Abstract

A linear Volterra difference equation of the form 𝑥(𝑛+1)=ğ‘Ž(𝑛)+𝑏(𝑛)𝑥(𝑛)+∑𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖), where 𝑥∶ℕ0→ℝ, ğ‘Žâˆ¶â„•0→ℝ, 𝐾∶ℕ0×ℕ0→ℝ and 𝑏∶ℕ0→ℝ⧵{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 ∏𝜔−1𝑗=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)=ğ‘Ž(𝑛)+𝑏(𝑛)𝑥(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖),(1.1) where 𝑛∈ℕ0∶={0,1,2,…}, ğ‘Žâˆ¶â„•0→ℝ, 𝐾∶ℕ0×ℕ0→ℝ, and 𝑏∶ℕ0→ℝ⧵{0} is 𝜔-periodic, 𝜔∈ℕ∶={1,2,…}. We will also adopt the customary notations 𝑘𝑖=𝑘+𝑠𝒪(𝑖)=0,𝑘𝑖=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 assumption𝜔−1𝑗=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 𝑦∶ℕ0→ℝ is called 𝜔-periodic if 𝑦(𝑛+𝜔)=𝑦(𝑛) for all 𝑛∈ℕ0. The sequence 𝑦 is called asymptotically 𝜔-periodic if there exist two sequences 𝑢,𝑣∶ℕ0→ℝ such that 𝑢 is 𝜔-periodic, limğ‘›â†’âˆžğ‘£(𝑛)=0, and 𝑦(𝑛)=𝑢(𝑛)+𝑣(𝑛)(1.4) for all 𝑛∈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 𝑦∶ℕ0→ℝ is called weighted asymptotically 𝜔-periodic if there exist two sequences 𝑢,𝑣∶ℕ0→ℝ such that 𝑢 is 𝜔-periodic and limğ‘›â†’âˆžğ‘£(𝑛)=0, and, moreover, if there exists a sequence 𝑤∶ℕ0→ℝ⧵{0} such that 𝑦(𝑛)𝑤(𝑛)=𝑢(𝑛)+𝑣(𝑛),(1.5) for all 𝑛∈ℕ0.

Apart from this, when we assume𝜔−1𝑘=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, [2–4, 10–13] 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)∶=𝑛−1𝑗=0𝑏(𝑗),𝑛∈ℕ0,(2.1)ℬ∶=𝛽(𝜔).(2.2) Moreover, we define 𝑛∗∶=𝑛−1âˆ’ğœ”î‚žğ‘›âˆ’1𝜔,(2.3) where ⌊⋅⌋ is the floor function (the greatest-integer function) and 𝑛∗ is the “remainder” of dividing 𝑛−1 by 𝜔. Obviously, {𝛽(𝑛∗)}, 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, 𝑏∶ℕ0→ℝ⧵{0} be 𝜔-periodic, ğ‘Žâˆ¶â„•0→ℝ, and 𝐾∶ℕ0×ℕ0→ℝ. Assume that âˆžî“ğ‘–=0||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||<∞,âˆžî“ğ‘—=0𝑗𝑖=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 ğ‘Ž weighted asymptotically 𝜔-periodic solution 𝑥∶ℕ0→ℝof (1.1) with 𝑢,𝑣∶ℕ0→ℝ and 𝑤∶ℕ0→ℝ⧵{0} in representation (1.5) such that 𝑤(𝑛)=ℬ⌊(𝑛−1)/𝜔⌋,𝑢(𝑛)∶=𝑐𝛽𝑛∗+1,limğ‘›â†’âˆžğ‘£(𝑛)=0,(2.5) that is, 𝑥(𝑛)ℬ⌊(𝑛−1)/𝜔⌋=𝑐𝛽𝑛∗+1+𝑣(𝑛),𝑛∈ℕ0.(2.6)

Proof. We will use a notation 𝑀∶=âˆžî“ğ‘—=0𝑗𝑖=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)}, 𝑛∈ℕ0. We set 𝛼(0)∶=âˆ‘âˆžğ‘–=0||ğ‘Ž(𝑖)/(𝛽(𝑖+1))||+ğ‘âˆ‘âˆžğ‘—=0∑𝑗𝑖=0||(𝐾(𝑗,𝑖)𝛽(𝑖))/(𝛽(𝑗+1))||1âˆ’âˆ‘âˆžğ‘—=0∑𝑗𝑖=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)∶=âˆžî“ğ‘–=𝑛||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))âˆžî“ğ‘—=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||,(2.9) for 𝑛≥1. It is easy to see that limğ‘›â†’âˆžğ›¼(𝑛)=0.(2.10) We show, moreover, that 𝛼(𝑛)≤𝛼(0),(2.11) for any 𝑛∈ℕ. Let us first remark that 𝛼(0)=âˆžî“ğ‘–=0||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))âˆžî“ğ‘—=0𝑗𝑖=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)=âˆžî“ğ‘–=0||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))âˆžî“ğ‘—=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||â‰¥âˆžî“ğ‘–=𝑛||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))âˆžî“ğ‘—=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=𝛼(𝑛)(2.13) obviously holds for every 𝑛∈ℕ and (2.11) is proved.
