 Research
 Open Access
 Published:
Asymptotically almost periodic solution to a class of Volterra difference equations
Advances in Difference Equations volume 2012, Article number: 199 (2012)
Abstract
This paper is concerned with an asymptotically almost periodic solution to a class of Volterratype difference equations. We establish a compactness criterion for the sets of asymptotically almost periodic sequences. Then, by using the compactness criterion and Schauder’s fixed point theorem, we present an existence theorem for an asymptotically almost periodic solution to the addressed Volterratype difference equation. Our existence theorem extends and complements a recent result due to (Ding et al. in Electron. J. Qual. Theory Differ. Equ. 6:113, 2012).
MSC:39A24, 34K14.
1 Introduction and preliminaries
In this paper, we consider the following nonlinear Volterratype difference equation:
where λ is a fixed positive integer and {f}_{i}:\mathbb{Z}\times \mathbb{R}\to \mathbb{R}, {a}_{i}:\mathbb{Z}\times \mathbb{Z}\to \mathbb{R}, {g}_{i}:\mathbb{Z}\times \mathbb{R}\to \mathbb{R} (i=1,2,\dots ,\lambda) satisfy some conditions recalled in Section 3.
For the background of discrete Volterra equations, we refer the reader to the wellknown monograph [1] by Agarwal. The first motivation for this paper is some recent work on asymptotical periodicity for Volterratype difference equations in [2–6] by Diblík et al. In fact, asymptotical behavior for Volterratype difference equations, including periodicity, asymptotical periodicity, etc., has been of great interest for many mathematicians. However, to the best of our knowledge, there is seldom literature available about asymptotically almost periodicity for Equation (1.1). Thus, in this paper, we will investigate this problem. In addition, it is needed to note that compared with asymptotically periodic sequences, in general, it is more difficult to obtain the compactness for a set of asymptotically almost periodic sequences.
On the other hand, in a recent work [7], by using the classical Schauder fixed point theorem, Ding et al. established an interesting existence theorem for the following functional integral equation:
In fact, the existence of almost periodic type solutions has been an interesting and important topic in the study of qualitative theory of difference equations. We refer the reader to [8–13] and references therein for some recent developments on this topic. Equation (1.1) can be seen as a discrete analogue (but more general) of Equation (1.2). That is another main motivation for this work.
Throughout the rest of this paper, we denote by ℤ ({\mathbb{Z}}^{+}) the set of (nonnegative) integers, by ℕ the set of positive integers, by ℝ ({\mathbb{R}}^{+}) the set of (nonnegative) real numbers, by Ω a subset of ℝ, and by X a Banach space.
First, let us recall some notations and basic results of almost periodic type sequences (for more details, see [11, 14, 15]).
Definition 1.1 [14]
A function f:\mathbb{Z}\to X is called almost periodic if ∀ε, \mathrm{\exists}N(\epsilon )\in \mathbb{N} such that among any N(\epsilon ) consecutive integers there exists an integer p with the property that
Denote by \mathit{AP}(\mathbb{Z},X) the set of all such functions. Moreover, we denote \mathit{AP}(\mathbb{Z},\mathbb{R}) by \mathit{AP}(\mathbb{Z}) for convenience.
Lemma 1.2 [[14], Theorem 1.26]
A necessary and sufficient condition for the sequence f:\mathbb{Z}\to \mathbb{R} to be almost periodic is that for any integer sequence \{{n}_{k}^{\mathrm{\prime}}\}, one can extract a subsequence \{{n}_{k}\} such that \{f(n+{n}_{k})\} converges uniformly with respect to n\in \mathbb{Z}.
Remark 1.3 Let f,g\in \mathit{AP}(\mathbb{Z}). By Lemma 1.2, it is not difficult to show that ∀ε, \mathrm{\exists}N(\epsilon )\in \mathbb{N} such that among any N(\epsilon ) consecutive integers there exists a common integer p with the property that
for all k\in \mathbb{Z}.
Next, we denote by {C}_{0}(\mathbb{Z},X) the space of all the functions f:\mathbb{Z}\to X such that {lim}_{n\to \mathrm{\infty}}\parallel f(n)\parallel =0.
