Asymptotically almost periodic solution to a class of Volterra difference equations
© Long and Pan; licensee Springer 2012
Received: 1 September 2012
Accepted: 3 November 2012
Published: 16 November 2012
This paper is concerned with an asymptotically almost periodic solution to a class of Volterra-type 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 Volterra-type difference equation. Our existence theorem extends and complements a recent result due to (Ding et al. in Electron. J. Qual. Theory Differ. Equ. 6:1-13, 2012).
1 Introduction and preliminaries
where λ is a fixed positive integer and , , () satisfy some conditions recalled in Section 3.
For the background of discrete Volterra equations, we refer the reader to the well-known monograph  by Agarwal. The first motivation for this paper is some recent work on asymptotical periodicity for Volterra-type difference equations in [2–6] by Diblík et al. In fact, asymptotical behavior for Volterra-type 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.
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 () the set of (nonnegative) integers, by ℕ the set of positive integers, by ℝ () the set of (nonnegative) real numbers, by Ω a subset of ℝ, and by X a Banach space.
Definition 1.1 
Denote by the set of all such functions. Moreover, we denote by for convenience.
Lemma 1.2 [, Theorem 1.26]
A necessary and sufficient condition for the sequence to be almost periodic is that for any integer sequence , one can extract a subsequence such that converges uniformly with respect to .
for all .
Next, we denote by the space of all the functions such that .
Definition 1.4 A function is called asymptotically almost periodic if it admits a decomposition , where and . Denote by the set of all such functions. Moreover, we denote by for convenience.
for all and . Denote by the set of all such functions.
Similarly, for each subset , we denote by the space of all the functions such that is continuous for each , and uniformly for x in any compact subset of Ω.
Definition 1.6 A function is called asymptotically almost periodic in n uniformly for if it admits a decomposition , where and . Denote by the set of all such functions.
implies that f is bounded.
implies that . Moreover, if .
E is a Banach space equipped with the supremum norm.
2 A compactness criterion
The following theorem is a well-known result for the continuous case (see, e.g., [, p.24, Theorem 2.5]). Here, we give a discrete version.
for all with and .
for all with and .
Now, let us show that . We divide the remaining proof into three steps.
for all with .
for all with , we know that . This completes the proof. □
for all with and .
for each , is bounded;
F is equi-asymptotically almost periodic.
Proof ‘only if’ part
By Remark 1.3, we can get that is equi-asymptotically almost periodic. Combing this with (2.2), we can show that F is equi-asymptotically almost periodic, i.e., (ii) holds.
which means that is uniformly convergent on ℤ, i.e., is convergent in . So, F is precompact in . □
3 Application to Volterra difference equations
For convenience, we first list some assumptions.
(H3) For each , , where , .
Theorem 3.1 Assume that (H1)-(H4) hold. Then Equation (1.1) has an asymptotically almost periodic solution.
It suffices to prove that ℳ has a fixed point in . We give the proof in three steps.
Step 1. and both map into , .
Since is Lipschitz, by Remark 1.3, we can first show that for each compact subset and each , is equi-asymptotically almost periodic. Then it is easy to show that for each .
we know that .
Noting that , has a unique fixed point in .
Step 3. ℳ has a fixed point in .
where is the unique fixed point of (see Step 2).
which is a contradiction.
Combining this with (3.2), we know that . So is continuous.
which yields that each is equi-asymptotically almost periodic since . Then, by Theorem 2.3, each is precompact in . Let . Then , if necessary going to a subsequence, is convergent in for each . By (3.1), we conclude that is convergent in . So, is precompact in .
which means that is a fixed point of ℳ. This completes the proof. □
Finally, we give a simple example to illustrate our result.
Thus, (H4) holds with . Then, by using Theorem 3.1, Equation (1.1) has an asymptotically almost periodic solution.
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 (2012-114).
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