Let 𝐵 be the Banach space of all real bounded sequences 𝑧∶ℕ0→ℝ equipped with the usual supremum norm ‖𝑧‖=sup𝑛∈ℕ0|𝑧(𝑛)| for 𝑧∈𝐵. We define a subset 𝑆⊂𝐵 as 𝑆∶=𝑧∈𝐵∶𝑐−𝛼(0)≤𝑧(𝑛)≤𝑐+𝛼(0),𝑛∈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 𝑇∶𝑆→𝐵 as follows: (𝑇𝑧)(𝑛)=ğ‘âˆ’âˆžî“ğ‘–=ğ‘›ğ‘Ž(𝑖)𝛽(𝑖+1)âˆ’âˆžî“ğ‘—=𝑛𝑗𝑖=0𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)𝑧(𝑖),(2.15) for any 𝑛∈ℕ0.
We will prove that the mapping 𝑇 has a fixed point in 𝑆.
We first show that 𝑇(𝑆)⊂𝑆. Indeed, if 𝑧∈𝑆, then |𝑧(𝑛)−𝑐|≤𝛼(0) for 𝑛∈ℕ0 and, by (2.11) and (2.15), we have ||(𝑇𝑧)(𝑛)−𝑐||â‰¤âˆžî“ğ‘–=𝑛||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))âˆžî“ğ‘—=𝑛𝑗𝑖=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 ğ‘â†’âˆž. Because 𝑆 is closed, 𝑧∈𝑆. Now, utilizing (2.15), we get ||𝑇𝑧(𝑝)(𝑛)−(𝑇𝑧)(𝑛)||=|||||âˆžî“ğ‘—=𝑛𝑗𝑖=0𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)𝑧(𝑝)(𝑖)−𝑧(𝑖)|||||≤𝑀sup𝑖≥0||𝑧(𝑝)(𝑖)−𝑧(𝑖)||=𝑀‖‖𝑧(𝑝)−𝑧‖‖,𝑛∈ℕ0.(2.17) Therefore, ‖‖𝑇𝑧(𝑝)−𝑇𝑧‖‖≤𝑀‖‖𝑧(𝑝)−𝑧‖‖,limğ‘â†’âˆžâ€–â€–ğ‘‡ğ‘§(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 ğ’žğœ€ of 𝑇(𝑆) is given. Then, from (2.10), we conclude that there exists an 𝑛𝜀∈ℕ such that 𝛼(𝑛)<𝜀/4 for 𝑛≥𝑛𝜀.
