Discrete Weighted Pseudo-Almost Automorphy and Applications

Abstract

We deal with discrete weighted pseudo almost automorphy which extends some classical concepts and systematically explore its properties in Banach space including a composition result. As an application, we establish some sufficient criteria for the existence and uniqueness of the discrete weighted pseudo almost automorphic solutions to the Volterra difference equations of convolution type and also to nonautonomous semilinear difference equations. Some examples are presented to illustrate the main findings.

1. Introduction

The notation of (continuous) almost automorphy, introduced by Bochner [1], is related to and more general than (continuous) almost periodicity. Since then, this concept has been attracting the attention of many researchers and the interest in this topic still increases. There is a lager literature on this topic. We refer to the recent books [2, 3], where the authors gave an important overview about the theory of (continuous) almost automorphic functions and their applications to differential equations. Recently, a new more general type of almost automorphy called (continuous) weighted pseudo almost automorphy is proposed by Blot et al. [4], which generalizes various extensions of (continuous) almost automorphy; one can refer to [5] for more details.

Discrete almost automorphic functions, a class of functions which are more general than discrete almost periodic ones, were considered in [6] in connection with the study of (continuous) almost automorphic bounded mild solutions of differential equations; see also [7–9]. Similar to (continuous) almost automorphic functions, discrete almost automorphic functions have made important applications to differential equations in Banach space. The range of applications of discrete almost automorphic functions include first order nonlinear difference equations [10], Volterra difference equations [11–13], nonautonomous difference equations [14, 15], and nonlinear stochastic difference equations [16]. On the other hand, recently, the concept of discrete weighted pseudo almost automorphic functions, which generalizes the notion of discrete almost automorphic functions, is introduced by Abbas in [17] and some basic properties of these functions are explored.

In this paper, we conduct further studies on discrete weighted pseudo almost automorphic functions; the main idea consists of enlarging the weighted ergodic space, with the help of two weighted functions, extending some results of [17]. We systematically explore its properties in Banach space including completeness, translation invariance, and composition results. As an application, the existence and uniqueness of the discrete weighted pseudo almost automorphic solutions to the Volterra difference equations of convolution type and nonautonomous semilinear difference equations are investigated. To the best of our knowledge, discrete weighted pseudo almost automorphy of Volterra difference equations and nonautonomous semilinear difference equations are an untreated topic and this is the main motivation of this paper.

The paper is organized as follows. In Section 2, some notations and preliminary results are presented. In Section 3, we propose a new class of functions called discrete weighted pseudo almost automorphic functions with the help of two weighted functions, explore its properties, and establish the composition theorem. Section 4 is divided into the existence and uniqueness of discrete weighted pseudo almost automorphic solutions to the Volterra difference equations of convolution type and nonautonomous semilinear difference equations, respectively. In Section 5, we provide some examples to illustrate our main results.

2. Preliminaries and Basic Results

Let (𝑋,‖⋅‖), (𝑌,‖⋅‖) be two Banach spaces and ℕ, ℤ, ℤ⁺ℝ, ℝ⁺, and ℂ stand for the set of natural numbers, integers, nonnegative integers, real numbers, nonnegative real numbers, and complex numbers, respectively. Let 𝐴 be bounded linear operator; 𝜎_𝑝(𝐴) denotes the point spectrum of 𝐴. 𝐵(𝑥,𝛿) stands for open balls with center 𝑥 and radius less than 𝛿. Let 𝑣∶ℤ⁺→ℂ; if ∑^∞_𝑘=0|𝑣(𝑘)|<∞, we call that 𝑣 is a summable function.

In order to facilitate the discussion below, we further introduce the following notations.(i)𝑆¹={𝜆∈ℂ,|𝜆|=1}.(ii)𝑙^∞(ℤ,𝑋)={𝑥∶ℤ→𝑋∶‖𝑥‖_𝑑=sup_𝑛∈ℤ‖𝑥(𝑛)‖<∞}. (iii)𝐿(𝑋,𝑌): the Banach space of bounded linear operators from 𝑋 to 𝑌 endowed with the operator topology. In particular, we write 𝐿(𝑋) when 𝑋=𝑌.(iv)𝒰𝒞(ℤ×𝑋,𝑋): the set of all functions 𝑓∶ℤ×𝑋→𝑋 satisfying that ∀𝜀>0, ∃𝛿>0 such that 𝑓(𝑘,𝑥)−𝑓𝑘,𝑦≤𝜀 (1)

for all 𝑘∈ℤand 𝑥,𝑦∈𝑋 with ‖𝑥−𝑦‖≤𝛿.

Next, we recall the so-called Matkowski's fixed point theorem [18] and exponential dichotomy on ℤ [19, 20] which will be used in the sequel.

Theorem 1 (Matkowski’s fixed point theorem [18]). Let (𝑋,𝑑) be a complete metric space and let ℱ∶𝑋→𝑋 be a map such that 𝑑ℱ𝑥,ℱ𝑦≤Φ𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋, (2) where Φ∶[0,+∞)→[0,+∞) is a nondecreasing function such that lim_𝑛→+∞Φ^𝑛(𝑡)=0 for all 𝑡>0. Then ℱ has a unique fixed point 𝑧∈𝑋.

Given a sequence {𝐴(𝑛)}_𝑛∈ℤ⊂𝐿(𝑋) of invertible operators, define 𝒜()=𝐴()(𝑛)𝐴𝑚,𝑛𝑚−1⋯𝐴,if𝑚>𝑛,Id,if𝑚=𝑛,₋₁(𝑚)⋯𝐴⁻¹(𝑛−1),if𝑚<𝑛, (3) where Id is the identity operator in 𝑋.

