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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 180672, 13 pages
http://dx.doi.org/10.1155/2012/180672
Research Article

The Existence of Positive Solutions for Fractional Differential Equations with Sign Changing Nonlinearities

College of Sciences, Hebei University of Science and Technology, Hebei, Shijiazhuang 050018, China

Received 23 March 2012; Revised 11 June 2012; Accepted 12 June 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Weihua Jiang 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.

Abstract

We investigate the existence of at least two positive solutions to eigenvalue problems of fractional differential equations with sign changing nonlinearities in more generalized boundary conditions. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. Some examples are given for the illustration of main results.

1. Introduction

The theory of fractional differential equations has become an important aspect of differential equations (see [18]). Boundary value problems of fractional differential equations have been investigated in many papers (see [946]). The existence of positive solutions to boundary value problems of fractional differential equations has been studied by many authors when nonlinearities are positive (see [924]). There are a few papers to study the existence of positive solutions of semipositone fractional differential equations. For example, using the Krasnoselskii fixed point theorem, Yuan et al. [9] discussed the existence of positive solutions for the singular positone and semipositone two-point boundary value problems where and are continuous. In [10], Wang et al. studied the existence of positive solutions for the singular semipositone two-point boundary value problems where , is continuous.

In [11], using Krasnoselskii fixed point theorem, Goodrich discussed the existence of at least one positive solutions for the system of fractional boundary value problems where , , , and are nonnegative for .

Motivated by the excellent results mentioned above, in this paper, we investigate the existence of at least two positive solutions for the problem where , , . The main tool is the Avery-Peterson theorem. To the best of our knowledge, this is the first paper dealing with eigenvalue problems of fractional differential equations with sign changing nonlinearities involving more general boundary conditions. Our results improve some of the earlier work presented in [10, 17, 46].

2. Preliminaries

For the convenience of the readers, we present here some necessary definitions and lemmas from fractional calculus theory. For more details see [1, 2].

Definition 2.1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

Definition 2.2. The Riemann-Liouville fractiontal derivative of order of a function is given by provided that the right-hand side is pointwise defined on , where .

Lemma 2.3. Assume , , then

Lemma 2.4. Assume , then(1)If , , , , then (2), .(3),  for a.e. , where .(4) if and only if for some , , where is the least integer greater than or equal to .

Lemma 2.5 (see[11]). Given , is a solution of the problem if and only if it satisfies where

Lemma 2.6 (see[11]). is continuous on and

Lemma 2.7. .

Proof. For , For ,

By simple calculation, we can get

By Lemma 2.5, we can easily get the following lemma.

Lemma 2.8. The boundary value problem has a unique solution
Obviously, satisfies

Lemma 2.9. is a solution of the following problem: if and only if is a positive solution of (1.5).

Proof. In fact, if is a positive solution of the problem (1.5), by Lemma 2.8, we get that satisfies Take . Then satisfies (2.16) and .
On the other hand, if is a solution of (2.16) and . Take . By Lemma 2.8, we can easily get that satisfies (1.5). Clearly, .

Define functions , and an operator by

Obviously, is a fixed point of the operator if and only if is a positive solution of the problem (1.5).

Take with norm . Define a cone by

Lemma 2.10. is a completely continuous operator.

Proof.. Take . By Lemmas 2.6 and 2.7, we get So, . Let be bounded. It follows from the continuity of , that there exist constants and such that and for . Thus, That is is bounded. For , By the uniform continuity of and , we get that is equicontinuous. Obviously, is continuous. By the Arzela-Ascoli theorem, we get that is completely continuous.

Definition 2.11. A map is said to be a nonnegative, continuous, and concave functional on a cone of a real Banach space if and only if is continuous and for all and .

Definition 2.12. A map is said to be a nonnegative, continuous, and convex functional on a cone of a real Banach space iff is continuous and for all and .

Let and be nonnegative, continuous, and convex functional on , a nonnegative, continuous, and concave functional on , and a nonnegative continuous functional on . Then, for positive numbers , , , and , we define the following sets:

We will use the following fixed point theorem of Avery and Peterson to study the problem (1.5).

Theorem 2.13 (see [47]). Let be a cone in a real Banach space . Let and be nonnegative, continuous, and convex functionals on , a nonnegative, continuous, and concave functional on , and a nonnegative continuous functional on satisfying for , such that for some positive numbers and , for all . Suppose that is completely continuous and there exist positive numbers , , with , such that the following conditions are satisfied:(S1) and for ;(S2) for with ;(S3) and for with .
Then has at least three fixed points , such that

3. Main Results

We define a concave function and convex functions .

Theorem 3.1. Assume that there exists a constant , such that for . In addition, suppose that there exist constants , , , , with , , such that the following conditions hold:(C1),  for ;(C2),  for ;(C3),  for .
Then the problem (1.5) has at least two positive solutions for

Proof. Take . By , , Lemma 2.6, (2.12), and (3.1), we have This means that .
It is easy to see that . implies , . It follows from (2.15) and (3.1) that . By , (2.12), (3.1), and Lemma 2.7, we get So, the condition of Theorem 2.13 holds.
Take with . By , we get Thus, holds.
By , we have . Take with . By , we get By Theorem 2.13, we get that has at least three fixed points such that , and
If and , by (2.15) and (3.1), we have
Obviously, and . So, are two positive solutions of (1.5). The proof is completed.

4. Example

For convenience, we define the following notations:

Example 4.1. Consider the following boundary value problem: where is a bounded variation function on with , Corresponding to the problem (1.5), we get that . Take .
By simple calculation, we can get that the conditions of Theorem 3.1 are satisfied. So, when , the problem (4.2) has at least two positive solutions.

Example 4.2. Consider the following boundary value problem: where is a bounded variation function on with , Corresponding to the problem (1.5), we get that . Take .
Obviously, . By simple calculation, we can get that satisfy ,, and
It is easy to see that , for , and , for . So, conditions and of Theorem 3.1 hold.
For . Therefore, condition of Theorem 3.1 holds. So, for , the problem (4.4) has at least two positive solutions.
Specially, in Example 4.2, we take where and all other conditions remain unchanged. Then . Clearly, . The problem (4.4) has at least two positive solutions for .

Acknowledgments

The authors are grateful to Editor Bashir Ahmad and anonymous referees for their constructive comments and suggestions which led to improvement of the paper. This work is supported by the Natural Science Foundation of China (11171088), the Doctoral Program Foundation of Hebei University of Science and Technology (QD201020), and the Foundation of Hebei University of Science and Technology (XL201236).

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