Definition 1.4 A function f:\mathbb{Z}\to X is called asymptotically almost periodic if it admits a decomposition f=g+h, where g\in \mathit{AP}(\mathbb{Z},X) and h\in {C}_{0}(\mathbb{Z},X). Denote by \mathit{AAP}(\mathbb{Z},X) the set of all such functions. Moreover, we denote \mathit{AAP}(\mathbb{Z},\mathbb{R}) by \mathit{AAP}(\mathbb{Z}) for convenience.
Definition 1.5 Let \mathrm{\Omega}\subset \mathbb{R} and f be a function from \mathbb{Z}\times \mathrm{\Omega} to ℝ such that f(n,\cdot ) is continuous for each n\in \mathbb{Z}. Then f is called almost periodic in n \in \mathbb{Z} uniformly for \omega \in \mathrm{\Omega} if for every \epsilon >0 and every compact \mathrm{\Sigma}\subset \mathrm{\Omega}, there corresponds an integer {N}_{\epsilon}(\mathrm{\Sigma})>0 such that among {N}_{\epsilon}(\mathrm{\Sigma}) consecutive integers there exists an integer p with the property that
for all k\in \mathbb{Z} and \omega \in \mathrm{\Sigma}. Denote by \mathit{AP}(\mathbb{Z}\times \mathrm{\Omega}) the set of all such functions.
Similarly, for each subset \mathrm{\Omega}\subset \mathbb{R}, we denote by {C}_{0}(\mathbb{Z}\times \mathrm{\Omega}) the space of all the functions f:\mathbb{Z}\times \mathrm{\Omega}\to \mathbb{R} such that f(n,\cdot ) is continuous for each n\in \mathbb{Z}, and {lim}_{n\to \mathrm{\infty}}f(n,x)=0 uniformly for x in any compact subset of Ω.
Definition 1.6 A function f:\mathbb{Z}\times \mathrm{\Omega}\to \mathbb{R} is called asymptotically almost periodic in n uniformly for x\in \mathrm{\Omega} if it admits a decomposition f=g+h, where g\in \mathit{AP}(\mathbb{Z}\times \mathrm{\Omega}) and h\in {C}_{0}(\mathbb{Z}\times \mathrm{\Omega}). Denote by \mathit{AAP}(\mathbb{Z}\times \mathrm{\Omega}) the set of all such functions.
Lemma 1.7 Let E\in \{\mathit{AP}(\mathbb{Z},X),\mathit{AAP}(\mathbb{Z},X)\}. Then the following hold true:

(a)
f\in E implies that f is bounded.

(b)
f,g\in E implies that f+g\in E. Moreover, f\cdot g\in E if X=\mathbb{R}.

(c)
E is a Banach space equipped with the supremum norm.
Proof The proof is similar to that of the continuous case (cf. [14, 15]). So, we omit the details. □
2 A compactness criterion
The following theorem is a wellknown result for the continuous case (see, e.g., [[16], p.24, Theorem 2.5]). Here, we give a discrete version.
Theorem 2.1 Let f be a function from ℤ to ℝ. Then f\in \mathit{AAP}(\mathbb{Z}) if and only if ∀ε, \mathrm{\exists}M(\epsilon ),N(\epsilon )\in \mathbb{N} such that among any N(\epsilon ) consecutive integers there exists an integer p with the property that
for all k\in \mathbb{Z} with k\ge M(\epsilon ) and k+p\ge M(\epsilon ).
Proof We first show the ‘only if’ part. Let f\in \mathit{AAP}(\mathbb{Z}). Then there exist g\in \mathit{AP}(\mathbb{Z}) and h\in {C}_{0}(\mathbb{Z},\mathbb{R}) such that f=g+h. By g\in \mathit{AP}(\mathbb{Z}), for each \epsilon >0, \mathrm{\exists}N(\epsilon )\in \mathbb{N} such that among any N(\epsilon ) consecutive integers there exists an integer p with the property that
In addition, since h\in {C}_{0}(\mathbb{Z},\mathbb{R}), for the above \epsilon >0, there exists M(\epsilon )\in \mathbb{N} such that h(k)<\frac{\epsilon}{3} for all k\in \mathbb{Z} with k\ge M(\epsilon ). Thus, we have
for all k\in \mathbb{Z} with k\ge M(\epsilon ) and k+p\ge M(\epsilon ).
