Abstract and Applied Analysis

Volume 2013 (2013), Article ID 181052, 6 pages

http://dx.doi.org/10.1155/2013/181052

Research Article

Infinitely Many Solutions for a Class of Fractional Boundary Value Problems with Nonsmooth Potential

Kaimin Teng

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

Received 4 June 2013; Accepted 8 August 2013

Academic Editor: Salvatore A. Marano

Copyright © 2013 Kaimin Teng. 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 establish the existence of infinitely many solutions for a class of fractional boundary value problems with nonsmooth potential. The technical approach is mainly based on a result of infinitely many critical points for locally Lipschitz functions.

1. Introduction

In the present paper, we are concerned with the existence of infinitely many solutions for a class of fractional boundary value problems with the following form:

where is a parameter, and and are the left and right Riemann-Liouville fractional derivatives of order , respectively. is a given function satisfying some assumptions, and is the generalized gradient in the sense of Clarke [1].

In particular, if , then problem (1) is reduced to the standard second-order boundary value problem

There are many excellent results that have been worked out on the existence of solutions for second-order BVP (we refer the reader to see [2, 3] and the references therein).

Recently, fractional differential equations and inclusions have attracted lots of people’s interests because of their applications in viscoelasticity, electrochemistry, control, porous media, and so forth. The existence and multiplicity of solutions for BVP of fractional differential equations and inclusions have been established by some fixed-point theorems; we refer the readers to see [4–7].

This paper is motivated by the recent papers [8] where several existence results concerning problem (1) under the smooth case are obtained by using variational methods. In their papers, the authors define a suitable space and find a variational functional for fractional differential equations with Dirichlet boundary conditions. The aim of the present paper is to establish the existence of infinitely many solutions for problem (1) by using a critical points theorem according to Bonanno and Bisci [9].

It is interesting that the existence of infinitely many solutions for differential equations can be established without the symmetry assumption. Recently, Bonanno and Bisci in [9] established a precise version of the infinitely many critical points theorem of Marano and Motreanu [10] which extended the results of Ricceri in [11] for the nondifferentiable functionals. In applying the theorem, we need to assume some appropriate oscillating behavior of the nonlinear term either at infinity or at zero. This methodology has been usefully used in obtaining the existence of multiple results for different kinds of problems, such as p-Laplacian problem [12], quasilinear elliptic system [13, 14], discrete BVP [15], double Sturm-Liouville problem [16], and elliptic problems with variable exponent [17]. By using this methodology, to the best of our knowledge, it seems that no similar results are obtained in the literature for fractional BVP. Therefore, the purpose of our paper is to establish the existence of infinitely many solutions for problem (1) by using this type of methodology.

Our main results are stated as follows. For this matter, put

Our first main result is the following theorem.

Theorem 1. Suppose satisfies the following conditions.

For all , the function is measurable. For almost all , the function is locally Lipschitz and  . There exist such that with for all , all , and almost all .   for almost all and all .  , where .

Then, for each , problem (1) admits a sequence of solutions which is unbounded in .

Next, we present the other main result. First, put

Theorem 2. Suppose satisfies the conditions , and

  , where .

Then, for each , problem (1) admits a sequence of pairwise distinct solutions which strongly converges to zero in .

In order to prove Theorems 1 and 2, we recall the critical point theorem in [9] here for the readers’ convenience.

Theorem 3. Let be a reflexive real Banach space, and let be two Lipschitz functions such that is sequentially weakly lower semicontinuous and coercive and is sequentially weakly upper semicontinuous. For every , one puts

Then,

(a)if , for each , the following alternative holds: either

(i) possesses a global minimum, or(ii)there is a sequence of critical points (local minimum) of such that .

(b)If , for each , the following alternative holds: either

(i)there is a global minimum of which is a local minimum of , or(ii)there is a sequence of pairwise distinct critical points (local minimum) of with , which weakly converges to a global minimum of .

The present paper is organized as follows. In Section 2 we present some basic definitions and facts from the nonsmooth analysis theory, and we prove a variational principle for problem (1). Section 3 is devoted to proving Theorems 1 and 2.