For the first order difference equation 𝑥(𝑛+1)=𝐴(𝑛)𝑥(𝑛),𝑛∈ℤ. (4)

Definition 2 (see [19]). Equation (4) is said to have an exponential dichotomy if there exist projections 𝑃(𝑛)∈𝐿(𝑋) for all 𝑛∈ℤ and positive constants 𝜂,𝜈,𝛼,𝛽 such that (i)𝑃(𝑚)𝒜(𝑚,𝑛)=𝒜(𝑚,𝑛)𝑃(𝑛),𝑚,𝑛∈ℕ,(ii)‖𝒜(𝑚,𝑛)𝑃(𝑛)‖≤𝜂𝑒^{−𝛼(𝑚−𝑛)},𝑚≥𝑛,(iii)‖𝒜(𝑚,𝑛)𝑄(𝑛)‖≤𝜈𝑒^{−𝛽(𝑛−𝑚)},𝑛≥𝑚,where 𝑄(𝑛)=Id−𝑃(𝑛) is the complementary projection of 𝑃(𝑛).
Finally, we recall the concept of discrete almost automorphic function.

Definition 3 (see [6]). A function 𝑓∈𝑙^∞(ℤ,𝑋) is said to be discrete almost automorphic if for every integer sequence (𝑘^′_𝑛), there exists a subsequence (𝑘_𝑛) such that lim_𝑛→∞𝑓(𝑘+𝑘_𝑛)∶=𝑔(𝑘) is well defined for each 𝑘∈ℤ, and lim_𝑛→∞𝑔(𝑘−𝑘_𝑛)=𝑓(𝑘) for each 𝑘∈ℤ.

Remark 4. (i) If 𝑓 is (continuous) almost automorphic function in ℝ, then 𝑓∣_ℤ is discrete almost automorphic.(ii) If the convergence in Definition 3 is uniform on ℤ, then we get discrete almost periodicity, so discrete almost automorphy is more general than discrete almost periodicity.(iii) Example of discrete almost automorphic functions which are not discrete almost periodicity was first constructed by Veech [21]. Note that the function 𝑓(𝑘)=12+cos𝑘+cos√2𝑘,𝑘∈ℤ (5) is discrete almost automorphic function but not discrete almost periodic (see [10] for more details).

Throughout the paper, we denote 𝐴𝐴_𝑑(ℤ,𝑋) the set of discrete almost automorphic functions. Note that if 𝑓∈𝐴𝐴_𝑑(ℤ,𝑋), then 𝐾∶={𝑓(𝑘)∶𝑘∈ℤ} is relative compact in 𝑋 and 𝑓 is a bounded function.

Definition 5 (see [6]). A function 𝑓∶ℤ×𝑋→𝑋 is said to be discrete almost automorphic in 𝑘∈ℤ for each 𝑥∈𝑋, if for every integer sequence (𝑘^′_𝑛), there exists a subsequence (𝑘_𝑛) such that lim_𝑛→∞𝑓(𝑘+𝑘_𝑛,𝑥)∶=𝑔(𝑘,𝑥) is well defined for each 𝑘∈ℤ, 𝑥∈𝑋 and lim_𝑛→∞𝑔(𝑘−𝑘_𝑛,𝑥)=𝑓(𝑘,𝑥) for each 𝑘∈ℤ, 𝑥∈𝑋. We denoted by 𝐴𝐴_𝑑(ℤ×𝑋,𝑋) the spaces of all discrete almost automorphic in 𝑘∈ℤ for each 𝑥∈𝑋.

Lemma 6 (see [11]). Let 𝑓∈𝐴𝐴_𝑑(ℤ×𝑋,𝑋)∩𝒰𝒞(ℤ×𝑋,𝑋); then 𝑓(⋅,ℎ(⋅))∈𝐴𝐴_𝑑(ℤ,𝑋) if ℎ∈𝐴𝐴_𝑑(ℤ,𝑋).

3. Discrete Weighted Pseudo Almost Automorphy

Let 𝑈 denote the collection of functions (weights) 𝜌∶ℤ→(0,+∞). For 𝜌∈𝑈 and 𝑇∈ℤ⁺={𝑛∈ℤ,𝑛≥0}, set 𝜇𝑇,𝜌∶=^𝑇_𝑘=−𝑇𝜌(𝑘). (6) Denote 𝑈_∞∶=𝜌∈𝑈∶lim_𝑇→+∞𝜇,𝑈𝑇,𝜌=+∞_𝐵∶=𝜌∈𝑈_∞∶0<inf_𝑘∈ℤ𝜌(𝑘)≤sup_𝑘∈ℤ𝜌(𝑘)<+∞. (7)

Definition 7. Let 𝜌₁,𝜌₂∈𝑈_∞. 𝜌₁ is said to be equivalent to 𝜌₂ (i.e., 𝜌₁∼𝜌₂) if 𝜌₁/𝜌₂∈𝑈_𝐵.

It is trivial to show that “∼” is a binary equivalence relation on 𝑈_∞. The equivalence class of a given weight 𝜌∈𝑈_∞ which is denoted by cl(𝜌)={𝜚∈𝑈_∞∶𝜌∼𝜚}. It is clear that 𝑈_∞=⋃_{𝜌∈𝑈∞}cl(𝜌).

For 𝜌₁∈𝑈_∞, define the weighted ergodic space [17] 𝑊𝑃𝐴𝐴₀(ℤ,𝑋)∶=𝑓∶ℤ→𝑋isbounded,lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓(𝑘)𝜌₁(𝑘)=0. (8) Particularly, for 𝜌₁,𝜌₂∈𝑈_∞, define 𝑊𝑃𝐴𝐴₀(ℤ,𝑋)∶=𝑓∶ℤ→𝑋isbounded,lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓(𝑘)𝜌₂(𝑘)=0. (9) Clearly, when 𝜌₁∼𝜌₂, 𝑊𝑃𝐴𝐴₀(ℤ,𝑋) coincide with 𝑊𝑃𝐴𝐴₀(ℤ,𝑋); that is, 𝑊𝑃𝐴𝐴₀(ℤ,𝑋)=𝑊𝑃𝐴𝐴₀(ℤ,𝑋), this fact suggests that 𝑊𝑃𝐴𝐴₀(ℤ,𝑋) are more interesting when 𝜌₁ and 𝜌₂ are not necessarily equivalent. So 𝑊𝑃𝐴𝐴₀(ℤ,𝑋) are general and richer than 𝑊𝑃𝐴𝐴₀(ℤ,𝑋).

