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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 370982, 14 pages
http://dx.doi.org/10.1155/2011/370982
Research Article

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 66237 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
3Department of Mathematics, University of Žilina, 01026 Žilina, Slovakia
4Faculty of Electrical Engineering, Institute of Mathematics, Poznań University of Technology, 60965 Poznań, Poland

Received 16 January 2011; Accepted 17 March 2011

Academic Editor: Elena Braverman

Copyright © 2011 Josef Diblík et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A linear Volterra difference equation of the form 𝑥(𝑛+1)=𝑎(𝑛)+𝑏(𝑛)𝑥(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖), where 𝑥0, 𝑎0, 𝐾0×0 and 𝑏0{0} is 𝜔-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on 𝜔1𝑗=0𝑏(𝑗) is assumed. The results generalize some of the recent results.

1. Introduction

In the paper, we study a linear Volterra difference equation𝑥(𝑛+1)=𝑎(𝑛)+𝑏(𝑛)𝑥(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖),(1.1) where 𝑛0={0,1,2,}, 𝑎0, 𝐾0×0, and 𝑏0{0} is 𝜔-periodic, 𝜔={1,2,}. We will also adopt the customary notations 𝑘𝑖=𝑘+𝑠𝒪(𝑖)=0,𝑘𝑖=𝑘+𝑠𝒪(𝑖)=1,(1.2) where 𝑘 is an integer, 𝑠 is a positive integer, and 𝒪 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 assumption𝜔1𝑗=0𝑏(𝑗)=1,(1.3) and gave sufficient conditions for the existence of asymptotically 𝜔-periodic solutions of (1.1) where the notion for an asymptotically 𝜔-periodic function has been given by the following definition.

Definition 1.1. Let 𝜔 be a positive integer. The sequence 𝑦0 is called 𝜔-periodic if 𝑦(𝑛+𝜔)=𝑦(𝑛) for all 𝑛0. The sequence 𝑦 is called asymptotically 𝜔-periodic if there exist two sequences 𝑢,𝑣0 such that 𝑢 is 𝜔-periodic, lim𝑛𝑣(𝑛)=0, and 𝑦(𝑛)=𝑢(𝑛)+𝑣(𝑛)(1.4) for all 𝑛0.

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 𝜔-periodic solution of (1.1). We give a definition of a weighted asymptotically 𝜔-periodic function.

Definition 1.2. Let 𝜔 be a positive integer. The sequence 𝑦0 is called weighted asymptotically 𝜔-periodic if there exist two sequences 𝑢,𝑣0 such that 𝑢 is 𝜔-periodic and lim𝑛𝑣(𝑛)=0, and, moreover, if there exists a sequence 𝑤0{0} such that 𝑦(𝑛)𝑤(𝑛)=𝑢(𝑛)+𝑣(𝑛),(1.5) for all 𝑛0.

Apart from this, when we assume𝜔1𝑘=0𝑏(𝑘)=1,(1.6) then, as a consequence of our main result (Theorem 2.2), the existence of an asymptotically 2𝜔-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 Kocić and Ladas [4]. Volterra difference equations were studied by many others, for example, by Appleby et al. [5], by Elaydi and Murakami [6], by Győri and Horváth [7], by Győri and Reynolds [8], and by Song and Baker [9]. For some results on periodic solutions of difference equations, see, for example, [24, 1013] and the related references therein.

2. Weighted Asymptotically Periodic Solutions

In this section, sufficient conditions for the existence of weighted asymptotically 𝜔-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 Ω be a Banach space and 𝑆 its nonempty, closed, and convex subset and let 𝑇 be a continuous mapping such that 𝑇(𝑆) is contained in 𝑆 and the closure 𝑇(𝑆) is compact. Then, 𝑇 has a fixed point in 𝑆.

We set𝛽(𝑛)=𝑛1𝑗=0𝑏(𝑗),𝑛0,(2.1)=𝛽(𝜔).(2.2) Moreover, we define 𝑛=𝑛1𝜔𝑛1𝜔,(2.3) where is the floor function (the greatest-integer function) and 𝑛 is the “remainder” of dividing 𝑛1 by 𝜔. Obviously, {𝛽(𝑛)}, 𝑛 is an 𝜔-periodic sequence.

