Abstract and Applied Analysis

Volume 2012 (2012), Article ID 963105, 13 pages

doi:10.1155/2012/963105

Research Article

Existence of Three Solutions for a Nonlinear Fractional Boundary Value Problem via a Critical Points Theorem

Chuanzhi Bai

Department of Mathematics, Huaiyin Normal University, Jiangsu, Huaian 223300, China

Received 13 May 2012; Revised 8 July 2012; Accepted 9 July 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Chuanzhi Bai. 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.

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: where , and is a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.

1. Introduction

Differential equations with fractional order have recently proved to be strong tools in the modeling of many physical phenomena in various fields of physical, chemical, biology, engineering, and economics. There has been significant development in fractional differential equations, one can see the monographs [1–5] and the papers [6–20] and the references therein.

Critical-point theory, which proved to be very useful in determining the existence of solution for integer-order differential equation with some boundary conditions, for example, one can refer to [21–25]. But till now, there are few results on the solution to fractional boundary value problem which were established by the critical-point theory, since it is often very difficult to establish a suitable space and variational functional for fractional boundary value problem. Recently, Jiao and Zhou [26] investigated the following fractional boundary value problem:

by using the critical point theory, where and are the left and right Riemann-Liouville fractional integrals of order , respectively, is a given function and is the gradient of at .

In this paper, by using the critical-points theorem established by Bonanno in [27], a new approach is provided to investigate the existence of three solutions to the following fractional boundary value problems:

where , and are the left and right Riemann-Liouville fractional integrals of order respectively, and are the left and right Caputo fractional derivatives of order respectively, is a positive real parameter, is a continuous function, and is a nonnegative continuous function with .

2. Preliminaries

In this section, we first introduce some necessary definitions and properties of the fractional calculus which are used in this paper.

Definition 2.1 (see [5]). Let be a function defined on . 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 [5]). Let and .

(i) If and , then the left and right Caputo fractional derivatives of order for function denoted by and , respectively, exist almost everywhere on , and are represented by

respectively.

(ii) If and , then and are represented by

With these definitions, we have the rule for fractional integration by parts, and the composition of the Riemann-Liouville fractional integration operator with the Caputo fractional differentiation operator, which were proved in [2, 5].

Property 1 (see [2, 5]). We have the following property of fractional integration:

provided that , , and , , or , , .

Property 2 (see [5]). Let and . If or , then

for . In particular, if and or , then

Remark 2.3. In view of Property 1 and Definition 2.2, it is obvious that is a solution of BVP (1.2) if and only if is a solution of the following problem:

where .

In order to establish a variational structure for BVP (1.2), it is necessary to construct appropriate function spaces.

Denote by the set of all functions with .

Definition 2.4 (see [26]). Let . The fractional derivative space is defined by the closure of with respect to the norm

Remark 2.5. It is obvious that the fractional derivative space is the space of functions having an -order Caputo fractional derivative and .

Proposition 2.6 (see [26]). Let . The fractional derivative space is reflexive and separable Banach space.

Lemma 2.7 (see [26]). Let . For all , one has the following:

(i)

(ii)

By (2.9), we can consider with respect to the norm

in the following analysis.

Lemma 2.8 (see [26]). Let , then for all any , one has

Our main tool is the critical-points theorem [27] which is recalled below.

Theorem 2.9 2.9(see [27]). Let be a separable and reflexive real Banach space; be a nonnegative continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on ; be a continuously Gateaux differentiable function whose Gateaux derivative is compact. Assume that there exists such that , and that

(i), for all . Further, assume that there are , such that(ii);(iii).

Then, for each

the equation

has at least three solutions in and, moreover, for each , there exists an open interval

and a positive real number such that, for each , (2.14) has at least three solutions in whose norms are less than .

3. Main Result

For given , we define functionals as follows:

where . Clearly, and are Gateaux differentiable functional whose Gateaux derivative at the point are given by

for every . By Definition 2.2 and Property 2, we have

Hence, . If is a critical point of , then

for . We can choose such that

The theory of Fourier series and (3.4) imply that

a.e. on for some . By (3.6), it is easy to know that is a solution of BVP (1.2).

By Lemma 2.7, if , we have for each that

where

Given two constants and , with , where as in (3.8).

For convenience, set

Theorem 3.1. Let be a continuous function,   be a nonnegative continuous function with , and . Put for every , and assume that there exist four positive constants , and , with and , such that

(H1),  for  all ;(H2) for all , and

where . Then, for each

where and denote and respectively,the problem (1.2) admits at least three solutions in and, moreover, for each , there exists an open interval

such that, for each , the problem (1.2) admits at least three solutions in whose norms are less that .

Proof. Let be the functionals defined in the above. By the Lemma 5.1 in [26], is continuous and convex, hence it is weakly sequentially lower semicontinuous. Moreover, is coercive, continuously Gateaux differentiable functional whose Gateaux derivative admits a continuous inverse on . The functional is well defined, continuously Gateaux differentiable and with compact derivative. It is well known that the critical point of the functional in is exactly the solution of BVP (1.2).

From (H1) and (2.12), we get

for all [. Put

It is easy to check that and . The direct calculation shows

That is, . Thus, . Moreover, the direct calculation shows

Let . Since , we obtain .

By (2.12) and (3.7), one has . Thus,

Moreover, we have

Hence, from (H2) one has

Now, taking into account that

Thus, by Theorem 2.9 it follows that, for each , BVP (1.2) admits at least three solutions, and there exists an open interval and a real positive number such that, for each , BVP (1.2) admits at least three solutions in whose norms are less than .

Finally, we give an example to show the effectiveness of the results obtained here.

Let , , , and . Then BVP (1.2) reduces to the following boundary value problem:

Example 3.2. Owing to Theorem 3.1, for each , BVP (3.21) admits at least three solutions. In fact, put and , it is easy to calculate that , , and

Since

we have that condition (H1) holds. Moreover, for each , and

which implies that condition (H2) holds. Thus, by Theorem 3.1, for each , the problem (3.21) admits at least three nontrivial solutions in . Moreover, for each , there exists an open interval and a real positive number such that, for each , the problem (3.21) admits at least three solutions in whose norms are less than .

Acknowledgments

The author thanks the reviewers for their suggestions and comments which improved the presentation of this paper. This work is supported by Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (10771212).

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