Lebesgue regularity for nonlocal time-discrete equations with delays

Abstract In this work we provide a new and effective characterization for the existence and uniqueness of solutions for nonlocal time-discrete equations with delays, in the setting of vector-valued Lebesgue spaces of sequences. This characterization is given solely in terms of the R-boundedness of the data of the problem, and in the context of the class of UMD Banach spaces.


Introduction
The recent technological innovations have caused a considerable interest in the study of dynamical processes that are of a mixed continuous and discrete nature. For instance, discrete-time linear models appear in the study of the solution to optimal control problems in dynamic programming [10]. Moreover, they are also used for modeling coal liquefaction mechanisms [34] and robust energy filtering in signal processing [25], among others fields of interest. In the biological context, qualitative behavior of discrete models with delays has been examined in [19] and [42]. A classical textbook is the monograph by Rugh [32]. Starting with the works of Weis [38], and Amann [6], characterizations of Lebesgue regularity using multiplier theorems for operator valued symbols have appeared in several papers in the last decade. See for instance the ones of Bu [12,13], Chill and Srivastava [15], the special volume [11] and references therein.
Lebesgue regularity of discrete time evolution equations in abstract spaces was first considered by Blünk [9] and Portal [30,31]. In [20], Kovács, Li and Lubich studied maximal regularity using Blunk's results for numerical schemes. In the same line, Kemmochi [22] introduced the notion of discrete maximal regularity for the finite difference method. Other recent contributions are [4,24] and [23]. A recent textbook on this topic is the monograph [3], where several applications in different contexts are given.
The analysis of p -maximal regularity for difference equations of fractional order α > 0 in the form where T is a bounded operator defined on a Banach space X was studied in [26] for the range 0 < α ≤ 1 and in [27] for 1 < α ≤ 2. Here Δ α denotes the fractional difference operator of order α > 0 in the Riemann-Liouville sense, see Definition 2.2 below. In [28] p -maximal regularity for the equation (1.1) with infinite delay was studied in Z for all α > 0 when T is an unbounded operator. Recently, in [29] the authors characterized the p -maximal regularity for the finite delayed equation 1) whenever 0 < α ≤ 1. However, the validity of such characterization for the case of 1 < α ≤ 2 was left as an open problem.
The main purpose of this work is to give a positive answer to this open problem.
An interesting feature that involves our model is that the fractional difference operator Δ α can be realized as sampling, by means of the Poisson distribution, of the classical fractional Riemann-Liouville operator. See [26,Theorem 3.5], where this remarkable connection has been discovered. This nonlocal operator has recently appeared in several research of increasing interest to different but related fields. For instance, in relation to the notion of Césaro operators of order α > 0 [2], chaos for fractional delayed logistic maps [39] and almost automorphic solutions of fractional difference equations [1]. Concerning applications, we note that fractional difference models A u t h o r ' s C o p y have been considered in areas such as nano-mechanics [35,36], economics [37], numerics [40,41] and stability [14] among others.
In order to present our main results, this paper is organized as follows: In Section 2 we provide the reader information about differences of fractional order and we show the main methods on operator-valued Fourier multipliers that we will use. In Section 3, we introduce the new concept of α τ resolvent operators in the range 1 < α ≤ 2, which is an important tool for the construction of the solution of (1.1). This family, denoted by {M α (n)} n≥−τ , incorporates directly the finite delay in its definition. Then, we will prove that a general solution for our model, with initial conditions u(j) = x j , j = −τ, . . . , 0, 1, can be written as Here, h α (n) = (α − 1) n and F α (n) = (M α * h α )(n). Note that in the case α = 2 and β = 0, the resolvent family M 2 (n) perfectly coincides with the notion of discrete cosine operator which was introduced and studied by Chojnacki [16] in the context of UMD Banach spaces. We remark that the representation (1.2) is not straightforward but it is one of the main tasks that we have overcome in order to achieve the solution of our problem.
Finally, in Section 4, we prove the main result of this work. We will show that if X is a UMD space and the condition sup n∈N M α (n) < ∞ is satisfied, then the maximal p -regularity of equation (1.1) and the Rboundedness of the sets are equivalent. This characterization coincides perfectly as the counterpart of the result achieved in the paper [29] where also an R-boundedness condition on two sets is needed. We note that in practice, tools to check this condition are generally not easy to find. However, the monograph [3] shows a way in the general case. For the case of Hilbert spaces, we observe that R-boundedness can be replaced merely by uniform boundedness. For such a case, we are able to provide a very simple criterion on T that ensures maximal p -regularity of equation (1.1), namely:

Preliminaries
In this section we define some preliminary concepts related to fractional differences, UMD spaces, Fourier multiplier theorems, discrete Fourier transforms, and R-boundedness. In what follows we denote by s(N 0 ; X) the vector space of all vector-valued sequences f : N 0 → X. The forward Euler operator is defined as and Δ 0 ≡ I, where I is the identity operator. The following fractional sum was introduced in [26, Formula 2.2]. This definition corresponds to a particular case of fractional sum proposed by Eloe and Atici in [8].
Definition 2.1. Let α > 0 and f ∈ s(N 0 ; X) be given. We define the fractional sum of order α as follows We also define k α (j) = 0 otherwise.

A u t h o r ' s C o p y
As a function of n, k α is increasing for α > 1, decreasing for 0 < α < 1 and k 1 (n) = 1 for n ∈ N ([43, Theorem III.
The following definition corresponds to an analogous version of fractional derivative in the sense of Riemann-Liouville, see [7].
2. Let f ∈ s(N 0 ; X) be given, we define the fractional difference operator of order α > 0 (in sense of Riemann-Liouville) as follows Recall that the finite convolution between two sequences f, g ∈ s(N 0 ; X) is defined by: where z is a complex number. Note that this series is convergent for |z| > R, for a sufficiently large R. The discrete time Fourier transform of a sequence f ∈ s(Z; X) is defined by whenever the right side of the above identity exists. The inverse transform is given by where C is a circle centered at the origin of the complex plane, that encloses all poles of u(z)z n−1 .
We finish this section with the following Fourier multiplier theorem established by Blunck [9] in the context of UMD Banach spaces (for more information, see [5, Section III.4.3-III.4.5]). Firstly, we recall the notion of R-boundedness.
Remark 2.2. Note that the Banach space B(X, Y ) is equipped with the uniform operator topology.
For more details about R-boundedness and its properties see [3,Section 2.2]. In what follows we denote by T := (π, π) \ {0}. (2.5) The converse of Blunck's Theorem also holds without any restriction on the Banach space X.
Theorem 2.2. [9, Proposition 1.3] Let p ∈ (1, ∞) and let X be a UMD space. Let M : T −→ B(X) be an operator-valued function. Suppose that there exists an operator T M ∈ B( p (Z; X)) such that the identity (2.5) holds. Then the set 3. Resolvent families with delay: 1 < α ≤ 2 In this section we study the existence and uniqueness of solutions for the following problem: where 1 < α ≤ 2 and T ∈ B(X). We start with the following definition.

A u t h o r ' s C o p y
Definition 3.1. Let T be a bounded linear operator defined in a Banach space X, and let 1 < α ≤ 2 and τ ∈ N be given. We say that T is a generator of an α τ -resolvent sequence if there exists a sequence of bounded and linear operators {M α (n)} n≥−τ ⊂ B(X) that satisfies the following properties: Remark 3.1. Note that in the case when β = 0, Definition 3.1 coincides with the definition of resolvent sequence defined in [27].
where ρ(T ) denotes the resolvent set of T and C is a circle centered at the origin that encloses all singularities of z n (z − (α − 1))(z 2−α (z − 1) α − βz −τ ) −1 in its interior. Then for any n ∈ N, n ≥ 2, the formula 2) defines an α τ -resolvent sequence of operators with generator T . This fact can be formally checked using the time discrete Fourier transform method to equation (3.1) and comparing it with the formula given in Theorem 3.1 below. Now, we recall the following Lemma proved in [29].

