A Hilbert space approach to fractional differential equations

We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on R, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolationand interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.


Introduction
The concept of a fractional derivative ∂ α 0 , α ∈ ]0, 1], which we utilize, will be based on inverting a suitable continuous extension of the Riemann-Liouville fractional integral of continuous functions f ∈ C c (R) with compact support given by as an apparently natural interpolation suggested by the iterated kernel formula for repeated integration. The choice of the lower limit as −∞ is determined by our wish to study dynamical processes, for which causality 1 should play an important role. It is a pleasant fact that the classical definition of ∂ α 0 in the sense 1 Other frequent choices such as for a ∈ R, would lose time-shift invariance (a suggestive choice is a = 0), which we consider undesirable. For our choice of the limit case a = −∞ it should be noted that the Riemann-Liouville and the Caputo fractional derivative essentially coincide.
of [1] coincides with the other natural choice of ∂ α 0 as a function of ∂ 0 in the sense of a spectral function calculus of a realization of ∂ 0 as a normal operator in a suitable Hilbert space setting. This is specified below. The Hilbert space framework is based on observations in [5] and has already been exploited for linear fractional partial differential equations in [2] and [6]. Since fractional differential equations are routinely discussed as integral equations, we shall establish the connection of typical Riemann-Liouville or Caputo type fractional differential equations to the above choice of fractional derivative, by working our way backward to the fractional differential equation it represents in our chosen terminology.

Fractional derivative in a Hilbert space setting
In the present section, we introduce the necessary operators to be used in the following. We will formulate all results in the vector-valued, more specifically, in the Hilbert space-valued situation. On a first read, one may think of scalar-valued functions.
To begin with, we introduce an L 2 -variant of the exponentially weighted space of continuous functions that proved useful in the proof of the Picard-Lindelöf Theorem and is attributed to Morgenstern, [3]. We denote by L p (R; H) and L 1 loc (R; H) the space of p-Bochner integrable functions and the space of locally Bochner integrable functions on a Hilbert space H, respectively.
Definition. Let H be a Hilbert space, ̺ ∈ R and p ∈ [1, ∞]. For f ∈ L 1 loc (R; H) we denote e −̺m f := (R ∋ t → e −̺t f (t)). We define the normed spaces Next, we introduce the time derivative.
Definition. Let H be a Hilbert space.
(a) Let f, g ∈ L 1 loc (R; H). We say that f ′ = g, if for all φ ∈ C ∞ c (R) The index 0 in ∂ 0,̺ shall indicate that the derivative is with respect to time. We will introduce the fractional derivatives and fractional integrals by means of a functional calculus for ∂ 0,̺ . For this, we introduce the Fourier-Laplace transform.
Definition. Let H be a complex Hilbert space. Let ̺ ∈ R.
(a) We define the Fourier transform of f ∈ L 1 (R; H) by From now on, H denotes a complex Hilbert space. With the latter notion at hand, we provide the spectral representation of ∂ 0,̺ as the multiplication-by-argument operator Proof. For the proof of (a), we observe that the equality holds on the Schwartz space S(R; H) of smooth, rapidly decaying functions. In fact, this is an easy application of integration by parts. The result thus follows from using that F is a bijection on S(R; H) and that S(R; H) is an operator core, for both m and ∂ 0,0 . For the statement (b) let f ∈ H 1 ̺ (R; H) and ϕ ∈ C ∞ c (R). Then e −̺m ϕ ∈ C ∞ c (R) and Hence e −̺m f ∈ H 1 0 (R; H) and ∂ 0,0 e −̺m f = e −̺m ∂ 0,̺ f − ̺e −̺m f . Next, we address (c). By part (a) and (b), we compute Theorem 2.2 tells us that ∂ 0,̺ is unitarily equivalent to a multiplication operator with spectrum equal to iR + ̺ = {z ∈ C; Re z = ̺}. In particular, we are now in the position to define functions of ∂ 0,̺ .
Definition. Let ̺ ∈ R and F : dom(F ) ⊆ {it + ̺ ; t ∈ R} → C be measurable such that {t ∈ R ; it + ̺ / ∈ dom(F )} has Lebesgue measure zero. We define We record an elementary fact on multiplication operators.
There is a sequence (f n ) n∈N ∈ L 2 (R; H) N with f n L 2 = 1 (n ∈ N) and´R |F (iξ + ̺)| 2 f n (ξ) 2 H dξ → ess sup |F (i · +̺)| 2 for n → ∞. This shows ∞ > F (im + ̺) ≥ F ̺,∞ . To construct (f n ) n∈N wlog. we may assume that ess sup |F (i · +̺)| > 0. Let x ∈ H with x H = 1 and let (c n ) n∈N ∈ R N be a positive sequence with c n ↑ ess sup |F (i · +̺)|. Then for n ∈ N set A n := [−n, n] ∩ {|F (i · +̺)| > c n }. By the definition of the essential supremum, we may assume that λ(A n ) > 0 and for n ∈ N we set One important class of operators that can be rooted to be of the form just introduced are fractional derivatives and fractional integrals: Then the fractional derivative of order α is given by and the fractional integral of order α is given by Note that both expressions are well-defined in the sense of functions of ∂ 0,̺ defined above and that ∂ −α We set ∂ 0 0,̺ as the identity operator on L 2 ̺ (R; H).
