3 Feb 2014 Fractional fundamental lemma and fractional integration ...

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Abstract. In the first part of the paper, we prove a fractional fundamental (du Bois-. Reymond) lemma and a fractional variant of the integration by parts formula.
arXiv:1402.0319v1 [math.OC] 3 Feb 2014

Fractional fundamental lemma and fractional integration by parts formula – Applications to critical points of Bolza functionals and to linear boundary value problems Lo¨ıc Bourdin Laboratoire de Math´ematiques et de leurs Applications UMR CNRS 5142, Pau Universit´e de Pau et des Pays de l’Adour, France e-mail : [email protected] Dariusz Idczak Faculty of Mathematics and Computer Science University of Lodz Banacha 22, 90-238 Lodz, Poland e-mail : [email protected] Abstract In the first part of the paper, we prove a fractional fundamental (du BoisReymond) lemma and a fractional variant of the integration by parts formula. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville fractional derivatives, derived in this paper too. In the second part of the paper, we use the previous results to give necessary optimality conditions of Euler-Lagrange type (with boundary conditions) for fractional Bolza functionals and to prove an existence result for solutions of linear fractional boundary value problems. In the last case we use a Hilbert structure and the Stampacchia theorem.

Keywords: fractional Riemann-Liouville derivative; fundamental lemma; integration by parts; Euler-Lagrange equation; boundary value problem. AMS Classification: 26A33; 49K99; 70H03.

Contents 1 Introduction

2 1

2 Fractional fundamental lemma 2.1 Preliminaries on fractional calculus . . . . . . . . . . . . . . . . . . . 2.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6

3 Fractional integration by parts formula 3.1 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . α,p α,p 3.2 Functional spaces ACa+ and ACb− . . . . . . . . . . . . . . . . . . . 3.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 9

4 Application to critical points of Bolza funtionals 10 4.1 Quasi-polynomially controlled growth for Lagrangian L . . . . . . . . 11 4.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Application to linear boundary value problems

1

15

Introduction

We first introduce some notations available in the whole paper. Let m be a nonzero integer and let k · k be the Euclidean norm of Rm . Let (a, b) ∈ R2 such that a < b. For any 1 ≤ p ≤ ∞, by Lp := Lp ([a, b], Rm ) (resp. W 1,p := W 1,p ([a, b], Rm )) we denote the usual p-Lebesgue space (resp. p-Sobolev space) endowed with its usual p we denote the usual adjoint of p. norm k · kLp (resp. k · kW 1,p ) and by p′ := p−1 ∞ ∞ m Finally, by Cc := Cc ([a, b], R ) we denote the classical space of smooth functions with compact supports contained in (a, b). Fractional calculus is the mathematical field that deals with the generalization of the classical notions of integral and derivative to any positive real order, see [19] for a detailed study. It has numerous applications in various areas of science, mainly to study some anomalous processes. Many physical applications can be found in the monograph [9]. The introduction of fractional operators in calculus of variations is due to F. Riewe in [18] where he aimed at giving a fractional variational structure for some non conservative systems in Physics. In the next years, numerous Euler-Lagrange equations were obtained for functionals depending on fractional integrals and/or derivatives. These functionals have different forms and are defined on different function spaces with different initial/boundary conditions. For example, the author of [2] considers a functional depending on left- and right-sided Riemann-Liouville derivatives, in the class of functions q possessing continuous leftand right-sided Riemann-Liouville derivatives and satisfying boundary conditions of the form q(a) = qa , q(b) = qb . (1) Such boundary conditions are also considered in [3]. In [7] the authors consider a functional depending on a general fractional derivative operator of order (α, β) (introduced in [6]) in a domain of complex piecewise smooth functions q satisfying conditions (1). Many other cases of necessary optimality conditions of Euler-Lagrange 2

type can be found in monographs [15, 17]. In [4], an existence result for solutions of fractional Euler-Lagrange equations is investigated: the functionals are defined α on Ia+ (Lp ) with 0 < 1/p < α < 1 and the existence of a minimizer is obtained under suitable assumptions of regularity, coercivity and convexity. In particular, the extremals q satisfy the initial conditions 1−α (Ia+ q)(a) = 0, q(a) = 0.