Definition 8. Let 𝜌₁,𝜌₂∈𝑈_∞. A function 𝑓∶ℤ→𝑋 is called discrete weighted pseudo almost automorphic if it can be expressed as 𝑓=𝑔+𝜑, where 𝑔∈𝐴𝐴_𝑑(ℤ,𝑋) and 𝜑∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). The set of such functions is denoted by 𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

Remark 9. If 𝜌₁∼𝜌₂, 𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) coincide with the discrete weighted pseudo almost automorphic functions defined in [17].

Throughout the rest of the paper, we denote by 𝑉_∞ the set of all the functions 𝜌₁,𝜌₂∈𝑈_∞ satisfying that there exists an unbounded set Ω⊂ℤ such that for all 𝑚∈ℤ, limsup_{|𝑘|→+∞,𝑘∈Ω}𝜌₂(𝑘+𝑚)𝜌₁(𝑘)∶=inf_𝑁∈ℕsup_{|𝑘|≥𝑁,𝑘∈Ω}𝜌₂(𝑘+𝑚)𝜌₁(𝑘)<+∞,lim_𝑇→+∞∑_{𝑘∈([−𝑇,𝑇]⧵Ω)+𝑚}𝜌₂(𝑘)𝜇𝑇,𝜌₁=0. (10) Next, we show some properties of the space 𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

Similar to the proof of [22], one has the following.

Lemma 10. Let 𝜌₁,𝜌₂∈𝑉_∞, then(i)for each 𝑚∈ℤ, one has limsup_𝑇→+∞𝜇𝑇+𝑚,𝜌₂𝜇𝑇,𝜌₁<+∞. (11)(ii)𝑊𝑃𝐴𝐴₀(ℤ,𝑋) is translation invariant; that is, 𝑓(⋅+𝑚)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋) for each 𝑚∈ℤ if 𝑓∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋).(iii)𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) is translation invariant.(iv)If 𝑓=𝑔+𝜑∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), where 𝑔∈𝐴𝐴_𝑑(ℤ,𝑋), 𝜑∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋), then 𝑔(𝑛)∶𝑛∈ℤ⊂𝑓(𝑛)∶𝑛∈ℤ. (12)(v)𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) is a Banach space under the supremum norm; that is, 𝑓_𝑑∶=sup_𝑘∈ℤ𝑓(𝑘). (13)

Lemma 11. If 𝐴∈𝐿(𝑋) and 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), then 𝐴𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

The proof is straightforward and is therefore omitted.

Similarly, define 𝑊𝑃𝐴𝐴₀(ℤ×𝑋,𝑋)∶=𝑓∶ℤ⟶𝑋isbounded,lim_𝑇→+∞1𝜇𝑇,𝜌₁×^𝑇_𝑘=−𝑇𝑓(𝑘,𝑥)𝜌₂(𝑘)=0uniformin𝑥∈𝑋. (14)

Definition 12. Let 𝜌₁,𝜌₂∈𝑈_∞. A function 𝑓∶ℤ×𝑋→𝑋 is said to be discrete weighted pseudo almost automorphic in 𝑘∈ℤ for each 𝑥∈𝑋, if it can be decomposed as 𝑓=𝑔+𝜑, where 𝑔∈𝐴𝐴_𝑑(ℤ×𝑋,𝑋) and 𝜑∈𝑊𝑃𝐴𝐴₀(ℤ×𝑋,𝑋). Denote by 𝑊𝑃𝐴𝐴_𝑑(ℤ×𝑋,𝑋) the set of such functions.

We will establish composition theorem for discrete weighted pseudo almost automorphic functions.

Lemma 13. Let 𝑓∶ℤ→𝑋 be bounded and 𝜌₁,𝜌₂∈𝑉_∞, then 𝑓∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋) if and only if for any 𝜀>0, lim_𝑇→+∞1𝜇𝑇,𝜌_{1𝑘∈𝐸𝑓()𝑇,𝜀}𝜌₂(𝑘)=0, (15)
where 𝐸_𝑓(𝑇,𝜀)={𝑘∈[−𝑇,𝑇]∩ℤ∶‖𝑓(𝑘)‖≥𝜀}.