Next, let us prove the ‘if’ part. First, let us show that f is bounded. Letting \epsilon =1, there exists M(1),N(1)\in \mathbb{N} such that among any N(1) consecutive integers there exists an integer p with the property that
for all k\in \mathbb{Z} with k\ge M(1) and k+p\ge M(1). Then, for each k\in \mathbb{Z} with k\ge M(1), there exists {p}_{k}\in [M(1)k,M(1)+N(1)k]\cap \mathbb{Z} such that
Noting that k+{p}_{k}\in [M(1),M(1)+N(1)], we get
for all k\in \mathbb{Z} with k\ge M(1). Thus,
Now, let us show that f\in \mathit{AAP}(\mathbb{Z}). We divide the remaining proof into three steps.
Step 1. Since f is bounded, we can choose a sequence \{{s}_{n}\}\subset \mathbb{N} such that {lim}_{n\to +\mathrm{\infty}}{s}_{n}=+\mathrm{\infty} and {lim}_{n\to +\mathrm{\infty}}f(k+{s}_{n}) exists for each k\in \mathbb{Z}. Let
For each \epsilon >0, among any N(\epsilon ) consecutive integers there exists an integer p with the property that
for all k\in \mathbb{Z} with k\ge M(\epsilon ) and k+p\ge M(\epsilon ). Then, for each fixed k\in \mathbb{Z}, we have
for sufficiently large n, which yields that
Thus, \overline{g}\in \mathit{AP}(\mathbb{Z}).
Step 2. Now fix \epsilon >0. Then, for each n\in \mathbb{N}, there exists {t}_{n}\in [{s}_{n}N(\epsilon ),{s}_{n}]\cap \mathbb{Z} such that
for all k\in \mathbb{Z} with k\ge M(\epsilon ) and k+{t}_{n}\ge M(\epsilon ). Let {r}_{n}={s}_{n}{t}_{n}. Then {r}_{n}\in \{0,1,2,\dots ,N(\epsilon )\}, which means that there exist a subsequence \{{r}_{n}^{\prime}\}\subset \{{r}_{n}\} and r(\epsilon )\in \{0,1,2,\dots ,N(\epsilon )\} such that
Thus, for all k\in \mathbb{Z} with k\ge M(\epsilon ), we have
Combining this with (2.1), {lim}_{n\to +\mathrm{\infty}}{t}_{n}^{\prime}=+\mathrm{\infty}, and
we conclude
for all k\in \mathbb{Z} with k\ge M(\epsilon ).
Step 3. By Step 2, we know that for each \epsilon >0, there exists r(\epsilon )\in \{0,1,2,\dots ,N(\epsilon )\} such that
for all k\in \mathbb{Z} with k\ge M(\epsilon ). Taking \epsilon =1,1/2,\dots , we get a sequence \{r(1/m)\}. On the other hand, it follows from Step 1 that \overline{g}\in \mathit{AP}(\mathbb{Z}). Thus, going to a subsequence, if necessary, we may assume that \overline{g}(\cdot r(1/m)) is uniformly convergent on ℤ. Let
Then g\in \mathit{AP}(\mathbb{Z}). In addition, noting that
for all k\in \mathbb{Z} with k\ge M(1/m), we know that fg\in {C}_{0}(\mathbb{Z}). This completes the proof. □
Definition 2.2 F\subseteq \mathit{AAP}(\mathbb{Z}) is said to be equiasymptotically almost periodic if for each \epsilon >0, there exist M(\epsilon ),N(\epsilon )\in \mathbb{N} such that among any N(\epsilon ) consecutive integers there exists an integer p with the property that
for all k\in \mathbb{Z} with k\ge M(\epsilon ) and k+p\ge M(\epsilon ).
Theorem 2.3 Let F\subseteq \mathit{AAP}(\mathbb{Z}). Then F is precompact in \mathit{AAP}(\mathbb{Z}) if and only if the following two conditions hold:

(i)
for each k\in \mathbb{Z}, \{f(k):f\in F\} is bounded;

(ii)
F is equiasymptotically almost periodic.