In [16], the authors obtain the existence of extremals in the case of a fractional functional associated with a quadratic Lagrangian L. These extremals satisfy boundary conditions of the form 1−α (Ia+ q)(a) = 0, q(b) = 0. The above results - necessary optimality conditions of Euler-Lagrange type and existence of solutions - both require the adaptation to the fractional case of two classical arguments of the calculus of variations. First of them is the classical fundamental lemma (also known as du Bois-Reymond lemma) and the second one is the classical theorem on the integration by parts. Let us recall them: Lemma 1 (fundamental lemma) If q1 , q2 ∈ L1 and Z b q1 (τ ) · h(τ ) + q2 (τ ) · h′ (τ ) dτ = 0 a

for any function h ∈ Cc∞ , then q2 is absolutely continuous (more precisely, it has an absolutely continuous representative) and q2′ (t) = q1 (t), t ∈ [a, b] a.e.. Theorem 1 (integration by parts) If q1 , q2 : [a, b] → Rm are absolutely continuous, then Z b Z b ′ q1 (τ ) · q2 (τ ) dτ = q1 (b) · q2 (b) − q1 (a) · q2 (a) − q1 (τ ) · q2′ (τ ) dτ. a

a

In the above papers, the fractional variants of these two previous results are considered in particular cases. In this paper, our first goal is to prove general fractional counterparts of Lemma 1 and Theorem 1 in the case of Riemann-Liouville derivatives. Our second aim is to give two applications of them in fractional variational frameworks: necessary optimality conditions of Euler-Lagrange type (with necessary boundary conditions) for fractional Bolza functionals and existence of a solution for linear fractional boundary value problems with the help of a Hilbert structure and the classical Stampacchia theorem. Precisely, in the first part of the paper, we prove a fractional counterpart of Lemma 1. Next, we prove a theorem on the integral representation of functions posα,p sessing Riemann-Liouville fractional derivatives and we consider the spaces ACa+ , 3

α,p ACb− being fractional counterparts of the classical space of absolutely continuous functions with p-integrable derivatives. Finally, we derive a fractional analogue of α,p α,r Theorem 1 for functions (q1 , q2 ) ∈ ACa+ × ACb− in the case of 0 ≤ 1p < α < 1 and 0 ≤ 1r < α < 1. In the second part of the paper, we consider Bolza functionals of type Z b

Φ(q) =

a

α 1−α L(τ, q(τ ), (Da+ q)(τ )) dτ + ℓ((Ia+ q)(a), q(b))

defined on the whole space of functions q possessing left-sided Riemann-Liouville α derivative Da+ q ∈ Lp with α ∈ ( 1p , 1). Here L : (t, x, v) ∈ [a, b] × Rm × Rm 7−→ L(t, x, v) ∈ R is a Lagrangian and ℓ : (x1 , x2 ) ∈ Rm × Rm 7−→ l(x1 , x2 ) ∈ R is a function describing a pointwise term of Bolza functional. We derive the associated Euler-Lagrange equation α α α Db− Lv (·, q, Da+ q) = −Lx (·, q, Da+ q), t ∈ [a, b] a.e.,

and the necessary boundary conditions: α 1−α Lv (a, q(a), (Da+ q)(a)) = ℓx1 ((Ia+ q)(a), q(b)),

(2)

1−α α 1−α (Ib− Lv (·, q, Da+ q))(b) = −ℓx2 ((Ia+ q)(a), q(b)).

(3)

Finally, we apply the fractional fundamental lemma and the Stampacchia theorem to prove the existence of a solution for the linear problem α α Db− Da+ q + q = f, t ∈ [a, b] a.e.,

where f ∈ L2 , with nonhomogeneous boundary conditions 1−α Ia+ q(a) = qa , q(b) = qb

where qa , qb ∈ Rn . The results of the paper (except Section 5) were presented during the 7th International Workshop on Multidimensional (nD) Systems, (Poitiers, France, 2011) and published - without proofs - in conference proceedings (cf. [11]) in the special case α ∈ ( 12 , 1). The aim of this paper is to give detailed proofs of these theorems and to improve them by removing the assumption α ∈ ( 12 , 1). In Section 5, we add an application of the obtained results to linear fractional boundary value problems. To our best knowledge, the results presented in the paper have not been obtained by other authors. In particular, note that the fractional integration by parts leads to the necessary boundary conditions (2)-(3). Finally, let us mention that [13] contains the following theorems: a fractional fundamental lemma for α ∈ (n − 1/2, n) with n ∈ N, n ≥ 2, a theorem on the fractional integration by parts for α ∈ (n − 1, n) 1 1 with n ∈ N, n ≥ 2 and α−n+1 < p < ∞, α−n+1 < q < ∞, as well as a theorem on the integral representation of functions possessing the fractional Riemann-Liouville derivative for α ∈ (n − 1, n) with n ∈ N, n ≥ 2. 4

