Abstract and Applied Analysis

Volume 2012 (2012), Article ID 468980, 16 pages

doi:10.1155/2012/468980

Research Article

Multiple Solutions for a Class of Fractional Boundary Value Problems

Ge Bin

Department of Mathematics, Harbin Engineering University, Harbin 150001, China

Received 6 March 2012; Accepted 19 September 2012

Academic Editor: Yong H. Wu

Copyright © 2012 Ge Bin. 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 study the multiplicity of solutions for the following fractional boundary value problem: where and are the left and right Riemann-Liouville fractional integrals of order , respectively, is a real number, is a given function, and is the gradient of at . The approach used in this paper is the variational method. More precisely, the Weierstrass theorem and mountain pass theorem are used to prove the existence of at least two nontrivial solutions.

1. Introduction

In this paper, we consider the fractional boundary value problem of the following form:

where and are the left and right Riemann-Liouville fractional integrals of order , respectively, is a real number, is a given function, and is the gradient of at .

In particular, if , the problem reduces to the standard second-order boundary value problem of the following form:

Fractional calculus and fractional differential equations can find many applications in various fields of physical science such as viscoelasticity, diffusion, control, relaxation processes, and modeling phenomena in engineering, see [1–12]. Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by use of techniques of nonlinear analysis, such as fixed-point theory (including Leray-Schauder nonlinear alternative) (see [13, 14]), topological degree theory (including coincidence degree theory) (see [15, 16]), and comparison method (including upper and lower solutions methods and monotone iterative method) (see [17, 18]). However, it seems that the popular methods mentioned above are not appropriate for discussing and , as the equivalent integral equation is not easy to be obtained.

In the past, there were investigations of the eigenvalue problems for fractional differential equations. For more detailed information on this topic, we refer to Zhang et al. [19–21], Wang et al. [22, 23], and Jiang et al. [24].

Recently, there are many papers dealing with the existence of solutions for problem . In [25], Jiao and Zhou obtained the existence of solutions for by mountain pass theorem under the Ambrosetti-Rabinowitz condition. Chen and Tang [26] studied the existence and multiplicity of solutions for the system when the nonlinearity is superquadratic, asymptotically quadratic, and subquadratic, respectively.

But so far, few papers discuss the two solutions of the system via critical point theory. The aim of the present paper is to study the existence of at least two solutions for the system as the parameter for some constant .

The paper is organized as follows. We first introduce some basic preliminary results and a well-known lemma in Section 2, including the fractional derivative space , where . In Section 3, we give the main result and its proof. In Section 4, we give the summary of this paper.

2. Preliminary

In this section, we recall some related preliminaries and display the variational setting which has been established for our problem.

Definition 2.1 (see [8]). Let be a function defined on and . The left and right Riemann-Liouville fractional integrals of order for function denoted by and , respectively, are defined by

provided the right-hand sides are pointwise defined on , where is the gamma function.

Definition 2.2 (see [8]). Let be a function defined on . The left and right Riemann-Liouville fractional derivatives of order for function denoted by and , respectively, are defined by

where , , and .

The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belong-ing to the space of absolutely continuous functions, which we denote by . is the space of functions such that . In particular, .

Definition 2.3 (see [8]). Let and . If and , then the left and right Caputo fractional derivatives of order for function denoted by and , respectively, which exist a.e. on . and are represented by

respectively, where .

Definition 2.4 (see [25]). Define and . The fractional derivative space is defined by the closure of with respect to the norm

where denotes the set of all functions with . It is obvious that the fractional derivative space is the space of functions having an -order Caputo fractional derivative and .

Proposition 2.5 (see [25]). Let and . The fractional derivative space is a reflexive and separable space.

Proposition 2.6 (see [25]). Let and . For all , one has

Moreover, if and , then

According to (2.6), one can consider with respect to the norm

Proposition 2.7 (see [25]). Define and . Assume that , and the sequence converges weakly to , that is, . Then in , that is, , as .

Making use of Definition 2.3, for any , problem is equivalent to the following problem:

where .

In the following, we will treat problem in the Hilbert space with the corresponding norm . It follows from [25, Theorem 4.1] that the functional given by

is continuously differentiable on . Moreover, for , we have

Definition 2.8 (see [25]). A function is called a solution of if

(i) is derivative for a.e. ,

(ii) satisfies , where .

Proposition 2.9 (see [25]). If , then for any , one has

Proposition 2.10 (see [25]). Let be satisfied. If , then the functional denoted by

is convex and continuous on .

In order to prove the existence of two solutions for problem , firstly, we recall some well-known results. Their proofs can be found in many books. Please refer to the references and its references therein.

Lemma 2.11 (see [27]). If is a Banach space, , , and , such that and

and if satisfies the PS condition, with

then is a critical value of .

3. The Main Result and Proof of the Theorem

In this part, we will prove that for there also exist two solutions for the general case.

Our hypotheses on nonsmooth potential are as follows.

: is a function such that a.e. on and satisfies the following facts:

(i)for all , is measurable,(ii)for a.a. , is continuously differentiable,(iii) there exist and , such that

(iv)there exist and , such that

(v)there exist , , and , such that

where .

