Asymptotic Almost-Periodicity for a Class of Weyl-Like Fractional Difference Equations

Abstract

This work deal with asymptotic almost-periodicity of mild solutions for a class of difference equations with a Weyl-like fractional difference in Banach space. Based on a combination of a decomposition technique and the Krasnoselskii’s fixed point theorem, we establish some new existence theorems of mild solutions with asymptotic almost-periodicity. Our results extend some related conclusions, since (locally) Lipschitz assumption on the nonlinear perturbation is not needed and with Lipschitz assumption becoming a special case. An example is presented to validate the application of our results.

1. Introduction

The original notion of discrete almost-periodic sequence was proposed by the famous mathematician Walther in 1928 [1,2] and the theory as well as applications of discrete almost-periodic sequences was further developed by Halanay [3] and Corduneanu [4]. Discrete almost-periodic sequence plays a key role in characterizing the phenomena that are similar to the periodic phenomena. As an important development of the well-known discrete almost-periodic sequence, the notion of a discrete asymptotically almost-periodic sequence, which is the key issue, will be investigated in this work and was introduced in the works of Fan [5] in 1943 based on the Fréchet concept from [6,7]. Since then, the investigation of the existence of solutions with asymptotically almost-periodicity has become one of the important as well as attractive subjects in the theory as well as applications of difference equations due to both the intensive development of the theorem of difference equations itself and the applications in various sciences such as computer science, chemistry, physics, engineering and so forth. For some meaningful and interesting works in this area, we refer readers to Thana [8] for periodic evolution equations, Agarwal, Cuevas and Dantas [9] for difference equations of Volterra type, Song [10] for nonlinear delay difference equations of Volterra type, Cuevas and Pinto [11] as well as Campo, Pinto and Vidal [12] for functional difference equations, Matkowski [13] for functional equations and the references therein.

On the other hand, the tools of fractional calculus are found to be of great utility in studying various scientific processes and systems. This has been mainly due to the ability of fractional order operators to describe long-memory effects of underlying processes. In particular, models described by fractional differential equations have gained significant importance and there has been a great interest in developing the theory and applications of fractional differential equations. For examples and details, we refer the reader to a series of recent research articles [14,15,16,17,18,19] and the references therein. Recently, fractional difference equations have gained more interest by many researchers. Besides their interest in terms of theory, the study of fractional difference equations has great importance in the aspect applications [20]. The study of modeling with fractional difference equations began with the works of Atici and Seng$\ddot{\mathrm{u}}$l [21] and then Atici and Eloe [22,23], Dassios and Baleanu [24,25], Goodrich [26,27] studied the initial value problems, boundary value problems and positivity as well as monotonicity properties [28] of solutions to fractional difference equations, Baleanu and Wu et al. [29,30] and Lizama [31] investigated the stability of fractional difference equations and Wu and Baleanu [32] studied the logistic map with fractional difference and its chaos. Especially in [33], Abadias and Lizama consider nonlinear partial difference-differential equations in Banach space X forms like

$$\begin{array}{c}\hfill {▵}^{\alpha }x\left(n\right)=Ax(n+1)+F(n,x\left(n\right)),n\in \mathbb{Z},\end{array}$$

(1)

where $\alpha \in (0,1)$, the operator A generates a $C_0$-semigroup defined on X, ${▵}^{\alpha }$ denotes the Weyl-like fractional difference operator, $F(n,x):\mathbb{Z}\times X\to X$ is a function satisfying some Lipschitz type assumptions. By using the operator theoretical method and the Banach fixed point theorem, they prove the existence and uniqueness of solutions with almost automorphy to Equation (1). Stimulated by the works of Abadias and Lizama [33], Cao and Zhou [19] established some criteria for the existence and uniqueness of mild solutions with asymptotical almost-periodicity to Equation (1) with the nonlinear perturbation $F(n,x)$ satisfying a Lipschitz assumption or locally Lipschitz assumption.

According to our knowledge, up to now, much less is known about the existence of mild solution with asymptotical almost-periodicity to Equation (1) when the nonlinear perturbation $F(n,x)$ loses the Lipschitz assumption with respect to x. Thus in this work, we will try to fill this gap. Combining a decomposition technique with the Krasnoselskii’s fixed point theorem, we establish some new existence theorems of mild solutions with asymptotic almost-periodicity to Equation (1). In our results, $F(n,x)$ does not have to satisfy a Lipschitz assumption or locally Lipschitz assumption with respect to x (see Remark 2) and $F(n,x)$ with Lipschitz assumption becomes a special case with our conditions (see Remark 4). Thus our results generalize some related conclusions on this topic. In particular, as an application, we prove the existence and uniqueness of mild solutions with asymptotically almost-periodicity to the fractional difference scalar equation forms like

$$\begin{array}{c}\hfill {▵}^{\alpha }x\left(n\right)=\lambda x(n+1)+F(n,x\left(n\right)),n\in \mathbb{Z},\end{array}$$

where $\alpha \in (0,1)$, $\lambda$ is a complex number satisfying $\mathrm{Re}\left(\lambda \right)<0$, $F:\mathbb{Z}\times \mathbb{C}\to \mathbb{C}$ is a function to be specified later.

An outline of this article is as follows. We introduce some basic concepts and recall some preliminaries in Section 2. Section 3 is concerned with some new existence theorems for mild solutions with asymptotically almost-periodicity to Equation (1). The last section deal with an example to validate the applications of our theoretical results.

2. Preliminaries

In this section, we introduce some basic concepts and recall some preliminaries.

In this paper, let $\mathbb{N}$, $\mathbb{Z}$, ${\mathbb{Z}}^{+}$, $\mathbb{R}$, ${\mathbb{R}}^{+}$, $\mathbb{C}$ be the sets of all natural numbers, integral numbers, positive integral numbers, real numbers, positive real numbers and complex numbers, respectively. For a Banach space $(X,\parallel ·\parallel )$, let $B_{\rho }\left(X\right)=\{x\in X:\parallel x\parallel \le \rho \}$. $s(\mathbb{Z},X)$ is a set consisting of sequences $f:\mathbb{Z}\to X$. $l^{\infty }(\mathbb{Z},X)$ is a set consisting of sequences

$$l^{\infty }(\mathbb{Z},X):=\left\{f:\mathbb{Z}\to X|f\mathrm{is}\mathrm{bounded}\mathrm{on}\mathbb{Z}\right\}.$$

The space $l^{\infty }(\mathbb{Z},X)$ is a Banach space under the norm

$${\parallel x\parallel }_d:=\underset{n\in \mathbb{Z}}{sup}\parallel x\left(n\right)\parallel .$$

$AA{P}_0(\mathbb{Z},X)$ is a set consisting of sequences

$$AA{P}_0(\mathbb{Z},X):=\{f\left(n\right)\in l^{\infty }(\mathbb{Z},X)|\underset{\left|n\right|\to +\infty }{lim}\parallel f\left(n\right)\parallel =0\}.$$

$AA{P}_0(\mathbb{Z},X)$ is also a Banach space under the norm ${\parallel x\parallel }_d$. $l(\mathbb{Z},X)$ is a set consisting of sequences

$$l(\mathbb{Z},X):=\left\{f:\mathbb{Z}\to {X|\parallel f\parallel }_l=\sum _{n=-\infty }^{+\infty }\parallel f\left(n\right)\parallel <+\infty \right\}.$$

Moreover when $X=\mathbb{R}$, we write $l\left(\mathbb{Z}\right)$ for short. $l_{\rho }(\mathbb{Z},X)$ is a set consisting of sequences

$$\begin{array}{cc}\hfill l_{\rho }(\mathbb{Z},X):=\{f:\mathbb{Z}\to X|{\parallel f\parallel }_{l_{\rho }}=& \sum _{n=-\infty }^{+\infty }\parallel f\left(n\right)\parallel \rho \left(n\right)<+\infty ,\hfill \\ & \rho :\mathbb{Z}\to {\mathbb{R}}^{+}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{sequence}\mathrm{weight}\}.\hfill \end{array}$$

$BC(\mathbb{R},X)$ is the space of all X-valued bounded continuous functions and $C_0(\mathbb{R},X)$ is the closed subspace of $BC(\mathbb{R},X)$ consisting of functions vanishing at infinity. Let $(Y,\parallel ·{\parallel }_Y)$ be another Banach space, $l^{\infty }(\mathbb{Z}\times Y,X)$ is a set consisting of functions

$$\begin{array}{cc}\hfill l^{\infty }(\mathbb{Z}\times Y,X):=\{G:\mathbb{Z}\times Y\to X|& G\mathrm{is}\mathrm{bounded}\mathrm{on}\mathbb{Z}\times Y\mathrm{and}\hfill \\ & G(n,·)\mathrm{is}\mathrm{continuous}\mathrm{on}Y\mathrm{for}\mathrm{each}\mathrm{fixed}n\in \mathbb{Z}\}.\hfill \end{array}$$

The space $l^{\infty }(\mathbb{Z}\times Y,X)$ is a Banach space under the norm

$$|\parallel G|\parallel :=\underset{n\in \mathbb{Z},x\in Y}{sup}\parallel G(n,x)\parallel .$$

$AA{P}_0(\mathbb{Z}\times Y,X)$ is a set consisting of functions

$$AA{P}_0(\mathbb{Z}\times Y,X):=\left\{G(n,x)\in l^{\infty }(\mathbb{Z}\times Y,X)|\underset{\left|n\right|\to +\infty }{lim}\parallel G(n,x)\parallel =0\mathrm{uniformly}\mathrm{for}x\in Y\right\}.$$

Let $\mathbb{L}(X,Y)$ be the collection of all bounded linear operators from X to Y. Under the uniform operator topology

$${\parallel \mathrm{Y}\parallel }_{\mathbb{L}(X,Y)}:={sup\{\parallel \mathrm{Y}f\parallel }_Y:f\in X,\parallel f\parallel =1\},$$

we denote by $\mathbb{L}\left(X\right)=\mathbb{L}(X,X)$. For $A\in \mathbb{L}\left(X\right)$, let $\rho \left(A\right)$ be the resolvent of A and $D\left(A\right)$ be the domain of A.

Firstly, we recall the definitions and related properties on discrete almost-periodic sequences as well as discrete asymptotically almost-periodic sequences.

Following Bohr, Walther has formulated the notion of discrete almost-periodic sequence.

Definition 1.

(Walther [1,2], Corduneanu [34]) Let ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ be a discrete sequence with values in X. If for each $\epsilon >0$, the collection

$$T(f,\epsilon ):=\{k\in \mathbb{Z}:\parallel f(n+k)-f(n)\parallel <\epsilon foreveryn\in \mathbb{Z}\}$$

is relatively dense in $\mathbb{Z}$, that is for any $\epsilon >0$, there is an integer $N=N\left(\epsilon \right)>0$, such that there exists at least one integer $k\in \Delta$, where Δ is a any collection consisting of N consecutive integers, satisfying

$$\parallel f(n+k)-f\left(n\right)\parallel <\epsilon ,n\in \mathbb{Z},$$

then ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is said to be a discrete almost-periodic sequence.

The integer $k\in T(k,\epsilon )$, with the property in Definition 1, is said to be an ε-translation number of the sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$.

By $AP(\mathbb{Z},X)$ we denote the collection of such sequences.