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

Theorem 2.2 (Main result). Let 𝜔 be a positive integer, 𝑏0{0} be 𝜔-periodic, 𝑎0, and 𝐾0×0. Assume that 𝑖=0||||𝑎(𝑖)𝛽(𝑖+1)||||<,𝑗=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||<1,(2.4) 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 𝑐, there exists 𝑎 weighted asymptotically 𝜔-periodic solution 𝑥0of (1.1) with 𝑢,𝑣0 and 𝑤0{0} in representation (1.5) such that 𝑤(𝑛)=(𝑛1)/𝜔,𝑢(𝑛)=𝑐𝛽𝑛+1,lim𝑛𝑣(𝑛)=0,(2.5) that is, 𝑥(𝑛)(𝑛1)/𝜔=𝑐𝛽𝑛+1+𝑣(𝑛),𝑛0.(2.6)

Proof. We will use a notation 𝑀=𝑗=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||,(2.7) whenever this is useful. Case 1. First assume 𝑐>0. We will define an auxiliary sequence of positive numbers {𝛼(𝑛)}, 𝑛0. We set 𝛼(0)=𝑖=0||𝑎(𝑖)/(𝛽(𝑖+1))||+𝑐𝑗=0𝑗𝑖=0||(𝐾(𝑗,𝑖)𝛽(𝑖))/(𝛽(𝑗+1))||1𝑗=0𝑗𝑖=0||(𝐾(𝑗,𝑖)𝛽(𝑖))/(𝛽(𝑗+1))||,(2.8) where the expression on the right-hand side is well defined due to (2.4). Moreover, we define 𝛼(𝑛)=𝑖=𝑛||||𝑎(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))𝑗=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||,(2.9) for 𝑛1. It is easy to see that lim𝑛𝛼(𝑛)=0.(2.10) We show, moreover, that 𝛼(𝑛)𝛼(0),(2.11) for any 𝑛. Let us first remark that 𝛼(0)=𝑖=0||||𝑎(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))𝑗=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||.(2.12) Then, due to the convergence of both series (see (2.4)), the inequality 𝛼(0)=𝑖=0||||𝑎(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))𝑗=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||𝑖=𝑛||||𝑎(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))𝑗=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=𝛼(𝑛)(2.13) obviously holds for every 𝑛 and (2.11) is proved.
Let 𝐵 be the Banach space of all real bounded sequences 𝑧0 equipped with the usual supremum norm 𝑧=sup𝑛0|𝑧(𝑛)| for 𝑧𝐵. We define a subset 𝑆𝐵 as 𝑆=𝑧𝐵𝑐𝛼(0)𝑧(𝑛)𝑐+𝛼(0),𝑛0.(2.14) It is not difficult to prove that 𝑆 is a nonempty, bounded, convex, and closed subset of 𝐵.
Let us define a mapping 𝑇𝑆𝐵 as follows: (𝑇𝑧)(𝑛)=𝑐𝑖=𝑛𝑎(𝑖)𝛽(𝑖+1)𝑗=𝑛𝑗𝑖=0𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)𝑧(𝑖),(2.15) for any 𝑛0.
We will prove that the mapping 𝑇 has a fixed point in 𝑆.
We first show that 𝑇(𝑆)𝑆. Indeed, if 𝑧𝑆, then |𝑧(𝑛)𝑐|𝛼(0) for 𝑛0 and, by (2.11) and (2.15), we have ||(𝑇𝑧)(𝑛)𝑐||𝑖=𝑛||||𝑎(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))𝑗=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=𝛼(𝑛)𝛼(0).(2.16) Next, we prove that 𝑇 is continuous. Let 𝑧(𝑝) be a sequence in 𝑆 such that 𝑧(𝑝)𝑧 as 𝑝. Because 𝑆 is closed, 𝑧𝑆. Now, utilizing (2.15), we get ||𝑇𝑧(𝑝)(𝑛)(𝑇𝑧)(𝑛)||=|||||𝑗=𝑛𝑗𝑖=0𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)𝑧(𝑝)(𝑖)𝑧(𝑖)|||||𝑀sup𝑖0||𝑧(𝑝)(𝑖)𝑧(𝑖)||=𝑀𝑧(𝑝)𝑧,𝑛0.(2.17) Therefore, 𝑇𝑧(𝑝)𝑇𝑧𝑀𝑧(𝑝)𝑧,lim𝑝𝑇𝑧(𝑝)𝑇𝑧=0.(2.18) This means that 𝑇 is continuous.