Remark 3.1. It is easy to verify that satisfies the whole assumption in [25, Theorem 4.1]. So, is a solution of the corresponding Euler equation , then is a solution of problem which, of course, corresponds to the solution of problem . For great details, please see [25, Theorem 4.2].

Theorem 3.2. Suppose that holds. Then there exists such that for each , the problem has at least two nontrivial solutions, which correspond to the two solutions of problem .

Proof. The proof is divided into four steps as follows.

Step 1. We will show that is coercive in this step.

Firstly, by (iii), (2.6), and (2.10), we have

where .

Step 2. We will show that the is weakly lower semicontinuous.

Let weakly in , and by Proposition 2.7, we obtain the following results:

By Fatou's lemma,

On the other hand, by Proposition 2.10, we have , that is,

Thus,

Hence, by the Weierstrass theorem, we deduce that there exists a global minimizer such that

Step 3. We will show that there exists such that for each , .

By the condition by (v), there exists such that , a.e. . It is clear that

Now we denote

where , and is given in the condition (v). A simple calculation shows that the function is positive whenever and . Thus, is well defined and .

We will show that for each , the problem has two nontrivial solutions. In order to do this, for , let us define

Then and

Hence,

By (iii) and (v), we have

For , by Proposition 2.9, we have

so that whenever .

Step 4. We will check the PS condition in the following.

Suppose that such that and .

Since is coercive and is bounded in and passed to a subsequence, which still denote , we may assume that there exists , such that weakly in ; thus, we have

as . Moreover, according to Proposition 2.7, we have , as . Observing that

combining this with (3.17), it is easy to verify that , as , and hence that in . Thus, satisfies the PS condition.

Step 5. We will show that there exists another nontrivial weak solution of problem .

From the mean value theorem and (v), we have

for all , a.e. .

It follows from the conditions (iii) and all and a.e. that

and this together with (3.19) yields that for all and a.e. ,

for some positive constant .

For all , , and , we have

So, for small enough, there exists a such that

and . So by the mountain pass theorem (cf. Lemma 2.11), we can get which satisfies

Therefore, is another nontrivial critical point of .

Remark 3.3. We can find a potential function satisfying the hypothesis of our Theorem 3.2. For great details, please see Section 4(B) in Summary.

So far, the results involved potential functions exhibiting sublinear. The next theorem concerns problems where the potential function is superlinear.

Our hypotheses on nonsmooth potential are as follows.

: is a function such that a.e. on and satisfies the following facts:

(i)for all , is measurable,(ii)for a.a. , is continuously differentiable,(iii)there exist and , such that

(iv)there exist and , such that

(v)there exist , , and , such that , a.e. ,where ,(vi)for a.a. and all , we have

Theorem 3.4. Suppose that holds. Then there exists a such that for each , the problem has at least two nontrivial solutions, which correspond to the two solutions of problem .

Proof. The steps are similar to those of Theorem 3.2. In fact, we only need to modify Step 1 and Step 4 as follows: 1′ shows that is coercive under the condition (vi); 4′ shows that there exists a second nontrivial solution under the conditions (iii) and (iv). Then from Steps 1′, 2, 3, and 4′ above, problem has at least two nontrivial solutions.

Step 1'. By (vi), for all , , we have

where .

Step 4'. Because of hypothesis (iii), we have

for a.e. and all with .

Combining (3.19) and (3.29), it follows that

for a.e. and all .

Thus, for all , and , we have

where and are positive constants.

So, for small enough, there exists a such that

and .

Arguing as in proof of Step 4 of Theorem 3.2, we conclude that satisfies the PS condition. So by the mountain pass theorem (cf. Lemma 2.11), we can get that satisfies

Therefore, is another nontrivial critical point of .

Remark 3.5. We will give some examples, which satisfy the hypothesis of our Theorem 3.4. For great details, please see Section 4(C) in Summary.

4. Summary

(A) If , then . Therefore, by Theorems 3.2 and 3.4, we actually obtain the existence of two weak solutions of the following eigenvalue problem:

where is a real number, is a given function, and is the gradient of at . Although many excellent results have been worked out on the existence of solutions for second-order system (e.g., [28, 29]), it seems that no similar results were obtained in the literature for fractional system .

(B) We give an example in the following to illustrate our viewpoint in Remark 3.1. We consider

where , , for all .

Obviously, hypotheses (i), (ii), and (v) are satisfied. Moreover,

Therefore,

So, condition (iii) holds.

On the other hand,

uniformly for a.a. , so condition (iv) holds.

(C) We can find the following potential functions satisfying the conditions stated in Theorem 3.4:

where , for all .

It is clear that () for a.e. , and hypotheses (i) and (ii) are satisfied. A direct verification shows that conditions (v) and (vi) are satisfied. Note that

So,

which shows that assumptions (iii) and (iv) are fulfilled.

Acknowlegments

This work was supported by the National Natural Science Foundation of China (nos. 11126286, 11001063, and 11201095), the Fundamental Research Funds for the Central Universities (no. 2012), China Postdoctoral Science Foundation funded project (no. 20110491032), and China Postdoctoral Science (Special) Foundation (no. 2012T50325).

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