The paper is organized as follows. In Section 2 we give some basic recalls on fractional calculus and we prove a fractional fundamental lemma. Section 3 contains a theorem on the integral representation of functions possessing fractional RiemannLiouville derivatives. Then, we derive a fractional integration by parts formula. The two last sections are devoted to applications of the previous results to EulerLagrange equations for fractional Bolza functionals (Section 4) as well as to existence of solutions for some linear boundary value problems (Section 5).

2

Fractional fundamental lemma

The aim of this section is to prove a fractional counterpart of Lemma 1. Section 2.1 gives some basic recalls on fractional calculus and the main result is derived in Section 2.2.

2.1

Preliminaries on fractional calculus

By the left- and right-sided Riemann-Liouville fractional integrals of order α > 0 of q ∈ L1 we mean the following functions Z t q(τ ) 1 α dτ, t ∈ [a, b] a.e., (Ia+ q)(t) := Γ(α) a (t − τ )1−α α (Ib− q)(t)

1 := Γ(α)

Z

t

b

q(τ ) dτ, t ∈ [a, b] a.e.. (τ − t)1−α

α α Ia+ , Ib−

Recall that are linear continuous operator from Lp to Lp for any 1 ≤ p ≤ ∞, see [19, Theorem 2.6]. The two following propositions have been proved in [19, Theorem 2.5] and [19, Corollary of Theorem 3.5]. α1 α2 α1 +α2 Proposition 1 If α1 , α2 > 0, then (Ia+ Ia+ q)(t) = (Ia+ q)(t), t ∈ [a, b] a.e. for any q ∈ L1 .

Remark 1 Analogous proposition for the right-sided integral holds true. Proposition 2 If α > 0, 1 ≤ p ≤ ∞, 1 ≤ r ≤ ∞ and 1p + 1r ≤ 1 + α (additionally, we assume that p > 1 and r > 1 when p1 + 1r = 1 + α), then Z

b a

α (Ia+ q1 )(τ )

· q2 (τ ) dτ =

Z

a

for any q1 ∈ Lp , q2 ∈ Lr .

5

b α q1 (τ ) · (Ib− q2 )(τ ) dτ

2.2

Main result

Recall that a function q : [a, b] −→ Rm is absolutely continuous if and only if there exist a constant c ∈ Rm and a function ϕ ∈ L1 such that 1 q(t) = c + (Ia+ ϕ)(t), t ∈ [a, b].

(4)

In this case, we have q(a) = c and q ′ (t) = ϕ(t), t ∈ [a, b] a.e.. Remark 2 A fractional counterpart of this integral representation is derived in Section 3.1. Recall the following definitions (see [12]): Definition 1 We say that q ∈ L1 possesses a left-sided Riemann-Liouville deriva1−α α tive Da+ q of order α ∈ (0, 1) if the function Ia+ q has an absolutely continuous 1−α representative. In this case, Ia+ q is identified to its absolutely continuous represen1−α α α q). tative and Da+ q is defined by Da+ q := dtd (Ia+ Definition 2 We say that q ∈ L1 possesses a right-sided Riemann-Liouville deriva1−α α tive Db− q of order α ∈ (0, 1) if the function Ib− q has an absolutely continuous 1−α representative. In this case, Ib− q is identified to its absolutely continuous represen1−α α α tative and Db− q is defined by Db− q := − dtd (Ib− q). Using the method of the proof presented in [4, proof of Theorem 2] we obtain the following fractional counterpart of Lemma 1. Lemma 2 (fractional fundamental lemma) If α ∈ (0, 1), q1 ∈ L1 , q2 ∈ L1 and Z b α q1 (τ ) · h(τ ) + q2 (τ ) · (Da+ h)(τ ) dτ = 0 a

for any h ∈ tive with

Cc∞ ,

then q2 possesses a right-sided Riemann-Liouville fractional derivaα (Db− q2 )(t) = −q1 (t), t ∈ [a, b] a.e..

