Variational Methods for a Fractional Dirichlet Problem Involving Jumarie’s Derivative

Abstract

We investigate a fractional Dirichlet problem involving Jumarie’s derivative. Using some variational methods a theorem on the existence and uniqueness of a solution to such problem is proved. In the proof of the main result we use a fractional counterpart of the du Bois-Reymond fundamental lemma.

1. Introduction

In the last time, fractional calculus plays an essential role in the fields of mathematics, physics, electronics, mechanics, engineering, and so forth (cf. [1–5]). Many processes in physics and engineering can be described accurately by using systems of differential equations containing different type of fractional derivatives. Among definitions of derivatives of fractional order we can pick the Riemann-Liouville and the Caputo derivatives out. Unfortunately, each of them has different unusual properties. For instance, the Riemann-Liouville derivative of a constant is not zero and the Caputo derivative is defined only for differentiable functions (alternatively, for such functions that have no first order derivative but then they might have fractional derivatives of all orders less than one, see [6]).

Recently, Jumarie proposed a new definition of the fractional derivative being a little modification of the Riemann-Liouville derivative (cf. [7–10]). His definition eliminates disadvantages of mentioned earlier derivatives, because the Jumarie derivative of a constant is equal to zero and it is defined for any continuous (nondifferentiable) functions.

In the paper we consider the following fractional boundary problem: where , , and denotes Jumarie’s derivative of a function The above problem is a generalization of the classical Dirichlet problem of the form We discuss the problem of the existence of solutions to above problem. In our investigations we use some variational method given in [11]. First, we consider some integral functional depending on the Jumarie derivative, for which (1) is the Euler-Lagrange equation. Next, we prove existence of a critical point of mentioned functional in an appropriate space of functions and under suitable assumptions of regularity, coercivity, and convexity. In order to do it, we use the following.

Proposition 1 (see [11]). If is a reflexive Banach space and the functional is coercive and sequentially weakly lower semicontinuos, then it possesses at least one minimum at

Let us remind that a functional defined on a Banach space is coercive if whenever , and is sequentially weakly lower semicontinuous at if for any sequence such that weakly in

Of course such critical point is a minimizer of this functional and it generates the solution to problem (1)-(2). In order to prove that a minimum point gives a solution it is sufficient to apply a fractional version of the du Bois-Reymond lemma obtained in Section 3. Results of a such type for the Dirichlet problem involving the Riemann-Liouville derivative have been obtained in [12].

The paper is organized as follows. In Section 2, we review some basic definitions and facts concerning fractional calculus that we need in the sequel. Moreover, we introduce the space of solutions to considered problem and give some useful properties of this space. In Section 3, we formulate and prove some fractional version of the du Bois-Reymond Lemma, which we use in the proof of the main result. Mentioned main results of the work, namely, a theorem on the existence and uniqueness of a solution to problem (1)-(2), are formulated and proved in Section 4.

2. Preliminaries

In the first part of this paper we recall some basic facts concerning fractional calculus (cf. [3, 7–10, 13]). Next, we introduce some function spaces, which will be used later.

2.1. Fractional Calculus

We will assume that is a bounded interval.

Let and . The left-sided Riemann-Liouville integral of the function of order is defined by In the rest of this paper we will assume that .

The left-sided Riemann-Liouville derivative of the function of order is defined in the following way: provided that has an absolutely continuous representant on (i.e., there exists an absolutely continuous function on which is equal a.e. on to ).

Now, let us assume that .

Jumarie’s modified Riemann-Liouville derivative of the function of order is defined by provided that has an absolutely continuous representant on .

Remark 2. It is easy to see that if , then defined above derivatives coincide. Moreover, Jumarie’s modified Riemann-Liouville derivative of a constant equals zero.

Remark 3. The definition of fractional derivative given by (6) is a consequence of the following fractional derivative via difference reads defined by Jumarie:

The integral of is given by

We have the following theorem on the integration by parts.

Theorem 4. Let and there exist derivatives and . Then

2.2. Space

Let .

Let us define the set with the norm We will identify two functions belonging to that coincide a.e. on .

It is easy to show that is a Banach space. In particular, the space , equipped with the inner product is a Hilbert space.

Now, we give some properties of the space .

Proposition 5. The injection is continuous.

Proof. Let . Then Consequently, The proof is completed.

Proposition 6 (Hölder inequality). Let , , and . Then and

Proof. From the Hölder inequality for the space we obtain

Lemma 7. The operator is bounded, and it means there exists a constant such that

Proof. Using Fubini’s Theorem and [14, Lemma ], we obtain where , , .
Since , it suffices to put .
The proof is completed.

2.3. Space

Let us define the set (shortly ) as follows: Functions belonging to and equal a.e. on are identified.

From Proposition 5, we immediately obtain the following.

Proposition 8. Consider where

From the above fact and [14, Proposition ] it follows that if then there exists the Riemann-Liouville derivative almost everywhere on .

Moreover, one can show that with the norm given by is complete and, consequently, is a Banach space. In particular, the space , equipped with the inner product is a Hilbert space.

