Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 267386, 9 pages

http://dx.doi.org/10.1155/2013/267386

Research Article

Existence Results for a Coupled System of Nonlinear Fractional Boundary Value Problems at Resonance

Ruijuan Liu,1,2 Chunhai Kou,³ and Xiujuan Xie³

¹College of Information Science and Technology, Donghua University, Shanghai 201620, China

²College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China

³Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Received 30 April 2013; Accepted 15 July 2013

Academic Editor: Yong-Kui Chang

Copyright © 2013 Ruijuan Liu 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.

Some new Banach spaces are established. Based on those new Banach spaces and by using the coincidence degree theory, we present the existence results for a coupled system of nonlinear fractional differential equations with multipoint boundary value conditions at resonance case.

1. Introduction

Fractional differential equations (FDEs) have been of great interest for the last three decades. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, and so forth. Therefore, the theory of FDEs has been developed very quickly. Many qualitative theories of FDEs have been obtained. Many important results can be found in [1–13] and the references cited therein.

However, there are few articles which consider the boundary value problems at resonance for a coupled system of nonlinear fractional differential equations. In [14], the authors investigated the existence and uniqueness of solutions for the multipoint boundary value problems for fractional differential equations of the form

where ,  ,  ,  , with and represents the standard Riemann-Liouville fractional derivative. And the analysis relies on the Schauder fixed point theorem and the Banach contraction principle.

In [15], the following coupled system of nonlinear fractional differential equation boundary value problem:

was considered. Where , ,  ,  , , , is the standard Riemann-Liouville fractional derivative and are continuous. For more works about coupled system involving fractional differential equations, for instance, see [16, 17].

In this paper, we investigate the existence of solutions for the boundary value problem with the following form:

where , , , , , , , , and  Moreover, at resonance case,

a sufficient condition for the existence of solutions is established by the coincidence degree continuation theorem.

2. Background Materials and Preliminaries

For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory. These definitions and lemmas can be found in [4–6].

Definition 1. The fractional integral of order of a function is given by

provided the right side is pointwise defined on .

Definition 2. The fractional derivative of order of a function is given by

where , provided the right side is pointwise defined on .

It can be directly verified that the Riemann-Liouville fractional integration and fractional differentiation operators of the power functions yield power functions of the same form. For ,  , there are

Lemma 3. Assume that , , then the differential equation has solutions

where ,  ,  .

Lemma 4. Assume that and , then , where ,  , .

Lemma 5. Let and , and then

The following definitions and coincidence degree theory are fundamental in the proofs of our main results.

Definition 6. Let and be Banach spaces. A linear mapping is called a Fredholm mapping if the following two conditions hold:

(1) has a finite dimension;(2) is closed and has a finite codimension.

If is a Fredholm mapping, its (Fredholm) index is the integer .

And if is a Fredholm mapping of index zero, there exist linear continuous projectors and such that , , and ,  . Then it follows that is invertible. We denote the inverse of this map by . The generalized inverse of this map is denoted by , .

Definition 7. Assume that is a Fredholm mapping, and if is an open bounded subset of , the map will be called -compact on if is bounded and is compact.

If is isomorphic to , there exists an isomorphism . We have the coincidence degree continuation theorem which is proved in [18].

Theorem 8. Assume that is a Fredholm mapping of index zero and let be on , where is an open bounded subset of . Suppose that the following conditions are satisfied:

(1) for each ;(2) for each ;(3), where is a continuous projection as mentioned above with and is any isomorphism.

Then the equation has at least one solution in .

Let with the norm . For , , we define a linear space

to be accompanied by

We can prove that is a Banach space with the , and is a Banach space, with the .

Let to be the linear operator from to with

And , . Let be the linear operator from to with

And ,  . Define to be the linear operator from to with

and . We define by setting

where ,   .

Then the coupled system of BVPs (3) can be written as .

Lemma 9 (see [19]). ,  , is a sequentially compact set if and only if is uniformly bounded and equicontinuous. Here uniformly bounded means that there exists such that for every ,

and equicontinuous means that for all ,  , for all ,  ,  , there hold ,  .

Definition 10 (see [19]). One says that satisfies the Carathéodory condition if (i) for each , the function is measurable on ; (ii) for a.e. , the function is continuous on ; (iii) for each , there exists such that for a.e. and all with .