1 h′ )(t), t ∈ [a, b]. From Proposition 1, Proof. For any h ∈ Cc∞ , h(t) = (Ia+ 1−α 1−α ′ 1−α ′ 1−α 1 (Ia+ h)(t) = (Ia+ Ia+ h )(t), t ∈ [a, b] a.e., where Ia+ h ∈ L∞ ⊂ L1 . Thus, Ia+ h has an absolutely continuous representative. Then, h possesses a left-sided Riemann1−α ′ α Liouville fractional derivative given by Da+ h = Ia+ h ∈ L∞ . Consequently, our assumption and Proposition 2 imply Z b 1−α q1 (τ ) · h(τ ) + (Ib− q2 )(τ ) · h′ (τ ) dτ = 0 (5) a

for any h ∈

Cc∞ .

Lemma 1 concludes the proof.

Remark 3 In the limit case of α = 1, the above lemma leads to the classical fundamental lemma (Lemma 1). 6

3

Fractional integration by parts formula

α α As usually, by Ia+ (Lp ), Ib− (Lp ) we respectively denote the ranges of the operators α α α α Ia+ , Ib− on Lp for any 1 ≤ p ≤ ∞. In particular, if q ∈ Ia+ (L1 ) with q = Ia+ ϕ, ϕ ∈ 1 α L , then q possesses a left-sided Riemann-Liouville derivative given by Da+ q = ϕ. Analogous result for the right-sided derivative holds true. From the previous remark and Proposition 2 the following result on the integration by parts for Riemann-Liouville fractional derivatives follows (cf. [19, Corollary 2 of Theorem 2.4]).

Proposition 3 If α ∈ (0, 1), 1 ≤ p ≤ ∞, 1 ≤ r ≤ ∞ and p1 + 1r ≤ 1 + α (additionally, we assume that p > 1 and r > 1 when p1 + 1r = 1 + α), then Z

b a

α (Da+ q1 )(τ )

· q2 (τ ) dτ =

Z

a

b α q1 (τ ) · (Db− q2 )(τ ) dτ

α α for any q1 ∈ Ia+ (Lp ), q2 ∈ Ib− (Lr ). α The aim of this section is to extend Proposition 3 from functions q1 ∈ Ia+ (Lp ), α q2 ∈ Ib− (Lr ) to all functions q1 , q2 possessing fractional Riemann-Liouville derivaα α tives such that Da+ q1 ∈ Lp and Db− q2 ∈ Lr . Our result will be valid under the additional assumption 0 ≤ p1 < α < 1 and 0 ≤ 1r < α < 1. Section 3.1 provides an integral representation for functions possessing fractional α,p Riemann-Liouville derivatives. Section 3.2 is devoted to the functional spaces ACa+ α,p and ACb− . The main result is stated in Section 3.3.

3.1

Integral representation

We first prove the fractional counterpart of the integral representation (4). Theorem 2 (integral representation) Let α ∈ (0, 1) and q ∈ L1 . Then, q has a α left-sided Riemann-Liouville derivative Da+ q of order α if and only if there exist a m 1 constant c ∈ R and a function ϕ ∈ L such that q(t) =

1 c α + (Ia+ ϕ)(t), t ∈ [a, b] a.e.. Γ(α) (t − a)1−α

(6)

1−α α In this case, we have (Ia+ q)(a) = c and (Da+ q)(t) = ϕ(t), t ∈ [a, b] a.e..

Proof. Let us assume that q ∈ L1 has a left-sided Riemann-Liouville derivative 1−α α Da+ q. This means that Ia+ q is (identified to) an absolutely continuous function. From the integral representation (4), there exist a constant c ∈ Rm and a function ϕ ∈ L1 such that 1−α 1 (Ia+ q)(t) = c + (Ia+ ϕ)(t), t ∈ [a, b], (7) 7

1−α 1−α α q)(t) = ϕ(t), t ∈ [a, b] a.e.. From Propowith (Ia+ q)(a) = c and Da+ q(t) = dtd (Ia+ α sition 1 and applying Ia+ on (7) we obtain 1 α 1 α (Ia+ q)(t) = (Ia+ c)(t) + (Ia+ Ia+ ϕ)(t), t ∈ [a, b] a.e..

(8)

The result follows from the differentiation of (8). Now, let us assume that (6) holds 1−α true. From Proposition 1 and applying Ia+ on (6) we obtain 1−α 1 (Ia+ q)(t) = c + (Ia+ ϕ)(t), t ∈ [a, b] a.e.

(9)

1−α and then, Ia+ q has an absolutely continuous representative and q has a left-sided α Riemann-Liouville derivative Da+ q. The proof is complete.