The following notion of a normal process is needed to formulate an important property of the discrete almost-periodic processes.

Definition 2.

(Corduneanu [34], Zhang, Liu and Gopalsamy [35]) A discrete sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is called a normal process, if for any sequence $\left\{\alpha \right(k\left)\right\}\in \mathbb{Z}$, there is a subsequence $\left\{\beta \right(k\left)\right\}\subset \left\{\alpha \right(k\left)\right\}$, for which $\left\{f\right(n+\beta \left(k\right)\left)\right\}$ converges uniformly with respect to $n\in \mathbb{Z}$, as $k\to \infty$. That is to say, for any $\epsilon >0$, there exist an integer $K\left(\epsilon \right)>0$ and a discrete process ${\left\{\overline{f}\left(n\right)\right\}}_{n\in \mathbb{Z}}$ such that

$$\parallel f(n+\beta \left(k\right))-\overline{f}\left(n\right)\parallel <\epsilon fork\ge K\left(\epsilon \right),n\in \mathbb{Z}.$$

Lemma 1.

(Corduneanu [34]) A discrete process is almost-periodic if and only if it is normal.

Definition 3.

(Bohr) [36,37] A continuous function $f:\mathbb{R}\to X$ is said to be (Bohr) almost-periodic in $t\in \mathbb{R}$ if for each $\epsilon >0$, the collection

$$T(f,\epsilon ):=\{\tau \in \mathbb{R}:\parallel f(t+\tau )-f(t)\parallel <\epsilon foreveryt\in \mathbb{R}\}$$

is relatively dense in $\mathbb{R}$; that is for every $\epsilon >0$ there exists $l\left(\epsilon \right)>0$ such that every interval of length $l\left(\epsilon \right)$ contains a number τ with the property that

$$\parallel f(t+\tau )-f\left(t\right)\parallel <\epsilon foreveryt\in \mathbb{R}.$$

The number τ is called an ε-translation number of $f\left(t\right)$ and the collection of those functions is denoted by $AP(\mathbb{R},X)$.

Lemma 2.

(Corduneanu [34], Zhang [38,39]) (I) For any almost-periodic sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$, there is a function $g\left(t\right)$, $t\in \mathbb{R}$ which is almost-periodic satisfying $g\left(n\right)=f\left(n\right)$ for $n\in \mathbb{Z}$.

(II) For any almost-periodic function $g\left(t\right)$, $t\in \mathbb{R}$, ${\left\{g\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is an almost-periodic sequence.

Remark 1.

(Campo, Pinto and Vidal [12]) The discretization of a periodic function may not lead to a periodic sequence. For instance, $\{cos(k\left)\right\}$, $k=1,2,3,\dots$, is not a periodic sequence, it is an almost-periodic sequence.

Lemma 3.

(Corduneanu [34], Zhang [38,39]) ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is bounded if ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is an almost-periodic sequence.

Lemma 4.

(Corduneanu [34], Long and Pan [40]) Under the norm ${\parallel ·\parallel }_d$, $AP(\mathbb{Z},X)$ forms a Banach space.

Definition 4.

(Abadias and Lizama [33]) An operator-valued sequence ${\left\{\mathrm{Y}\left(n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{L}\left(X\right)$ is summable if

$${\parallel \mathrm{Y}\parallel }_1:=\sum _{n=0}^{+\infty }{\parallel \mathrm{Y}\left(n\right)\parallel }_{\mathbb{L}\left(X\right)}<\infty .$$

The following lemma, which comes from Gohberg and Feldman [41], is the essential property to study almost-periodicity and asymptotic almost-periodicity of difference equations.

Lemma 5.

(Gohberg and Feldman [41]) Assume ${\left\{\mathrm{Y}\left(n\right)\right\}}_{n\in \mathbb{N}}$ is a summable sequence. Then for any discrete sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ which is almost-periodic, the sequence ${\left\{g\left(n\right)\right\}}_{n\in \mathbb{Z}}$ defined by

$$g\left(n\right)=\sum _{k=0}^{+\infty }\mathrm{Y}\left(k\right)f(n-k),n\in \mathbb{Z}$$

is also an almost-periodic sequence.

Definition 5.

(Song [10]) Let $G:\mathbb{Z}\times Y\to X$ and Ω be any compact set in Y. If for any $\epsilon >0$, the collection

$$T(G,\epsilon ,\Omega ):=\{k\in \mathbb{Z}:\parallel G(n+k,x)-G(n,x)\parallel <\epsilon foreachn\in \mathbb{Z}andx\in \Omega \}$$

is relatively dense in $\mathbb{Z}$, that is for any $\epsilon >0$, there is an integer $N=N(\epsilon ,\Omega )$ such that there exists at least one integer $k\in \Delta$, where Δ is any collection consisting of N consecutive integers, satisfying

$$\parallel G(n+k,x)-G(n,x)\parallel <\epsilon \forall n\in \mathbb{Z},x\in \Omega ,$$

then $G(n,x)$ is said to be almost-periodic in $n\in \mathbb{Z}$ uniformly for $x\in Y$. The integer $k\in T(G,\epsilon ,\Omega )$, with the property in Definition 4, is said to be the ε-translation number of $G(n,x)$.

By $AP(\mathbb{Z}\times Y,X)$ we denote the collection of such functions.

Lemma 6.

(Campo, Pinto and Vidal [12], Song [10]) Let Ω be any compact set in Y and assume $G\in AP(\mathbb{Z}\times Y,X)$. Then $G(n,·)$ is continuous on Ω uniformly for $n\in \mathbb{Z}$, that is for any $\epsilon >0$, there is a constant $\delta >0$ such that

$$\parallel G(n,x)-G(n,y)\parallel <\epsilon \forall n\in \mathbb{Z}x,y\in \Omega with\parallel x-y\parallel <\delta ,$$

and $G(\mathbb{Z}\times \Omega )$ is relatively compact in X.

Lemma 7.

The following statements are equivalent.

(i) $G\in AP(\mathbb{Z}\times Y,X)$;

(ii) $G(·,f)\in AP(\mathbb{Z},X)$ for each $f\in Y$ and G is continuous in $f\in \Omega$ uniformly in $n\in \mathbb{Z}$, i.e., for $\forall \epsilon >0$, $\exists \delta >0$, s.t.

$${\parallel G(n,f)-G(n,g)\parallel <\epsilon ,\forall n\in \mathbb{Z},f,g\in \Omega with\parallel f-g\parallel }_Y<\delta ,$$

where Ω is any compact set in Y.

Proof. 

The proof of $\left(i\right)\Rightarrow \left(ii\right)$ follows from Lemma 6. In the following, we prove $\left(ii\right)\Rightarrow \left(i\right)$.

In fact, if G is continuous in $f\in \Omega$ uniformly in $n\in \mathbb{Z}$, then G is uniformly continuous in f on $\Omega$, thus $\forall \epsilon >0$, $\exists \delta =\delta (\epsilon /3)>0$, s.t.

$$\parallel G(n,f)-G(n,g)\parallel <\frac{\epsilon }{3},\forall n\in \mathbb{Z},f,g\in \Omega \mathrm{with}{\parallel f-g\parallel }_Y<\delta .$$

Since $\Omega$ is any compact set in Y, then there are $f_1,f_2,\dots ,f_n\in \Omega$ so that for each $f\in \Omega$, $\parallel f-f_i\parallel <\delta$ for some i. It follows from $G(n,f_i)\in AP(\mathbb{Z},X)$ that for the above $\epsilon$, there is an integer $N=N(\epsilon ,\Omega )$ such that there exists at least one integer $k\in \Delta$, where $\Delta$ is a any collection consisting of N consecutive integers, satisfying

$$\parallel G(n+k,f_i)-G(n,f_i)\parallel <\frac{\epsilon }{3},\forall n\in \mathbb{Z}.$$

Thus for each $f\in \Omega$, choose some $f_i$ such that $\parallel f-f_i\parallel <\delta$, then

$$\begin{array}{cc}& \parallel G(n+k,f)-G(n,f)\parallel \hfill \\ \hfill \le & \parallel G(n+k,f)-G(n+k,f_i)\parallel +\parallel G(n+k,f_i)-G(n,f_i)\parallel +\parallel G(n,f_i)-G(n,f)\parallel \hfill \\ \hfill <& \frac{\epsilon }{3}+\frac{\epsilon }{3}+\frac{\epsilon }{3}=\epsilon ,\hfill \end{array}$$

which yields $G\in AP(\mathbb{Z}\times Y,X)$. □

Lemma 8.

(Song [10]) Assume $f\in AP(\mathbb{Z},X)$. Then for any integer sequence $\left\{{\alpha }_k\right\}$, there are a subsequence $\left\{{\beta }_k\right\}\subset \left\{{\alpha }_k\right\}$ and a sequence $g:\mathbb{Z}\to X$ satisfying

$$f(n+{\beta }_k)\to g\left(n\right)$$

uniformly on $\mathbb{Z}$ as $k\to \infty$. Moreover $g\in AP(\mathbb{Z},X)$.

Lemma 9.

(Song [10]) Assume $G\in AP(\mathbb{Z}\times Y,X)$ and Ω is any compact set in Y. Then for any integer sequence $\left\{{\alpha }_k\right\}$, there are a subsequence $\left\{{\beta }_k\right\}\subset \left\{{\alpha }_k\right\}$ and a function $H:\mathbb{Z}\times Y\to X$ satisfying

$$G(n+{\beta }_k,x)\to H(n,x)$$

uniformly on $\mathbb{Z}\times \Omega$ as $k\to \infty$. Moreover $H\in AP(\mathbb{Z}\times Y,X)$.

We now give a composition theorem of discrete almost-periodic functions.

Lemma 10.

Assume $G\in AP(\mathbb{Z}\times Y,X)$ satisfying $G(n,·)$ is continuous in each bounded subset of Y uniformly in $n\in \mathbb{Z}$, i.e., for $\forall \epsilon >0$ and any bounded subset K of Y, $\exists \delta =\delta (\epsilon ,K)>0$ s.t.

$$\parallel G(n,f)-G(n,g)\parallel \le {\epsilon ,\forall n\in \mathbb{Z},f,g\in Kwith\parallel f-g\parallel }_Y\le \delta .$$

Suppose $f:\mathbb{Z}\to Y$ is a discrete almost-periodic sequence. Then ${\left\{G(n,f\left(n\right))\right\}}_{n\in \mathbb{Z}}$ belongs to $AP(\mathbb{Z},X)$.

Proof. 