Proof ‘only if’ part
Let F\subset \mathit{AAP}(\mathbb{Z}) be precompact. Then F is bounded in \mathit{AAP}(\mathbb{Z}). So, (i) obviously holds. In addition, \mathrm{\forall}\epsilon >0, there exists N\in \mathbb{N} and {f}_{1},{f}_{2},\dots ,{f}_{N}\in F such that
By Remark 1.3, we can get that \{{f}_{1},{f}_{2},\dots ,{f}_{N}\} is equiasymptotically almost periodic. Combing this with (2.2), we can show that F is equiasymptotically almost periodic, i.e., (ii) holds.
‘if part’
Let \{{f}_{n}\}\subset F. Since \{{f}_{n}(k)\} is bounded for each k\in \mathbb{Z}, we can assume that (if necessary going to a subsequence) \{{f}_{n}(k)\} is convergent for each k\in \mathbb{Z}. On the other hand, since F is equiasymptotically almost periodic, for each \epsilon >0, there exist M(\epsilon ),N(\epsilon )\in \mathbb{N} such that among any N(\epsilon ) consecutive integers there exists an integer p with the property that
for all k\in \mathbb{Z} with k\ge M(\epsilon ) and k+p\ge M(\epsilon ). For the above \epsilon >0, there exists a positive integer K such that for all n,m>K, the following hold:
For all k\in \mathbb{Z} with k\ge M(\epsilon ), taking p\in [k+M(\epsilon ),k+M(\epsilon )+N(\epsilon )]\cap \mathbb{Z}, by (2.3) and (2.4), we get
also, for all k\in \mathbb{Z} with k<M(\epsilon ), by (2.4), we have
Thus, we get
which means that \{{f}_{n}(k)\} is uniformly convergent on ℤ, i.e., \{{f}_{n}\} is convergent in \mathit{AAP}(\mathbb{Z}). So, F is precompact in \mathit{AAP}(\mathbb{Z}). □
3 Application to Volterra difference equations
In this section, we discuss the existence of an asymptotically almost periodic solution to Volterra difference equation (1.1). Throughout the rest of this paper, p,q\ge 1 are two fixed real numbers and
In addition, we denote by {l}^{p}(\mathbb{Z}) (resp. {l}^{q}(\mathbb{Z})) the space of all the functions f:\mathbb{Z}\to \mathbb{R} satisfying
For convenience, we first list some assumptions.
(H1) For each i\in \{1,2,\dots ,\lambda \}, {f}_{i}(\cdot ,x)\in \mathit{AAP}(\mathbb{Z}) for any fixed x\in \mathbb{R}, and there exists a constant {L}_{i}\ge 0 such that
(H2) For each i\in \{1,2,\dots ,\lambda \}, {g}_{i}(k,\cdot ) is continuous for each k\in \mathbb{Z}, and for each r>0, there exists a sequence \{{\mu}_{i}^{r}\}\subset {l}^{p}(\mathbb{Z}) such that
(H3) For each i\in \{1,2,\dots ,\lambda \}, {\tilde{a}}_{i}\in \mathit{AAP}(\mathbb{Z},{l}^{q}(\mathbb{Z})), where [{\tilde{a}}_{i}(k)](l)={a}_{i}(k,l), \mathrm{\forall}k,l\in \mathbb{Z}.
(H4) There exists a constant M>0 such that
where {\alpha}_{i}={sup}_{n\in \mathbb{Z}}{\parallel {\tilde{a}}_{i}(n)\parallel}_{q}; and
Theorem 3.1 Assume that (H1)(H4) hold. Then Equation (1.1) has an asymptotically almost periodic solution.
Proof We denote
and
It suffices to prove that ℳ has a fixed point in \mathit{AAP}(\mathbb{Z}). We give the proof in three steps.
Step 1. {A}_{i} and {B}_{i} both map \mathit{AAP}(\mathbb{Z}) into \mathit{AAP}(\mathbb{Z}), i=1,2,\dots ,\lambda.
