Lebesgue regularity for differential difference equations with fractional damping

We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences ℓp(Z,X) for equations that can be modeled in the form Δαu(n)+λΔβu(n)=Au(n)+G(u)(n)+f(n),n∈Z,α,β>0,λ≥0, where X is a Banach space, f∈ℓp(Z,X), A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δγ denotes the fractional difference operator of order γ>0 in the sense of Grünwald‐Letnikov. Our class of models includes the discrete time Klein‐Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term.


INTRODUCTION
This paper is concerned with a wide class of mixed evolution equations that can be considered either as models for partial differential equations that are continuous in space but discrete in time, 1 or systems of difference equations. 2, Chapter 3;3 An additional feature of these models is that they admit the possibility of two fractional orders in the discrete variable.
Typical models that are included in this article correspond to the discrete time Klein-Gordon equation where Δ 2 u(n, x) ∶= u(n + 2, x) − 2u(n + 1, x) + u(n, x), and the discrete time telegraph equation Δ 2 u(n, x) + Δu(n, x) = u xx (n, x), n ∈ Z, ≥ 0, > 0, x ∈ J ⊂ R, (2) as well as fractional versions of them. 4,5 The discrete version of the Basset equation 6,7 Δ 2 u(n) + Δ 3∕2 u(n) + bu(n) = (n), n ∈ Z, ,b > 0 will be also included in our framework. The study of the uniqueness and causality of p-summable solutions 8 suggests to consider the above equations on Z. More precisely, given a Banach space X, we ask the following problem: Is it possible to characterize solely in terms of the data of a given mixed evolution equation, the existence, and uniqueness of solutions that belong to the vector-valued space of sequences (Z; X) ?
We success solving this open problem for the following abstract model Δ u(n) + Δ u(n) = Au(n) + (n), n ∈ Z, , > 0, ≥ 0, (4) where ∈ (Z, X), A is a closed linear operator with domain D(A) defined on X and Δ denotes the fractional difference operator of order > 0 as defined recently in Abadías and Lizama. 9 Roughly speaking, it corresponds to a slight variant of the Grünwald-Letnikov derivative. Compare Definition 2.1 below with Ortigueira et al. 10 , Section 3.3 formula (27). It is worthwhile to observe that, for instance, the model (1) includes the Basset Equation 3 taking X = C, A = bI, = 2, and = 3∕2, whereas it also includes the linearized Klein-Gordon Equation 1 choosing X = L 2 (Ω), A = xx − bI, = 2, and = 0. Modeling with fractional difference equations is a recent and promising area of research that has been developed from different sides of interest. For instance, Atici and Sengul 11 develop some basics results of discrete fractional calculus. These authors introduce and solve Gompertz fractional difference equation for tumor growth models. See also Atici and Eloe 12 for related results in this direction. The methodology used in such discrete fractional calculus was extended in Lizama 13 to the context of abstract models, including in this way the handling of difference differential equations by methods of functional analysis and operator theory. Studies on qualitative properties, as for example, the existence of positive solutions for discrete fractional systems, have been provided by Goodrich. [14][15][16] Other interesting contributions are due to Ferreira, 17 Holm, 18 Kovács, Li, and Lubich, 19 Dassios,20,21 Wu, Baleanu et al, [22][23][24][25] and Tarasov et al. [26][27][28] Starting with the work of Blunck, 29 the existence and uniqueness of solutions for discrete systems that belong to the Lebesgue space of vector-valued sequences began to be considered by many authors. [30][31][32][33] Some of the studies correspond to a numerical point of view. 19 However, none of them have considered causal solutions, ie, solutions with domain on Z. On the other hand, some abstract models that are less general than ours have been recently reported in the literature and analyzed from diverse perspectives. For instance, in Abadias and Lizama,9 it is shown that the semilinear problem admits almost automorphic solutions whenever the operator A is the generator of a C 0 -semigroup and f satisfies Lipschitz conditions of global and local type. We note that in such paper, an application to a model of population of cells is also given. This paper is organized as follows: In Section 2, we first recall the notions of UMD-spaces, R-boundedness, and the discrete time Fourier transform defined over the space of distributions. These concepts allow us to formulate Blunck's Fourier multiplier theorem for operator-valued symbols on UMD-spaces. 29 In Section 3, we prove our main result, namely, if where (A) denotes the resolvent set of A, then the following assertions are equivalent: (i) For all ∈ (Z, X), the problem Δ u(n) + Δ u(n) = Au(n) + (n), n ∈ Z has a unique solution in (Z, [D(A)]); Furthermore, in the context of Hilbert spaces, a simpler criterion is also provided, replacing the condition (iii) above by As a consequence, we analyze the nonlinear equation where g ∈ (Z, X) and G ∶ (Z, X) → (Z, X) are given. We show that if G(0) = G ′ (0) = 0 and g is small enough, then the nonlinear equation has at least one solution in (Z, X). Finally, in Section 4, we prove, as an application of our characterization, that for all 0 < , < 2 and b > 2 + 2 , we can find * > 0 such that for all ∈ (0, * ), there exists u ∈ (Z, L 2 (R)) that solves the problem