2. Preliminaries

2.1. Nonsmooth Analysis

Let be a real Banach space and its conjugate space; we denote by and , respectively, the norm and the duality pairing between and .

For a locally Lipschitz function , we define the generalized directional derivative of at point in the direction as follows:

The generalized gradient of a locally Lipschitz function at the point , denoted by , is the set . If , then for all .

A point is said to be a critical point of a locally Lipschitz function if . Clearly, if is a minimum of a locally Lipschitz function , then ; that is, is a critical point of .

2.2. Fractional Derivative Space

Throughout this paper, we denote the norm of the space for as and .

Definition 4. Let and . The fractional derivative space is defined by the closure of with respect to the norm

Remark 5. It is obvious that this fractional derivative space is the space of functions having an -order Caputo fractional derivative and .

The properties of the fractional derivative spaces are listed as the following lemma.

Lemma 6 (see [8]). Let and .

(1)The fractional derivative spaces are a reflexive and separable Banach space.(2)If , for any , one has , for any .(3)If or , one has is compact and

(4)Assume that and the sequence converges weakly to in ; that is, . Then converges strongly to in ; that is, , as . Moreover, if , one has

According to (8), we can consider with respect to the following norm:

In this paper, the work space for problem (1) is . The space is a Hilbert space with respect to the norm given by (10), and the corresponding inner product is defined by the following:

2.3. Variational Framework

We first give the definition for the solution of problem (1).

Definition 7. A function is called a solution of problem (1) if and are derivatives for almost all and satisfies (1).

The functional corresponding to problem (1) is defined by the following:

By the conditions , it is easy to check that is locally Lipschitz on . Moreover, we can get the variational principle as follows.

Proposition 8. Every critical point of is a solution of problem (1).

Proof. We assume that is a critical point of ; that is ; then,

with some . Noting that , then . Let , . Hence, by the formula of integration by parts for the left and right Riemann-Liouville fractional derivatives, we have the following:

By (13), for every and hence for every , we have

Since , we have ′. By the formula of integration by parts for the left and right Riemann-Liouville fractional derivatives, we get

Since , the standard Fourier series theory implies that

for some constant . Using the properties of the left and right Riemann-Liouville fractional derivatives, we have

Since , we can identify the equivalence class and its continuous representant

Firstly, we notice that is derivative for almost every and as . On the other hand, is derivative a.e. on and .

By (19), we have

Moreover, implies that . The proof is completed.

3. Proof of Main Results

Throughout this section, for , we denote , where , and

Clearly, is Gâteaux differentiable and sequentially weakly lower semicontinuous and coercive; is locally Lipschitz continuous on ; by standard argument, is sequentially weakly continuous. We denote

Proof of Theorem 1. First, we verify that . By , let be a real sequence such that and

Put for all . From (9), one has for all such that . Take into account that and , where for all . For all , we have

Since from assumption one has , then we have

Now fix . We claim that the functional is unbounded from below.

By , let be a sequence of such that and

For each , we define a sequence as follows:

It is easy to check that and . Moreover, is Lipschitz continuous on , and hence is absolutely continuous on . By calculations, we get

Obviously, is continuous on and

where depends on and .

By condition , we have

for all . Then,

for every .

If , let ; by (26) there exists such that

for all . Hence, by (31) and (32), we obtain

for all . Choosing suitable , we have

On the other hand, if , we fix , and again from (26) there exists such that

for all . Therefore, from (31) and (35), we have

for all . From the choice of , we have

Hence, our claim is proved.

Since all the assumptions of the case  (a) of Theorem 3 are verified, for each , the functional admits an unbounded sequence of critical points. The conclusion follows from Proposition 8.

Proof of Theorem 2. The proof is the same as Theorem 1 by using the case   (b) of Theorem 3 instead of the case   (a).

Example 9. We give an example to illustrate Theorem 1.

Set

for every . Define the nonnegative function as follows:

Obviously, satisfies the conditions . Next, we show that is true. Indeed, by direct computation, we get

Hence, we see that

Therefore, is verified.

Acknowledgments

The author is supported by the NSFC under Grant 11226117 and the Shanxi Province Science Foundation for Youths under Grant 2013021001-3.

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