From $G\in AP(\mathbb{Z}\times Y,X)$ and $f\in AP(\mathbb{Z},Y)$, together with Lemmas 8 and 9, it follows that for any integer sequence $\left\{{\alpha }_k\right\}$, there exist a subsequence $\left\{{\beta }_k\right\}\subset \left\{{\alpha }_k\right\}$ and two functions $H:\mathbb{Z}\times Y\to X$, $g:\mathbb{Z}\to Y$ satisfying

$$G(n+{\beta }_k,x)\to H(n,x)$$

uniformly on $\mathbb{Z}\times \Omega$ as $k\to \infty$ and

$$f(n+{\beta }_k)\to g\left(n\right)$$

uniformly on $\mathbb{Z}$ as $k\to \infty$. Moreover $H\in AP(\mathbb{Z}\times Y,X)$ and $g\in AP(\mathbb{Z},Y)$. It follows from $f\in AP(\mathbb{Z},Y)$ and $g\in AP(\mathbb{Z},Y)$, together with Lemma 3, that there exists a bounded subset K of Y such that $f\left(n\right)\in K$ and $g\left(n\right)\in K$ for each $n\in \mathbb{Z}$. As $G(n,x)$ is continuous on K uniformly for $n\in \mathbb{Z}$, thus for any $\epsilon >0$, one can find a constant $\delta =\delta \left(\frac{\epsilon }{2}\right)>0$ satisfying

$$\parallel G(n,f)-G(n,g)\parallel <\frac{\epsilon }{2},\forall n\in \mathbb{Z},f,g\in K\mathrm{with}{\parallel x-y\parallel }_Y<\delta .$$

Moreover for every compact set $\Omega \subset Y$, there exists an $N=N\left(\frac{\epsilon }{2}\right)$ such that for all $k>N$,

$$\parallel G(n+{\beta }_k,f)-H(n,f)\parallel <\frac{\epsilon }{2},\forall n\in \mathbb{Z},x\in \Omega ,$$

$$\parallel f(n+{\beta }_k)-g\left(n\right)\parallel <\delta ,\forall n\in \mathbb{Z}.$$

Thus for all $k>N$,

$$\begin{array}{cc}& \parallel G(n+{\beta }_k,f(n+{\beta }_k))-H(n,f\left(n\right))\parallel \hfill \\ \hfill \le & \parallel G(n+{\beta }_k,f(n+{\beta }_k))-G(n+{\beta }_k,f\left(n\right))\parallel +\parallel G(n+{\beta }_k,f\left(n\right))-H(n,f\left(n\right))\parallel <\frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon ,\hfill \end{array}$$

which implies ${\left\{G(n,f\left(n\right))\right\}}_{n\in \mathbb{Z}}$ is a normal process, thus $G(n,f(n\left)\right)$ is almost-periodic which follows from Lemma 1.

Let $G(n,f):\mathbb{Z}\times X\to X$ satisfy a Lipschitz assumption in $f\in X$ uniformly in $n\in \mathbb{Z}$, we obtain the following result as an immediate consequence of the previous lemma, Lemma 10. □

Lemma 11.

Assume $G(n,f)\in AP(\mathbb{Z}\times X,X)$ and satisfies a Lipschitz assumption in $f\in X$ uniformly for $n\in \mathbb{Z}$, i.e.,

$$\begin{array}{c}\hfill \parallel G(n,f)-G(n,g)\parallel \le L\parallel f-g\parallel ,for\forall x,y\in X,n\in \mathbb{Z}andsomeconstantL.\end{array}$$

(2)

Then, the conclusion of Lemma 10 is true.

Lemma 11 admits a new version with local Lipschitz assumption on the function G.

Lemma 12.

Assume $G(n,f)\in AP(\mathbb{Z}\times X,X)$ and satisfies a local Lipschitz condition in $f\in X$ uniformly for $n\in \mathbb{Z}$, i.e., for each $r>0$ and $\forall x,y\in X$ with $\parallel x\parallel \le r$, $\parallel y\parallel \le r$,

$$\begin{array}{c}\hfill \parallel G(n,x)-G(n,y)\parallel \le L_G\left(r\right)\parallel x-y\parallel ,\forall n\in \mathbb{Z},\end{array}$$

(3)

where $L_G:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a function. Then, the conclusion of Lemma 10 is true.

As an important development of the well-known discrete almost-periodic sequence, the notion of discrete asymptotically almost-periodic sequence, which is based upon the Fréchet concept from [6,7], was introduced in the literature [5] by Fan.

Definition 6.

(Fan [5], Song [10]) If a sequence $f\left(n\right)=g\left(n\right)+h\left(n\right)$ with $g\left(n\right)\in AP(\mathbb{Z},X)$ and $h\left(n\right)\in AA{P}_0(\mathbb{Z},X)$, then the sequence $f:\mathbb{Z}\to X$ is said to be asymptotically almost-periodic.

The sequences ${\left\{g\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is called the almost-periodic component of ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$ and ${\left\{h\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is called the ergodic perturbation of ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$.

By $AAP(\mathbb{Z},X)$ we denote the collection of such sequences.

Definition 7.

(Fréchet) [6,7] A continuous function $f:\mathbb{R}\to X$ is said to be asymptotically almost-periodic if it can be decomposed as $f\left(t\right)=g\left(t\right)+h\left(t\right)$, where

$$g\left(t\right)\in AP(\mathbb{R},X),h\left(t\right)\in C_0(\mathbb{R},X).$$

Lemma 13.

(Zhang [38,39]) (I) For any discrete asymptotically almost-periodic sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$, there is a function $g\left(t\right)$, $t\in \mathbb{R}$ which is asymptotically almost-periodic satisfying $g\left(n\right)=f\left(n\right)$ for $n\in \mathbb{Z}$.

(II) For any asymptotically almost-periodic function $g\left(t\right)$, $t\in \mathbb{R}$, ${\left\{g\left(n\right)\right\}}_{n\in \mathbb{Z}}$ is a discrete asymptotically almost-periodic sequence.

Lemma 14.

(Zhang [38,39]) The decomposition of an asymptotically almost-periodic sequence ${\left\{f\left(n\right)\right\}}_{n\in \mathbb{Z}}$

$$f\left(n\right)=g\left(n\right)+h\left(n\right)$$

with $g\left(n\right)\in AP(\mathbb{Z},X)$ and $h\left(n\right)\in AA{P}_0(\mathbb{Z},X)$, is unique.

Lemma 15.

(Long and Pan [40]) Under the norm ${\parallel ·\parallel }_d$, $AAP(\mathbb{Z},X)$ also forms a Banach space.

Lemma 16.

Assume $g\left(n\right)$ is the almost-periodic component of the sequence $f\left(n\right)\in AAP(\mathbb{Z},X)$. Then $g\left(\mathbb{Z}\right)\subset \overline{f\left(\mathbb{Z}\right)}$.

Proof. 

Denote by $h\left(n\right)=f\left(n\right)-g\left(n\right)$, the ergodic perturbation of $f\left(n\right)$. If $g\left(\mathbb{Z}\right)$ is not contained in $\overline{f\left(\mathbb{Z}\right)}$, then there are ${\epsilon }_0>0$ and $n_0\in \mathbb{Z}$ such that

$$\underset{n\in \mathbb{Z}}{inf}\parallel f\left(n\right)-g\left(n_0\right)\parallel \ge {\epsilon }_0.$$

Let us take $k\in T(g,\frac{{\epsilon }_0}{2})$. Then,

$$\begin{array}{cc}& \parallel h(n_0+k)\parallel =\parallel f(n_0+k)-g(n_0+k)\parallel \hfill \\ \hfill \ge & \parallel f(n_0+k)-g\left(n_0\right)\parallel -\parallel g(n_0+k)-g\left(n_0\right)\parallel \ge \frac{{\epsilon }_0}{2},\hfill \end{array}$$

which is a contradiction with $h\left(n\right)\in AA{P}_0(\mathbb{Z},X)$. □

Definition 8.

(Campo, Pinto and Vidal [12], Long and Pan [40]) A function $G(n,x):\mathbb{Z}\times Y\to X$ is said to be a discrete asymptotically almost-periodic function in $n\in \mathbb{Z}$ for each $x\in Y$ if

$$G(n,x)=H(n,x)+W(n,x)$$

with $H(n,x)\in AP(\mathbb{Z}\times Y,X)$ and $W(n,x)\in AA{P}_0(\mathbb{Z}\times Y,X)$.

By $AAP(\mathbb{Z}\times Y,X)$ we denote the collection of such functions.

Lemma 17.

Assume $G(n,x)\in AAP(\mathbb{Z}\times Y,X)$ and Ω is any compact set in Y. Then $G(n,x)$ is bounded on $\mathbb{Z}\times \Omega$.

Proof. 

As $G(n,x)\in AAP(\mathbb{Z}\times Y,X)$, then

$$G(n,x)=H(n,x)+W(n,x)$$

with $H(n,x)\in AP(\mathbb{Z}\times Y,X)$ and $W(n,x)\in AA{P}_0(\mathbb{Z}\times Y,X)$. From Lemma 6, it follows that $H(n,x)$ is bounded on $\mathbb{Z}\times \Omega$. On the other hand, as $W(n,x)\in AA{P}_0(\mathbb{Z}\times Y,X)$, then $W(n,x)$ is bounded on $\mathbb{Z}\times Y$. Thus $G(n,x)=H(n,x)+W(n,x)$ is bounded on $\mathbb{Z}\times \Omega$.

In the following, we give the definitions of the discrete $\alpha$-resolvent family, the fractional sum and fractional difference of $\alpha$.

Let $\alpha >0$, defined a sequence ${\left\{k_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}$ by

$$k_{\alpha }\left(n\right):=\frac{\mathrm{\Gamma }(n+\alpha )}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(n+1)},$$

where $\mathrm{\Gamma }$ is the Gamma function. Note that $k_{\alpha }$ satisfies the following semigroup property

$$(k_{\alpha }\ast k_{\beta })\left(n\right)=\sum _{j=0}^n{k}_{\alpha }(n-j)k_{\beta }\left(j\right)=k_{\alpha +\beta }\left(n\right),\forall n\in \mathbb{N},\alpha ,\beta >0.$$

Define the forward Euler operator $▵:s(\mathbb{Z},X)\to s(\mathbb{Z},X)$ as

$$▵\gamma \left(n\right):=\gamma (n+1)-\gamma \left(n\right),n\in \mathbb{Z}.$$

Recursively, for each $k\in \mathbb{N}$, define ${▵}^{k+1}=▵{▵}^k={▵}^k▵$ and ${▵}^0=I$ is the identity operator. □

Definition 9.

(Abadias and Lizama [33]) Assume $\alpha >0$ and A is a closed linear operator with domain $D\left(A\right)\subset X$. An operator sequence ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{L}\left(X\right)$ is said to be a discrete α-resolvent family generated by A if for any $n\in \mathbb{N}$ and $x\in D\left(A\right)$

$$S_{\alpha }\left(n\right)Ax=A{S}_{\alpha }\left(n\right)x,S_{\alpha }\left(n\right)x=k_{\alpha }\left(n\right)x+A(k_{\alpha }\ast S_{\alpha })\left(n\right)x.$$

Definition 10.

(Abadias and Lizama [33]) For any $\alpha >0$, let $\rho \left(n\right)={\left|n\right|}^{\alpha -1}$, $n\in \mathbb{Z}$ and $f\in l_{\rho }(\mathbb{Z},X)$ be a sequence. The fractional sum of f is given by

$${▵}^{-\alpha }f\left(n\right):=\sum _{j=-\infty }^n{k}_{\alpha }(n-j)f\left(j\right),n\in \mathbb{Z}.$$

Definition 11.

(Abadias and Lizama [33]) For any $\alpha >0$, let $\rho \left(n\right)={\left|n\right|}^{\alpha -1}$, $n\in \mathbb{Z}$ and $f\in l_{\rho }(\mathbb{Z},X)$ be a sequence. The fractional difference of f is given by

$${▵}^{\alpha }f\left(n\right):={▵}^p{▵}^{-(p-\alpha )}f\left(n\right),n\in \mathbb{Z},$$

where $p=\left[\alpha \right]+1$, $[·]$ is the largest integer function.