Suppose that 𝑥1𝑇∈𝑇(𝑆) is one of the elements generating the 𝜀-cover ğ’žğœ€ of 𝑇(𝑆). Then (as follows from (2.16)), for an arbitrary 𝑥𝑇∈𝑇(S), ||𝑥1𝑇(𝑛)−𝑥𝑇(𝑛)||<𝜀(2.19) if 𝑛≥𝑛𝜀. In other words, the 𝜀-neighborhood of 𝑥1𝑇−𝑐∗: ‖‖𝑥1𝑇−𝑐∗‖‖<𝜀,(2.20) where 𝑐∗={𝑐,𝑐,…}∈𝑆 covers the set 𝑇(𝑆) on an infinite interval 𝑛≥𝑛𝜀. It remains to cover the rest of 𝑇(𝑆) on a finite interval for 𝑛∈{0,1…,𝑛𝜀−1} by a finite number of 𝜀-neighborhoods of elements generating 𝜀-cover ğ’žğœ€. Supposing that 𝑥1𝑇 itself is not able to generate such cover, we fix 𝑛∈{0,1,…,𝑛𝜀−1} and split the interval [𝑐−𝛼(𝑛),𝑐+𝛼(𝑛)](2.21) into a finite number ℎ(𝜀,𝑛) of closed subintervals 𝐼1(𝑛),𝐼2(𝑛),…,ğ¼â„Ž(𝜀,𝑛)(𝑛)(2.22) each with a length not greater then 𝜀/2 such that ℎ(𝜀,𝑛)𝑖=1𝐼𝑖(𝑛)=[𝑐−𝛼(𝑛),𝑐+𝛼(𝑛)],int𝐼𝑖(𝑛)∩int𝐼𝑗(𝑛)=∅,𝑖,𝑗=1,2,…,ℎ(𝜀,𝑛),𝑖≠𝑗.(2.23) Finally, the set 𝑛𝜀−1𝑛=0[𝑐−𝛼(𝑛),𝑐+𝛼(𝑛)](2.24) equals 𝑛𝜀−1𝑛=0ℎ(𝜀,𝑛)𝑖=1𝐼𝑖(𝑛)(2.25) and can be divided into a finite number 𝑀𝜀∶=𝑛𝜀−1𝑛=0ℎ(𝜀,𝑛)(2.26) of different subintervals (2.22). This means that, at most, 𝑀𝜀 of elements generating the cover ğ’žğœ€ are sufficient to generate a finite 𝜀-subcover of 𝑇(𝑆) for 𝑛∈{0,1,…,𝑛𝜀−1}. We remark that each of such elements simultaneously plays the same role as 𝑥1𝑇(𝑛) for 𝑛≥𝑛𝜀. Since 𝜀>0 can be chosen as arbitrarily small, 𝑇(𝑆) is compact.
By Schauder's fixed point theorem, there exists a 𝑧∈𝑆 such that 𝑧(𝑛)=(𝑇𝑧)(𝑛) for 𝑛∈ℕ0. Thus, 𝑧(𝑛)=ğ‘âˆ’âˆžî“ğ‘–=ğ‘›ğ‘Ž(𝑖)𝛽(𝑖+1)âˆ’âˆžî“ğ‘—=𝑛𝑗𝑖=0𝛽(𝑖)𝛽(𝑗+1)𝐾(𝑗,𝑖)𝑧(𝑖),(2.27) for any 𝑛∈ℕ0.
Due to (2.10) and (2.16), for fixed point 𝑧∈𝑆 of 𝑇, we have limğ‘›â†’âˆž||𝑧(𝑛)−𝑐||=limğ‘›â†’âˆž||(𝑇𝑧)(𝑛)−𝑐||≤limğ‘›â†’âˆžğ›¼(𝑛)=0,(2.28) or, equivalently, limğ‘›â†’âˆžğ‘§(𝑛)=𝑐.(2.29) Finally, we will show that there exists a connection between the fixed point 𝑧∈𝑆 and the existence of a solution of (1.1) which divided by ℬ⌊(𝑛−1)/𝜔⌋ provides an asymptotically 𝜔-periodic sequence. Considering (2.27) for 𝑧(𝑛+1) and 𝑧(𝑛), we get Δ𝑧(𝑛)=ğ‘Ž(𝑛)𝛽(𝑛+1)+𝑛𝑖=0𝛽(𝑖)𝛽(𝑛+1)𝐾(𝑛,𝑖)𝑧(𝑖),(2.30) where 𝑛∈ℕ0. Hence, we have 𝑧(𝑛+1)−𝑧(𝑛)=ğ‘Ž(𝑛)𝛽(𝑛+1)+1𝛽(𝑛+1)𝑛𝑖=0𝛽(𝑖)𝐾(𝑛,𝑖)𝑧(𝑖),𝑛∈ℕ0.