Abstract and Applied Analysis

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Recent Progress in Differential and Difference Equations

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Volume 2011 |Article ID 370982 | https://doi.org/10.1155/2011/370982

Josef Diblík, Miroslava Růžičková, Ewa Schmeidel, Małgorzata Zbąszyniak, "Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations", Abstract and Applied Analysis, vol. 2011, Article ID 370982, 14 pages, 2011. https://doi.org/10.1155/2011/370982

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

Academic Editor: Elena Braverman
Received16 Jan 2011
Accepted17 Mar 2011
Published22 May 2011

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.

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