#### Abstract

A linear Volterra difference equation of the form where , , and is -periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on is assumed. The results generalize some of the recent results.

#### 1. Introduction

In the paper, we study a linear Volterra difference equation where , , , and is -periodic, . We will also adopt the customary notations where is an integer, is a positive integer, and ββ denotes the function considered independently of whether it is defined for the arguments indicated or not.

In [1], the authors considered (1.1) under the assumption and gave sufficient conditions for the existence of asymptotically -periodic solutions of (1.1) where the notion for an asymptotically -periodic function has been given by the following definition.

*Definition 1.1. *Let be a positive integer. The sequence is called -periodic if for all . The sequence is called asymptotically -periodic if there exist two sequences such that is -periodic, , and
for all .

In this paper, in general, we do not assume that (1.3) holds. Then, we are able to derive sufficient conditions for the existence of a weighted asymptotically -periodic solution of (1.1). We give a definition of a weighted asymptotically -periodic function.

*Definition 1.2. *Let be a positive integer. The sequence is called weighted asymptotically -periodic if there exist two sequences such that is -periodic and , and, moreover, if there exists a sequence such that
for all .

Apart from this, when we assume then, as a consequence of our main result (Theorem 2.2), the existence of an asymptotically -periodic solution of (1.1) is obtained.

For the reader's convenience, we note that the background for discrete Volterra equations can be found, for example, in the well-known monograph by Agarwal [2], as well as by Elaydi [3] or KociΔ and Ladas [4]. Volterra difference equations were studied by many others, for example, by Appleby et al. [5], by Elaydi and Murakami [6], by GyΕri and HorvΓ‘th [7], by GyΕri and Reynolds [8], and by Song and Baker [9]. For some results on periodic solutions of difference equations, see, for example, [2β4, 10β13] and the related references therein.

#### 2. Weighted Asymptotically Periodic Solutions

In this section, sufficient conditions for the existence of weighted asymptotically -periodic solutions of (1.1) will be derived. The following version of Schauder's fixed point theorem given in [14] will serve as a tool used in the proof.

Lemma 2.1. *Let be a Banach space and its nonempty, closed, and convex subset and let be a continuous mapping such that is contained in and the closure is compact. Then, has a fixed point in .*

We set Moreover, we define where is the floor function (the greatest-integer function) and is the βremainderβ of dividing by . Obviously, , is an -periodic sequence.

Now, we derive sufficient conditions for the existence of a weighted asymptotically -periodic solution of (1.1).

Theorem 2.2 (Main result). * Let be a positive integer, be -periodic, , and . Assume that
**
and that at least one of the real numbers in the left-hand sides of inequalities (2.4) is positive. **Then, for any nonzero constant , there exists weighted asymptotically -periodic solution of (1.1) with and in representation (1.5) such that
**
that is,
*

*Proof. *We will use a notation
whenever this is useful. *Case 1. *First assume . We will define an auxiliary sequence of positive numbers , . We set
where the expression on the right-hand side is well defined due to (2.4). Moreover, we define
for . It is easy to see that
We show, moreover, that
for any . Let us first remark that
Then, due to the convergence of both series (see (2.4)), the inequality
obviously holds for every and (2.11) is proved.

Let be the Banach space of all real bounded sequences equipped with the usual supremum norm for . We define a subset as
It is not difficult to prove that is a nonempty, bounded, convex, and closed subset of .

Let us define a mapping as follows:
for any .

We will prove that the mapping has a fixed point in .

We first show that . Indeed, if , then for and, by (2.11) and (2.15), we have
Next, we prove that is continuous. Let be a sequence in such that as . Because is closed, . Now, utilizing (2.15), we get
Therefore,
This means that is continuous.

Now, we show that is compact. As is generally known, it is enough to verify that every -open covering of contains a finite -subcover of , that is, finitely many of these open sets already cover ([15], page 756 (12)). Thus, to prove that is compact, we take an arbitrary and assume that an open -cover of is given. Then, from (2.10), we conclude that there exists an such that for .

Suppose that is one of the elements generating the -cover of . Then (as follows from (2.16)), for an arbitrary ,
if . In other words, the -neighborhood of :
where covers the set on an infinite interval . It remains to cover the rest of on a finite interval for by a finite number of -neighborhoods of elements generating -cover . Supposing that itself is not able to generate such cover, we fix and split the interval
into a finite number of closed subintervals
each with a length not greater then such that
Finally, the set
equals
and can be divided into a finite number
of different subintervals (2.22). This means that, at most, of elements generating the cover are sufficient to generate a finite -subcover of for . We remark that each of such elements simultaneously plays the same role as for . Since can be chosen as arbitrarily small, is compact.

By Schauder's fixed point theorem, there exists a such that for . Thus,
for any .