PRELIMINARIES
In this section, we recall some concepts about fractional derivatives, the discrete time Fourier transform, in short, DTFT, and operator-valued Fourier multipliers theorems defined on UMD spaces. For more details, see Agarwal et al, Denk et al 34,35 and the references therein. Let X be a Banach space. X is said to be a UMD space if for each p > 1, there exists a constant C p > 0 such that for any ( n ) n≥0 ⊂ L (Ω, Σ, ; X) and any choice of signs ( n ) n≥0 ⊂ (−1, 1) and any N ∈ Z + , we have the following estimate In what follows, we denote by (X, Y ) the space of bounded linear operators between Banach spaces X and Y endowed with the uniform operator topology; when X = Y, we denote it by (X).
Let X and Y be a Banach spaces.
Given u ∈ (Z; X) and v ∈ 1 (Z), we define the convolution product For any ∈ R, we set Some properties related to the special kernel k can be found in Abadías et al 36 , Section 2 and Lizama. 13 Definition 2.1. Let > 0 be given and ∶ Z → X a vector-valued sequence. We define the fractional sum of order as follows and the fractional difference of order is defined by where Γ is the Euler function. see Ortigueira et al 10, formula (27), with h = 1 . The numbers k are known as Cesàro numbers of order . They were introduced by Zygmund 37, p. 77 and rediscovered in several instances. The above definition of fractional difference operator of order (for n ∈ Z) was first introduced by Abadías and Lizama, 9 after previous work of Lizama, 13 as follows: where m ∶= [ ] + 1. In the above cited references, it is also called Weil difference operator of order and is denoted by W instead of Δ. Their equivalence with (2.1) was recently proved.
In what follows, we detail the definition and properties of the DTFT in the vector-valued Lebesgue space of sequences (Z, X). We denote by (Z; X) the space of all vector-valued sequences ∶ Z → X such that for each k ∈ N 0 , there exists We also denote by C n er (R; X), n ∈ N 0 , the space of all 2 -periodic X-valued and n-times continuously differentiable functions defined in R.
Let T ∶= (− , ) and T 0 ∶= (− , )∖{0}. We introduce the space of test functions as C ∞ er (T; X) ∶= ⋂ n∈N 0 C n er (R; X) endowed with the topology induced by the countable family of seminorms: We also consider the following spaces of vector-valued distributions Observe that we can identify (Z; X) with a subspace of  ′ (Z; X) via the mapping and we have T ∈  ′ (Z, X). The space C ∞ er (T; X) can be also identified with a subspace of  ′ (T; X) via the linear map and we get L S ∈  ′ (T; X). The discrete time Fourier transform  ∶ (Z; X) → C ∞ er (T; X) is given by It is an isomorphism whose inverse is defined by where ∈ C ∞ er (T; X). This isomorphism allows to define the discrete time Fourier transform (DTFT) between the spaces of distributions  ′ (Z; X) and  ′ (T; X) as follows whose inverse  −1 ∶  ′ (T; X) →  ′ (Z; X) is given by In particular, we get The convolution of a distribution T ∈  ′ (Z, X) with a function a ∈ 1 (Z) is defined by From Lizama, 13 the following generation formula holds , ∈ R, |z| < 1, see also Zygmund. 37, p.42 formulae (1) and (8) In particular, for all ∈ R + , we have that the radial limit exists and , t ∈ T 0 .
Observe that k − ∈ 1 (Z) (see also Zygmund 37, p.42 formula (2) ). We also recall the following lemma stated in Lizama and Murillo-Arcila, 38 which will be used in the proof of our main result.