Remark 4 The above theorem can also be deduced from [14, Corollary 2.1, Lemma 2.5 (b), Lemma 2.6 (b)] but, to our best knowledge, it has not been formulated by other authors. Of course, in an analogous way one can prove: Theorem 3 Let α ∈ (0, 1) and q ∈ L1 . Then, q has the right-sided Riemannα Liouville derivative Db− q of order α if and only if there exist a constant d ∈ Rm and 1 a function ψ ∈ L such that q(t) =

d 1 α + (Ib− ψ)(t), t ∈ [a, b] a.e.. 1−α Γ(α) (b − t)

(10)

1−α α In this case, we have (Ib− q)(b) = d and (Db− q)(t) = ψ(t), t ∈ [a, b] a.e..

3.2

α,p α,p Functional spaces ACa+ and ACb−

Let us consider the following spaces. α,p Definition 3 For every α ∈ (0, 1) and every 1 ≤ p ≤ ∞, we denote by ACa+ := α,p m 1 ACa+ ([a, b], R ) the set of all functions q ∈ L that have a left-sided Riemannα Liouville derivative Da+ q ∈ Lp . α,p Definition 4 For every α ∈ (0, 1) and every 1 ≤ p ≤ ∞, we denote by ACb− := α,p m 1 ACb− ([a, b], R ) the set of all functions q ∈ L that have a right-sided Riemannα Liouville derivative Db− q ∈ Lp . α,p In other words, ACa+ is the set of all functions that have integral represenα,p p tation (6) with ϕ ∈ L and ACb− is the set of all functions that have integral representation (10) with ψ ∈ Lp .

8

α Remark 5 For any α ∈ (0, 1) and any q ∈ L1 , recall that Ia+ q ∈ Lr for every α,p 1 1 ≤ r < 1−α , see [19, Theorem 21.4]. Then, the inclusions ACa+ ⊂ Lr and α,p 1 and any 1 ≤ p ≤ ∞. ACb− ⊂ Lr hold for any 1 ≤ r < 1−α

In what follows, we are specially interested in the case 0 ≤ 1p < α < 1. In such a α case, for any ϕ ∈ Lp , Ia+ ϕ can be identified to a continuous function (more precisely to a H¨olderian continuous function with exponent α − 1p ) vanishing at t = a, see [19, α Theorem 3.6]. Similarly, Ib− ψ can be identified to a continuous function vanishing p at t = b for any ψ ∈ L . In the sequel, in the case 0 ≤ p1 < α < 1, we consider the following identification α,p for every q ∈ ACa+ : q(t) =

1 c α + (Ia+ ϕ)(t), t ∈ (a, b] Γ(α) (t − a)1−α

(11)

1−α α where ϕ = Da+ q ∈ Lp and c = (Ia+ q)(a). In particular, q is defined at t = b. The α,p analogous identification is considered for every q ∈ ACb− :

q(t) =

1 d α + (Ib− ψ)(t), t ∈ [a, b) Γ(α) (b − t)1−α

(12)

1−α α where ψ = Db− q ∈ Lp and d = (Ib− q)(b). In particular, q is defined at t = a.

3.3

Main result

The following theorem on the integration by parts for Riemann-Liouville fractional derivatives holds. Theorem 4 (fractional integration by parts) If 0 ≤ α < 1, then Z b Z b α α (Da+ q1 )(τ ) · q2 (τ ) dτ = q1 (τ ) · Db− q2 (τ ) dτ a

1 p

< α < 1 and 0 ≤

1 r


0 such that a(x, x) ≥ β kxk2H , x ∈ H. Theorem 6 (Stampacchia) Let a : H × H −→ R be a continuous and coercive bilinear form and let K ⊂ H be a nonempty closed convex set. Then, for any continuous linear form φ : H −→ R, there exists exactly one point x ∈ H such that a(x, x − y) ≥ φ(x − y), y ∈ K. α,2 In what follows, we assume that p = 2 and α ∈ ( 21 , 1). In particular, ACa+ ⊂ L2 . α,2 We endow ACa+ with the scalar product Z b Z b α α hq1 , q2 iAC α,2 = q1 (τ ) · q2 (τ ) dτ + (Da+ q1 )(τ ) · (Da+ q2 )(τ ) dτ a+