Maximal p -regularity
Let T ∈ B(X) and f ∈ s(N 0 ; X) be given. In this section, we consider the following nonlocal time-discrete equation with delay τ ∈ N : where 1 < α ≤ 2 and β ∈ R. Assume that T is a generator of an α τresolvent sequence M α (n). Since u(j) = 0 for all j = −τ, ..., 1 we obtain by Theorem 3.1, that the solution of (4.1) can be represented by Furthermore, from Lemma 3.1, we have the representation 2) This motivates the following definition.
Definition 4.1. Let 1 < p < ∞, 1 < α ≤ 2 and T ∈ B(X) be given and suppose that T is a generator of an α τ -resolvent sequence M α (n). We say that the equation (4.1) has maximal p -regularity if the operators K α and P α , defined by

A u t h o r ' s C o p y
are linear bounded operators in p (N 0 ; X) for some p > 1.
Remark 4.1. Observe that, in contrast with the continuous context, the discrete maximal p -regularity ensures the stability of the solution and its fractional difference in the sense that |u(n)| → 0 and |Δ α u(n)| → 0 as n → ∞.
In what follows we need the following hypothesis: Now, we prove the main result of this paper.
Theorem 4.1. Let 1 < p < ∞, 1 < α ≤ 2 and let X be a UMD space. Let T ∈ B(X) be given such that T is a generator of an α τ -resolvent sequence M α (n) and the hypothesis (H) α is satisfied. Then the following assertions are equivalent: (i) Equation (4.1) has maximal p -regularity.
(ii) The sets gives us Then,

and (4.3), we obtain that the left hand side in the identity
Observe that the Z-transform of M α * h α exists by hypothesis (H) α and definition of h α , and (z 2−α (z − 1) α − βz −τ − T ) M α * h α (z) = z 2 I. Then, from the identity (4.3), we have that the discrete Fourier transform of K α f (n − 2) coincides with the discrete Fourier transform of R α f (n) for n ≥ 2. So, K α f (n − 2) = R α f (n) for each n ≥ 2 by uniqueness. On the other hand, we define Using again the identity (4.3), we obtain that the discrete Fourier transform of P α f (n − 2) coincides with the discrete Fourier transform of U α f (n). So, P α f (n − 2) = U α f (n) for each n ≥ 2 by uniqueness. This proves (i). Now, we suppose that (i) is satisfied. We define the following operators Remark 4.3. With the same proof and obvious modifications, the theorem is also true when we consider a finite number of delays in the equation (4.1).
We immediately obtain the following corollary (compare with [29]).
Corollary 4.1. If the hypothesis of Theorem 4.1 hold, then we have u, Δ α u, T u ∈ p (N 0 ; X) and there exists a constant C > 0 (independent of f ∈ p (N 0 ; X)) such that the following inequality holds

A u t h o r ' s C o p y
where C is a circle centered at the origin that encloses the roots λ 1 , λ 2 , λ 3 of the equation z 3 − 2z 2 + qz + r = 0 in its interior. It follows from the Schur-Cohn criterion (see [18,Theorem 5.1]) or the Samuelson criterion (see for example [33]) that all these roots lie inside of the unitary disc D if and only if |r −2| < 1+q and |q +2r| < 1−r 2 which, in turn, is equivalent to 1 < q < 2 and 1 − q < r < −1 + √ 2 − q. See Figure  1 below. Then, under this restriction on the parameters of equation (5.2), we obtain that sup n∈N |M 2 (n)| < ∞. That means that the first part of the condition (H) 2 hold. In particular, we also have z 3 − 2z 2 + qz + r = 0 for |z| = 1 and consequently, sup Therefore all the conditions given in Theorem 4.1 holds and we conclude that whenever 1 < q < 2 and 1 − q < r < −1 + √ 2 − q and f ∈ p (N 0 ), there exists a unique u ∈ p (N 0 ) solving (5.2). In order to handle fractional models, the following result will be useful.
P r o o f. We first prove that T is the generator of an α τ -resolvent sequence M α (n) and the hypothesis (H) α is satisfied. Indeed, by hypothesis and an application of the minimum principle, we obtain that f α,β,τ (z) ∈ ρ(T ) and T n (f α,β,τ (z)) n+1 , whenever |z| ≤ 1. Hence there exists a circle Γ centered at the origin of radius R < 1 such that for any n ∈ N, n ≥ 2, for different values of 1 < α ≤ 2 and r ∈ R. Observe that given α and r, there are cases where there exists a number q satisfying the hypothesis of Corollary 5.1. See Figures 2 and 3 below. Moreover, the graphs show that ω α,−r,1 → 0 as α → 1 for some values of r (for instance when r = 0.6).  Figure 2. α = 1.5 and r = 0.6. Observe that the minimum value ω 1.5,−0.6,1 is attained approximately at 0.5 and consequently 0.5 < q < 1.5.