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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 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 Ε½ilina, 01026 Ε½ilina, Slovakia
4Faculty of Electrical Engineering, Institute of Mathematics, PoznaΕ„ University of Technology, 60965 PoznaΕ„, Poland

Received 16 January 2011; Accepted 17 March 2011

Academic Editor: ElenaΒ Braverman

Copyright Β© 2011 Josef DiblΓ­k 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 π‘₯(𝑛+1)=π‘Ž(𝑛)+𝑏(𝑛)π‘₯(𝑛)+βˆ‘π‘›π‘–=0𝐾(𝑛,𝑖)π‘₯(𝑖), where π‘₯βˆΆβ„•0→ℝ, π‘ŽβˆΆβ„•0→ℝ, πΎβˆΆβ„•0Γ—β„•0→ℝ and π‘βˆΆβ„•0→ℝ⧡{0} is πœ”-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on βˆπœ”βˆ’1𝑗=0𝑏(𝑗) is assumed. The results generalize some of the recent results.

1. Introduction

In the paper, we study a linear Volterra difference equationπ‘₯(𝑛+1)=π‘Ž(𝑛)+𝑏(𝑛)π‘₯(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)π‘₯(𝑖),(1.1) where π‘›βˆˆβ„•0∢={0,1,2,…}, π‘ŽβˆΆβ„•0→ℝ, πΎβˆΆβ„•0Γ—β„•0→ℝ, and π‘βˆΆβ„•0→ℝ⧡{0} is πœ”-periodic, πœ”βˆˆβ„•βˆΆ={1,2,…}. We will also adopt the customary notations π‘˜ξ“π‘–=π‘˜+𝑠π’ͺ(𝑖)=0,π‘˜ξ‘π‘–=π‘˜+𝑠π’ͺ(𝑖)=1,(1.2) where π‘˜ is an integer, 𝑠 is a positive integer, and β€œπ’ͺ” 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 assumptionπœ”βˆ’1𝑗=0𝑏(𝑗)=1,(1.3) and gave sufficient conditions for the existence of asymptotically πœ”-periodic solutions of (1.1) where the notion for an asymptotically πœ”-periodic function has been given by the following definition.

Definition 1.1. Let πœ” be a positive integer. The sequence π‘¦βˆΆβ„•0→ℝ is called πœ”-periodic if 𝑦(𝑛+πœ”)=𝑦(𝑛) for all π‘›βˆˆβ„•0. The sequence 𝑦 is called asymptotically πœ”-periodic if there exist two sequences 𝑒,π‘£βˆΆβ„•0→ℝ such that 𝑒 is πœ”-periodic, limπ‘›β†’βˆžπ‘£(𝑛)=0, and 𝑦(𝑛)=𝑒(𝑛)+𝑣(𝑛)(1.4) for all π‘›βˆˆβ„•0.

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 πœ”-periodic solution of (1.1). We give a definition of a weighted asymptotically πœ”-periodic function.

Definition 1.2. Let πœ” be a positive integer. The sequence π‘¦βˆΆβ„•0→ℝ is called weighted asymptotically πœ”-periodic if there exist two sequences 𝑒,π‘£βˆΆβ„•0→ℝ such that 𝑒 is πœ”-periodic and limπ‘›β†’βˆžπ‘£(𝑛)=0, and, moreover, if there exists a sequence π‘€βˆΆβ„•0→ℝ⧡{0} such that 𝑦(𝑛)𝑀(𝑛)=𝑒(𝑛)+𝑣(𝑛),(1.5) for all π‘›βˆˆβ„•0.

Apart from this, when we assumeπœ”βˆ’1ξ‘π‘˜=0𝑏(π‘˜)=βˆ’1,(1.6) then, as a consequence of our main result (Theorem 2.2), the existence of an asymptotically 2πœ”-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 GyΕ‘ri and HorvΓ‘th [7], by GyΕ‘ri 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 πœ”-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 Ξ© 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)∢=π‘›βˆ’1𝑗=0𝑏(𝑗),π‘›βˆˆβ„•0,(2.1)β„¬βˆΆ=𝛽(πœ”).(2.2) Moreover, we define π‘›βˆ—βˆΆ=π‘›βˆ’1βˆ’πœ”ξ‚žπ‘›βˆ’1πœ”ξ‚Ÿ,(2.3) where βŒŠβ‹…βŒ‹ is the floor function (the greatest-integer function) and π‘›βˆ— is the β€œremainder” of dividing π‘›βˆ’1 by πœ”. Obviously, {𝛽(π‘›βˆ—)}, π‘›βˆˆβ„• is an πœ”-periodic sequence.