In the following, we recall the definitions and properties of Mittag-Lefer functions [42]

$$E_{\alpha }\left(t\right)=\sum _{k=0}^{\infty }\frac{t^k}{\mathrm{\Gamma }(\alpha k+1)},E_{\alpha ,\alpha }\left(t\right)=\sum _{k=0}^{\infty }\frac{t^k}{\mathrm{\Gamma }\left(\alpha \right(k+1\left)\right)},t\in \mathbb{C}.$$

These functions have the following properties for $\alpha \in (0,1)$ and $t\in \mathbb{R}$.

Lemma 18.

(1) [43] $E_{\alpha }\left(t\right)>0$, $E_{\alpha ,\alpha }\left(t\right)>0$.

(2) [44] ${\left(E_{\alpha }\left(t\right)\right)}^{\prime }=(1/\alpha )E_{\alpha ,\alpha }\left(t\right)$.

(3) [45,46] $\underset{t\to -\infty }{lim}{E}_{\alpha }\left(t\right)=\underset{t\to -\infty }{lim}{E}_{\alpha ,\alpha }\left(t\right)=0$.

Finally, we give a compactness criterion and the so-called Krasnoselskii’s fixed point theorem.

Lemma 19.

A set $D\subset C_0(\mathbb{Z},X)$ is relatively compact if

(1) $\underset{\left|n\right|\to +\infty }{lim}x\left(n\right)=0$ uniformly for $x\in D$,

(2) the set $D\left(n\right):=\left\{x\right(n):x\in D\}$ is relatively compact in X for every $n\in \mathbb{Z}$.

Proof. 

For any $n_1,n_2\in \mathbb{Z}$ with $n_1<n_2$, set

$$D(n_1,n_2)=\left\{(x\left(n_1\right),x(n_1+1),\dots ,x\left(n_2\right))|x\in D\right\}.$$

For any $x*=(x\left(n_1\right),x(n_1+1),\dots ,x\left(n_2\right)),y*=(y\left(n_1\right),y(n_1+1),\dots ,y\left(n_2\right))\in D(n_1,n_2)$, define

$$\rho (x*,y*)=max\left\{\parallel x\left(m\right)-y\left(m\right)\parallel |n_1\le m\le n_2\right\},$$

it is clear that $(D(n_1,n_2),\rho )$ is a metric space. Let $\left\{x_k^{*}\right\}$ be a sequence in $D(n_1,n_2)$ with $x_k^{*}=(x_k\left(n_1\right),x_k(n_1+1),\dots ,x_k\left(n_2\right))$, it follows from the condition (2) that $\left\{x_k\left(n_1\right)\right\}$ is relatively compact in X. Then there exists subsequence $\{x_{1,k}^{*}\}\subset \left\{x_k^{*}\right\}$ such that $\left\{x_{1,k}\left(n_1\right)\right\}$ is convergent in X. Then from the condition (2), it follows that $\left\{x_{1,k}(n_1+1)\right\}$ is relatively compact in X, thus there exists subsequence $\{x_{2,k}^{*}\}\subset \{x_{1,k}^{*}\}$ such that $\left\{x_{2,k}(n_1+1)\right\}$ is convergent in X. Note that $\left\{x_{2,k}\left(n_1\right)\right\}$ is also convergent in X. Repeating like this, we can obtain a subsequence $\{x_{k_j}^{*}\}$ such that $\left\{x_{k_j}\left(m\right)\right\}$$(n_1\le m\le n_2)$ is convergent in X, that is $\{x_{k_j}^{*}\}$ is relatively compact in $D(n_1,n_2)$. Then $\left\{x_k^{*}\right\}$ is relatively compact in $D(n_1,n_2)$. Since $\left\{x_k^{*}\right\}$ is arbitrary, then $(D(n_1,n_2),\rho )$ is relatively compact.

From the condition (1), it follows that for any $\epsilon >o$, there exists $n_0\in {\mathbb{Z}}^{+}$ such that

$$\parallel x\left(n\right)\parallel <\epsilon ,\mathrm{for}\mathrm{all}\left|n\right|>n_0,x\in D.$$

As $(D(-n_0,n_0),\rho )$ is relatively compact, then it has a finite $\epsilon$-net A. Let

$$\tilde{A}=\left\{(\dots ,0,x(-n_0),\dots ,x\left(n_0\right),0,\dots |(x(-n_0),\dots ,x\left(n_0\right))\in A\right\}.$$

For any $x\in D$, assume that $x^{*}\in D(-n_0,n_0)$ is a corresponding point of x and taking $y^{*}\in A$ such that $\rho (x^{*},y^{*})<\epsilon$. Assume that ${\tilde{y}}^{*}$ is a corresponding point of $y^{*}$ in $\tilde{A}$, then

$$\parallel x-{\tilde{y}}^{*}{\parallel }_d<\epsilon .$$

This implies that $\tilde{A}$ is a finite $\epsilon$-net of D, thus D is totally bounded; this together with $C_0(\mathbb{Z},X)$ is a complete space and yields that D is is relatively compact. □

Lemma 20.

[47,48]. Let S be a subset of X and Let ${\Pi }_1,{\Pi }_2:S\to X$ be maps satisfying

$${\Pi }_{1}x+{\Pi }_{2}y\in S,\forall x,y\in S,$$

where $S\subset X$ is a bounded closed and convex collection. If ${\Pi }_1$ is completely continuous and ${\Pi }_2$ is a contraction, then the equation ${\Pi }_{1}x+{\Pi }_{2}x=x$ has a solution on S.

3. Asymptotically Almost-Periodic Mild Solutions

In this section, we will state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of the nonhomogeneous nonlinear difference equations of fractional order given by

$$\begin{array}{c}\hfill {▵}^{\alpha }x\left(n\right)=Ax(n+1)+F(n,x\left(n\right)),n\in \mathbb{Z},\end{array}$$

(4)

where $\alpha \in (0,1)$, the operator A generates a $C_0$-semigroup on X, ${▵}^{\alpha }$ denotes the Weyl-like fractional difference operator, $F(n,x):\mathbb{Z}\times X\to X$ is a function to be specified later.

In [33], Abadias and Lizama obtained the following remarkable result.

Lemma 21.

(Abadias and Lizama [33]) Assume A generates a $C_0$-semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on X, which is exponentially stable, i.e.,

$$\begin{array}{c}\hfill {\parallel T\left(t\right)\parallel }_{\mathbb{L}\left(X\right)}\le M{e}^{-\delta t}for\forall t>0andsomeconstantsM>0,\delta >0.\end{array}$$

(5)

Then there exists a discrete α-resolvent family ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}$ generated by A, which is given by

$$S_{\alpha }\left(n\right)x:={\int }_0^{\infty }{\int }_0^{\infty }{e}^{-t}\frac{t^n}{n!}{f}_{s,\alpha }\left(t\right)T\left(s\right)xdsdt,n\in \mathbb{N},x\in X,$$

where $f_{s,\alpha }\left(t\right)$ is a function given by

$$f_{s,\alpha }\left(t\right)=\frac{1}{2\pi i}{\int }_{\sigma -i\infty }^{\sigma +i\infty }{e}^{z\lambda -t{z}^{\alpha }}dz,\sigma >0,s>0,t\ge 0,0<\alpha <1.$$

Moreover, ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}$ satisfies

$$\parallel S_{\alpha }{\left(n\right)\parallel }_{\mathbb{L}\left(X\right)}\le M{\int }_0^{\infty }{e}^{-t}\frac{t^n}{n!}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt,$$

and furthermore ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}$ is summable with

$$\parallel S_{\alpha }{\parallel }_1=\sum _{k=0}^{\infty }\parallel S_{\alpha }\left(n\right)\parallel \le \frac{1}{\delta }.$$

The following definition of mild solutions to Equation (4), which is given in [33], is essential for us.

Definition 12.

(Abadias and Lizama [33]) Assume ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{L}\left(X\right)$ is a discrete α-resolvent family generated by A. A sequence $x\in s(\mathbb{Z},X)$ is said to be a mild solution of Equation (4) if for each $n\in \mathbb{Z}$, $m\to S_{\alpha }\left(m\right)F(n-1-m,x(n-1-m))$ is summable on $\mathbb{N}$ and x satisfies

$$\begin{array}{c}\hfill x\left(n\right)=\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F(k,x\left(k\right)),n\in \mathbb{Z}.\end{array}$$

(6)

The following auxiliary result plays a key role in the proofs of our main results.

Lemma 22.

Let ${\left\{S\left(n\right)\right\}}_{n\in \mathbb{N}}$ be summable. Given sequence $F\left(n\right)\in AP(\mathbb{Z},X)$, $G\left(n\right)\in AA{P}_0(\mathbb{Z},X)$. Let

$$\Phi \left(n\right):=\sum _{k=-\infty }^{n-1}S(n-1-k)F\left(k\right),\Psi \left(n\right):=\sum _{k=-\infty }^{n-1}S(n-1-k)G\left(k\right),n\in \mathbb{Z}.$$

Then $\Phi \left(n\right)\in AP(\mathbb{Z},X)$ and $\Psi \left(n\right)\in AA{P}_0(\mathbb{Z},X)$.

Proof. 

From $F\left(n\right)\in AP(\mathbb{Z},X)$, combining with Lemma 3, it follows that $F\left(n\right)$ is bounded on X. In addition, note that

$$\begin{array}{cc}\hfill \parallel \Phi \left(n\right)\parallel \le & \sum _{k=-\infty }^{n-1}\parallel S(n-1-k)\parallel \parallel F\left(k\right)\parallel =\sum _{k=0}^{+\infty }\parallel S\left(k\right)\parallel \parallel F(n-1-k)\parallel \hfill \\ \hfill \le & {\parallel F\parallel }_d\sum _{k=0}^{+\infty }\parallel S\left(k\right)\parallel \le {\parallel F\parallel }_d{\parallel S\parallel }_1<+\infty ,\hfill \end{array}$$

hence $\Phi \left(n\right)$ is well defined.

Similarly, $\Psi \left(n\right)$ is also well defined. □

From Lemma 5, together with

$$\begin{array}{c}\hfill \Phi \left(n\right)=\sum _{k=-\infty }^{n-1}\parallel S(n-1-k)F\left(k\right)=\sum _{k=0}^{+\infty }S\left(k\right)F(n-1-k),\end{array}$$

it follows that $\Phi \left(n\right)\in AP(\mathbb{Z},X)$.

As $G\left(n\right)\in AA{P}_0(\mathbb{Z},X)$, then for $\forall \epsilon >0$, $\exists N=N\left(\epsilon \right)>0$ s.t.

$$\parallel G\left(k\right)\parallel \le \epsilon ,\forall k>N.$$

Thus

$$\begin{array}{cc}\hfill \parallel \Psi \left(n\right)\parallel =& ∥\sum _{k=-\infty }^{n-1}S(n-1-k)G\left(k\right)∥\le \sum _{k=-\infty }^{n-1}\parallel S(n-1-k)\parallel \parallel G\left(k\right)\parallel \hfill \\ \hfill =& \sum _{k=0}^{+\infty }\parallel S\left(k\right)\parallel \parallel G(n-1-k)\parallel \le \epsilon \sum _{k=0}^{+\infty }\parallel S\left(k\right)\parallel \le {\epsilon \parallel S\parallel }_1,\forall n>N+k\hfill \end{array}$$

which implies $\Psi \left(n\right)\in AA{P}_0(\mathbb{Z},X)$.

