing that , we have Therefore, which implies
On the other hand, we have Therefore, , which implies Then (ii) When , we have On the other hand, clearly we have The inequalities (2.23)–(2.28) imply that (3) holds.
By Lemma 2.7, we have the following results.
Lemma 2.8. Assume holds, then the Green function defined by (2.6) satisfies(1), for all ;(2), for all ;(3), for all ,where
Lemma 2.9. Assume holds, then the function satisfies(1), for all ;(2), for all ;(3), for all .
For convenience, we list here four more assumptions to be used later: satisfies where , .(), . There exists such that There exists such that
Remark 2.10. The second limit of implies where
Proof. By , for any , there exists , such that for any we have Let , for any we have Therefore, .
Lemma 2.11. Assume holds and is nonnegative, then the BVP has a unique solution with , where
Let be endowed with the maximum norm and define a cone by and then set , , .
Next we consider the following singular nonlinear BVP: where , , is defined in Lemma 2.11.
Clearly, if is a fixed point of , then is a positive solution of (2.41).
Lemma 2.12. Suppose that ()–() hold. Then is a completely continuous operator.
Proof. It is clear that is well defined on . For any , Lemma 2.9 implies
On the other hand,
Therefore, . Noticing that
we have .
Using the Ascoli-Arzela theorem, we can then get that is a completely continuous operator.
Lemma 2.13 (see ). Let be a real Banach space and let be a cone. Assume that and are two-bounded open subsets of with , a completely continuous operator such that either(1), and , , or(2), and , .Then has a fixed point in .
3. Existence of Positive Solutions
Theorem 3.1. Suppose that ()–() hold. Then there exists such that the BVP (1.1) has at least one positive solution for any .
Proof . Choose . Let
In the following of the proof, we suppose .
For any , noticing and Lemma 2.11, we have Therefore, Thus,
Now choose a real number By , there exists a constant such that Let Then for any , we have Thus, for any , we have . Hence, we get Therefore, By Lemma 2.13, has a fixed point such that . Let . Since , by (3.3) we have on and . Notice that is the solution of (2.37) and is the solution of (2.41). Thus, is a positive solution of the BVP (1.1).
Theorem 3.2. Suppose that – and hold. Then there exists such that the BVP (1.1) has at least one positive solution for any .
Proof. By the first limit of , there exists such that
Let In the following part of the proof, we suppose .
Let Then for any , we have Therefore, , for any and . Then This implies
On the other hand, is continuous on , and thus from the second limit of , we have where is defined by (3.2). For there exists such that for any and .
Let For any , by (3.16) we can get , for all . Therefore, Thus, By Lemma 2.13, has a fixed point such that . Let . Since , by (3.16) we have on and . Notice that is a solution of (2.37) and is a solution of (2.41). Thus, is a positive solution of the BVP (1.1).
By the proof of Theorem 3.2, we have the following corollary.
Corollary 3.3. The conclusion of Theorem 3.2 is valid if is replaced by . There exist and such that for any and ,
Example 4.1 (a 4-point BVP with coefficients of both signs). Consider the following problem:
We have By direct calculations, we have and , which implies that holds.
Let , , . It is easy to see that and hold. Moreover, Therefore, the assumptions of Theorem 3.1 are satisfied. Thus, Theorem 3.1 ensures that there exists such that the BVP (4.1) has at least one positive solution for any .
Remark 4.2. Noticing that does not satisfy , therefore, the work in the present paper improves and generalizes the main results of .
The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11101237) and the Natural Science Foundation of Shandong Province of China (ZR2011AQ008, ZR2011AL018). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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