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

Theorem 2.2 (Main result). Let πœ” be a positive integer, π‘βˆΆβ„•0→ℝ⧡{0} be πœ”-periodic, π‘ŽβˆΆβ„•0→ℝ, and πΎβˆΆβ„•0Γ—β„•0→ℝ. Assume that βˆžξ“π‘–=0||||π‘Ž(𝑖)𝛽(𝑖+1)||||<∞,βˆžξ“π‘—=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 π‘Ž weighted asymptotically πœ”-periodic solution π‘₯βˆΆβ„•0→ℝof (1.1) with 𝑒,π‘£βˆΆβ„•0→ℝ and π‘€βˆΆβ„•0→ℝ⧡{0} in representation (1.5) such that 𝑀(𝑛)=β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹,𝑒(𝑛)∢=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ,limπ‘›β†’βˆžπ‘£(𝑛)=0,(2.5) that is, π‘₯(𝑛)β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ+𝑣(𝑛),π‘›βˆˆβ„•0.(2.6)

Proof. We will use a notation π‘€βˆΆ=βˆžξ“π‘—=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)}, π‘›βˆˆβ„•0. We set 𝛼(0)∢=βˆ‘βˆžπ‘–=0||π‘Ž(𝑖)/(𝛽(𝑖+1))||+π‘βˆ‘βˆžπ‘—=0βˆ‘π‘—π‘–=0||(𝐾(𝑗,𝑖)𝛽(𝑖))/(𝛽(𝑗+1))||1βˆ’βˆ‘βˆžπ‘—=0βˆ‘π‘—π‘–=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)∢=βˆžξ“π‘–=𝑛||||π‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))βˆžξ“π‘—=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||,(2.9) for 𝑛β‰₯1. It is easy to see that limπ‘›β†’βˆžπ›Ό(𝑛)=0.(2.10) We show, moreover, that 𝛼(𝑛)≀𝛼(0),(2.11) for any π‘›βˆˆβ„•. Let us first remark that 𝛼(0)=βˆžξ“π‘–=0||||π‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))βˆžξ“π‘—=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)=βˆžξ“π‘–=0||||π‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))βˆžξ“π‘—=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||β‰₯βˆžξ“π‘–=𝑛||||π‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))βˆžξ“π‘—=𝑛𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=𝛼(𝑛)(2.13) obviously holds for every π‘›βˆˆβ„• and (2.11) is proved.
Let 𝐡 be the Banach space of all real bounded sequences π‘§βˆΆβ„•0→ℝ equipped with the usual supremum norm ‖𝑧‖=supπ‘›βˆˆβ„•0|𝑧(𝑛)| for π‘§βˆˆπ΅. We define a subset π‘†βŠ‚π΅ as π‘†βˆΆ=ξ€½π‘§βˆˆπ΅βˆΆπ‘βˆ’π›Ό(0)≀𝑧(𝑛)≀𝑐+𝛼(0),π‘›βˆˆβ„•0ξ€Ύ.(2.14) It is not difficult to prove that 𝑆 is a nonempty, bounded, convex, and closed subset of 𝐡.
Let us define a mapping π‘‡βˆΆπ‘†β†’π΅ as follows: (𝑇𝑧)(𝑛)=π‘βˆ’βˆžξ“π‘–=π‘›π‘Ž(𝑖)𝛽(𝑖+1)βˆ’βˆžξ“π‘—=𝑛𝑗𝑖=0𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)𝑧(𝑖),(2.15) for any π‘›βˆˆβ„•0.
We will prove that the mapping 𝑇 has a fixed point in 𝑆.