Proof. Sufficiency. It is clear that 𝑀=sup_𝑘∈ℤ‖𝑓(𝑘)‖<+∞ and 𝜇(𝑇,𝜌₂)/𝜇(𝑇,𝜌₁)<+∞ by Lemma 10. It follows from (15) that ∀𝜀>0; there exists 𝑁∈ℕ such that 1𝜇𝑇,𝜌_{1𝑘∈𝐸𝑓()𝑇,𝜀}𝜌₂(𝑘)<𝜀2𝑀,𝑇>𝑁; (16) then 1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓(𝑘)𝜌₂(𝑘)=1𝜇𝑇,𝜌_{1𝑘∈𝐸𝑓()𝑇,𝜀}𝑓(𝑘)𝜌₂(𝑘)+1𝜇𝑇,𝜌_{1])𝑘∈([−𝑇,𝑇∩ℤ⧵𝐸𝑓()𝑇,𝜀}𝑓(𝑘)𝜌₂(𝑘)1≤𝑀⋅𝜇𝑇,𝜌_{1𝑘∈𝐸𝑓()𝑇,𝜀}𝜌₂(𝑘)1+𝜀⋅𝜇𝑇,𝜌_{1])𝑘∈([−𝑇,𝑇∩ℤ⧵𝐸𝑓()𝑇,𝜀}𝜌₂(𝑘)≤𝜀2𝜇+𝜀⋅𝑇,𝜌₂𝜇𝑇,𝜌₁, (17) so lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓(𝑘)𝜌₂(𝑘)=0. (18) That is, 𝑓∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋).
Necessity. Suppose the contrary, that there exists 𝜀₀>0, such that (1/𝜇(𝑇,𝜌₁))∑_{𝑘∈𝐸𝑓(𝑇,𝜀)}𝜌₂(𝑘) does not converge to 0 as 𝑇→+∞. That is, there exists 𝛿>0, such that for each 𝑚∈ℤ, 1𝜇𝑇_𝑚,𝜌_{1𝑘∈𝐸𝑓(𝑇𝑚,𝜀0)}𝜌₂(𝑘)≥𝛿,forsome𝑇_𝑚>𝑚. (19) Then 1𝜇𝑇_𝑚,𝜌₁^𝑇𝑚_{𝑘=−𝑇𝑚}𝑓(𝑘)𝜌₂(𝑘)=1𝜇𝑇_𝑚,𝜌_{1𝑘∈𝐸𝑓𝑇𝑚,𝜀0}𝑓(𝑘)𝜌₂(𝑘)+1𝜇𝑇_𝑚,𝜌_{1𝑘∈−𝑇𝑚,𝑇𝑚∩ℤ⧵𝐸𝑓𝑇𝑚,𝜀0}𝑓(𝑘)𝜌₂(𝑘)≥1𝜇𝑇_𝑚,𝜌_{1𝑘∈𝐸𝑓𝑇𝑚,𝜀0}𝑓(𝑘)𝜌₂(𝑘)≥𝜀₀⋅1𝜇𝑇_𝑚,𝜌_{1𝑘∈𝐸𝑓𝑇𝑚,𝜀0}𝜌₂(𝑘)≥𝛿𝜀₀, (20) which contradicts the fact that lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓(𝑘)𝜌₂(𝑘)=0. (21) Thus (15) holds.

Remark 14. If 𝜌₁=𝜌₂=1 and 𝑋=ℝ, the result of Lemma 13 is obtained by [23, Lemma 2.9].

Lemma 15. Let 𝑓∈𝑊𝑃𝐴𝐴₀(ℤ×𝑋,𝑋) and 𝐾⊆𝑋 be compact; if 𝑓∈𝒰𝒞(ℤ×𝐾,𝑋), then 𝑓(⋅)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋), where 𝑓(𝑘)=sup_𝑥∈𝐾𝑓(𝑘,𝑥),𝑘∈ℤ. (22)

Proof. Since 𝑓∈𝒰𝒞(ℤ×𝐾,𝑋), ∀𝜀>0, there exists 𝛿>0 such that 𝑓(𝑘,𝑢)−𝑓(𝑘,𝑣)<𝜀 (23) for all 𝑘∈ℤ and 𝑢,𝑣∈𝐾 with ‖𝑢−𝑣‖<𝛿. Note that 𝐾 is compact; for the above 𝛿>0, there exists 𝑥₁,𝑥₂,…,𝑥_𝑚∈𝐾 such that 𝐾⊂^𝑚_𝑖=1𝐵𝑥_𝑖,𝛿. (24) Since 𝑓∈𝑊𝑃𝐴𝐴₀(ℤ×𝑋,𝑋), for the above 𝜀>0, there exists 𝑇₀∈ℕ such that for all 𝑇>𝑇₀, 1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓𝑘,𝑥_𝑖𝜌₂(𝑘)<𝜀𝑚,𝑖=1,2…,𝑚. (25)
For each 𝑥∈𝐾, there exists 𝑖∈{1,2,…,𝑚} such that ‖𝑥−𝑥_𝑖‖<𝛿; hence 𝑓(𝑘,𝑥)≤𝑓(𝑘,𝑥)−𝑓𝑘,𝑥_𝑖+𝑓𝑘,𝑥_𝑖≤𝜀+𝑓𝑘,𝑥_𝑖,𝑘∈ℤ, (26) which implies that 𝑓(𝑘)=sup_𝑥∈𝐾𝑓(𝑘,𝑥)≤^𝑚_𝑖=1𝑓𝑘,𝑥_𝑖+𝜀,𝑘∈ℤ. (27) Thus 𝑓 is bounded and 1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓(𝑘)𝜌₂(𝑘)≤^𝑚_𝑖=11𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑓𝑘,𝑥_𝑖𝜌₂(𝑘)+𝜀≤2𝜀,𝑇>𝑇₀, (28) which means that 𝑓(⋅)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋).

Theorem 16. Assume that 𝜌₁,𝜌₂∈𝑉_∞, 𝑓∈𝑊𝑃𝐴𝐴_𝑑(ℤ×𝑋,𝑋)∩𝒰𝒞(ℤ×𝑋,𝑋); then 𝑓(⋅,ℎ(⋅))∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) if ℎ∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