(2.31) Putting 𝑧(𝑛)=𝑥(𝑛)𝛽(𝑛),𝑛∈ℕ0(2.32) in (2.31), we get (1.1) since 𝑥(𝑛+1)𝛽(𝑛+1)−𝑥(𝑛)𝛽(𝑛)=ğ‘Ž(𝑛)𝛽(𝑛+1)+1𝛽(𝑛+1)𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖),𝑛∈ℕ0(2.33) yields 𝑥(𝑛+1)=ğ‘Ž(𝑛)+𝑏(𝑛)𝑥(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖),𝑛∈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 ğ‘›â†’âˆž (where 𝑜(1) is the Landau order symbol). Hence, 𝑥(𝑛)=𝛽(𝑛)(𝑐+𝑜(1)),ğ‘›âŸ¶âˆž.(2.36) It is easy to show that the function 𝛽 defined by (2.1) can be expressed in the form 𝛽(𝑛)=𝑛−1𝑗=0𝑏(𝑗)=ℬ⌊(𝑛−1)/𝜔⌋⋅𝛽𝑛∗+1,(2.37) for 𝑛∈ℕ0. Then, as follows from (2.36), 𝑥(𝑛)=ℬ⌊(𝑛−1)/𝜔⌋⋅𝛽𝑛∗+1(𝑐+𝑜(1)),ğ‘›âŸ¶âˆž,(2.38) or 𝑥(𝑛)ℬ⌊(𝑛−1)/𝜔⌋=𝑐𝛽𝑛∗+1+𝛽𝑛∗+1𝑜(1),ğ‘›âŸ¶âˆž.(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 𝛽𝑛∗+1𝑜(1)=𝑜(1),ğ‘›âŸ¶âˆž,(2.40) and it is easy to see that the choice 𝑢(𝑛)∶=𝑐𝛽𝑛∗+1,𝑤(𝑛)∶=ℬ⌊(𝑛−1)/𝜔⌋,𝑛∈ℕ0,(2.41) and an appropriate function 𝑣∶ℕ0→ℝ such that limğ‘›â†’âˆžğ‘£(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)=âˆ’ğ‘Ž(𝑛)+𝑏(𝑛)𝑦(𝑛)+𝑛𝑖=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)ℬ⌊(𝑛−1)/𝜔⌋=𝑐𝛽𝑛∗+1+𝑣(𝑛),𝑛∈ℕ0.(2.44) Or, in other words, (1.1) has a solution that can be represented by formula (2.44) as 𝑥(𝑛)ℬ⌊(𝑛−1)/𝜔⌋=𝑐0𝛽𝑛∗+1+𝑣∗(𝑛),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 ğ‘Ž(𝑛)=(−1)𝑛+11−13𝑛+1,𝑏(𝑛)=3(−1)𝑛,𝐾(𝑛,𝑖)=(−1)𝑛+(𝑖(𝑖−1))/2132𝑖,(2.46) that is, the equation 𝑥(𝑛+1)=(−1)𝑛+11−13𝑛+1+3(−1)𝑛𝑥(𝑛)+𝑛𝑖=0(−1)𝑛+(𝑖(𝑖−1))/2132𝑖𝑥(𝑖).(2.47) The sequence 𝑏(𝑛) is 2-periodic and 𝛽(𝑛)=𝑛−1𝑗=0𝑏(𝑗)=(−1)𝑛(𝑛−1)/23𝑛,ℬ=𝛽(𝜔)=𝛽(2)=−9,𝛽𝑛∗+1=−3+6(−1)𝑛+1,ğ‘Ž(𝑛)𝛽(𝑛+1)=(−1)(−𝑛2+𝑛+2)/213𝑛+1−132(𝑛+1),âˆžî“ğ‘–=0||||ğ‘Ž(𝑖)𝛽(𝑖+1)||||<∞,âˆžî“ğ‘—=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||<âˆžî“ğ‘—=0âˆžî“ğ‘–=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=âˆžî“ğ‘—=0âˆžî“ğ‘–=013𝑖+𝑗+1=13âŽ›âŽœâŽâˆžî“ğ‘—=013ğ‘—âŽžâŽŸâŽ âŽ›âŽœâŽâˆžî“ğ‘–=013ğ‘–âŽžâŽŸâŽ =13⋅11−1/3⋅11−1/3=13⋅32⋅32=34<1.