Now, we state and prove our main results. The following assumptions are required:

($H_1$) The $C_0$-semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ generated by A on X is exponentially stable.

($H_2$) $F(n,x)=F_1(n,x)+F_2(n,x)\in AAP(\mathbb{Z}\times X,X)$ with

$$F_1(n,x)\in AP(\mathbb{Z}\times X,X),F_2(n,x)\in C_0(\mathbb{Z}\times X,X)$$

and it is bounded on $\mathbb{Z}\times X$. Moreover

$$\begin{array}{c}\hfill \parallel F_1(n,x)-F_1(n,y)\parallel \le L\parallel x-y\parallel ,\mathrm{for}\forall n\in \mathbb{Z},x,y\in X\mathrm{and}\mathrm{some}\mathrm{constant}L.\end{array}$$

(7)

($H_3$) There is a function $\beta \left(n\right)\in C_0(\mathbb{Z},{\mathbb{R}}^{+})$ satisfying

$${\mathrm{Y}}_n\left(t\right)=\sum _{k=0}^{+\infty }\beta (n-1-k)e^{-t}\frac{t^k}{k!}\in C_0(\mathbb{Z},{\mathbb{R}}^{+})\mathrm{uniformly}\mathrm{for}t\in {\mathbb{R}}^{+},$$

and a nondecreasing function $\Phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying

$$\begin{array}{c}\hfill \parallel F_2(n,x)\parallel \le \beta \left(n\right)\Phi \left(r\right)\mathrm{and}\underset{r\to +\infty }{lim\; inf}\frac{\Phi \left(r\right)}{r}={\rho }_1,\mathrm{for}\forall n\in \mathbb{Z},x\in X\mathrm{with}\parallel x\parallel \le r.\end{array}$$

(8)

($H_4$) For all $n_1,n_2\in \mathbb{Z}$, $n_1\le n_2$ and $\eta >0$, the collection $\{F_2(n,x):n_1\le n\le n_2,\parallel x\parallel \le \eta \}$ is relatively compact in X.

Remark 2.

Note that in $\left(H_2\right)$, $F(n,x)$ does not need to meet the Lipschitz assumption with respect to x. Such classes of functions $F(n,x)$ are more complicated than those with the Lipschitz assumption (see Remark 3).

Remark 3.

In Lemma 17, we prove that a discrete asymptotically almost-periodic function $F(n,x):\mathbb{Z}\times X\to X$ is bounded only on $\mathbb{Z}\times \Omega$ for any compact set Ω in X, so the condition boundedness in $\left(H_2\right)$ does not conflict with the condition $F(n,x)\in AAP(\mathbb{Z}\times X,X)$.

Firstly, we give some important Lemmas.

Lemma 23.

Let $G(n,f)=G_1(n,f)+G_2(n,f)\in AAP(\mathbb{Z}\times X,X)$ with

$$G_1(n,f)\in AP(\mathbb{Z}\times X,X),G_2(n,f)\in C_0(\mathbb{Z}\times X,X).$$

Then we have

$$\begin{array}{c}\hfill \underset{n\in \mathbb{Z}}{sup}\parallel G_1(n,f)-G_1(n,g)\parallel \le \underset{n\in \mathbb{Z}}{sup}\parallel G(n,f)-G(n,g)\parallel ,f,g\in X.\end{array}$$

(9)

Proof. 

To obtain this result, it suffices to show

$$\{G_1(n,f)-G_1(n,g):n\in \mathbb{Z}\}\subset \overline{\left\{G\right(n,f)-G(n,g):n\in \mathbb{Z}\}},f,g\in X.$$

If this is not true, then there are $n_0\in \mathbb{Z}$ and $\epsilon >0$ s.t.

$$\parallel G{(}_1(n_0,f)-G_1(n_0,g))-(G(n,f)-G(n,g))\parallel \ge 3\epsilon ,\forall n\in \mathbb{Z}\mathrm{and}\mathrm{fixed}x,y\in X.$$

As $G_2(n,f)\in C_0(\mathbb{Z}\times X,X)$, then

$$\underset{n\to +\infty }{lim}\parallel F_2(n,x)-F_2(n,y)\parallel =0,$$

this yields that $\exists N>0$ s.t.

$$\begin{array}{c}\hfill \parallel G_2(n,f)-G_2(n,g)\parallel <\epsilon ,\forall n\ge N.\end{array}$$

(10)

From $F_1(n,x)\in AP(\mathbb{Z}\times X,X)$, it follows that for the above $\epsilon$ and every compact $\Omega \subset X$, there is an integer $N=N\left(\epsilon \right)>0$, such that there exists at least one integer $k\in \Delta$, where $\Delta$ is a any collection consisting of N consecutive integers, satisfying

$$\parallel G_1(n_0+k,f)-G_1(n_0,f)\parallel <\epsilon ,\parallel G_1(n_0+k,g)-G_1(n_0,g)\parallel <\epsilon ,$$

Then

$$\begin{array}{cc}& \parallel G_2(n_0+k,f)-G_2(n_0+k,g)\parallel \hfill \\ \hfill \ge & \parallel G(n_0+k,f)-G(n_0+k,g)-G_1(n_0,f)+G_1(n_0,g)\parallel -\parallel G_1(n_0+k,f)-G_1(n_0,f)\parallel \hfill \\ & -\parallel G_1(n_0+k,g)-G_1(n_0,g)\parallel \ge \epsilon ,\hfill \end{array}$$

which contradicts Equation (10). □

Remark 4.

From Equation (9) it follows that if $G(n,x)$ meets the Lipschitz assumption of Equation (2), then $G_1(n,x)$ satisfies Equation (7). Then Lipschitz assumptions become a special case of our condition. Note that in [19,33], a Lipschitz assumption (Equation (2)) or a locally Lipschitz assumption (Equation (3)) for $F(n,x)$ of Equation (4) is needed. Thus our condition in ($H_2$) is weaker than those of [19,33] and our results generalize some related conclusions of [19,33].

Let $\beta \left(n\right)$ be the sequence in ($H_3$). Set

$$\begin{array}{cc}\hfill \sigma \left(n\right):=& M\sum _{k=-\infty }^{n-1}{\int }_0^{\infty }\beta \left(k\right)e^{-t}\frac{t^{n-1-k}}{(n-1-k)!}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt\hfill \\ \hfill =& M\sum _{k=0}^{+\infty }{\int }_0^{\infty }\beta (n-1-k)e^{-t}\frac{t^k}{k!}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt\hfill \\ \hfill =& M{\int }_0^{\infty }{\mathrm{Y}}_n\left(t\right)t^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt,n\in \mathbb{Z}.\hfill \end{array}$$

Lemma 24.

$\sigma \left(n\right)\in C_0(\mathbb{Z},{\mathbb{R}}^{+})$.

Proof. 

From ${\mathrm{Y}}_n\left(t\right)\in C_0(\mathbb{Z},{\mathbb{R}}^{+})$ uniformly for $t\in {\mathbb{R}}^{+}$, it follows that for any $\epsilon >0$, one can choose a $N_1>0$ such that $|{\mathrm{Y}}_n\left(t\right)|<\epsilon$, for all $\left|n\right|>N_1$. This combining with Lemma 18 and $E_{\alpha }\left(0\right)=1$, implies that

$$\begin{array}{cc}\hfill \sigma \left(n\right)=& M{\int }_0^{\infty }{\mathrm{Y}}_n\left(t\right)t^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt\le M\epsilon {\int }_0^{\infty }{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt\hfill \\ \hfill =& M\epsilon E_{\alpha }(-\delta t^{\alpha }){|}_0^{+\infty }=M\epsilon \mathrm{for}\forall \left|n\right|>N_1,\hfill \end{array}$$

which implies $\underset{\left|n\right|\to +\infty }{lim}\sigma \left(n\right)=0$. □

Firstly, we state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of Equation (4) when the almost-periodic component $F_1$ of F satisfies Equation (7).

Theorem 1.

Let $\left(H_1\right)$–$\left(H_4\right)$ hold. Put ${\rho }_2:=\underset{n\in \mathbb{Z}}{sup}\sigma \left(n\right)$. Then there is a mild solution of Equation (4) whenever

$$\begin{array}{c}\hfill L\parallel S_{\alpha }{\parallel }_1+{\rho }_1{\rho }_2<1.\end{array}$$

(11)

Moreover the mild solution is asymptotically almost-periodic.

Proof. 

We divide into five steps to complete the proof. □

Step 1. Let $\Lambda$ be a mapping defined on $AP(\mathbb{Z},X)$ given by

$$\begin{array}{cc}\hfill (\Lambda v)\left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_1(k,v\left(k\right)),n\in \mathbb{Z}.\hfill \end{array}$$

There is a unique fixed point $v\left(n\right)\in AP(\mathbb{Z},X)$ of $\Lambda$.

Firstly, according to $\left(H_1\right)$, there is a discrete $\alpha$-resolvent family ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{L}\left(X\right)$ generated by A, which is summable (by Lemma 21) and, combining with the discrete function $k\to F_1(k,v\left(k\right))$, is bounded on $\mathbb{Z}$, which follows from Lemma 3, and one has

$$\begin{array}{cc}\hfill \parallel (\Lambda v)\left(n\right)\parallel \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel F(k,v\left(k\right))\parallel =\sum _{k=0}^{+\infty }\parallel S_{\alpha }\left(k\right)\parallel \parallel F(n-1-k,v(n-1-k))\parallel \hfill \\ \hfill \le & |\parallel F|\parallel \sum _{k=0}^{+\infty }\parallel S_{\alpha }\left(k\right)\parallel =|\parallel F|\parallel \parallel S_{\alpha }{\parallel }_1<+\infty \mathrm{for}\mathrm{all}n\in \mathbb{Z},\hfill \end{array}$$

which yields that $(\Lambda v)\left(n\right)$ exists. Furthermore from $F_1(n,x)\in AP(\mathbb{Z}\times X,X)$ satisfying Equation (5), combined with Lemmas 3 and 11, it follows that

$$F_1(·,v(·))\in AP(\mathbb{Z},X)\forall v(·)\in AP(\mathbb{Z},X).$$

This, combined with Lemma 2, yields that $\Lambda$ is well defined.

Then, we show $\Lambda$ is continuous.

Let $v_j\left(n\right),v\left(n\right)\in AP(\mathbb{Z},X)$ satisfy $v_j\left(n\right)\to v\left(n\right)$ as $j\to \infty$, then

$$\begin{array}{cc}\hfill \parallel [\Lambda v_j]\left(n\right)-[\Lambda v]\left(n\right)\parallel =& \parallel \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)\left[F_1(k,v_j\left(k\right))-F_1(k,v\left(k\right))\right]\parallel \hfill \\ \hfill \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel F_1(k,v_j\left(k\right))-F_1(k,v\left(k\right))\parallel \hfill \\ \hfill \le & L\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel v_j\left(k\right)-v\left(k\right)\parallel \hfill \\ \hfill \le & L\parallel v_j{-v\parallel }_d\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill =& L\parallel S_{\alpha }{\parallel }_1{\parallel v_j-v\parallel }_d.\hfill \end{array}$$

Therefore, as $j\to \infty$, $\Lambda v_j\to \Lambda v$, which yields that $\Lambda$ is continuous.

