Existence of (N,λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N,\lambda )$$\end{document}-Periodic Solutions for Abstract Fractional Difference Equations

We establish sufficient conditions for the existence and uniqueness of (N,λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N,\lambda )$$\end{document}-periodic solutions for the following abstract model: Δαu(n)=Au(n+1)+f(n,u(n)),n∈Z,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^{\alpha }u(n)=Au(n+1)+f(n,u(n)), \quad n\in {\mathbb {Z}}, \end{aligned}$$\end{document}where 0<α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \alpha \le 1 $$\end{document}, A is a closed linear operator defined in a Banach space X, Δα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{\alpha }$$\end{document} denotes the fractional difference operator in the Weyl-like sense, and f satisfies appropriate conditions.


Introduction
In this article, we investigate the existence of a class of solutions for the abstract fractional difference equation Δ α u(n) = Au(n + 1) + f (n, u(n)), n ∈ Z, (1.1) called (N, λ)-periodic solutions. In (1.1), A is a possibly unbounded operator defined on a Banach space X and f : Z × X → X is given. This class of (N, λ)-periodic functions was introduced in the reference [6] as the discrete counterpart of the notion of (ω, c)-periodic functions [10], a notion that has been studied by various authors, see, e.g., [8,9,[15][16][17][18][19]26] and [30]. It is worth noting that class of (N, λ)-periodic functions contains the classes of discrete periodic (λ = 1), discrete anti-periodic (λ = −1), discrete Bloch-periodic (λ = e ikN , k ∈ Z fixed), and unbounded functions. Existence and uniqueness of (N, λ)-periodic solutions for scalar models, such as Volterra difference equations with infinite delay, were recently investigated in [6]. Anticipating a growing theoretical and practical interest in this class of solutions, we study in this article the existence and uniqueness for all x, y ∈ X and all n ∈ Z. If then Eq. (1.1) has a unique (N, λ)-periodic solution in a mild sense. This article is organized as follows: Section 2 is devoted to the preliminaries about the notion of fractional order difference operator Δ α that we will use. We will also remember the notion of Mittag-Leffler sequence and of (μ, ν)-resolvent sequence of operators. We end this section by remembering the definition of (N, λ)-periodic sequence. Section 3 introduces the notion of a scaled Wright sequence and lists some of its main properties. Also in this section, we establish the important Theorem 3.1, showing that the condition 1 ∈ ρ(A) is sufficient for the existence of (μ, ν)-resolvent sequence of operators. Then, we prove the Theorem 3.4 which says that under the condition (1.2), the summability of (ν, ν)-resolvent sequences can be ensured. Using this relevant fact, we solve in Section 4 the problem of existence and uniqueness of (N, λ)-periodic solutions for the equation (1.1). See Theorem 4.5. Finally, an example is given where A is the one-dimensional Laplacian in X = L 2 (0, 1).

Preliminaries
Let X be a complex Banach space with norm · and B(X) denotes the Banach space of all bounded operators defined on X. For a real number a, we denote N a :={a, a + 1, a + 2, . . .}, and when a = 1, we write N . We recall that the finite discrete convolution * of two sequences f, g : N 0 → X is defined by We denote by s(Z, X) the vector space consisting of all vector-valued sequences f : Z → X. For f ∈ s(Z, X), we recall that the forward difference operator Δ : s(Z, X) → s(Z, X) is defined by On the other hand, for an arbitrary α ∈ C\{0, −1, −2, ...} the Cesàro sequence {k α (n)} n∈N0 , introduced in [21] (see also [33]), is defined by In case α = 0, we define k 0 (n):=δ 0 (n), the Kronecker delta.
The following equality and estimate holds: for α > 0 Furthermore, given α, β > 0, the sequence k α satisfies the semigroup property in N 0 , that is see [3,Sect. 2]. Given α > 0, we define the set If α = 1, then we simply write 1 (Z, X). Now, suppose that 0 < α ≤ 1. Observe that, if f ∈ 1 (Z, X), then Hence, 1 (Z, X) ⊂ 1 α (Z, X) for 0 < α ≤ 1. The theory and applications of operators defined by means of the Cesàro sequences defined on N 0 have been worked in different investigations (see, for example, [3,7,11,20,21,27]). In this paper, we will work with the following fractional sum operator defined on Z in the reference [2]. Definition 2.1. [2] Given 0 < α < 1 the α-th fractional sum operator Δ −α : 1 (Z, X) → s(Z, X) is defined by means of the formula The next definition about fractional differences operators in the sense of Riemman-Liouville and Caputo was introduced by Abadias and Lizama in [2]. Definition 2.3. Let 0 < α < 1 and f ∈ 1 (Z, X). The Caputo fractional difference operator of order α is defined by , and the Riemann-Liouville fractional difference operator of order α is defined by (2.5) Given f ∈ 1 α (Z, X), it was proved in [2] that , n ∈ Z. Therefore, from now on, we will simply denote by Δ α either R Δ α or c Δ α . Now, we recall the notion of Mittag-Leffler sequence defined and studied in the references [7,21,23,27]. Let α, β > 0 and σ ∈ C be such that |σ| < 1. We define Note that the series on the right-hand side of (2.6) converges by (2.2). Furthermore, the Z-transform of the Mittag-Leffler sequence exists for |z| > 1 (see [7]), and is given by (2.7) Motivated by [7, (the proof of) Proposition 4.6], we get the following result regarding the asymptotic behavior of the Mittag-Leffler sequence.
Next, we recall the concept of discrete (α, )-resolvent sequence defined in [7,Sect. 4,Definition 4.4]. Also, useful results related with this definition are given. Definition 2.5. Let , α > 0 be given and A be a closed linear operator with domain D(A) defined on a Banach space X. An operator-valued sequence {S α, (n)} n∈N0 ⊂ B(X) is called a discrete (α, )-resolvent sequence generated by A if it satisfies the following conditions: We finish this section recalling the notion of (N, λ)-periodic sequences and their main properties. The notion of (N, λ)-periodic sequences was introduced in [6] as a discrete counterpart of the concept of (ω, c)-periodic functions defined in [10].
The collection of those sequences with the same λ-period N will be denoted by P Nλ (Z, X).
The following result is central for the theory.
Proposition 2.7 [6]. A function f is (N, λ)-periodic discrete function if and only if there exists u ∈ P N (Z, X), such that for all n ∈ Z, The vector-valued space of sequences P Nλ (Z, X) is a Banach space with the norm (2.9)

