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

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 RiemannLiouville 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 RiemannLiouville 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:


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][2][3][4][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][8][9][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: ( () ) () () =   (,  ()) , for a.e. ∈ [, ] , where  ∈ (1/2, 1),  : [, ] × R  → R, and  () denotes Jumarie's derivative of a function .The above problem is a generalization of the classical Dirichlet problem of the form   () =   (,  ()) ,  () =  () = 0. (3) 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 L :  → R is coercive and sequentially weakly lower semicontinuos, then it possesses at least one minimum at  0 ∈ .
Let us remind that a functional L defined on a Banach space  is coercive if L() → ∞ whenever ‖‖ → ∞, and L is sequentially weakly lower semicontinuous at  0 ∈  if lim inf  → ∞ L(  ) ≥ L( 0 ) for any sequence {  } ⊂  such that   ⇀  0 weakly in .

Fractional Calculus. We will assume that
Let  > 0 and  ∈  1 ([, ], R  ).The left-sided Riemann-Liouville integral of the function  of order  is defined by In the rest of this paper we will assume that  ∈ (0, 1).The left-sided Riemann-Liouville derivative   +  of the function  of order  is defined in the following way: provided that  1− +  has an absolutely continuous representant on [, ] (i.e., there exists an absolutely continuous function on [, ] which is equal a.e. on [, ] to  1− + ).Now, let us assume that  ∈ ([, ], R  ).Jumarie's modified Riemann-Liouville derivative of the function  of order  is defined by provided that  Remark 2. It is easy to see that if () = 0, 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.

Du Bois-Reymond Lemma
In this section, we will prove the du Bois-Reymond lemma for nondifferentiable functions.
We have the following.
Proof.First, let us note that from the Hölder inequality (cf.Proposition 6) it follows that the integral (28) is well-defined.
Remark 13.In [16] result of such a type, but for Caputo derivative (for differentiable functions ℎ), 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.
Let us notice that since  ∈ (1/2, 1), from Remark 9 it follows that  and  () are continuous and  satisfies the initial condition () = 0.
We say that L possesses the first variation L(, ℎ) at the point  ∈  2, 0 in the direction ℎ ∈  2, 0 (cf.[17]) if there exists a finite limit We will prove that, under assumptions (A1) and (A2), L possesses its minimum at a point  0 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 L is well-defined on  2, 0 and possesses the first variation L(, ℎ) at any point  ∈  2, 0 and in any direction ℎ ∈  2, 0 given by Proof.The fact that L and L are well-defined follows directly from (A1)-(A2) and the Hölder inequality (cf.Proposition 6).Let us fix  ∈  2, 0 and ℎ ∈  2, 0 and write the functional L as where It is clear that L 1 and L 2 are well-defined and L 2 is linear.Moreover, so L 2 is continuous.Consequently, it is differentiable in the sense of Frechet on  1, ([, ], R) and the differential at any point  ∈  1, ([, ], R) is equal to L 2 .Using the Lebesque dominated convergence theorem and the mean value theorem, we assert that the mapping L 1 has the first variation L 1 (, ℎ) at any point  ∈  2, 0 and in any direction ℎ ∈  2, 0 given by This means (cf.[17, Section 2.2.2]) that there exists the first variation of the mapping L given by equality (49).The proof is completed.
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 ,  ∈  2, 0 ,  ̸ = , and  ∈ (0, 1) we have If  1 ≤ 0 then This means that the solution  0 to problem (1)-( 2) is a minimum point of L, so it is unique.The proof is completed.