Lemma 2.2.
Let u, v ∈ (Z; X) and a ∈ 1 (Z). The following assertions are equivalent: We can now introduce the following notion of p -multiplier.
for all ∈ (Z; X) and all ∈ C ∞ er (T). Here, We finally recall the following Fourier multiplier theorem for operator-valued symbols due to Blunck, see previous studies 29, 34 for more details. This theorem will be crucial for showing our main characterization. Blunck's theorem and its converse establish an equivalence between R-bounded sets and l p -multipliers. The converse of Blunck's theorem holds without any restriction on the Banach spaces X, Y in the following sense. Let p ∈ (1, ∞) and let X, Y be Banach spaces. Let M ∶ T → (X; Y ) be an operator valued function. Suppose that there is a bounded operator T M ∶ l (Z; X) → l (Z; Y ) such that (11) holds. Then the set {M(t) ∶ t ∈ T} is R-bounded.

A CHARACTERIZATION OF MAXIMAL REGULARITY
In this section, we first provide a characterization on the existence and uniqueness of solutions in (Z; [D(A)]) for the general model where , > 0, ≥ 0, A is a closed linear operator defined on a Banach space X and ∶ Z → X is a vector-valued sequence. Recall that the above model is an abbreviated form to write a partial differential equation, which is continuous in space but discrete in time. For example, the equation where n ∈ Z, x ∈ Ω ⊂ R N , fits in the abstract setting of the model (12) with = 2, = 1, and A = xx . We introduce the following definition, also called p -well-posedness in the literature. We are ready to prove our main result in this paper. Proof. We first show (iii) ⇒ (ii). Let {M(t) ∶ t ∈ T} be R-bounded. We will show that the set {(1 − e it )(1 + e it )M(t) ∶ t ∈ T} is also R-bounded. Defining for each t ∈ T, f (t) ∶= (1 − e −it ) and f (t) ∶= (1 − e −it ) , it can be shown that

Theorem 3.2. Let A be a closed linear operator defined on an UMD space X. Set
. 34, Proposition 2.2.5 we deduce that the set {(1 − e it )(1 + e it )M ′ (t) ∶ t ∈ T} is R-bounded and the claim is proved. Consequently, by Theorem 2.4, we obtain (i). The implication (ii) ⇒ (iii) follows immediately from Theorem 2.5. Let us now show that (i) ⇒ (ii). Let ∈ (Z, X) be given. Then there exists a unique u ∈ (Z, [D(A)]) solution of (12). We define T , ∶ (Z, X) → (Z, [D(A)]) the linear operator given by T , (f) = u f . By the Closed Graph Theorem, we get that T , is bounded. Let ∈ C ∞ er (T), ∈ (Z; X) and u = T , f. Since k − ∈ 1 (Z), we obtain the following identity,
The following statement follows from the closed graph theorem and Theorem 3.2.
As a consequence of Theorem 3.2, we easily have a corresponding one in the case of Hilbert spaces, where R-boundedness is equivalent to norm boundedness. 35 (i) For all ∈ (Z, H), there exists a unique u ∈ (Z, H) such that u(n) ∈ D(A) for all n ∈ Z and u satisfies (12); Now, we can consider the nonlinear perturbed version of (12) given by where > 0, ∈ (Z, X) and G ∶ (Z, X) → (Z, X). We can show a result concerning the existence of (Z, X)-solutions of (21) in terms of the symbol of the equation and the regularity of G. Then there exists * such the Equation 21 has a solution u = u ∈ (Z, X) for each ∈ [0, * ).

EXAMPLES
We verify the conditions provided in Theorem 3.5 to show the existence and uniqueness of (Z, X) solutions for the following equation where ≥ 0 is fixed, b is a real number, > 0 and ∈ (Z, X) is an external force whose size is controlled by . Note that the linear part of the Equation 22 corresponds to the discrete time Telegraph equation when = 2, = 1, b = 0, = 1 , and A = xx . Also, it coincides with the discrete time Klein-Gordon equation for = 2, = 0, and A = xx − bI.
Remark 4.1. In particular, this example shows that the discrete Klein-Gordon Equation 1 admits nontrivial square-summable solutions defined on Z, for small and square-summable external forcing terms whenever b > 4. In the case of the generalized discrete Basset equation Δ 2 u(n) + Δ u(n) + bu(n) = (n), n ∈ Z, ,b > 0, > 0, we obtain that for any ∈ (Z), there exist p-summable solutions whenever b > 4 and < b−4 2 .