Now, we show that 𝑇(𝑆) is compact. As is generally known, it is enough to verify that every 𝜀-open covering of 𝑇(𝑆) contains a finite 𝜀-subcover of 𝑇(𝑆), that is, finitely many of these open sets already cover 𝑇(𝑆) ([15], page 756 (12)). Thus, to prove that 𝑇(𝑆) is compact, we take an arbitrary 𝜀>0 and assume that an open 𝜀-cover 𝒞𝜀 of 𝑇(𝑆) is given. Then, from (2.10), we conclude that there exists an 𝑛𝜀 such that 𝛼(𝑛)<𝜀/4 for 𝑛𝑛𝜀.
Suppose that 𝑥1𝑇𝑇(𝑆) is one of the elements generating the 𝜀-cover 𝒞𝜀 of 𝑇(𝑆). Then (as follows from (2.16)), for an arbitrary 𝑥𝑇𝑇(S), ||𝑥1𝑇(𝑛)𝑥𝑇(𝑛)||<𝜀(2.19) if 𝑛𝑛𝜀. In other words, the 𝜀-neighborhood of 𝑥1𝑇𝑐: 𝑥1𝑇𝑐<𝜀,(2.20) where 𝑐={𝑐,𝑐,}𝑆 covers the set 𝑇(𝑆) on an infinite interval 𝑛𝑛𝜀. It remains to cover the rest of 𝑇(𝑆) on a finite interval for 𝑛{0,1,𝑛𝜀1} by a finite number of 𝜀-neighborhoods of elements generating 𝜀-cover 𝒞𝜀. Supposing that 𝑥1𝑇 itself is not able to generate such cover, we fix 𝑛{0,1,,𝑛𝜀1} and split the interval [𝑐𝛼(𝑛),𝑐+𝛼(𝑛)](2.21) into a finite number (𝜀,𝑛) of closed subintervals 𝐼1(𝑛),𝐼2(𝑛),,𝐼(𝜀,𝑛)(𝑛)(2.22) each with a length not greater then 𝜀/2 such that (𝜀,𝑛)𝑖=1𝐼𝑖(𝑛)=[𝑐𝛼(𝑛),𝑐+𝛼(𝑛)],int𝐼𝑖(𝑛)int𝐼𝑗(𝑛)=,𝑖,𝑗=1,2,,(𝜀,𝑛),𝑖𝑗.(2.23) Finally, the set 𝑛𝜀1𝑛=0[𝑐𝛼(𝑛),𝑐+𝛼(𝑛)](2.24) equals 𝑛𝜀1𝑛=0(𝜀,𝑛)𝑖=1𝐼𝑖(𝑛)(2.25) and can be divided into a finite number 𝑀𝜀=𝑛𝜀1𝑛=0(𝜀,𝑛)(2.26) of different subintervals (2.22). This means that, at most, 𝑀𝜀 of elements generating the cover 𝒞𝜀 are sufficient to generate a finite 𝜀-subcover of 𝑇(𝑆) for 𝑛{0,1,,𝑛𝜀1}. We remark that each of such elements simultaneously plays the same role as 𝑥1𝑇(𝑛) for 𝑛𝑛𝜀. Since 𝜀>0 can be chosen as arbitrarily small, 𝑇(𝑆) is compact.
By Schauder's fixed point theorem, there exists a 𝑧𝑆 such that 𝑧(𝑛)=(𝑇𝑧)(𝑛) for 𝑛0. Thus, 𝑧(𝑛)=𝑐𝑖=𝑛𝑎(𝑖)𝛽(𝑖+1)𝑗=𝑛𝑗𝑖=0𝛽(𝑖)𝛽(𝑗+1)𝐾(𝑗,𝑖)𝑧(𝑖),(2.27) for any 𝑛0.
Due to (2.10) and (2.16), for fixed point 𝑧𝑆 of 𝑇, we have lim𝑛||𝑧(𝑛)𝑐||=lim𝑛||(𝑇𝑧)(𝑛)𝑐||lim𝑛𝛼(𝑛)=0,(2.28) or, equivalently, lim𝑛𝑧(𝑛)=𝑐.(2.29) Finally, we will show that there exists a connection between the fixed point 𝑧𝑆 and the existence of a solution of (1.1) which divided by (𝑛1)/𝜔 provides an asymptotically 𝜔-periodic sequence. Considering (2.27) for 𝑧(𝑛+1) and 𝑧(𝑛), we get Δ𝑧(𝑛)=𝑎(𝑛)𝛽(𝑛+1)+𝑛𝑖=0𝛽(𝑖)𝛽(𝑛+1)𝐾(𝑛,𝑖)𝑧(𝑖),(2.30) where 𝑛0. Hence, we have 𝑧(𝑛+1)𝑧(𝑛)=𝑎(𝑛)𝛽(𝑛+1)+1𝛽(𝑛+1)𝑛𝑖=0𝛽(𝑖)𝐾(𝑛,𝑖)𝑧(𝑖),𝑛0.(2.31) Putting 𝑧(𝑛)=𝑥(𝑛)𝛽(𝑛),𝑛0(2.32) in (2.31), we get (1.1) since 𝑥(𝑛+1)𝛽(𝑛+1)𝑥(𝑛)𝛽(𝑛)=𝑎(𝑛)𝛽(𝑛+1)+1𝛽(𝑛+1)𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖),𝑛0(2.33) yields 𝑥(𝑛+1)=𝑎(𝑛)+𝑏(𝑛)𝑥(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖),𝑛0.(2.34) Consequently, 𝑥 defined by (2.32) is a solution of (1.1). From (2.29) and (2.32), we obtain 𝑥(𝑛)𝛽(𝑛)=𝑧(𝑛)=𝑐+𝑜(1),(2.35) for 𝑛 (where 𝑜(1) is the Landau order symbol). Hence, 𝑥(𝑛)=𝛽(𝑛)(𝑐+𝑜(1)),𝑛.(2.36) It is easy to show that the function 𝛽 defined by (2.1) can be expressed in the form 𝛽(𝑛)=𝑛1𝑗=0𝑏(𝑗)=(𝑛1)/𝜔𝛽𝑛+1,(2.37) for 𝑛0. Then, as follows from (2.36), 𝑥(𝑛)=(𝑛1)/𝜔𝛽𝑛+1(𝑐+𝑜(1)),𝑛,(2.38) or 𝑥(𝑛)(𝑛1)/𝜔=𝑐𝛽𝑛+1+𝛽𝑛+1𝑜(1),𝑛.(2.39)