Finally, we show there is a unique fixed point $v\left(n\right)\in AP(\mathbb{Z},X)$ of $\Lambda$.

Let $v_1\left(n\right),v_2\left(n\right)\in AP(\mathbb{Z},X)$, similar to the proof of the continuity of $\Lambda$, and we have

$$\begin{array}{c}\hfill \parallel [\Lambda v_1]\left(n\right)-[\Lambda v_2]\left(n\right)\parallel \le L\parallel S_{\alpha }{\parallel }_1{\parallel v_1-v_2\parallel }_d,\end{array}$$

which implies

$$\begin{array}{c}\hfill \parallel \Lambda v_1-\Lambda v_2{]\parallel }_d\le L\parallel S_{\alpha }{\parallel }_1{\parallel v_1-v_2\parallel }_{\infty }.\end{array}$$

From Equation (11), it follows that $\Lambda$ is a contraction on $AP(\mathbb{Z},X)$. Thus, there is a unique fixed point $v\left(n\right)\in AP(\mathbb{Z},X)$ of $\Lambda$.

Step 2. Set ${\Omega }_r:=\{\omega \left(n\right)\in C_0{(\mathbb{Z},X):\parallel \omega \parallel }_d\le r\}$. For the above $v\left(n\right)$, define $\mathrm{\Gamma }:={\mathrm{\Gamma }}^1+{\mathrm{\Gamma }}^2$ on $C_0(\mathbb{Z},X)$ as

$$\begin{array}{cc}\hfill \left({\mathrm{\Gamma }}^1\omega \right)\left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)[F_1(k,v\left(k\right)+\omega \left(k\right))-F_1(k,v\left(k\right))],n\in \mathbb{Z},\hfill \\ \hfill \left({\mathrm{\Gamma }}^2\omega \right)\left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+\omega \left(k\right)),n\in \mathbb{Z}.\hfill \end{array}$$

(12)

There is a constant $k_0$ s.t. $\mathrm{\Gamma }$ maps ${\Omega }_{k_0}$ into itself.

Firstly, according to Equation (7), one has

$$\begin{array}{c}\hfill \parallel F_1(k,v\left(k\right)+\omega \left(k\right))-F_1(k,v\left(k\right))\parallel \le L\parallel \omega \left(k\right)\parallel ,\forall k\in \mathbb{Z},\omega \left(k\right)\in X,\end{array}$$

which yields

$$F_1(·,v(·)+\omega (·))-F_1(·,v(·))\in C_0(\mathbb{Z},X),\forall \omega (·)\in C_0(\mathbb{Z},X).$$

According to Equation (8), one has

$$\begin{array}{c}\hfill \parallel F_2(k,v\left(k\right)+\omega \left(k\right))\parallel \le \beta \left(k\right)\Phi \left(r+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right),\forall k\in \mathbb{Z},\omega \left(k\right)\in X\mathrm{with}\parallel \omega \left(k\right)\parallel \le r.\end{array}$$

Then

$$F_2(·,v(·)+\omega (·))\in C_0(\mathbb{Z},X)\mathrm{as}\beta (·)\in C_0(\mathbb{Z},{\mathbb{R}}^{+}).$$

Thus $\mathrm{\Gamma }$ is well-defined and maps $C_0(\mathbb{Z},X)$ into itself, which follows from Lemma 22.

From Equations (8) and (11), it easily follows that there is a constant $k_0>0$ s.t.

$$\begin{array}{c}\hfill L\parallel S_{\alpha }{\parallel }_1{k}_0+{\rho }_2\Phi \left(k_0+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\le k_0.\end{array}$$

Then for any $n\in \mathbb{Z}$ and ${\omega }_1\left(n\right),{\omega }_2\left(n\right)\in {\Omega }_{k_0}$, one has

$$\begin{array}{cc}& \parallel \left({\mathrm{\Gamma }}^1{\omega }_1\right)\left(n\right)+\left({\mathrm{\Gamma }}^2{\omega }_2\right)\left(n\right)\parallel \hfill \\ \hfill \le & ∥\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)[F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))]∥\hfill \\ & +∥\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+{\omega }_2\left(k\right))∥\hfill \\ \hfill \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)[F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))]\parallel \hfill \\ & +\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)F_2(k,v\left(k\right)+{\omega }_2\left(k\right))\parallel \hfill \\ \hfill \le & L\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k){\omega }_1\left(k\right)\parallel +\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\sum _{k=-\infty }^{n-1}\parallel \beta \left(k\right)S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill \le & L\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k){\omega }_1\left(k\right)\parallel \hfill \\ & +\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)M\sum _{k=-\infty }^{n-1}{\int }_0^{\infty }\beta \left(k\right)e^{-t}\frac{t^{n-1-k}}{(n-1-k)!}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & L\parallel {\omega }_1{\parallel }_d\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel +\sigma \left(n\right)\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\hfill \\ \hfill \le & L\parallel S_{\alpha }{\parallel }_1{\parallel {\omega }_1\parallel }_d+{\rho }_2\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\hfill \\ \hfill \le & L\parallel S_{\alpha }{\parallel }_1{k}_0+{\rho }_2\Phi \left(k_0+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\le k_0.\hfill \end{array}$$

This indicates that

$$\left({\mathrm{\Gamma }}^1{\omega }_1\right)\left(n\right)+\left({\mathrm{\Gamma }}^2{\omega }_2\right)\left(n\right)\in {\Omega }_{k_0}.$$

Thus $\mathrm{\Gamma }$ maps ${\Omega }_{k_0}$ into itself.

Step 3.${\mathrm{\Gamma }}^1$ is a contraction on ${\Omega }_{k_0}$.

Let ${\omega }_1\left(n\right),{\omega }_2\left(n\right)\in {\Omega }_{k_0}$, according to Equation (7), one has

$$\begin{array}{c}\hfill \parallel [F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))]-[F_1(k,v\left(k\right)+{\omega }_2\left(k\right))-F_1(k,v\left(k\right))]\parallel \le L\parallel {\omega }_1\left(k\right)-{\omega }_2\left(k\right)\parallel .\end{array}$$

Thus

$$\begin{array}{cc}\hfill \parallel \left({\mathrm{\Gamma }}^1{\omega }_1\right)\left(n\right)-\left({\mathrm{\Gamma }}^1{\omega }_2\right)\left(n\right)\parallel =& \parallel \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)[\left(F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))\right)\hfill \\ & -\left(F_1(k,v\left(k\right)+{\omega }_2\left(k\right))-F_1(k,v\left(k\right))\right)]\parallel \hfill \\ \hfill \le & L\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)[{\omega }_1\left(k\right)-{\omega }_2\left(k\right)]\parallel \hfill \\ \hfill \le & L\parallel {\omega }_1-{\omega }_2{\parallel }_d\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill =& L\parallel S_{\alpha }{\parallel }_1{\parallel {\omega }_1-{\omega }_2\parallel }_d,\hfill \end{array}$$

which indicates

$$\begin{array}{c}\hfill \parallel {\mathrm{\Gamma }}^1{\omega }_1-{\mathrm{\Gamma }}^1{\omega }_2{\parallel }_d\le L\parallel S_{\alpha }{\parallel }_1{\parallel {\omega }_1-{\omega }_2\parallel }_d.\end{array}$$

Thus ${\mathrm{\Gamma }}^1$ is a contraction on ${\Omega }_{k_0}$ by Equation (11).

Step 4.${\mathrm{\Gamma }}^2$ is completely continuous on ${\Omega }_{k_0}$.

Firstly, ${\mathrm{\Gamma }}^2$ is continuous on ${\Omega }_{k_0}$.

In fact, $\forall \epsilon >0$, let ${\left\{{\omega }_j\right\}}_{j=1}^{+\infty }\subset {\Omega }_{k_0}$ with ${\omega }_j\to {\omega }_0$ as $j\to +\infty$. From $\sigma \left(n\right)\in C_0(\mathbb{Z},{\mathbb{R}}^{+})$ (from Lemma 24), one may choose an $n_1>0$ big enough s.t.

$$\Phi \left(k_0+{\parallel v\parallel }_d\right)\sigma \left(n\right)<\frac{\epsilon }{3},\forall n\ge n_1.$$

In addition, $\left(H_2\right)$ implies that

$$F_2(k,v\left(k\right)+{\omega }_j\left(k\right))\to F_2(k,v\left(k\right)+{\omega }_0\left(k\right))\mathrm{for}\mathrm{all}k\in (-\infty ,n_1]\mathrm{as}j\to +\infty ,$$

and

$$\parallel F_2(·,v(·)+{\omega }_j(·))-F_2(·,v(·)+{\omega }_0(·))\parallel \le 2\Phi \left(k_0+{\parallel v\parallel }_d\right)\beta (·)\in L^1(-\infty ,n_1].$$

Hence, according to the Lebesgue dominated convergence theorem, there exists an $N>0$ such that for any $j\ge N$,

$$\begin{array}{c}\hfill \sum _{k=-\infty }^{n_1-1}\parallel S_{\alpha }(n-1-k)[F_2(k,v\left(k\right)+{\omega }_j\left(k\right))-F_2(k,v\left(k\right)+{\omega }_0\left(k\right))]\parallel \le \frac{\epsilon }{3}.\end{array}$$

Thus when $j\ge N$,

$$\begin{array}{cc}& \parallel \left({\mathrm{\Gamma }}^2{\omega }_j\right)\left(n\right)-\left({\mathrm{\Gamma }}^2{\omega }_0\right)\left(n\right)\parallel \hfill \\ \hfill =& ∥\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+{\omega }_j\left(k\right))-\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+{\omega }_0\left(k\right))∥\hfill \\ \hfill \le & \sum _{k=-\infty }^{n_1-1}\parallel S_{\alpha }(n-1-k)[F_2(k,v\left(k\right)+{\omega }_j\left(k\right))-F_2(k,v\left(k\right)+{\omega }_0\left(k\right))]\parallel \hfill \\ & +\sum _{n_1}^{max\{n,n_1\}}\parallel S_{\alpha }(n-1-k)[F_2(k,v\left(k\right)+{\omega }_j\left(k\right))-F_2(k,v\left(k\right)+{\omega }_0\left(k\right))]\parallel \hfill \\ \hfill \le & \sum _{k=-\infty }^{n_1-1}\parallel S_{\alpha }(n-1-k)[F_2(k,v\left(k\right)+{\omega }_j\left(k\right))-F_2(k,v\left(k\right)+{\omega }_0\left(k\right))]\parallel \hfill \\ & +2\Phi \left(k_0+{\parallel v\parallel }_d\right)\sum _{n_1}^{max\{n,n_1\}}\parallel \beta \left(k\right)S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill \le & \sum _{k=-\infty }^{n_1-1}\parallel S_{\alpha }(n-1-k)[F_2(k,v\left(k\right)+{\omega }_j\left(k\right))-F_2(k,v\left(k\right)+{\omega }_0\left(k\right))]\parallel \hfill \\ & +2\Phi \left(k_0+{\parallel v\parallel }_d\right)\sigma (n-1)\hfill \\ \hfill \le & \frac{\epsilon }{3}+\frac{2\epsilon }{3}=\epsilon .\hfill \end{array}$$

Accordingly, ${\mathrm{\Gamma }}^2$ is continuous on ${\Omega }_{k_0}$.