a

a

and by k · kAC α,2 we denote the generated norm. In particular a+

α,2 α . qk2L2 )1/2 , q ∈ ACa+ kqkAC α,2 = (kqk2L2 + kDa+ a+

α,2 Lemma 5 The functional space (ACa+ , k · kAC α,2 ) is a Hilbert space. a+

α,2 α,2 Proof. Let us prove that ACa+ is complete. Let (qn )n∈N ⊂ ACa+ be a Cauchy α sequence. Then (qn )n∈N and (Da+ qn )n∈N are Cauchy sequences in the complete space (L2 , k · kL2 ). By q and u we denote their respective limits in (L2 , k · kL2 ). Our aim α,2 α is to prove that q ∈ ACa+ and Da+ q = u. Theorem 4 leads to Z b α α (Da+ qn )(τ ) · h(τ ) − qn · (Db− h)(τ ) dτ = 0 a

15

for any n ∈ N and any h ∈ Cc∞ . Passing to the limit on n, we obtain Z

a

b α u(τ ) · h(τ ) − q · (Db− h)(τ ) dτ = 0

for any h ∈ Cc∞ . The counterpart of Lemma 2 for the right fractional derivative concludes the proof. Lemma 6 The set α,2 1−α K = {q ∈ ACa+ ; (Ia+ q)(a) = qa , q(b) = qb } α,2 is a nonempty closed and convex subset of ACa+ .

Proof. The convexity of K is obvious. To prove that K is nonempty, it is sufficient to consider 1 qa α q(t) = + (Ia+ θ)(t), t ∈ (a, b] 1−α Γ(α) (t − a) with θ =

Γ(α+1) q (b−a)α b



Γ(α+1) q . Γ(α)(b−a) a

Let us prove that K is closed. Let (qn )n∈N be a

α,2 1−α sequence of K tending to q in ACa+ . Our aim is to prove that (Ia+ q)(a) = qa and 1−α 1−α q(b) = qb . Since (qn )n∈N tends to q in (L2 , k·kL2 ), we have (Ia+ qn )n∈N tends to Ia+ q 1−α 1−α d d α α 2 in (L , k · kL2 ). Since (Da+ qn )n∈N = ( dt (Ia+ qn ))n∈N tends to Da+ q = dt (Ia+ q) in 1−α 1−α (L2 , k · kL2 ), we conclude that (Ia+ qn )n∈N tends to Ia+ q in (W 1,2 , k · kW 1,2 ). From the compact embedding of (W 1,2 , k·kW 1,2 ) in the set of continuous functions endowed 1−α with the usual uniform norm (cf. [5]), we conclude that ((Ia+ qn )(a))n∈N tends to 1−α 1−α (Ia+ q)(a) and then (Ia+ q)(a) = qa . Finally, the integral representation of qn at t = b gives Z b α Da+ qn (τ ) 1 1 qa qb = + dτ Γ(α) (b − a)1−α Γ(α) a (b − τ )1−α

for any n ∈ N. Passing to the limit on n leads to q(b) = qb and concludes the proof. Finally, we prove α,2 Theorem 7 The linear boundary value problem (20)-(21) has a solution q ∈ ACa+ . α,2 α,2 Proof. Let us consider the continuous coercive bilinear form a : ACa+ × ACa+ −→ R given by α,2 a(q1 , q2 ) = hq1 , q2 iAC α,2 , q1 , q2 ∈ ACa+ , a+

the continuous linear form φ : φ(q) =

α,2 ACa+

Z

b

−→ R given by

α,2 f (τ ) · q(τ ) dτ, q ∈ ACa+

a

16

and the nonempty closed convex set K defined in the previous lemma. Stampacchia theorem gives the existence of a function q ∈ K such that Z

b

q(τ ) · (q(τ ) − q1 (τ )) dτ +

a

Z

a

b α α α (Da+ q)(τ ) · ((Da+ q)(τ ) − (Da+ q1 )(τ )) dτ Z b ≥ f (τ ) · (q(τ ) − q1 (τ )) dτ (22) a

for any q1 ∈ K. Considering the above inequality with functions q1 = q ± h ∈ K with h ∈ Cc∞ we assert that Z

b a

α α (q(τ ) − f (τ )) · h(τ ) + Da+ q(τ ) · Da+ h(τ ) dτ = 0

for any h ∈ Cc∞ . The fractional fundamental lemma (Lemma 2) concludes. Acknowledgement 8 The project was partially financed with funds of National Science Centre, granted on the basis of decision DEC-2011/01/B/ST7/03426.

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