In order to provide the connections to the more commonly known integral representation formulas for the fractional integrals, we recall the multiplication theorem, that is, for f ∈ L 1 (R) and g ∈ L 2 (R; H).
We recall the cut-off function Lemma 2.5. For all ̺, α > 0, and ξ ∈ R, we have Proof. We start by defining the function Then we have where we have used integration by parts. By separation of variables, it follows that Since the left hand side of (2.1) equals 1 Γ(α) f (ξ), the assertion follows.
Next, we draw the connection from our fractional integral to the one used in the literature.
Using the convolution property of the Fourier transform we obtain √ 2πL ̺ g · L ̺ f = L ̺ (g * f ).
Using Lemma 2.5 we compute Proof. We use Theorem 2.6 and obtain for t ∈ R Thus, if t > 0, we obtain For t ≤ 0, we infer χ R>0 (t − s)χ R>0 (s) = 0 for s ∈ R, i.e. (∂ −α 0,̺ χ R>0 h)(t) = 0. Remark 2.8. It seems to be hard to determine analog formulas for the case ̺ < 0, although the operator ∂ −α 0,̺ for ̺ < 0, α > 0 is bounded. The reason for this is that the corresponding multiplier (im + ̺) −α is not defined in 0 and has a jump there. In particular, it cannot be extended to an analytic function on some right half plane of C. This, however, corresponds to the causality or anticausality of the operator ∂ −α 0,̺ by a Paley-Wiener result ( [4] or [7,19.2 Theorem]) and hence, we cannot expect to get a convolution formula as in the case ̺ > 0.

A reformulation of classical Riemann-Liouville and Caputo differential equations
As it has been slightly touched in the introduction, there are two main concepts of fractional differentiation (or integration). In this section we shall start to identify both these notions as being part of the same solution theory. More precisely, equipped with the results from the previous section, we will consider the initial value problems for the Riemann-Liouville and for the Caputo derivative. In order to avoid subtelties as much as possible, we will consider the associated integral equations for both the Riemann-Liouville differential equation and the Caputo differential equation and reformulate these equivalently with the description of the time-derivative from the previous section. Well-posedness results for these equations are postponed to Section 6.
To start off, we recall the Caputo differential equation. In [1], the author treated the following initial value problem of Caputo type for 1 ≥ α > 0: where y 0 ∈ C n is a given initial value; f : R >0 × C n → C n is continuous, satisfying for some c ≥ 0 and all y 1 , y 2 ∈ C n , t > 0. For definiteness, we shall also assume that for some ̺ 0 ∈ R. In order to circumvent discussions of how to interpret the initial condition, we shall rather put [1, Equation (6)] into the perspective of the present exposition. In fact, this equation reads First of all, we remark that in contrast to the setting in the previous section, the differential equation just discussed 'lives' on R >0 , only. To this end we put with the apparent meaning that f vanishes for negative times t. We note that by (3.1) and (3.2) it follows that which in turn can be (trivially) stated for all t ∈ R. Next, we present the desired rewriting of equation (3.4).
. Then the following statements are equivalent: Proof. The assertion follows trivially from Theorem 2.6.
Remark 3.2. For a real valued-function g : R >0 × R n → R n we may consider the Caputo differential equation with f : R >0 × C n → C n , (t, z) → g(t, Re(z)).
Next we introduce Riemann-Liouville differential equations. Using the exposition in [8], we want to discuss the Riemann-Liouville fractional differential equation given by where as before f satisfies (3.1) and (3.2) and y 0 ∈ R, and α ∈ (0, 1]. Again, not hinging on too much of an interpretation of this equation, we shall rather reformulate the equivalent integral equation related to this initial value problem. According to [8,Chapter 42] this initial value problem can be formulated as Let us assume that α > 1/2. Invoking the cut-off function χ R>0 and defining f as before, we may provide a reformulation of the Riemann-Liouville equation on the space By a formal calculation and when applying Corollary 2.7, i.e. ∂ −α 0,̺ χ R>0 y 0 = g α y 0 , we would obtain when understood distributionally, the delta function y 0 δ 0 and we could reformulate the Riemann-Liouville equation by ∂ α 0,̺ y = y 0 δ 0 + f (·, y(·)).