Then we consider the compactness of ${\mathrm{\Gamma }}^2$.

Let $\Delta ={\mathrm{\Gamma }}^2\left({\Omega }_{k_0}\right)$ and $z\left(n\right)={\mathrm{\Gamma }}^2\left(u\left(n\right)\right)$ for $u\left(n\right)\in {\Omega }_{k_0}$. First, for all $\omega \left(n\right)\in {\Omega }_{k_0}$ and $n\in \mathbb{Z}$,

$$\begin{array}{cc}\hfill \parallel \left({\mathrm{\Gamma }}^2\omega \right)\left(n\right)\parallel =& \parallel \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(s,v\left(s\right)+\omega \left(s\right))ds\parallel \hfill \\ \hfill \le & \Phi \left(k_0+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\sum _{k=-\infty }^{n-1}\parallel \beta \left(k\right)S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill =& M\sigma \left(n\right)\Phi \left(k_0+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right),\hfill \end{array}$$

in view of $\sigma \left(n\right)\in C_0(\mathbb{Z},{\mathbb{R}}^{+})$, which follows from Lemma 24, one has

$$\underset{\left|n\right|\to +\infty }{lim}\left({\mathrm{\Gamma }}^2\omega \right)\left(n\right)=0\mathrm{uniformly}\mathrm{for}\omega \left(n\right)\in {\Omega }_{k_0}.$$

From

$$\begin{array}{cc}\hfill \left({\mathrm{\Gamma }}^2\omega \right)\left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+\omega \left(k\right))\hfill \\ \hfill =& \sum _{k=0}^{+\infty }{S}_{\alpha }\left(k\right)F_2(n-1-k,v(n-1-k)+\omega (n-1-k)),\hfill \end{array}$$

it follows that for given ${\epsilon }_0>0$, one can choose an $n_2>0$ such that

$$\begin{array}{c}\hfill \parallel \sum _{k=n_2}^{+\infty }{S}_{\alpha }\left(k\right)F_2(n-1-k,v(n-1-k)+\omega (n-1-k))\parallel <{\epsilon }_0.\end{array}$$

Thus we get

$$\begin{array}{c}\hfill z\left(n\right)\in n_2\overline{c\left(\right\{S_{\alpha }\left(k\right)F_2(\lambda ,v\left(\lambda \right)+\omega \left(\lambda \right)):0\le \tau \le n_2,n-n_2\le \lambda \le n_2{,\parallel \omega \parallel }_d\le r\left\}\right)}+B_{{\epsilon }_0}\left(X\right),\end{array}$$

where $c\left(K\right)$ is the convex hull of K. Using $\left(H_4\right)$, the collection

$$K=\{S_{\alpha }\left(k\right)F_2(\lambda ,v\left(\lambda \right)+\omega \left(\lambda \right)):0\le \tau \le n_2,n-n_2\le \lambda \le n_2,\parallel \omega {\parallel }_d\le r\}$$

is relatively compact, combing with $\Delta \subset n_2\overline{c\left(K\right)}+B_{{\epsilon }_0}\left(X\right)$, one obtains that $\Delta$ is a relatively compact subset of X.

Thus ${\mathrm{\Gamma }}^2$ is compact by Lemma 19, which further implies ${\mathrm{\Gamma }}^2$ is completely continuous on ${\Omega }_{k_0}$.

Step 5. Show Equation (4) has at least one mild solution, which is asymptotically almost periodic.

Firstly, from the results of step 4, combining with the results of steps 2 and 3 as well as the Krasnoselskii’s fixed point theorem (see Lemma 20), it follows that $\mathrm{\Gamma }$ has at least one fixed point $\omega \left(n\right)\in {\Omega }_{k_0}$; furthermore, $\omega \left(n\right)\in C_0(\mathbb{Z},X)$. Then, consider the following coupled system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill v\left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_1(k,v\left(k\right)),n\in \mathbb{Z},\hfill \\ \hfill \omega \left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)[F_1(k,v\left(k\right)+\omega \left(k\right))-F_1(k,v\left(k\right))]\hfill \\ & +\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+\omega \left(k\right)),n\in \mathbb{Z}.\hfill \end{array}\right.\end{array}$$

(13)

Combing the unique fixed point $v\left(n\right)\in AP(\mathbb{Z},X)$ in step 1 with the fixed point $\omega \left(n\right)\in C_0(\mathbb{Z},X)$ in Step 4, it follows that $(v\left(n\right),\omega \left(n\right))\in AP(\mathbb{Z},X)\times C_0(\mathbb{Z},X)$ is a solution to System (13); then $x\left(n\right):=v\left(n\right)+\omega \left(n\right)\in AAP(\mathbb{Z},X)$ and it is a solution to the equation

$$\begin{array}{cc}\hfill x\left(t\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F(k,x\left(k\right)),n\in \mathbb{Z},\hfill \end{array}$$

that is $x\left(n\right)$ is a mild solution to Equation (4), which is asymptotically almost-periodic

In the following, we state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of Equation (4) when the almost-periodic component $F_1$ satisfies a locally Lipschitz assumption

($H_2^{{}^{\prime }}$) $F(n,x)=F_1(n,x)+F_2(n,x)\in AAP(\mathbb{Z}\times X,X)$ with

$$F_1(n,x)\in AP(\mathbb{Z}\times X,X),F_2(n,x)\in C_0(\mathbb{Z}\times X,X)$$

and it is bounded on $\mathbb{Z}\times X$. Moreover, for each $r>0$,

$$\begin{array}{c}\hfill \parallel F_1(n,x)-F_1(n,y)\parallel \le L\left(r\right)\parallel x-y\parallel ,\forall n\in \mathbb{Z},x,y\in X\mathrm{with}\parallel x\parallel \le r,\parallel y\parallel \le r,\end{array}$$

(14)

where $L:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a nondecreasing function.

Theorem 2.

Assume that $\left(H_1\right)$, $\left(H_2^{{}^{\prime }}\right)$, $\left(H_3\right)$ and $\left(H_4\right)$ hold and if there exist $r_0>0$, $r_1>0$ such that

$$\parallel S_{\alpha }{\parallel }_1\left(L(r_0+r_1)+\frac{1}{min\{r_0,r_1\}}\underset{k\in \mathbb{Z}}{sup}\parallel F_1(k,0)\parallel \right)<1.$$

Put ${\rho }_2:=\underset{n\in \mathbb{Z}}{sup}\sigma \left(n\right)$. Then there is a mild solution of Equation (4) whenever

$$\begin{array}{c}\hfill L(r_0+r_1){\parallel S_{\alpha }\parallel }_1+{\rho }_1{\rho }_2<1.\end{array}$$

(15)

Moreover the mild solution is asymptotically almost-periodic.

Proof. 

We also divide into five steps to complete the proof. □

Step 1. Define a mapping

$$\Lambda :B_{r_0}\left(AAP(\mathbb{Z},X)\right)\to B_{r_0}\left(AAP(\mathbb{Z},X)\right)$$

by

$$\begin{array}{cc}\hfill (\Lambda v)\left(n\right)=& \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_1(k,v\left(k\right)),n\in \mathbb{Z}.\hfill \end{array}$$

$\Lambda$ has a unique fixed point $v\left(n\right)\in B_{r_0}\left(AAP(\mathbb{Z},X)\right)$.

Firstly, according to $\left(H_1\right)$, there is a discrete $\alpha$-resolvent family ${\left\{S_{\alpha }\left(n\right)\right\}}_{n\in \mathbb{N}}\subset \mathbb{L}\left(X\right)$ generated by A, which is summable (by Lemma 21) and, combined with the discrete function $k\to F_1(k,v\left(k\right))$, is bounded on $\mathbb{Z}$, which follows from Lemma 3, and one has

$$\begin{array}{cc}\hfill \parallel (\Lambda v)\left(n\right)\parallel \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel F_1(k,v\left(k\right))\parallel =\sum _{k=0}^{+\infty }\parallel S_{\alpha }\left(k\right)\parallel \parallel F_1(n-1-k,v(n-1-k))\parallel \hfill \\ \hfill \le & |\parallel F_1|\parallel \sum _{k=0}^{+\infty }\parallel S_{\alpha }\left(k\right)\parallel =|\parallel F_1|\parallel \parallel S_{\alpha }{\parallel }_1<+\infty \mathrm{for}\mathrm{all}n\in \mathbb{Z},\hfill \end{array}$$

which yields that $(\Lambda v)\left(n\right)$ exists. Furthermore from $F_1(n,x)\in AP(\mathbb{Z}\times X,X)$ satisfying Equation (14), combined with Lemmas 3 and 12, it follows that

$$F_1(·,v(·))\in AP(\mathbb{Z},X)\mathrm{for}\mathrm{every}v(·)\in AP(\mathbb{Z},X).$$

Furthermore it follows from Lemma 22 that $(\Lambda v)\left(n\right)\in AP(\mathbb{Z},X)$. Let $v\in B_{r_0}\left(AP(\mathbb{Z},X)\right)$, and one has

$$\begin{array}{cc}\hfill \parallel (\Lambda v)\left(n\right)\parallel =& ∥\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_1(k,v\left(k\right))∥\hfill \\ \hfill \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel F_1(k,v\left(k\right))-F_1(k,0)\parallel +\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel F_1(k,0)\parallel \hfill \\ \hfill \le & L\left(r_0\right)\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel v\left(k\right)\parallel +\parallel S_{\alpha }{\parallel }_1\underset{k\in \mathbb{Z}}{sup}\parallel F_1(k,0)\parallel \hfill \\ \hfill \le & \parallel S_{\alpha }{\parallel }_1\left(L\left(r_0\right)+\frac{\underset{k\in \mathbb{Z}}{sup}\parallel F_1(k,0)\parallel }{r_0}\right)r_0\hfill \\ \hfill \le & \parallel S_{\alpha }{\parallel }_1\left(L(r_0+r_1)+\frac{\underset{k\in \mathbb{Z}}{sup}\parallel F_1(k,0)\parallel }{min\{r_0,r_1\}}\right)r_0<r_0.\hfill \end{array}$$

Hence $\Lambda v\in B_{r_0}\left(AP(\mathbb{Z},X)\right)$, which yields that $\Lambda$ is well defined.

Then, we show $\Lambda$ is continuous.

Let $v_j\left(n\right),v\left(n\right)\in AP(\mathbb{Z},X)$ satisfy $v_j\left(n\right)\to v\left(n\right)$ as $j\to \infty$, then one has

$$\begin{array}{cc}\hfill \parallel [\Lambda v_j]\left(n\right)-[\Lambda v]\left(n\right)\parallel =& \parallel \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)\left[F_1(k,v_j\left(k\right))-F_1(k,v\left(k\right))\right]\parallel \hfill \\ \hfill \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel F_1(k,v_j\left(k\right))-F_1(k,v\left(k\right))\parallel \hfill \\ \hfill \le & L\left(r_0\right)\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \parallel v_j\left(k\right)-v\left(k\right)\parallel \hfill \\ \hfill \le & L\left(r_0\right)\parallel v_j{-v\parallel }_d\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill =& L\left(r_0\right)\parallel S_{\alpha }{\parallel }_1{\parallel v_j-v\parallel }_d.\hfill \end{array}$$

Therefore, as $j\to \infty$, $\Lambda v_j\to \Lambda v$, hence $\Lambda$ is continuous.