3. Main Results

In this section, we will give and prove our main results.

Lemma 11. Let ,  ,  ,  . Then are, respectively, the solutions of the following boundary value problems:

if and only if satisfied

Proof. in are the solutions of the following boundary value problem:

According to Lemma 4, we have

By , we get , and then , .

Using (7) and Lemma 5, we get that

By the boundary conditions, ,  , we have

Conversely, suppose that (23) and (24) hold. In view of Lemma 5, we can easily verify that are the solutions of (17) and (18), respectively. Then, we complete the proof.

Lemma 12. The mapping is a Fredholm operator of index zero.

Proof. Since , , it can be directly verified that , . It is clear that .

Let ; then there exists such that , that is, , and , . From these results, we can easily deduce the results

Let , , . Assume that

Consider the auxiliary mapping,

It is obvious that are continuous linear mappings. Also we have that , . Take the mapping defined by . , , where . Note that implies . Evidently,

and is a continuous linear projector. In fact, for any , we have

that is to say, is idempotent.

Observe that leads to and , where is the element in . Conversely, if , we can have that , that is to say, . So, .

Let , where is an arbitrary element. Since and , we obtain that . Take , can be written as , . For then since , by (27), we get that

which implies that . Therefore, , and thus .

Now, , and observing that is closed in , so is a Fredholm mapping of index zero.

Let be defined by

and it is clear that , , are linear continuous projectors and . Also, proceeding as the proof of Lemma 12, we can show that .

Lemma 13. Assume that satisfies the Carathéodory conditions

Then is completely continuous.

Here we omit reasoning process, and it is easy to prove.

Consider the mapping ,

For , we have .

, for all . Thus, , where ; hence, for each and , we have

where

The operator is continuous in view of the continuity of the function and . Let be bounded; that is to say, there exists a positive constant such that for all .

We can derive that

Here,

By Lemma 5,

Similarly, we can prove that .

Hence,

that is to say, is bounded. On the other hand, let

For every , ,   for all , there is , and such that

This ensures that is equicontinuous too. Then an application of Ascoli-Arezela theorem ensures that is completely continuous.

Theorem 14. Assume that satisfy the Carathéodory conditions, ,  , ,  ,  ,  ,  ,  , and , . Assume that all of the following hypotheses hold.

  There exist nonnegative functions , , such that for all ,

 , , where is defined by (35) and , , .  There exist constants such that , , for each , satisfying , ,  for all .  There exist constants such that for any , if , , either

or else

Then the problem (3) has at least one solution in .

Proof. The proof consists of four main steps.

Step 1. Set , and prove is bounded. In fact, for , then and , so and . Hence, , that is, . From , we have that such that , .

Again, for , we get

In view of Lemma 5, we have

Similarly, we can also obtain .

Thus,

According to Lemma 13, we have

Combining (48) and (49), we can obtain

From , for each , we have

Also, we can obtain that

where . Thus, by and (50)–(52) we can derive that

which clearly states that is bounded.

Step 2. Set and prove that is bounded. Let ; then

Since , we have , that is, , . Taking account of , , , which implies that is bounded.

Step 3. Set , ; if (35) holds, we will set

if (44) holds, where is a linear isomorphism defined as

And then we will prove that is bounded. Without loss of generality, in the following part of the proof, we assume that (35) holds in , for ; we have , , ,  , and Thus,

Therefore, via and (35), we have , , which show that is bounded.

Step 4. Let be an bounded open set such that and prove that . The operator is -compact on due to the fact that is bounded and is compact by Lemma 13; then by Step 1 and Step 2, we have

(i) for each ;(ii) for each .

Define , where is the identity operator in . According to the arguments in Step 3, we have , for all , and therefore, via the homotopy property of degree, we obtain that

which certifies the condition of Theorem 8. Then, applying Theorem 8, we conclude that the problem (3) has at least one solution in . The proof is complete.

4. Conclusion

In this paper, we obtain the existence results of muti-point boundary value problem at resonance for a coupled system of nonlinear fractional differential equations by means of the coincidence degree theory.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Ruijuan Liu completed the proof and wrote the initial draft. Chunhai Kou provided the problem and gave some suggestions of amendment. Ruijuan Liu then finalized the paper. All authors read and approved the final paper.

Acknowledgment

This work was supported by the Natural Science Foundation of China under Grant no. 11271248.

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