The Discrete Scaled Wright Function and Summability of Resolvent Sequences
In [7], the authors introduced a discrete version of the Lévy α-stable distribution which can be defined as The sequence l α is a probability density function in n, which means that This representation of the discrete Lévy function allowed to establish a subordination principle which relates a discrete (α, )-resolvent sequence with a C-semigroup generated by a given closed linear operator A defined on a Banach space X (see [7]). The following result is a consequence of this fact. Theorem 3.1. Let 0 < α ≤ ≤ 1 be given. Let A be a closed and linear operator defined on a Banach space X, such that 1 ∈ ρ(A). Then, the family is a discrete (α, )-resolvent sequence generated by A.
The concept of scaled Wright function in the continuous case was introduced by Abadias and Miana in [1]. Motivated by the above theorem, we propose in this paper the following definition. Some properties of the discrete scaled Wright function can be deduced from those properties of the Lévy α-stable distribution. They are stated in the following remark.
We recall that an operator-valued sequence {S(n)} n∈N0 ∈ B(X) is said to be summable if We finish this section with a useful result which is a direct consequence of the above considerations.
Proof. Since 1 ∈ ρ(A), then by Theorem 3.1, the family is a discrete (α, α)-resolvent sequence generated by A. We will prove that it is summable. Indeed, since 0 ≤ ϕ α,0 (n, j) ≤ 1 for j ∈ N 0 , then The following example provide concrete conditions on A under which the condition (3.8) holds.
Example. Let A be the generator of a C 0 -semigroup strictly contractive. For instance, on X:=L 1 (R), we define We deduce that 1 ∈ ρ(A) and (I − A) −1 < 1. Indeed, The last part of the earlier example shows the following result.

(N, λ)-Periodic Solutions for Fractional Difference Equations on Z
In this section, we study regularity of solutions to the linear fractional difference equation Definition 4.1 [2]. Let A be the generator of a discrete (α, α)-resolvent family {S α,α (n)} n∈N0 and g : Z −→ X. The sequence is called a mild solution for Eq.
Note that if g ∈ 1 (Z, D(A)), then each mild solution is a strong one; see [2,Theorem 4.2].
In the following theorem, we establish the existence of (N, λ)-periodic mild solutions for Eq. (4.1).
Proof. By Theorem 3.4, A generates a summable discrete (α, α)-resolvent sequence {S α,α (n)} n∈N given by Since g is bounded and {S α,α (n)} n∈N0 is summable, it follows that the sequence u is a mild solution of (4.1). It remains to prove that u ∈ P Nλ (Z, X). Indeed, getting that u ∈ P Nλ (Z, X).

The Semilinear Case
In this subsection, we consider the following fractional difference equation: Δ α u(n) = Au(n + 1) + f (n, u(n)), n ∈ Z, (4.4) where 0 < α ≤ 1, A satisfies the hypotheses in Theorem 3.4 and f satisfies suitable conditions. Inspired in the solution of the linear case, we give the following definition of mild solution for the semilinear case.
To prove the main theorem, we will need to recall the following.
(ii) f is N -periodic in the first variable and homogeneous in the second variable, that is f (n + N, λx) = λf (n, x) for all (n, x) ∈ Z × X.
Theorem 4.5. Let f : Z × X → X be given and let A be a closed linear operator defined on a Banach space X, such that 1 ∈ ρ(A) and Assume the following conditions: for all x, y ∈ X and all n ∈ Z. H 3 . The constant L in H 2 is such that Then, Eq. Proof. First, let us define the operator G : P Nλ (Z, X) → P Nλ (Z, X) by Let u ∈ P Nλ (Z, X) and g(n):=f (n, u(n)). By H 1 and Theorem 4.4 we get that g ∈ P N,λ (Z, X). As in the linear case, we can see that G(u) ∈ P Nλ (Z, X). It follows that G is well defined. Now, for u, v ∈ P Nλ (Z, X) where we have by H 2 that where S α,α (n) = λ ∧ (−n)S α,α (n). Then where by Theorem 3.1, Remark 3.3 (iv), and (2.7), we have Therefore, the conclusion follows from H 3 . For the above, it follows that there exists a unique function u ∈ P Nλ (Z, X), such that Gu = u. Hence, u is the unique (N, λ)-periodic mild solution of equation (4.4). Regarding condition H 3 , we observe that it is enough to have the weaker condition L S α,α 1 < 1 where S ∼ α,α (n):=λ ∧ (−n)S α,α (n) and {S α,α (n)} n∈N0 is the (α, α)-resolvent sequence generated by A.
We finally finish with an application of the main result presented in this paper.
Finally, observe that in case |λ| = 1, we have that and therefore condition H 3 independent of α. This happens precisely in the standard cases of discrete periodic, discrete anti-periodic, and discrete Blochperiodic functions.