The proof is completed since the sequence {𝛽(𝑛+1)} is 𝜔-periodic, hence bounded and, due to the properties of Landau order symbols, we have 𝛽𝑛+1𝑜(1)=𝑜(1),𝑛,(2.40) and it is easy to see that the choice 𝑢(𝑛)=𝑐𝛽𝑛+1,𝑤(𝑛)=(𝑛1)/𝜔,𝑛0,(2.41) and an appropriate function 𝑣0 such that lim𝑛𝑣(𝑛)=0(2.42) finishes this part of the proof. Although for 𝑛=0, there is no correspondence between formula (2.36) and the definitions of functions 𝑢 and 𝑤, we assume that function 𝑣 makes up for this.
Case 2. If 𝑐<0, we can proceed as follows. It is easy to see that arbitrary solution 𝑦=𝑦(𝑛) of the equation 𝑦(𝑛+1)=𝑎(𝑛)+𝑏(𝑛)𝑦(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑦(𝑖)(2.43) defines a solution 𝑥=𝑥(𝑛) of (1.1) since a substitution 𝑦(𝑛)=𝑥(𝑛) 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 𝑐>0, (2.43) has a solution that can be represented by formula (2.6), that is, 𝑦(𝑛)(𝑛1)/𝜔=𝑐𝛽𝑛+1+𝑣(𝑛),𝑛0.(2.44) Or, in other words, (1.1) has a solution that can be represented by formula (2.44) as 𝑥(𝑛)(𝑛1)/𝜔=𝑐0𝛽𝑛+1+𝑣(𝑛),𝑛0,(2.45) with 𝑐0=𝑐 and 𝑣(𝑛)=𝑣(𝑛). In (2.45), 𝑐0<0 and the function 𝑣(𝑛) has the same properties as the function 𝑣(𝑛). Therefore, formula (2.6) is valid for an arbitrary negative 𝑐 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 𝑎(𝑛)=(1)𝑛+1113𝑛+1,𝑏(𝑛)=3(1)𝑛,𝐾(𝑛,𝑖)=(1)𝑛+(𝑖(𝑖1))/2132𝑖,(2.46) that is, the equation 𝑥(𝑛+1)=(1)𝑛+1113𝑛+1+3(1)𝑛𝑥(𝑛)+𝑛𝑖=0(1)𝑛+(𝑖(𝑖1))/2132𝑖𝑥(𝑖).(2.47) The sequence 𝑏(𝑛) is 2-periodic and 𝛽(𝑛)=𝑛1𝑗=0𝑏(𝑗)=(1)𝑛(𝑛1)/23𝑛,=𝛽(𝜔)=𝛽(2)=9,𝛽𝑛+1=3+6(1)𝑛+1,𝑎(𝑛)𝛽(𝑛+1)=(1)(𝑛2+𝑛+2)/213𝑛+1132(𝑛+1),𝑖=0||||𝑎(𝑖)𝛽(𝑖+1)||||<,𝑗=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||<𝑗=0𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=𝑗=0𝑖=013𝑖+𝑗+1=13𝑗=013𝑗𝑖=013𝑖=13111/3111/3=133232=34<1.(2.48) By virtue of Theorem 2.2, for any nonzero constant 𝑐, there exists a solution 𝑥0 of (1.1) which is weighed asymptotically 2-periodic. Let, for example, 𝑐=2/3. Then, 𝑤(𝑛)=(9)(𝑛1)/2,𝑢(𝑛)=𝑐𝛽𝑛+1=233+6(1)𝑛+1=2+4(1)𝑛+1,(2.49) and the sequence 𝑥(𝑛) given by 𝑥(𝑛)(9)(𝑛1)/2=2+4(1)𝑛+1+𝑣(𝑛),𝑛0,(2.50) or, equivalently, 𝑥(𝑛)=(9)(𝑛1)/22+4(1)𝑛+1+𝑣(𝑛),𝑛0(2.51) 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 𝑥(𝑛) obtained from (2.51) if 𝑣(𝑛)=0, 𝑛0, that is, 𝑥(𝑛)=(9)(𝑛1)/22+4(1)𝑛+1=23(1)𝑛(𝑛1)/23𝑛,𝑛0(2.52) 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, ||<1, then every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6) satisfies lim𝑛𝑥(𝑛)=0.(3.1) If ||>1, then, for every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6), one has liminf𝑛𝑥(𝑛)=(3.2) or/and limsup𝑛𝑥(𝑛)=.(3.3) Finally, if >1, then, for every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6), one has lim𝑛𝑥(𝑛)=,(3.4) and if <1, then, for every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6), one has lim𝑛𝑥(𝑛)=.(3.5)