Due to (2.10) and (2.16), for fixed point of , we have
or, equivalently,
Finally, we will show that there exists a connection between the fixed point and the existence of a solution of (1.1) which divided by provides an asymptotically -periodic sequence. Considering (2.27) for and , we get
where . Hence, we have
Putting
in (2.31), we get (1.1) since
yields
Consequently, defined by (2.32) is a solution of (1.1). From (2.29) and (2.32), we obtain
for (where is the Landau order symbol). Hence,
It is easy to show that the function defined by (2.1) can be expressed in the form
for . Then, as follows from (2.36),
or

The proof is completed since the sequence is -periodic, hence bounded and, due to the properties of Landau order symbols, we have
and it is easy to see that the choice
and an appropriate function such that
finishes this part of the proof. Although for , there is no correspondence between formula (2.36) and the definitions of functions and , we assume that function makes up for this.*Case 2. *If , we can proceed as follows. It is easy to see that arbitrary solution of the equation
defines a solution of (1.1) since a substitution in (2.43) turns (2.43) into (1.1). If the assumptions of Theorem 2.2 hold for (1.1), then, obviously, Theorem 2.2 holds for (2.43) as well. So, for an arbitrary , (2.43) has a solution that can be represented by formula (2.6), that is,
Or, in other words, (1.1) has a solution that can be represented by formula (2.44) as
with and . In (2.45), and the function has the same properties as the function . Therefore, formula (2.6) is valid for an arbitrary negative as well.

Now, we give an example which illustrates the case where there exists a solution to equation of the type (1.1) which is weighted asymptotically periodic, but is not asymptotically periodic.

*Example 2.3. *We consider (1.1) with
that is, the equation
The sequence is 2-periodic and
By virtue of Theorem 2.2, for any nonzero constant , there exists a solution of (1.1) which is weighed asymptotically 2-periodic. Let, for example, . Then,
and the sequence given by
or, equivalently,
is such a solution. We remark that such solution is not asymptotically 2-periodic in the meaning of Definition 1.1.

It is easy to verify that the sequence obtained from (2.51) if , , that is,
is a true solution of (2.47).

#### 3. Concluding Remarks and Open Problems

It is easy to prove the following corollary.

Corollary 3.1. *Let Theorem 2.2 be valid. If, moreover, , then every solution of (1.1) described by formula (2.6) satisfies
**
If , then, for every solution of (1.1) described by formula (2.6), one has
**
or/and
**
Finally, if , then, for every solution of (1.1) described by formula (2.6), one has
**
and if , then, for every solution of (1.1) described by formula (2.6), one has
*

Now, let us discuss the case when (1.6) holds, that is, when

Corollary 3.2. *Let Theorem 2.2 be valid. Assume that . Then, for any nonzero constant , there exists an asymptotically -periodic solution , of (1.1) such that
**
with
*

*Proof. *Putting in Theorem 2.2, we get
with

Due to the definition of , we see that the sequence
is an -periodic sequence. Since
for , we have
Therefore, the sequence
is a -periodic sequence. Set
Then,
The proof is completed.

*Remark 3.3. *From the proof, we see that Theorem 2.2 remains valid even in the case of . Then, there exists an βasymptotically weighted -periodic solutionβ of (1.1) as well. The formula (2.6) reduces to
since . In the light of Definition 1.2, we can treat this case as follows. We set (as a singular case) with an arbitrary (possibly other than ) period and with , .

*Remark 3.4. *The assumptions of Theorem 2.2 [1] are substantially different from those of the present Theorem 2.2. However, it is easy to see that Theorem 2.2 [1] is a particular case of the present Theorem 2.2 if (1.3) holds, that is, if . Therefore, our results can be viewed as a generalization of some results in [1].

In connection with the above investigations, some open problems arise.

*Open Problem 1. *The results of [1] are extended to systems of linear Volterra discrete equations in [16, 17]. It is an open question if the results presented can be extended to systems of linear Volterra discrete equations.

*Open Problem 2. *Unlike the result of Theorem 2.2 [1] where a parameter can be arbitrary, the assumptions of the results in [16, 17] are more restrictive since the related parameters should satisfy certain inequalities as well. Different results on the existence of asymptotically periodic solutions were recently proved in [8]. Using an example, it is shown that the results in [8] can be less restrictive. Therefore, an additional open problem arises if the results in [16, 17] can be improved in such a way that the related parameters can be arbitrary and if the expected extension of the results suggested in Open Problem 1 can be given in such a way that the related parameters can be arbitrary as well.

#### Acknowledgments

The first author has been supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government MSM 00216 30519, and by the project FEKT/FSI-S-11-1-1159. The second author has been supported by the Grant VEGA 1/0090/09 of the Grant Agency of Slovak Republic and by the Grant APVV-0700-07 of the Slovak Research and Development Agency.