Proof. Let 𝑓=𝑔+𝜑,ℎ=𝑥+𝑦, (29) where 𝑔∈𝐴𝐴_𝑑(ℤ×𝑋,𝑋), 𝜑∈𝑊𝑃𝐴𝐴₀(ℤ×𝑋,𝑋), 𝑥∈𝐴𝐴_𝑑(ℤ,𝑋) and 𝑦∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). The function 𝑓 can be decomposed as 𝑓((𝑛((𝑛((𝑛((𝑛((𝑛((𝑛((𝑛((𝑛.𝑛,ℎ))=𝑔𝑛,𝑥))+𝑓𝑛,ℎ))−𝑔𝑛,𝑥))=𝑔𝑛,𝑥))+𝑓𝑛,ℎ))−𝑓𝑛,𝑥))+𝜑𝑛,𝑥)) (30) Set 𝐹(𝑛)((𝑛,𝐺(𝑛)((𝑛((𝑛,𝐻(𝑛)((𝑛=𝑔𝑛,𝑥))=𝑓𝑛,ℎ))−𝑓𝑛,𝑥))=𝜑𝑛,𝑥)),𝑛∈ℤ. (31)
By Lemma 10, {𝑔(𝑛)∶𝑛∈ℤ}⊂{𝑓(𝑛)∶𝑛∈ℤ}, it is not difficult to see that 𝑔∈𝒰𝒞(ℤ×𝑋,𝑋); hence 𝐹(⋅)∈𝐴𝐴_𝑑(ℤ,𝑋) by Lemma 6.
We claim that 𝐺(⋅)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). In fact, since 𝑓∈𝒰𝒞(ℤ×𝑋,𝑋), ∀𝜀>0, ∃𝛿>0 such that ‖𝐺(𝑛)‖=𝑓(𝑛,ℎ(𝑛))−𝑓(𝑛,𝑥(𝑛))<𝜀 (32) for all 𝑛∈ℤ and ‖𝑦(𝑛)‖=‖ℎ(𝑛)−𝑥(𝑛)‖<𝛿; hence 𝐸_𝐺(𝑛,𝜀)⊂𝐸_𝑦(𝑛,𝛿). (33) Since 𝑦∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋), by Lemma 13, one has lim_𝑇→+∞1𝜇𝑇,𝜌_{1𝑘∈𝐸𝑦()𝑇,𝛿}𝜌₂(𝑘)=0, (34) which implies that lim_𝑇→+∞1𝜇𝑇,𝜌_{1𝑘∈𝐸𝐺(𝑇,𝜀)}𝜌₂(𝑘)=0. (35) By Lemma 13, 𝐺(⋅)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋).
Since 𝑓∈𝒰𝒞(ℤ×𝑋,𝑋), then 𝜑∈𝒰𝒞(ℤ×𝑋,𝑋). Let 𝐾={𝑥(𝑛)∶𝑛∈ℤ}; by Lemma 15, one has lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇sup_𝑧∈𝐾𝜑(𝑘,𝑧)𝜌₂(𝑘)=0, (36) so lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝜑(𝑘,𝑥(𝑘))𝜌₂(𝑘)≤lim_𝑇→+∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇sup_𝑧∈𝐾𝜑(𝑘,𝑧)𝜌₂(𝑘)=0. (37) That is 𝐻(⋅)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). Then 𝐹(⋅)∈𝐴𝐴_𝑑(ℤ,𝑋), 𝐺(⋅)+𝐻(⋅)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). Hence 𝑓(⋅,ℎ(⋅))∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

Corollary 17. Let 𝜌₁,𝜌₂∈𝑉_∞, 𝑓∈𝑊𝑃𝐴𝐴_𝑑(ℤ×𝑋,𝑋) and satisfies the Lipschitz condition 𝑓(𝑘,𝑢)−𝑓(𝑘,𝑣)≤𝐿_𝑓‖𝑢−𝑣‖,∀𝑘∈ℤ,𝑢,𝑣∈𝑋, (38) then 𝑓(⋅,ℎ(⋅))∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) if ℎ∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

The following Lemma is the essential property to study the existence of 𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) solutions of Volterra difference equations of convolution type.

Lemma 18. Let 𝑣∶ℤ⁺→ℂ be a summable function; if 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), 𝜌₁,𝜌₂∈𝑉_∞, then Λ(⋅)∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), where Λ(𝑘)=^𝑘_𝑙=−∞|𝑣(𝑘−𝑙)|𝑢(𝑙),𝑘∈ℤ. (39)

Proof. Note that ‖Λ(𝑘)‖≤^∞_𝑙=0|𝑣(𝑙)|‖𝑢(𝑘−𝑙)‖≤‖𝑢‖_𝑑⋅^∞_𝑙=0|𝑣(𝑙)|<∞; (40) hence Λ is bounded.
Let 𝑢=𝑢₁+𝑢₂, where 𝑢₁∈𝐴𝐴_𝑑(ℤ,𝑋), 𝑢₂∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋); then Λ=Λ₁+Λ₂, (41) where Λ₁(𝑘)=^𝑘_𝑙=−∞|𝑣(𝑘−𝑙)|𝑢₁(𝑙),Λ₂(𝑘)=^𝑘_𝑙=−∞|𝑣(𝑘−𝑙)|𝑢₂(𝑙),𝑘∈ℤ. (42) The almost automorphy of Λ₁ follows from [10].
Next, we show that Λ₂∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). In fact, 1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Λ₂(𝑘)𝜌₂(𝑘)≤1𝜇𝑇,𝜌₁^𝑇𝑘_{𝑘=−𝑇𝑙=−∞}|𝑣(𝑘−𝑙)|𝑢₂(𝑙)𝜌₂(𝑘)=1𝜇𝑇,𝜌₁^𝑇∞_{𝑘=−𝑇𝑙=0}|𝑣(𝑙)|𝑢₂(𝑘−𝑙)𝜌₂(𝑘)=^∞_𝑙=0|𝑣(𝑙1)|𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑢₂(𝑘−𝑙)𝜌₂(𝑘). (43) Since 𝑢₂∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋), 𝑢₂(⋅−𝑙)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋) for each 𝑙∈ℤ by Lemma 10, then lim_𝑇→∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑢₂(𝑘−𝑙)𝜌₂(𝑘)=0; (44) by Lebesgue dominated convergence theorem, one has lim_𝑇→∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Λ₂(𝑘)𝜌₂(𝑘)≤lim^∞_{𝑇→∞𝑙=0}|𝑣(𝑙1)|𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇𝑢₂(𝑘−𝑙)𝜌₂(𝑘)=0, (45) which implies that Λ₂∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). The proof is completed.