(3.5)
However, the calculation indicates that we have to extend the L 2 ̺ (R; H) calculus to understand Riemann-Liouville differential equations. This will be done in the coming sections.
and equip it with the natural inner product We shall use X ֒→ Y to denote the mapping X ∋ x → x ∈ Y , if X ⊆ Y (under a canonical identification, which will always be obvious from the context).
which proves the continuity of the embedding j β→α and the asserted norm estimate. The density follows, Definition. Let ̺ = 0 and α ∈ R. We consider the space and set H α ̺ (R; H) as its completion with respect to the norm induced by ·, · ̺,α .
has a unique unitary extension, which will again be denoted by L ̺ .
(d) For each β > 0 and α ∈ R the operator has a unique unitary extension, which will again be denoted by ∂ β 0,̺ . Proof.
by Example 2.4. Moreover, since Hence, the continuous extension of L ̺ to H α ̺ (R; H) is onto and, thus, unitary.
We conclude this section by providing an alternative perspective to elements lying in H α ̺ (R; H) for some α ∈ R (with a particular focus on α < 0). In particular, we aim for a definition of a support for those elements which coincides with the usual support of L 2 functions in the case α ≥ 0.
extends to a unitary operator. Moreover, for f ∈ H α ̺ (R; H) we have which proves the isometry of σ −1 . Moreover, σ −1 has dense range, since In particular, for α = 0 Note that the operator Proof. Let f ∈ H α ̺ (R; H). We first prove that the expression f, · is indeed a distribution. Due to Lemma 4.2(c) it suffices to prove this for and hence, for ϕ ∈ C ∞ c (R; H) we obtain using Hölder's inequality and the fact that which proves that f, · is indeed a distribution. Next, we prove the asserted formula. For this, we note the following elementary equality In particular, in the case α = 0 we obtain Moreover, we can now compare elements in H α ̺ (R; H) and H β µ (R; H) by saying that those elements are equal if they are equal as distributions. We shall further elaborate on this matter in Proposition 4.9. In particular, we shall show that f → f, · is injective. We shall also mention that the notation f, ϕ is justified, as it does not depend on ̺ nor α.
Proof. We first note that it suffices to prove the assertion for ̺ > 0, since the operator σ −1 from Lemma . By Lemma 4.7(a) we have ∂ −α 0,̺ ψ n ∈ C ∞ (R; H) ∩ H ∞ ̺ (R; H) and by Lemma 4.7(b) we find (ϕ n ) n∈N ∈ C ∞ c (R; H) N with ∂ −α 0,̺ ψ n − ϕ n ̺,α → 0 (n → ∞). Then With this result at hand, we can characterize those distributions, which belong to H α ̺ (R; H) for some α ∈ R, ̺ = 0, in the following way. Proof. Assume first that there is f ∈ H α ̺ (R; H) representing ψ. Then we estimate In the next proposition, we shall also obtain the announced uniqueness statement, that is, the injectivity of the mapping f → f, · .
which completes the proof.

A unified solution theory -well-posedness and causality of fractional differential equations
We are now able to study abstract fractional differential equations of the form In order to obtain well-posedness of the latter problem, we need to restrict the class of admissible righthand sides F in the latter equation.
Definition. Let ̺ 0 > 0 and β, γ ∈ R. We call a function F : H) and there exists ν ≥ ̺ 0 such that for each ̺ ≥ ν the function F has a Lipschitz continuous extension satisfying sup ̺≥ν |F ̺ | Lip < ∞. Moreover, we call F eventually (β, γ)-contracting, if F is eventually (β, γ)-Lipschitz continuous and lim sup ̺→∞ |F ̺ | Lip < 1. Here, we denote by | · | Lip the smallest Lipschitz constant of a Lipschitz continuous function: Note that by Lemma 4.8, any eventually Lipschitz continuous function is densely defined. Thus, the Lipschitz continuous extension F ̺ is unique. H) and g ∈ H β µ (R; H) generate the same distribution, we have that Indeed, by Proposition 4.10 there exists a sequence (ϕ n ) n∈N in C ∞ c (R; H) with ϕ n → f and ϕ n → g in H β ̺ (R; H) and H β µ (R; H), respectively. We infer that with convergence in H γ ̺ (R; H) and H γ µ (R; H) respectively. Consequently with convergence in L 2 ̺ (R; H) and hence almost everywhere for a suitable subsequence of (ϕ n ) n∈N . The assertion follows from Proposition 4.10. (b) We shall need the following elementary observation later on. Let F be evenutally (β, γ)-Lipschitz continuous, α ∈ R. Let ̺ ≥ ̺ 0 . Then is eventually (β + α, γ)-Lipschitz continuous. Indeed, the assertion follows from part (a) and for µ ≥ ν, f, g ∈ C ∞ c (R; H).
