Abstract and Applied Analysis

Volume 2013 (2013), Article ID 531038, 12 pages

http://dx.doi.org/10.1155/2013/531038

Research Article

The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applications

Fuquan Jiang,¹ Xiaojie Xu,² and Zhongwei Cao³

¹The Department of Foundation, Harbin Finance University, Harbin 150030, China

²College of Science, China University of Petroleum (East China), Qingdao 266580, China

³Department of Applied Mathematics, Changchun Taxation College, Changchun 130117, China

Received 13 July 2012; Revised 10 December 2012; Accepted 24 December 2012

Academic Editor: Dumitru Baleanu

Copyright © 2013 Fuquan 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 consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem: , where , is the standard Riemann-Liouville derivative. Here our nonlinearity may be singular at . As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.

1. Introduction

Fractional differential equations have been of great interest recently. This is due to the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in selfsimilar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science. For details, see [1–10].

It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. Recently, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional differential equations boundary value problems, see [11–17].

Bai [14] investigated the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation

by contraction map principle and fixed-point index theory, where , , , is the standard Riemann-Liouville derivative. The function is continuous on .

Li et al. [17] investigated the the existence and multiplicity results of positive solutions for the nonlinear differential equation of fractional order

by using some fixed-point theorems, where , , , , , , is the standard Riemann-Liouville derivative.

Xu and Fei [18] considered the properties of Green’s function for the nonlinear fractional differential equation boundary value problem

where , , , , , is the standard Riemann-Liouville derivative. Here the nonlinearity may be singular at . As applications of Green’s function, they give some existence of positive solutions for singular boundary value problems by means of Schauder fixed-point theorem. Here they consider the case: , , .

In this paper, we consider the singular boundary value problem

where , , , is a real constant, is the standard Riemann-Liouville fractional derivative. We will deduce a property of Green’s function. The result we establish in Section 2 can be stated as follow.

Theorem 1. The Function defined by (12) is continuous and satisfies

where .

In this paper, we give some existence of positive solutions for singular boundary value problems by means of Schauder fixed-point theorem for the case: , , , .

The paper is organized as follows. In Section 2, we state some known results and give a property of Green’s function. In Section 3, using Schauder fixed-point theorem, the existence of positive solutions to singular problems are obtained.

2. Background Materials

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory.

Definition 2 (see [7]). The Riemann-Liouville fractional integral of order of a function is given by

provided the right side is pointwise defined on .

Definition 3 (see [7]). The Riemann-Liouville fractional derivative of order of a continuous function is given by

where denotes the integer part of number , provided that the right side is pointwise defined on .

From the definition of Riemann-Liouville’s derivative, we can obtain the statement.

Lemma 4 (see [7]). Let , if one assumes that , then the fractional differential equation

has , , , as unique solutions, where is the smallest integer greater than or equal to .

Lemma 5 (see [7]). Assume that with a fractional derivative of order that belongs to . Then,

for some , , is the smallest integer greater than or equal to .

Lemma 6. Given the problem

is equivalent to

where

Proof. We can apply Lemma 5 to reduce (10) to an equivalent integral equation

for some . Consequently, the general solution of (10) is

By ,  this is .

On the other hand, combining with

yields

Therefor, the unique solution of problem (10) is

For , we have

For , we have

The proof is complete.

Proof of Theorem 1. It is easy to prove that is continuous on , here we omit it. In the following, we consider . When , , let

We have

When , let

We have

When , let , we have

When , , we have

It is easy to see that . Thus, the proof is complete.

Let us fix some notations to be used in the following. “For a.e.” means “for almost every”. Given , we write if for a.e. , and it is positive in a set of positive measure, we write if is a -caratheodory function, that is, the map is continuous for a.e. , and the map is measurable for all .

Let us define

Then,

3. Main Results

In this section, we establish the existence of positive solutions for equation

where ,  ,  ,  , , , is the standard Riemann-Liouville derivative. The following is the first main result in this section.

Theorem 7. Suppose that the following conditions are satisfied.

(H₁)For each , there exists a function such that for a.e. , all .(H₂) There exist , , and , such that

here

(H₃) There exist two positive constants such that

and here

Then, (28) has at least one positive solution.

Proof. Let , and is a closed convex set defined as

here is the Banach space of continuous functions defined on with the norm

and are positive constants to be given below.

Now, we define an operator by

Then, (28) is equivalent to the fixed-point problem

Let be the positive constant satisfying (H₃) and

Then, we have . Now, we prove .

In fact, for each and for all , by (H₁) and (H₃)

On the other hand, by conditions (H₂) and (H₃), we have

In conclusion, .

Finally, it is standard that is a continuous and completely continuous operator. By a direct application of Schauder’s fixed-point theorem, (28) has at least one positive solution , the proof is finished.

Case 1 (). As an application of Theorem 7, we consider the case . The following corollary is a direct result of Theorem 7 with .

Corollary 8. Suppose that satisfies conditions -. Furthermore, assume the following.

() There exists a positive constant such that

and here

Then, (28) has at least one positive solution.

From now on, let us define

Example 9. Suppose that the nonlinearity in (28) is

where , and

If , then (28) has at least one positive solution.