Remark 19. If 𝜌₁∼𝜌₂, the results of Corollary 17 and Lemma 18 are obtained by [17, Theorem 2.8] and [17, Theorem 2.9], respectively.

4. Applications in Difference Equations

As an application, the main goal of this section is to establish some sufficient criteria for the existence and uniqueness of 𝑊𝑃𝐴𝐴_𝑑 solution to the Volterra difference equations and nonautonomous semilinear difference equations.

4.1. Volterra Difference Equation

This subsection is devoted to establish some sufficient criteria for the existence and uniqueness of 𝑊𝑃𝐴𝐴_𝑑 solutions of (46).

Consider the Volterra difference equations of convolution type 𝑢(𝑛+1)=𝜆^𝑛_𝑗=−∞𝑎𝑛−𝑗𝑢𝑗+𝑓(𝑛,𝐴𝑢(𝑛)),𝑛∈ℤ, (46) where 𝜆∈ℂ, 𝑎(⋅) is a summable function; 𝐴∈𝐿(𝑋) and 𝑓∈𝑊𝑃𝐴𝐴_𝑑(ℤ×𝑋,𝑋). Its associated homogeneous linear equation is given by 𝑢(𝑛+1)=𝜆^𝑛_𝑗=−∞𝑎𝑛−𝑗𝑢𝑗+𝑓(𝑛),𝑛∈ℤ, (47) where 𝜆∈ℂ, 𝑎(⋅) is a summable function.

For a given 𝜆∈ℂ, let 𝑠(𝜆,𝑘)∈ℂ be the solution of the difference equation 𝑠(𝜆,𝑘+1)=𝜆^𝑘_𝑗=0𝑎𝑠𝑠()𝑘−𝑗𝜆,𝑗,𝑘=0,1,2,…𝜆,0=1. (48) In this case, 𝑠(𝜆,𝑘) is called the fundamental solution to (47) generated by 𝑎(⋅). We define the set Ω_𝑠=𝜆∈ℂ,‖𝑠(𝜆,⋅)‖₁∶=^∞_𝑘=0|𝑠(𝜆,𝑘)|<∞. (49) By [13], if 𝜆∈Ω_𝑠, the solution of (47) is given by 𝑢(𝑛+1)=^𝑛_𝑘=−∞𝑠(𝜆,𝑛−𝑘)𝑓(𝑘). (50)

To establish our results, we introduce the following condition.(H₁)𝜆∈Ω_𝑠, 𝐴∈𝐿(𝑋).(H₂)𝑓∈𝑊𝑃𝐴𝐴_𝑑(ℤ×𝑋,𝑋), 𝜌₁,𝜌₂∈𝑉_∞.(H₃₁)There exists a constant 𝐿_𝑓>0 such that 𝑓(𝑘,𝑢)−𝑓(𝑘,𝑣)≤𝐿_𝑓‖𝑢−𝑣‖,∀𝑘∈ℤ,𝑢,𝑣∈𝑋. (51)(H₃₂)There exists a linear nondecreasing function Φ∶[0,∞)→[0,∞) and 𝑓 satisfies 𝑓(𝑘,𝑢)−𝑓(𝑘,𝑣)≤Φ(‖𝑢−𝑣‖),∀𝑘∈ℤ,𝑢,𝑣∈𝑋. (52)

Theorem 20. Assume that (𝐻₁), (𝐻₂), and (𝐻₃₁) hold and 𝐿_𝑓‖𝐴‖|𝑠(𝜆,⋅)|₁<1; then (46) has a unique solution 𝑢(𝑛)∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) which is given by 𝑢(𝑛+1)=^𝑛_𝑘=−∞𝑠(𝜆,𝑛−𝑘)𝑓(𝑘,𝐴𝑢(𝑘)). (53)

Proof. Similar to the proof of [13], it can be shown that a solution of (53) is the solution of (46).
Define the operator ℱ∶𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋)→𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) by (ℱ𝑢)(𝑛)=^𝑛−1_𝑘=−∞𝑠(𝜆,𝑛−1−𝑘)𝑓(𝑘,𝐴𝑢(𝑘)). (54) Since 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) and (𝐻₃₁) holds, 𝑓(⋅,𝐴𝑢(⋅))∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) by Lemma 11, Corollary 17. By Lemma 18, ℱ𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋). Hence ℱ is well-defined.
For 𝑢,𝑣∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), ‖ℱ𝑢−ℱ𝑣‖_𝑑≤sup_𝑛∈ℤ^𝑛−1_𝑘=−∞|𝑠(𝜆,𝑛−1−𝑘)|𝑓(𝑘,𝐴𝑢(𝑘))−𝑓(𝑘,𝐴𝑣(𝑘))≤𝐿_𝑓‖𝐴‖sup_𝑛∈ℤ^𝑛−1_𝑘=−∞|𝑠(𝜆,𝑛−1−𝑘)|‖𝑢(𝑘)−𝑣(𝑘)‖≤𝐿_𝑓‖𝐴‖‖𝑢−𝑣‖_𝑑sup_𝑛∈ℤ^𝑛−1_𝑘=−∞|𝑠(𝜆,𝑛−1−𝑘)|≤𝐿_𝑓‖𝐴‖|𝑠(𝜆,⋅)|₁‖𝑢−𝑣‖_𝑑. (55) By the Banach contraction mapping principle, ℱ has a unique fixed point 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), which is the unique 𝑊𝑃𝐴𝐴_𝑑 solution to (46).