We first show that 𝑇(𝑆)βŠ‚π‘†. Indeed, if π‘§βˆˆπ‘†, then |𝑧(𝑛)βˆ’π‘|≀𝛼(0) for π‘›βˆˆβ„•0 and, by (2.11) and (2.15), we have ||(𝑇𝑧)(𝑛)βˆ’π‘||β‰€βˆžξ“π‘–=𝑛||||π‘Ž(𝑖)𝛽(𝑖+1)||||+(𝑐+𝛼(0))βˆžξ“π‘—=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 π‘β†’βˆž. Because 𝑆 is closed, π‘§βˆˆπ‘†. Now, utilizing (2.15), we get ||𝑇𝑧(𝑝)ξ€Έ(𝑛)βˆ’(𝑇𝑧)(𝑛)||=|||||βˆžξ“π‘—=𝑛𝑗𝑖=0𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)𝑧(𝑝)(𝑖)βˆ’π‘§(𝑖)ξ€Έ|||||≀𝑀sup𝑖β‰₯0||𝑧(𝑝)(𝑖)βˆ’π‘§(𝑖)||=𝑀‖‖𝑧(𝑝)βˆ’π‘§β€–β€–,π‘›βˆˆβ„•0.(2.17) Therefore, ‖‖𝑇𝑧(𝑝)βˆ’π‘‡π‘§β€–β€–β‰€π‘€β€–β€–π‘§(𝑝)βˆ’π‘§β€–β€–,limπ‘β†’βˆžβ€–β€–π‘‡π‘§(𝑝)βˆ’π‘‡π‘§β€–β€–=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 πœ€-open covering of 𝑇(𝑆) contains a finite πœ€-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 πœ€>0 and assume that an open πœ€-cover π’žπœ€ of 𝑇(𝑆) is given. Then, from (2.10), we conclude that there exists an π‘›πœ€βˆˆβ„• such that 𝛼(𝑛)<πœ€/4 for 𝑛β‰₯π‘›πœ€.
Suppose that π‘₯1π‘‡βˆˆπ‘‡(𝑆) is one of the elements generating the πœ€-cover π’žπœ€ of 𝑇(𝑆). Then (as follows from (2.16)), for an arbitrary π‘₯π‘‡βˆˆπ‘‡(S), ||π‘₯1𝑇(𝑛)βˆ’π‘₯𝑇(𝑛)||<πœ€(2.19) if 𝑛β‰₯π‘›πœ€. In other words, the πœ€-neighborhood of π‘₯1π‘‡βˆ’π‘βˆ—: β€–β€–π‘₯1π‘‡βˆ’π‘βˆ—β€–β€–<πœ€,(2.20) where π‘βˆ—={𝑐,𝑐,…}βˆˆπ‘† covers the set 𝑇(𝑆) on an infinite interval 𝑛β‰₯π‘›πœ€. It remains to cover the rest of 𝑇(𝑆) on a finite interval for π‘›βˆˆ{0,1…,π‘›πœ€βˆ’1} by a finite number of πœ€-neighborhoods of elements generating πœ€-cover π’žπœ€. Supposing that π‘₯1𝑇 itself is not able to generate such cover, we fix π‘›βˆˆ{0,1,…,π‘›πœ€βˆ’1} and split the interval [π‘βˆ’π›Ό(𝑛),𝑐+𝛼(𝑛)](2.21) into a finite number β„Ž(πœ€,𝑛) of closed subintervals 𝐼1(𝑛),𝐼2(𝑛),…,πΌβ„Ž(πœ€,𝑛)(𝑛)(2.22) each with a length not greater then πœ€/2 such that β„Ž(πœ€,𝑛)ξšπ‘–=1𝐼𝑖(𝑛)=[π‘βˆ’π›Ό(𝑛),𝑐+𝛼(𝑛)],int𝐼𝑖(𝑛)∩int𝐼𝑗(𝑛)=βˆ…,𝑖,𝑗=1,2,…,β„Ž(πœ€,𝑛),𝑖≠𝑗.(2.23) Finally, the set π‘›πœ€βˆ’1ξšπ‘›=0[π‘βˆ’π›Ό(𝑛),𝑐+𝛼(𝑛)](2.24) equals π‘›πœ€βˆ’1ξšπ‘›=0β„Ž(πœ€,𝑛)ξšπ‘–=1𝐼𝑖(𝑛)(2.25) and can be divided into a finite number π‘€πœ€βˆΆ=π‘›πœ€βˆ’1𝑛=0β„Ž(πœ€,𝑛)(2.26) of different subintervals (2.22). This means that, at most, π‘€πœ€ of elements generating the cover π’žπœ€ are sufficient to generate a finite πœ€-subcover of 𝑇(𝑆) for π‘›βˆˆ{0,1,…,π‘›πœ€βˆ’1}. We remark that each of such elements simultaneously plays the same role as π‘₯1𝑇(𝑛) for 𝑛β‰₯π‘›πœ€. Since πœ€>0 can be chosen as arbitrarily small, 𝑇(𝑆) is compact.