(2.48) By virtue of Theorem 2.2, for any nonzero constant 𝑐, there exists a solution 𝑥∶ℕ0→ℝ of (1.1) which is weighed asymptotically 2-periodic. Let, for example, 𝑐=2/3. Then, 𝑤(𝑛)=(−9)⌊(𝑛−1)/2⌋,𝑢(𝑛)=𝑐𝛽𝑛∗+1=23−3+6(−1)𝑛+1=−2+4(−1)𝑛+1,(2.49) and the sequence 𝑥(𝑛) given by 𝑥(𝑛)(−9)⌊(𝑛−1)/2⌋=−2+4(−1)𝑛+1+𝑣(𝑛),𝑛∈ℕ0,(2.50) or, equivalently, 𝑥(𝑛)=(−9)⌊(𝑛−1)/2⌋−2+4(−1)𝑛+1+𝑣(𝑛),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, 𝑛∈ℕ0, that is, 𝑥∗(𝑛)=(−9)⌊(𝑛−1)/2⌋−2+4(−1)𝑛+1=23⋅(−1)𝑛(𝑛−1)/2⋅3𝑛,𝑛∈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, |ℬ|<1, then every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6) satisfies limğ‘›â†’âˆžğ‘¥(𝑛)=0.(3.1) If |ℬ|>1, then, for every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6), one has liminfğ‘›â†’âˆžğ‘¥(𝑛)=−∞(3.2) or/and limsupğ‘›â†’âˆžğ‘¥(𝑛)=∞.(3.3) Finally, if ℬ>1, then, for every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6), one has limğ‘›â†’âˆžğ‘¥(𝑛)=∞,(3.4) and if ℬ<−1, then, for every solution 𝑥=𝑥(𝑛) of (1.1) described by formula (2.6), one has limğ‘›â†’âˆžğ‘¥(𝑛)=−∞.(3.5)

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

Corollary 3.2. Let Theorem 2.2 be valid. Assume that ℬ=−1. Then, for any nonzero constant 𝑐, there exists an asymptotically 2𝜔-periodic solution 𝑥=𝑥(𝑛), 𝑛∈ℕ0 of (1.1) such that 𝑥(𝑛)=(−1)⌊(𝑛−1)/𝜔⌋𝑢(𝑛)+𝑧(𝑛),𝑛∈ℕ0,(3.7) with 𝑢(𝑛)∶=𝑐𝛽𝑛∗+1,limğ‘›â†’âˆžğ‘§(𝑛)=0.(3.8)

Proof. Putting ℬ=−1 in Theorem 2.2, we get 𝑥(𝑛)=(−1)⌊(𝑛−1)/𝜔⌋𝑢(𝑛)+(−1)⌊(𝑛−1)/𝜔⌋𝑣(𝑛),(3.9) with 𝑢(𝑛)∶=𝑐𝛽𝑛∗+1,limğ‘›â†’âˆžğ‘£(𝑛)=0.(3.10)
Due to the definition of 𝑛∗, we see that the sequence 𝛽𝑛∗+1={𝛽(𝜔),𝛽(1),𝛽(2),…,𝛽(𝜔),𝛽(1),𝛽(2),…,𝛽(𝜔),…},(3.11) is an 𝜔-periodic sequence. Since î‚†î‚žğ‘›âˆ’1𝜔=⎧⎪⎨⎪⎩−1,0,…,0𝜔,1,…,1𝜔,2,…⎫⎪⎬⎪⎭,(3.12) for 𝑛∈ℕ0, we have (−1)⌊(𝑛−1)/𝜔⌋=⎧⎪⎨⎪⎩−1,1,…,1𝜔,−1,…,−1𝜔,1,…⎫⎪⎬⎪⎭.(3.13) Therefore, the sequence (−1)⌊(𝑛−1)/𝜔⌋𝑢(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)⌊(𝑛−1)/𝜔⌋𝑣(𝑛).(3.15) Then, limğ‘›â†’âˆžğ‘§(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)=ℬ⌊(𝑛−1)/𝜔⌋𝑣(𝑛)=𝑜(1),𝑛∈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) 𝑢≡0 with an arbitrary (possibly other than 𝜔'') period and with 𝑣=𝑜(1), ğ‘›â†’âˆž.

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 ℬ=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.