Theorem 21. Assume that (𝐻₁), (𝐻₂), and (𝐻₃₂) hold; then (46) has a unique solution 𝑢(𝑛)∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) if (‖𝐴‖|𝑠(𝜆,⋅)|₁Φ)^𝑛(𝑡)→0 as 𝑛→∞ for each 𝑡>0.

Proof. Define the operator ℱ as in (54), so ℱ is well defined. For 𝑢,𝑣∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), one has 𝑛)−(𝑛≤‖(ℱ𝑢)(ℱ𝑣)()‖_{𝑛−1𝑘=−∞}|𝑠(𝑓((𝑘((𝑘≤𝜆,𝑛−1−𝑘)|𝑘,𝐴𝑢))−𝑓𝑘,𝐴𝑣))_{𝑛−1𝑘=−∞}|𝑠(Φ(𝑘)(𝑘≤‖𝐴𝑠(𝜆,𝑛−1−𝑘)|(‖𝐴𝑢−𝐴𝑣)‖)‖|𝜆,⋅)|₁Φ(‖𝑢(𝑘)−𝑣(𝑘)‖). (56) Since (‖𝐴‖|𝑠(𝜆,⋅)|₁Φ)^𝑛(𝑡)→0 as 𝑛→∞ for each 𝑡>0, by Matkowski fixed point theorem (Theorem 1), ℱ has a unique fixed point 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), which is the unique 𝑊𝑃𝐴𝐴_𝑑 solution to (46).

4.2. Nonautonomous Semilinear Difference Equations

In this subsection, consider the following nonautonomous semilinear difference equations 𝑢(𝑛+1)=𝐴(𝑛)𝑢(𝑛)+𝑓(𝑛,𝑢(𝑛)),𝑛∈ℤ. (57) Its associated homogeneous linear difference equation is given by 𝑢(𝑛+1)=𝐴(𝑛)𝑢(𝑛),𝑛∈ℤ. (58)

To establish our results, we introduce the following condition.(A₁)𝐴∈𝐴𝐴_𝑑(ℤ,𝑋).(A₂)𝑓∈𝑊𝑃𝐴𝐴_𝑑(ℤ×𝑋,𝑋), 𝜌₁,𝜌₂∈𝑉_∞.(A₃)There exists a constant 𝐿_𝑓>0 such that 𝑓(𝑘,𝑢)−𝑓(𝑘,𝑣)≤𝐿_𝑓‖𝑢−𝑣‖,∀𝑘∈ℤ,𝑢,𝑣∈𝑋. (59)(A₄)Equation (58) admits an exponential dichotomy on ℤ with positive constants 𝜂,𝜈,𝛼,𝛽.

Theorem 22. Assume that (𝐴₁)--(𝐴₄) hold and (𝜂/(1−𝑒^−𝛼)+𝜈𝑒^−𝛽/(1−𝑒^−𝛽))𝐿_𝑓<1; then (57) has a unique solution 𝑢(𝑛)∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) which is given by 𝑢(𝑛)=^𝑛−1_𝑗=−∞𝒜𝑃𝑓𝑗−𝑛,𝑗+1𝑗+1𝑗,𝑢_∞𝑗=𝑛𝒜𝑛,𝑗+1𝑄𝑗+1𝑓𝑗,𝑢𝑗,𝑛∈ℤ. (60)