Proof. We will apply Corollary 8. To this end, we take

then (H₁) and (H₂) are satisfied since , and the existence condition () becomes

for some . Since , we can choose large enough such that (46) is satisfied, and the proof is finished.

Example 10. Suppose that the nonlinearity in (28) is

where , and is a nonnegative parameter. For each with ,

(i)if , then (28) has at least one positive solution for each .(ii)If , then (28) has at least one positive solution for each , where is some positive constant.

Proof. We will apply Corollary 8. To this end, we take

Then, (H₁)-(H₂) are satisfied since . Now, the existence condition () becomes , and

for some with . So, (28) has at least one positive solution for

Note that if and if , set

then, we have

Let the function possess a maximum at , then

so we have

it is easy to find that since , and . Finally, it would remain to prove . This is easily verified through elementary computations since . We have the desired results (i) and (ii).

Case 2 (). The next result explores the case when . In this case .

Corollary 11. Suppose that satisfies . Furthermore, assume the following.

(H₄) There exists , , such that

If , then (28) has at least one positive solution.

Example 12. Suppose that the nonlinearity in (28) be (43) with , . If , , then (28) has at least one positive solution.

Proof. We will apply Corollary 11. Take , , and as the same in the proof of Example 9. Then, (H₂) is satisfied, and the existence condition (H₄) is satisfied if we take with , and .

Example 13. Let the nonlinearity in (28) be (47) with and . For each with , ,

(i)if , then (28) has at least one positive solution for each .(ii)If , then (28) has at least one positive solution for each , where is some positive constant.

Proof. We will apply Corollary 11. To this end, we take , , and as the same in the proof of Example 10, then (H₂) is satisfied, and the existence condition (H₄) becomes ,

for some . So, (28) has at least one positive solution for

Note that if and if , and if set

The function possesses a maximum at

then . We have the desired results (i) and (ii).

Case 3 (). The next result considers the case .

Corollary 14. Suppose that satisfies -. Furthermore, assume the following.

(H₅) There exist two positive constants such that

here

Then, (28) has at least one positive solution.

Example 15. Suppose that the nonlinearity in (28) be (43) with , . If , ,

then (28) has at least one positive solution.

Proof. We will apply Corollary 14. Take , as the same in the proof of Example 9. Then, (H₂) is satisfied, and the existence condition (H₅) is satisfied if we take with

and . If we fix , then the first inequality holds if satisfies

or equivalently

The function possesses a minimum at

Taking , then the first inequality in (63) holds if , which is just condition (62). The second inequality holds directly from the choice of , so it remains to prove that . This is easily verified through elementary computations.

Example 16. Let the nonlinearity in (28) be (47) with and . If ,

here is the unique solution of the equation

Then (28) has at least one positive solution.

Proof. We will apply Corollary 14. To this end, we take ,  , and   as the same in the proof of Example 10, then (H₂) is satisfied, and the existence condition (H₅) is satisfied if we take with

and . If we fix , then the first inequality holds if satisfies

Let , then

Then, we have

Let , then we have

Now, let us define by

It is easy to see that is a nondecreasing function for and , as . Thus, has a unique solution such that

and .

So, it remains to prove that , that is,

In fact, by (75), we have

that is,

Also we have

that is,

Thus, we have

since , and .

We have the desired results.

Case 4 ().  

Example 17. Suppose that the nonlinearity in (28) be (43) with . If ,

and here is the unique solution of the equation

then (28) has at least one positive solution.

Proof. We will apply Theorem 7. Take as the same in the proof of Example 9. Then, (H₂) is satisfied, and the existence condition (H₃) is satisfied if we take with

and . If we fix , then the first inequality holds if satisfies

Taking , then for , we have

Then, we have

Let , then we have

Now, let us define by

It is easy to see that is a nondecreasing function for , and , as . Thus, has a unique solution such that

and .

So, it remains to prove that , that is,

In fact, by (90), we have

that is,

Thus, we have

The proof is complete.

Example 18. Let the nonlinearity in (28) be (47) with and . If ,

here is the unique solution of the equation

Then (28) has at least one positive solution.

Proof. We will apply Theorem 7. To this end, we take ,  , and as the same in the proof of Example 10, then (H₂) is satisfied, and the existence condition (H₃) is satisfied if we take with

and . If we fix , then the first inequality holds if

Let , then

Then, we have

Let , then we have

Now, let us define by

It is easy to see that is a nondecreasing function for , and , as . Thus, has a unique solution such that

and .

So, it remains to prove that , that is,

In fact, by (103), we have

that is,

Also we have

that is,

Thus, we have

since and .

Thus, the proof is complete.

Remark 19. It is easy to find that analogous results to Examples 10, 13, 16, and 18 for the general equation with the nonlinearity in (28) are

with , but the notation becomes cumbersome. Here we consider the nonlinearity (47) only for the simplicity.

Acknowledgments

The work was supported by NSFC of China (no. 10971021), the Science and Technology Research Project of Jilin Provincial Department of Education of China (no. 2011176), and the Science and Technology Research Project of Jilin Provincial Department of Education of China (no. 2010391). Supported by Scientific Research Fund of Heilongjiang Provincial Education Department.

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