Proof. This is a simple consequence of the contraction mapping theorem. Indeed, choosing ν ≥ ̺ 0 large enough, such that |F ̺ | Lip < 1 for each ̺ ≥ ν, we obtain that Hence, the mapping ∂ −α 0,̺ F ̺ admits a unique fixed point u ̺ ∈ H β ̺ (R; H), which is equivalent to u ̺ being a solution of (5.1).
For doing so, we need the concept of causality, which will be addressed in the next propositions.
Lemma 5.4. Let ̺ > 0, α ∈ R and a ∈ R. Let f ∈ H α ̺ (R; H) with spt f ⊆ R ≥a . Then there is a sequence (ϕ n ) n∈N ∈ C ∞ c (R; H) N with spt ϕ n ⊆ R ≥a for n ∈ N and ϕ n → f in H α ̺ (R; H) as n → ∞.
The proof of the following theorem outlining causality of ∂ −α 0,̺ F ̺ , is in spirit similar to the approach in [2,Theorem 4.5]. However, one has to adopt the distributional setting and the (different) definition of eventually Lipschitz continuity here accordingly.
Theorem 5.6. Let the assumptions of Theorem 5.2 be satisfied. Then, for each ̺ ≥ ν, where ν is chosen according to Theorem 5.2, the mapping Here, the support is meant in the sense of distributions.
Proof. First of all, we shall show the result for u, v ∈ C ∞ c (R; H). So, let u, v ∈ C ∞ c (R; H) with spt(u−v) ⊆ R ≥a . Take ϕ ∈ C ∞ c (R; H) with spt ϕ ⊆ R <a . Let µ ≥ ̺. Then F ̺ (u) = F µ (u) and . According to Proposition 5.5 we have that spt ∂ −β 0,µ σ −1 ϕ ⊆ R >−a and hence, we compute On the other hand and consequently, by dominated convergence. Summarizing, we have shown that spt( Before we conclude the proof, we show that if (w n ) n∈N is a convergent sequence in H β ̺ (R; H) with spt w n ⊆ R ≥a for each n ∈ N, then its limit w also satisfies spt w ⊆ R ≥a . For doing so, let ϕ ∈ C ∞ c (R; H) with spt ϕ ⊆ R <a . Then We set u n := ϕ n +v n . Then u n → u in H α ̺ (R; H) and spt(u n −v n ) ⊆ R ≥a . By the already proved result for C ∞ c (R; H), we infer that spt Finally, we prove that our solution is independent of the particular choice of the parameter ̺ > ν in Theorem 5.2. The precise statement is as follows.
Then u µ = u µ as distributions in the sense of Proposition 4.4.
Proof. We note that it suffices to show v µ := ∂ β 0,µ u µ = ∂ β 0, µ u µ =: v µ as L 1 loc (R; H) functions by Proposition 4.10. We consider the function given by Note that the expression on the right hand side of (5.2) does not depend on the particular choice of ̺ ≥ ̺ 0 by Proposition 4.10. Clearly, F is eventually (0, β − α)-contracting (see also Remark 5.1(b)) and In particular, Let now a ∈ R and assume without loss of generality that µ < µ. We note that spt We obtain, applying Theorem 5.6, that i.e. χ R ≤a v µ is a fixed point of χ R ≤a ∂ β−α 0,µ F µ . However, since χ R ≤a v µ is also a fixed point of this mapping, which is strictly contractive, we derive and since a ∈ R was arbitrary, the assertion follows.
6 Riemann-Liouville and Caputo differential equations revisited In this section, we shall consider the differential equations introduced in Section 3 and prove their wellposedness and causality. First of all, we gather some results ensuring the Lipschitz continuity property needed to apply either of the well-posedness theorems presented in the previous section. As in Section 3 we fix α ∈ (0, 1].
Then the mapping F : C ∞ c (R; C n ) → C(R; C n ) given by is eventually (0, 0)-Lipschitz continuous.
Proof. Let ̺ ≥ ̺ 0 . In order to prove that F attains values in L 2 ̺ (R; C n ), we shall show F (0) ∈ L 2 ̺ (R; C n ) first. For this we computê Here we used that L 2 ̺ (R >0 ; H) ֒→ L 2 ̺0 (R >0 ; H) as contraction. Next, let ϕ, ψ ∈ C ∞ c (R; R n ). Then we obtainˆR Since F (0) ∈ L 2 ̺ (R; C n ), the shown estimate yields F (ϕ) ∈ L 2 ̺ (R; C n ) for each ϕ ∈ C ∞ c (R; C n ) as well as the eventual (0,0)-Lipschitz continuity of F . The next result is concerned with the well-posedness for Caputo fractional differential equations. We shall use the characterization of the Caputo differential equation outlined in Theorem 3.1.