Since {f}_{i} is Lipschitz, by Remark 1.3, we can first show that for each compact subset K\subset \mathbb{R} and each i\in \{1,2,\dots ,\lambda \}, \{{f}_{i}(\cdot ,x):x\in K\} is equiasymptotically almost periodic. Then it is easy to show that {A}_{i}x\in \mathit{AAP}(\mathbb{Z}) for each x\in \mathit{AAP}(\mathbb{Z}).
Since {\tilde{a}}_{i}\in \mathit{AAP}(\mathbb{Z},{l}^{q}(\mathbb{Z})), there exist {b}_{i}\in \mathit{AP}(\mathbb{Z},{l}^{q}(\mathbb{Z})) and {c}_{i}\in {C}_{0}(\mathbb{Z},{l}^{q}(\mathbb{Z})) such that {\tilde{a}}_{i}={b}_{i}+{c}_{i}. For each x\in \mathit{AAP}(\mathbb{Z}), noting that for n,p\in \mathbb{Z},
and
we know that {B}_{i}x\in \mathit{AAP}(\mathbb{Z}).
Step 2. For each y\in \mathit{AAP}(\mathbb{Z}) with \parallel y\parallel \le M, there exists a unique {x}_{y}\in \mathit{AAP}(\mathbb{Z}) such that
Let
Then, by Step 1, \mathcal{Y} maps \mathit{AAP}(\mathbb{Z}) into \mathit{AAP}(\mathbb{Z}). For all {x}_{1},{x}_{2}\in \mathit{AAP}(\mathbb{Z}) and n\in \mathbb{Z}, we have
which yields that
Noting that {\sum}_{i=1}^{\lambda}{\alpha}_{i}{L}_{i}{\parallel {\mu}_{i}^{M}\parallel}_{p}<1, \mathcal{Y} has a unique fixed point {x}_{y} in \mathit{AAP}(\mathbb{Z}).
Step 3. ℳ has a fixed point in \mathit{AAP}(\mathbb{Z}).
Let E=\{y\in \mathit{AAP}(\mathbb{Z}):\parallel y\parallel \le M\} and
where {x}_{y} is the unique fixed point of \mathcal{Y} (see Step 2).
We claim that \mathcal{N}(E)\subset E. In fact, if there exists {y}_{0}\in E such that \parallel \mathcal{N}{y}_{0}\parallel >M, then by (H4), we have
which is a contradiction.
Next, let us show that \mathcal{N}:E\to E is continuous. For all {y}_{1},{y}_{2}\in E, we have
which gives that
where
Letting {y}_{k}\to y in E, by (3.1), we have
For each i=1,2,\dots ,\lambda, noting that
{g}_{i}(m,\cdot ) is continuous for each m\in \mathbb{Z}, and {y}_{k}(m)\to y(m) for each m\in \mathbb{Z}, we conclude that
Combining this with (3.2), we know that \mathcal{N}{y}_{k}\to \mathcal{N}y. So \mathcal{N}:E\to E is continuous.
Now, let us show that \mathcal{N}(E) is precompact in \mathit{AAP}(\mathbb{Z}). In order to show that, we first prove each {B}_{i}(E) is precompact in \mathit{AAP}(\mathbb{Z}). By a direct calculation, we can get
for all y\in E and n\in \mathbb{Z}. In addition, for all {n}_{1},{n}_{2}\in \mathbb{Z} and y\in E, we have
which yields that each {B}_{i}(E) is equiasymptotically almost periodic since . Then, by Theorem 2.3, each {B}_{i}(E) is precompact in \mathit{AAP}(\mathbb{Z}). Let \{{y}_{k}\}\subset E. Then \{{B}_{i}{y}_{k}\}, if necessary going to a subsequence, is convergent in \mathit{AAP}(\mathbb{Z}) for each i\in \{1,2,\dots ,\lambda \}. By (3.1), we conclude that \{\mathcal{N}{y}_{k}\} is convergent in \mathit{AAP}(\mathbb{Z}). So, \mathcal{N}(E) is precompact in \mathit{AAP}(\mathbb{Z}).
By applying Schauder’s fixed point theorem, there exists a fixed point {y}_{\ast} of \mathcal{N} in E. Then we have
which means that {y}_{\ast} is a fixed point of ℳ. This completes the proof. □
Finally, we give a simple example to illustrate our result.