Now, let us discuss the case when (1.6) holds, that is, when =𝜔1𝑗=0𝑏(𝑗)=1.(3.6)

Corollary 3.2. Let Theorem 2.2 be valid. Assume that =1. Then, for any nonzero constant 𝑐, there exists an asymptotically 2𝜔-periodic solution 𝑥=𝑥(𝑛), 𝑛0 of (1.1) such that 𝑥(𝑛)=(1)(𝑛1)/𝜔𝑢(𝑛)+𝑧(𝑛),𝑛0,(3.7) with 𝑢(𝑛)=𝑐𝛽𝑛+1,lim𝑛𝑧(𝑛)=0.(3.8)

Proof. Putting =1 in Theorem 2.2, we get 𝑥(𝑛)=(1)(𝑛1)/𝜔𝑢(𝑛)+(1)(𝑛1)/𝜔𝑣(𝑛),(3.9) with 𝑢(𝑛)=𝑐𝛽𝑛+1,lim𝑛𝑣(𝑛)=0.(3.10)
Due to the definition of 𝑛, we see that the sequence 𝛽𝑛+1={𝛽(𝜔),𝛽(1),𝛽(2),,𝛽(𝜔),𝛽(1),𝛽(2),,𝛽(𝜔),},(3.11) is an 𝜔-periodic sequence. Since 𝑛1𝜔=1,0,,0𝜔,1,,1𝜔,2,,(3.12) for 𝑛0, we have (1)(𝑛1)/𝜔=1,1,,1𝜔,1,,1𝜔,1,.(3.13) Therefore, the sequence (1)(𝑛1)/𝜔𝑢(𝑛)=𝑐{𝛽(𝜔),𝛽(1),𝛽(2),,𝛽(𝜔),𝛽(1),𝛽(2),,𝛽(𝜔),}(3.14) is a 2𝜔-periodic sequence. Set 𝑧(𝑛)=(1)(𝑛1)/𝜔𝑣(𝑛).(3.15) Then, lim𝑛𝑧(𝑛)=0.(3.16) The proof is completed.

Remark 3.3. From the proof, we see that Theorem 2.2 remains valid even in the case of 𝑐=0. Then, there exists an “asymptotically weighted 𝜔-periodic solution” 𝑥=𝑥(𝑛) of (1.1) as well. The formula (2.6) reduces to 𝑥(𝑛)=(𝑛1)/𝜔𝑣(𝑛)=𝑜(1),𝑛0,(3.17) since 𝑢(𝑛)=0. In the light of Definition 1.2, we can treat this case as follows. We set (as a singular case) 𝑢0 with an arbitrary (possibly other than 𝜔'') period and with 𝑣=𝑜(1), 𝑛.

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 =1. 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 c 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.

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