By Schauder's fixed point theorem, there exists a π‘§βˆˆπ‘† such that 𝑧(𝑛)=(𝑇𝑧)(𝑛) for π‘›βˆˆβ„•0. Thus, 𝑧(𝑛)=π‘βˆ’βˆžξ“π‘–=π‘›π‘Ž(𝑖)𝛽(𝑖+1)βˆ’βˆžξ“π‘—=𝑛𝑗𝑖=0𝛽(𝑖)𝛽(𝑗+1)𝐾(𝑗,𝑖)𝑧(𝑖),(2.27) for any π‘›βˆˆβ„•0.
Due to (2.10) and (2.16), for fixed point π‘§βˆˆπ‘† of 𝑇, we have limπ‘›β†’βˆž||𝑧(𝑛)βˆ’π‘||=limπ‘›β†’βˆž||(𝑇𝑧)(𝑛)βˆ’π‘||≀limπ‘›β†’βˆžπ›Ό(𝑛)=0,(2.28) or, equivalently, limπ‘›β†’βˆžπ‘§(𝑛)=𝑐.(2.29) Finally, we will show that there exists a connection between the fixed point π‘§βˆˆπ‘† and the existence of a solution of (1.1) which divided by β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹ provides an asymptotically πœ”-periodic sequence. Considering (2.27) for 𝑧(𝑛+1) and 𝑧(𝑛), we get Δ𝑧(𝑛)=π‘Ž(𝑛)𝛽(𝑛+1)+𝑛𝑖=0𝛽(𝑖)𝛽(𝑛+1)𝐾(𝑛,𝑖)𝑧(𝑖),(2.30) where π‘›βˆˆβ„•0. Hence, we have 𝑧(𝑛+1)βˆ’π‘§(𝑛)=π‘Ž(𝑛)𝛽(𝑛+1)+1𝛽(𝑛+1)𝑛𝑖=0𝛽(𝑖)𝐾(𝑛,𝑖)𝑧(𝑖),π‘›βˆˆβ„•0.(2.31) Putting 𝑧(𝑛)=π‘₯(𝑛)𝛽(𝑛),π‘›βˆˆβ„•0(2.32) in (2.31), we get (1.1) since π‘₯(𝑛+1)𝛽(𝑛+1)βˆ’π‘₯(𝑛)𝛽(𝑛)=π‘Ž(𝑛)𝛽(𝑛+1)+1𝛽(𝑛+1)𝑛𝑖=0𝐾(𝑛,𝑖)π‘₯(𝑖),π‘›βˆˆβ„•0(2.33) yields π‘₯(𝑛+1)=π‘Ž(𝑛)+𝑏(𝑛)π‘₯(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)π‘₯(𝑖),π‘›βˆˆβ„•0.(2.34) Consequently, π‘₯ defined by (2.32) is a solution of (1.1). From (2.29) and (2.32), we obtain π‘₯(𝑛)𝛽(𝑛)=𝑧(𝑛)=𝑐+π‘œ(1),(2.35) for π‘›β†’βˆž (where π‘œ(1) is the Landau order symbol). Hence, π‘₯(𝑛)=𝛽(𝑛)(𝑐+π‘œ(1)),π‘›βŸΆβˆž.(2.36) It is easy to show that the function 𝛽 defined by (2.1) can be expressed in the form 𝛽(𝑛)=π‘›βˆ’1𝑗=0𝑏(𝑗)=β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹β‹…π›½ξ€·π‘›βˆ—+1ξ€Έ,(2.37) for π‘›βˆˆβ„•0. Then, as follows from (2.36), π‘₯(𝑛)=β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹β‹…π›½ξ€·π‘›βˆ—+1ξ€Έ(𝑐+π‘œ(1)),π‘›βŸΆβˆž,(2.38) or π‘₯(𝑛)β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ+π›½ξ€·π‘›βˆ—+1ξ€Έπ‘œ(1),π‘›βŸΆβˆž.(2.39)
The proof is completed since the sequence {𝛽(π‘›βˆ—+1)} is πœ”-periodic, hence bounded and, due to the properties of Landau order symbols, we have π›½ξ€·π‘›βˆ—+1ξ€Έπ‘œ(1)=π‘œ(1),π‘›βŸΆβˆž,(2.40) and it is easy to see that the choice 𝑒(𝑛)∢=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ,𝑀(𝑛)∢=β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹,π‘›βˆˆβ„•0,(2.41) and