Example 3.2 Let \lambda =2, p=1, q=\mathrm{\infty},
and
It is easy to see that (H1) holds with {L}_{1}=\frac{3}{10} and {L}_{2}=\frac{1}{20}. Also, (H2) holds with {\mu}_{1}^{r}(n)\equiv \frac{1}{2(1+{n}^{2})} and {\mu}_{2}^{r}(n)\equiv \frac{\pi}{2}\cdot \frac{1}{1+{n}^{2}}. In addition, (H3) can be easily verified. By a direct calculation, we can get
and
Letting M=1, we have
and
Thus, (H4) holds with M=1. Then, by using Theorem 3.1, Equation (1.1) has an asymptotically almost periodic solution.
References
 1.
Agarwal RP Monographs and Textbooks in Pure and Applied Mathematics. In Difference Equations and Inequalities. Theory, Methods, and Applications. 2nd edition. Marcel Dekker, New York; 2000.
 2.
Diblík J, Schmeidel E, Ružičková M: Existence of asymptotically periodic solutions of system of Volterra difference equations. J. Differ. Equ. Appl. 2009, 15: 1165–1177. 10.1080/10236190802653653
 3.
Diblík J, Ružičková M, Schmeidel E: Existence of asymptotically periodic solutions of scalar Volterra difference equations. Tatra Mt. Math. Publ. 2009, 43: 51–61.
 4.
Diblík J, Schmeidel E, Ružičková M: Asymptotically periodic solutions of Volterra system of difference equations. Comput. Math. Appl. 2010, 59: 2854–2867. 10.1016/j.camwa.2010.01.055
 5.
Diblík J, Ružičková M, Schmeidel E, Zbaszyniak M: Weighted asymptotically periodic solutions of linear Volterra difference equations. Abstr. Appl. Anal. 2011., 2011: Article ID 370982. doi:10.1155/2011/370982
 6.
Diblík J, Schmeidel E: On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence. Appl. Math. Comput. 2012, 218: 9310–9320. 10.1016/j.amc.2012.03.010
 7.
Ding HS, Chen YY, N’Guérékata GM:{C}^{n}almost periodic and almost periodic solutions for some nonlinear integral equations. Electron. J. Qual. Theory Differ. Equ. 2012, 6: 1–13.
 8.
Araya D, Castro R, Lizama C: Almost automorphic solutions of difference equations. Adv. Differ. Equ. 2009., 2009: Article ID 591380
 9.
Blot J, Pennequin D: Existence and structure results on almost periodic solutions of difference equations. J. Differ. Equ. Appl. 2001, 7: 383–402. 10.1080/10236190108808277
 10.
Cuevas, C, Henríquez, H, Lizama, C: On the existence of almost automorphic solutions of Volterra difference equations. J. Differ. Equ. Appl. (in press)
 11.
Ding HS, Fu JD, N’Guérékata GM: Positive almost periodic type solutions to a class of nonlinear difference equations. Electron. J. Qual. Theory Differ. Equ. 2011, 25: 1–16.
 12.
Hamaya Y: Existence of an almost periodic solution in a difference equation with infinite delay. J. Differ. Equ. Appl. 2003, 9: 227–237. 10.1080/1023619021000035836
 13.
Pennequin D: Existence of almost periodic solutions of discrete time equations. Discrete Contin. Dyn. Syst. 2001, 7: 51–60.
 14.
Corduneanu C: Almost Periodic Functions. 2nd edition. Chelsea, New York; 1989.
 15.
Fink AM Lecture Notes in Mathematics. In Almost Periodic Differential Equations. Springer, Berlin; 1974.
 16.
Zhang C: Almost Periodic Type Functions and Ergodicity. Kluwer Academic, Dordrecht; 2003.
Acknowledgements
The work was supported by the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province, the Foundation of Jiangxi Provincial Education Department (GJJ12205), and the Research Project of Jiangxi Normal University (2012114).
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WL completed the main study, carried out the results of this article and drafted the manuscript. WP checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Long, W., Pan, WH. Asymptotically almost periodic solution to a class of Volterra difference equations. Adv Differ Equ 2012, 199 (2012). https://doi.org/10.1186/168718472012199
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718472012199
Keywords
 asymptotically almost periodic
 Volterra difference equation