Proof. Similar to the proof of [14, 15], it can be shown that 𝑢(⋅) given by (60) is the solution of (57).
Define an operator Γ∶𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋)→𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) as follows: (Γ𝑢)(𝑛)∶=^𝑛−1_𝑗=−∞𝒜𝑃𝑓𝑗−𝑛,𝑗+1𝑗+1𝑗,𝑢_∞𝑗=𝑛𝒜𝑛,𝑗+1𝑄𝑗+1𝑓𝑗,𝑢𝑗,𝑛∈ℤ. (61) Since 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) and (A₃) holds, Ψ(⋅)=𝑓(⋅,𝑢(⋅))∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋) by Corollary 17. Let Ψ=Ψ₁+Ψ₂, where Ψ₁∈𝐴𝐴_𝑑(ℤ,𝑋), Ψ₂∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋); then (Γ𝑢)(𝑛)∶=Γ₁𝑢(𝑛)+Γ₂𝑢(𝑛), (62) where Γ₁𝑢(𝑛)=^𝑛−1_𝑗=−∞𝒜𝑛,𝑗+1𝑃𝑗+1Ψ₁𝑗−^∞_𝑗=𝑛𝒜𝑛,𝑗+1𝑄𝑗+1Ψ₁𝑗,Γ₂𝑢(𝑛)=^𝑛−1_𝑗=−∞𝒜𝑛,𝑗+1𝑃𝑗+1Ψ₂𝑗−^∞_𝑗=𝑛𝒜𝑛,𝑗+1𝑄𝑗+1Ψ₂𝑗. (63) Similar to the proof of [14, 15], Γ₁𝑢∈𝐴𝐴_𝑑(ℤ,𝑋).
Next, we show that Γ₂𝑢∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). In fact, let Γ₂𝑢(𝑛)∶=Γ₂₁𝑢(𝑛)+Γ₂₂𝑢(𝑛), (64) where Γ₂₁𝑢(𝑛)=^𝑛−1_𝑗=−∞𝒜𝑛,𝑗+1𝑃𝑗+1Ψ₂𝑗,Γ₂₂𝑢(𝑛)=^∞_𝑗=𝑛𝒜𝑛,𝑗+1𝑄𝑗+1Ψ₂𝑗. (65) Then, 1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Γ₂₁𝑢(𝑘)𝜌₂(𝑘)=1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇^𝑘−1_𝑗=−∞𝒜𝑘,𝑗+1𝑃𝑗+1Ψ₂𝑗𝜌₂(𝑘)≤1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇^𝑘−1_𝑗=−∞𝜂𝑒^{−𝛼(𝑘−𝑗−1)}Ψ₂𝑗𝜌₂(𝑘)=1𝜇𝑇,𝜌₁^𝑇∞_{𝑘=−𝑇𝑗=0}𝜂𝑒^−𝛼𝑗Ψ₂𝑘−1−𝑗𝜌₂(𝑘)=^∞_𝑗=0𝜂𝑒^−𝛼𝑗1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Ψ₂𝑘−1−𝑗𝜌₂(𝑘). (66) Since Ψ₂∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋), Ψ₂(⋅−1−𝑗)∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋) for each 𝑗∈ℤ by Lemma 10; then lim_𝑇→∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Ψ₂𝑘−1−𝑗𝜌₂(𝑘)=0; (67) by Lebesgue dominated convergence theorem, one has lim_𝑇→∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Γ₂₁𝑢(𝑘)𝜌₂(𝑘)≤^∞_𝑗=0𝜂𝑒^−𝛼𝑗lim_𝑇→∞1𝜇𝑇,𝜌₁^𝑇_𝑘=−𝑇Ψ₂𝑘−1−𝑗𝜌₂(𝑘)=0; (68) hence Γ₂₁𝑢∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). Similarly, one can prove Γ₂₂𝑢∈𝑊𝑃𝐴𝐴₀(ℤ,𝑋). So Γ is well defined.
For 𝑢,𝑣∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), by the exponential dichotomy and Lipschitz condition, one has 𝑛)−(𝑛=‖(Γ𝑢)(Γ𝑣)()‖_{𝑛−1𝑗=−∞}𝒜𝑃𝑓𝑗𝑗−𝑛,𝑗+1𝑗+1𝑗,𝑢−𝑓𝑗,𝑣_∞𝑗=𝑛𝒜𝑄×𝑓𝑗𝑗≤𝑛,𝑗+1𝑗+1𝑗,𝑢−𝑓𝑗,𝑣_{𝑛−1𝑗=−∞}𝜂𝑒^{−𝛼(𝑛−𝑗−1)}𝐿_𝑓𝑢𝑗𝑗+−𝑣_{+∞𝑗=𝑛}𝜈𝑒^{−𝛽(𝑗+1−𝑛)}𝐿_𝑓𝑢𝑗−𝑣𝑗≤𝜂𝐿_𝑓‖𝑢−𝑣‖_𝑑^+∞_𝑗=0𝑒^−𝛼𝑗+𝜈𝐿_𝑓‖𝑢−𝑣‖_𝑑^+∞_𝑗=1𝑒^−𝛽𝑗≤𝜂𝐿_𝑓1−𝑒^−𝛼‖𝑢−𝑣‖_𝑑+𝜈𝐿_𝑓𝑒^−𝛽1−𝑒^−𝛽‖𝑢−𝑣‖_𝑑=𝜂1−𝑒^−𝛼+𝜈𝑒^−𝛽1−𝑒^−𝛽𝐿_𝑓‖𝑢−𝑣‖_𝑑; (69) hence Γ is a contraction. By the Banach contraction mapping principle, Γ has a unique fixed point 𝑢∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋), which is the unique 𝑊𝑃𝐴𝐴_𝑑 solution to (57). The proof is completed.

5. Examples

In this section, we provide some examples to illustrate our main results.

Example 1. For 𝑎(𝑘)=𝑝^𝑘, where |𝑝|<1, after a calculation using in (48) the unilateral-𝑍 transform, that 𝑠(𝜆,𝑘)=𝜆(𝜆+𝑝)^𝑘−1, 𝑘≥1, and define 𝔻−𝑝,1∶=𝑧∈ℂ∶𝑧+𝑝<1⊆Ω_𝑠. (70)
Consider the following difference equation: 𝑢(𝑛+1)=𝜆^𝑛_𝑘=−∞𝑝^𝑛−𝑘𝑢(𝑘)+𝜅𝑔(𝑘)𝑢(𝑘),𝑛∈ℤ, (71) where |𝑝|<1, 𝜆∈𝔻(−𝑝,1), 𝑔∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋). It is easy to see that (H₁), (H₂), and (H₃₁) hold with 𝐿_𝑓=|𝜅|‖𝑔‖_𝑑. By Theorem 20, if |𝜆||𝜅|‖𝑔‖_𝑑∑^∞_𝑘=0|𝜆+𝑝|^𝑘−1<1, then (71) has a unique solution 𝑢(𝑛)∈𝑊𝑃𝐴𝐴_𝑑(ℤ,𝑋).

Example 2. Consider the system 𝑢(𝑘+1)=𝐴𝑢(𝑘)+ℎ(𝑘)𝑔(𝑢),𝑘∈ℤ, (72) where 𝐴 is a nonsingular 𝑛×𝑛 matrix such that 𝜎_𝑝(𝐴)∩𝑆¹=∅, ℎ∈𝑊𝑃𝐴𝐴_𝑑(ℤ,ℝ^𝑛), and there exists a constant 𝐿_𝑓>0 such that 𝑔(𝑢)−𝑔(𝑣)≤𝐿_𝑔‖𝑢−𝑣‖,𝑢,𝑣∈ℝ^𝑛. (73)
Since 𝜎_𝑝(𝐴)∩𝑆¹=∅, the system 𝑢(𝑘+1)=𝐴𝑢(𝑘),𝑘∈ℤ (74) admits an exponential dichotomy with positive constants 𝜂,𝜈,𝛼,𝛽 [19] and (A₃) holds with 𝐿_𝑓=𝐿_𝑔‖ℎ‖_𝑑. By Theorem 22, If we suppose that (𝜂/(1−𝑒^−𝛼)+𝜈𝑒^−𝛽/(1−𝑒^−𝛽))𝐿_𝑔‖ℎ‖_𝑑<1, then (72) has a unique discrete weighted pseudo almost automorphic solution.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ13A010015.