Remark 9. Let us note that in the case , from Proposition 8 and [15, Property ], it follows that where . Consequently, if then possesses Jumarie’s modified Riemann-Liouville derivative and . Of course, then

Remark 10. From Proposition 8, monography [3, Lemma ] and Remarks 2 and 9 it follows that if with then

From the above remark and Lemma 7, we immediately obtain the following.

Lemma 11 (fractional Poincaré Inequality). Let . Then for , where is the constant from Lemma 7.

3. Du Bois-Reymond Lemma

In this section, we will prove the du Bois-Reymond lemma for nondifferentiable functions.

We have the following.

Lemma 12 (du Bois-Reymond lemma). Let , , and for any function such that . Then there exists a constant such that for a.e. and, consequently, for a.e. .

Proof. First, let us note that from the Hölder inequality (cf. Proposition 6) it follows that the integral (28) is well-defined. Let for , where . Then, for any function such that (in view of Remark 9 the condition is satisfied also), from assumption (28) and Theorem 4, we obtain Thus, Let us consider the function for a.e. . It is easy to see that and, in view of Remark 9, . We will show that . Indeed, we have Consequently, the function satisfies equality (30). So, It means that The proof is completed.

Remark 13. In [16] result of such a type, but for Caputo derivative (for differentiable functions ), had been proved.

Using Lemma 12, we will prove the next lemma, which will play a key role in the next section. We have the following.

Lemma 14. Let , , , and for any function such that . Then, where is the constant from Lemma 12, and consequently

Proof. Using the Hölder inequality for spaces and , we check that integral (34) exists. Let us put . Then and . From Theorem 4 and assumption (34), we obtain Thus From Lemma 12 it follows that there exists a constant such that Consequently, It is well known that the function possesses the left-sided Riemann-Liouville derivative and Since , from [15, Property ] it follows that the function is continuous and . Consequently, it possesses also the Jumarie modified Riemann-Liouville derivative, wchich equals . It means that the function has the Jumarie modified Riemann-Liouville derivative and (using the second part of Remark 2) The proof is completed.

4. Main Result

Let us consider Dirichlet problem (1)-(2). By a solution to such problem we will mean a function , satysfying condition , such that

Let us notice that since , from Remark 9 it follows that and are continuous and satisfies the initial condition .

In order to prove the existence of solutions to problem (1)-(2), we use variational methods.

Let us consider a functional of the form defined on the following space

We impose the following assumption on the function :(A1)The function  is measurable on for any and the function  is of class on for a.e. .(A2)There exist constants and functions and such that  for almost every and all .

We say that possesses the first variation at the point in the direction (cf. [17]) if there exists a finite limit

We will prove that, under assumptions (A1) and (A2), possesses its minimum at a point which is a solution to (1).

To begin with, we will prove the following.

Theorem 15. Let us assume that conditions (A1)-(A2) are satisfied. Then the functional is well-defined on and possesses the first variation at any point and in any direction given by

Proof. The fact that and are well-defined follows directly from (A1)-(A2) and the Hölder inequality (cf. Proposition 6). Let us fix and   and write the functional as where It is clear that and are well-defined and is linear. Moreover, so is continuous. Consequently, it is differentiable in the sense of Frechet on and the differential at any point is equal to . Using the Lebesque dominated convergence theorem and the mean value theorem, we assert that the mapping has the first variation at any point and in any direction given by This means (cf. [17, Section ]) that there exists the first variation of the mapping given by equality (49).
The proof is completed.

Using the same arguments as in [15, Proposition ], we can obtain the following.

Theorem 16. Let and . If weakly in (with topology induced from ), then uniformly on .

Now, we will prove the main result of this paper, namely, a theorem on the existence of a unique solution to problem (1)-(2). We have the following.

Theorem 17. Let and assume that assumptions (A1)-(A2) are satisfied. If there are constants , , such that then problem (1)-(2) possesses at least one solution which minimizes functional . Moreover, if the function is convex for a.e. , then the solution is unique.

Proof. Let . Then from condition (54), Proposition 6, and Lemma 11 it follows that where is the constant from Lemma 7.
Consequently, since , ; so is coercive.
Now, Let weakly in From Theorem 16 it follows that , on . Thus and from assumption (A2), using the dominated convergence theorem, we get that Moreover, is a Hilbert space as the closed subspace of . Consequently, the mapping is weakly lower semicontinuous, so This means that the functional is sequentially weakly lower semicontinuous and, by the virtue of Proposition 1, we conclude that it possesses minimum at the point
From Theorem 15 and Fermat lemma it follows that for any Applying Lemma 14, we get Since , boundary conditions are satisfied.
The proof of the existence part is completed.
Now, we will show that, under assumption (55), the solution to problem (1)-(2) is unique. First, let us note that for ,  , and we have If then otherwise, from Lemma 11 and assumption , it follows that It means that the functional is strictly convex. Consequently, using assumption (55), we assert that the functional is strictly convex, so its minimum point is unique. On the other hand if is a solution to (1)-(2) then Thus, for any and from Theorem 4 we get This means that the solution to problem (1)-(2) is a minimum point of , so it is unique. The proof is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The project was financially supported by the Faculty of Mathematics and Computer Science, University of Lodz, under Grant no. B1411600000451.02 for young researchers and participants of a grad school.