Finally, we show there is a unique fixed point $v\left(n\right)\in AP(\mathbb{Z},X)$ of $\Lambda$.

Let $v_1\left(n\right),v_2\left(n\right)\in B_{r_0}\left(AP(\mathbb{Z},X)\right)$ and, similar to the proof of the continuity of $\Lambda$, one has

$$\begin{array}{c}\hfill \parallel (\Lambda v_1)\left(n\right)-(\Lambda v_2)\left(n\right)\parallel \le \parallel S_{\alpha }{\parallel }_{1}L\left(r_0\right){\parallel v_1-v_2\parallel }_d,\end{array}$$

which implies

$$\begin{array}{c}\hfill \parallel (\Lambda v_1)-(\Lambda v_2){\parallel }_d\le \parallel S_{\alpha }{\parallel }_{1}L\left(r_0\right){\parallel v_1-v_2\parallel }_d.\end{array}$$

From Equation (15), it follows that $\Lambda$ is a contraction on $AP(\mathbb{Z},X)$. Thus, there is a unique fixed point $v\left(n\right)\in AP(\mathbb{Z},X)$ of $\Lambda$ and $\parallel v\left(n\right)\parallel \le r_0$.

Step 2. Set ${\Omega }_r:=\{\omega \left(n\right)\in C_0{(\mathbb{Z},X):\parallel \omega \parallel }_d\le r\}$. For the above $v\left(n\right)\in B_{r_0}\left(AAP(\mathbb{Z},X)\right)$, define $\mathrm{\Gamma }:={\mathrm{\Gamma }}^1+{\mathrm{\Gamma }}^2$ on $C_0(\mathbb{Z},X)$ as Equation (12). There exists a constant $r_1$ s.t. $\mathrm{\Gamma }$ maps ${\Omega }_{r_1}$ into itself.

Firstly, as $\omega (·)\in C_0(\mathbb{Z},X)$, then $\exists r>0$ s.t. $\parallel \omega \left(k\right)\parallel \le r$, $\forall k\in \mathbb{Z}$. This combined with Equation (14) implies

$$\begin{array}{c}\hfill \parallel F_1(k,v\left(k\right)+\omega \left(k\right))-F_1(k,v\left(k\right))\parallel \le L(r_0+r)\parallel \omega \left(k\right)\parallel ,\end{array}$$

which yields

$$F_1(·,v(·)+\omega (·))-F_1(·,v(·))\in C_0(\mathbb{Z},X).$$

According to Equation (8), one has

$$\begin{array}{c}\hfill \parallel F_2(k,v\left(k\right)+\omega \left(k\right))\parallel \le \beta \left(k\right)\Phi \left(r+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right),\forall k\in \mathbb{Z},\omega \left(k\right)\in X\mathrm{with}\parallel \omega \left(k\right)\parallel \le r,\end{array}$$

which yields

$$F_2(·,v(·)+\omega (·))\in C_0(\mathbb{Z},X)\mathrm{as}\beta (·)\in C_0(\mathbb{Z},{\mathbb{R}}^{+}).$$

Thus $\mathrm{\Gamma }$ is well-defined and maps $C_0(\mathbb{Z},X)$ into itself, which follows from Lemma 22.

From Equations (8) and (15), it easily follows that there is a constant $r_1>0$ s.t.

$$\begin{array}{c}\hfill L(r_0+r_1){\parallel S_{\alpha }\parallel }_1{r}_1+{\rho }_2\Phi \left(r_1+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\le r_1.\end{array}$$

Then for any $n\in \mathbb{Z}$ and ${\omega }_1\left(n\right),{\omega }_2\left(n\right)\in {\Omega }_{r_1}$, one has

$$\begin{array}{cc}& \parallel \left({\mathrm{\Gamma }}^1{\omega }_1\right)\left(n\right)+\left({\mathrm{\Gamma }}^2{\omega }_2\right)\left(n\right)\parallel \hfill \\ \hfill \le & ∥\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)[F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))]∥\hfill \\ & +∥\sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)F_2(k,v\left(k\right)+{\omega }_2\left(k\right))∥\hfill \\ \hfill \le & \sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)[F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))]\parallel \hfill \\ & +\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)F_2(k,v\left(k\right)+{\omega }_2\left(k\right))\parallel \hfill \\ \hfill \le & L(r_0+r_1)\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k){\omega }_1\left(k\right)\parallel +\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\sum _{k=-\infty }^{n-1}\parallel \beta \left(k\right)S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill \le & L(r_0+r_1)\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k){\omega }_1\left(k\right)\parallel \hfill \\ & +\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)M\sum _{k=-\infty }^{n-1}{\int }_0^{\infty }\beta \left(k\right)e^{-t}\frac{t^{n-1-k}}{(n-1-k)!}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\delta t^{\alpha })dt\hfill \\ \hfill \le & L(r_0+r_1)\parallel {\omega }_1{\parallel }_d\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel +\sigma \left(n\right)\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\hfill \\ \hfill \le & L(r_0+r_1)\parallel S_{\alpha }{\parallel }_1{\parallel {\omega }_1\parallel }_d+{\rho }_2\Phi \left(\parallel {\omega }_2{\parallel }_d+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\hfill \\ \hfill \le & L(r_0+r_1){\parallel S_{\alpha }\parallel }_1{r}_1+{\rho }_2\Phi \left(r_1+\underset{k\in \mathbb{Z}}{sup}\parallel v\left(k\right)\parallel \right)\le r_1,\hfill \end{array}$$

this indicates that

$$\left({\mathrm{\Gamma }}^1{\omega }_1\right)\left(n\right)+\left({\mathrm{\Gamma }}^2{\omega }_2\right)\left(n\right)\in {\Omega }_{r_1}.$$

Thus $\mathrm{\Gamma }$ maps ${\Omega }_{r_1}$ into itself.

Step 3.${\mathrm{\Gamma }}^1$ is a contraction on ${\Omega }_{r_1}$.

Let ${\omega }_1\left(n\right),{\omega }_2\left(n\right)\in {\Omega }_{r_1}$ and, from Equation (7), one has

$$\begin{array}{cc}& \parallel [F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))]-[F_1(k,v\left(k\right)+{\omega }_2\left(k\right))-F_1(k,v\left(k\right))]\parallel \hfill \\ \hfill \le & L(r_0+r_1)\parallel {\omega }_1\left(k\right)-{\omega }_2\left(k\right)\parallel .\hfill \end{array}$$

Thus

$$\begin{array}{cc}\hfill \parallel \left({\mathrm{\Gamma }}^1{\omega }_1\right)\left(n\right)-\left({\mathrm{\Gamma }}^1{\omega }_2\right)\left(n\right)\parallel =& \parallel \sum _{k=-\infty }^{n-1}{S}_{\alpha }(n-1-k)[\left(F_1(k,v\left(k\right)+{\omega }_1\left(k\right))-F_1(k,v\left(k\right))\right)\hfill \\ & -\left(F_1(k,v\left(k\right)+{\omega }_2\left(k\right))-F_1(k,v\left(k\right))\right)]\parallel \hfill \\ \hfill \le & L(r_0+r_1)\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)[{\omega }_1\left(k\right)-{\omega }_2\left(k\right)]\parallel \hfill \\ \hfill \le & L(r_0+r_1)\parallel {\omega }_1-{\omega }_2{\parallel }_d\sum _{k=-\infty }^{n-1}\parallel S_{\alpha }(n-1-k)\parallel \hfill \\ \hfill =& L(r_0+r_1)\parallel S_{\alpha }{\parallel }_1{\parallel {\omega }_1-{\omega }_2\parallel }_d,\hfill \end{array}$$

which implies

$$\begin{array}{c}\hfill \parallel {\mathrm{\Gamma }}^1{\omega }_1-{\mathrm{\Gamma }}^1{\omega }_2{\parallel }_d\le L(r_0+r_1)\parallel S_{\alpha }{\parallel }_1{\parallel {\omega }_1-{\omega }_2\parallel }_d.\end{array}$$

Thus ${\mathrm{\Gamma }}^1$ is a contraction on ${\Omega }_{r_1}$ by Equation (15).

Step 4.${\mathrm{\Gamma }}^2$ is completely continuous on ${\Omega }_{k_0}$.

The proof of this step is the same as step 4 of Theorem 1.

Step 5. Equation (4) has at least one mild solution which is asymptotically almost-periodic.

The proof of this step is the same as step 5 of Theorem 1.

4. Applications

In this section, an example is provide to demonstrate the effectiveness of our abstract results.

Consider the different scalar equation of fractional order forms like

$$\begin{array}{c}\hfill {▵}^{\alpha }x\left(n\right)=\lambda x(n+1)+F(n,x\left(n\right)),n\in \mathbb{Z},\end{array}$$

(16)

where $\alpha \in (0,1)$, $\lambda$ is a complex number satisfying $\mathrm{Re}\left(\lambda \right)<0$ and $F:\mathbb{Z}\times \mathbb{C}\to \mathbb{C}$ is a function. It is clear that there is an exponentially stable $C_0$-semigroup generated by $\lambda$ given by

$$T\left(t\right)=e^{\lambda t},\forall t\ge 0.$$

Thus $\left(H_1\right)$ holds. From Abadias and Lizama [33], it follows that there is a discrete $\alpha$-resolvent family $\left\{S\alpha \right(n\left)\right\}n\in \mathbb{N}$ generated by $\lambda$, which is summable and given by

$$S_{\alpha }\left(n\right)={\int }_0^{\infty }{e}^{-t}\frac{t^n}{n!}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\lambda t^{\alpha })\mathrm{d}t.$$

Let

$$F_1(n,x):=\mu (sinn+sin\sqrt{2}n)sinx,F_2(n,x):=\nu e^{-\left|n\right|}xsin{x}^2.$$

Then, according to Lemma 13, one has $F_1(n,x)\in AP(\mathbb{Z}\times \mathbb{C},\mathbb{C})$ satisfying

$$\parallel F_1(n,x)-F_1(n,y)\parallel \le 2\mu \parallel x-y\parallel ,\forall n\in \mathbb{Z},x,y\in \mathbb{C},$$

and

$$\parallel F_2(n,x)\parallel \le \nu e^{-\left|n\right|}\parallel x\parallel ,\forall n\in \mathbb{Z},x\in \mathbb{C}.$$

This indicates $F_2(n,x)\in C_0(\mathbb{R}\times \mathbb{C},\mathbb{C})$. Furthermore

$$F(n,x)=F_1(n,x)+F_2(n,x)\in AAP(\mathbb{R}\times \mathbb{C},\mathbb{C}).$$

Thus $\left(H_2\right)$–$\left(H_4\right)$ hold with

$$L=2\mu ,\Phi \left(r\right)=r,\beta \left(n\right)=\nu e^{-\left|n\right|},{\rho }_1=1,{\rho }_2\le \nu .$$

From Theorem 1 it follows that there exists at least one mild solution to Equation (4) whenever $2\mu +\nu <1$. Moreover it is asymptotically almost-periodic.