an appropriate function π‘£βˆΆβ„•0→ℝ such that limπ‘›β†’βˆžπ‘£(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 𝑒 and 𝑀, 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)=βˆ’π‘Ž(𝑛)+𝑏(𝑛)𝑦(𝑛)+𝑛𝑖=0𝐾(𝑛,𝑖)𝑦(𝑖)(2.43) defines a solution π‘₯=π‘₯(𝑛) of (1.1) since a substitution 𝑦(𝑛)=βˆ’π‘₯(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)β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ+𝑣(𝑛),π‘›βˆˆβ„•0.(2.44) Or, in other words, (1.1) has a solution that can be represented by formula (2.44) as π‘₯(𝑛)β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹=𝑐0π›½ξ€·π‘›βˆ—+1ξ€Έ+π‘£βˆ—(𝑛),π‘›βˆˆβ„•0,(2.45) with 𝑐0=βˆ’π‘ and π‘£βˆ—(𝑛)=βˆ’π‘£(𝑛). In (2.45), 𝑐0<0 and the function π‘£βˆ—(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 π‘Ž(𝑛)=(βˆ’1)𝑛+1ξ‚€1βˆ’13𝑛+1,𝑏(𝑛)=3(βˆ’1)𝑛,𝐾(𝑛,𝑖)=(βˆ’1)𝑛+(𝑖(π‘–βˆ’1))/2132𝑖,(2.46) that is, the equation π‘₯(𝑛+1)=(βˆ’1)𝑛+1ξ‚€1βˆ’13𝑛+1+3(βˆ’1)𝑛π‘₯(𝑛)+𝑛𝑖=0(βˆ’1)𝑛+(𝑖(π‘–βˆ’1))/2132𝑖π‘₯(𝑖).(2.47) The sequence 𝑏(𝑛) is 2-periodic and 𝛽(𝑛)=π‘›βˆ’1𝑗=0𝑏(𝑗)=(βˆ’1)𝑛(π‘›βˆ’1)/23𝑛,ℬ=𝛽(πœ”)=𝛽(2)=βˆ’9,π›½ξ€·π‘›βˆ—+1ξ€Έ=βˆ’3+6(βˆ’1)𝑛+1,π‘Ž(𝑛)𝛽(𝑛+1)=(βˆ’1)(βˆ’π‘›2+𝑛+2)/2ξ‚€13𝑛+1βˆ’132(𝑛+1),βˆžξ“π‘–=0||||π‘Ž(𝑖)𝛽(𝑖+1)||||<∞,βˆžξ“π‘—=0𝑗𝑖=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||<βˆžξ“π‘—=0βˆžξ“π‘–=0||||𝐾(𝑗,𝑖)𝛽(𝑖)𝛽(𝑗+1)||||=βˆžξ“π‘—=0βˆžξ“π‘–=013𝑖+𝑗+1=13βŽ›βŽœβŽβˆžξ“π‘—=013π‘—βŽžβŽŸβŽ βŽ›βŽœβŽβˆžξ“π‘–=013π‘–βŽžβŽŸβŽ =13β‹…11βˆ’1/3β‹…11βˆ’1/3=13β‹…32β‹…32=34<1.(2.48) By virtue of Theorem 2.2, for any nonzero constant 𝑐, there exists a solution π‘₯βˆΆβ„•0→ℝ of (1.1) which is weighed asymptotically 2-periodic. Let, for example, 𝑐=2/3. Then, 𝑀(𝑛)=(βˆ’9)⌊(π‘›βˆ’1)/2βŒ‹,𝑒(𝑛)=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ=23ξ€·βˆ’3+6(βˆ’1)𝑛+1ξ€Έ=βˆ’2+4(βˆ’1)𝑛+1,(2.49) and the sequence π‘₯(𝑛) given by π‘₯(𝑛)(βˆ’9)⌊(π‘›βˆ’1)/2βŒ‹=βˆ’2+4(βˆ’1)𝑛+1+𝑣(𝑛),π‘›βˆˆβ„•0,(2.50) or, equivalently, π‘₯(𝑛)=(βˆ’9)⌊(π‘›βˆ’1)/2βŒ‹ξ€·βˆ’2+4(βˆ’1)𝑛+1ξ€Έ+𝑣(𝑛),π‘›βˆˆβ„•0(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 π‘₯βˆ—(𝑛) obtained from (2.51) if 𝑣(𝑛)=0, π‘›βˆˆβ„•0, that is, π‘₯βˆ—(𝑛)=(βˆ’9)⌊(π‘›βˆ’1)/2βŒ‹ξ€·βˆ’2+4(βˆ’1)𝑛+1ξ€Έ=23β‹…(βˆ’1)𝑛(π‘›βˆ’1)/2β‹…3𝑛,π‘›βˆˆβ„•0(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 π‘₯=π‘₯(𝑛) of (1.1) described by formula (2.6) satisfies limπ‘›β†’βˆžπ‘₯(𝑛)=0.(3.1) If |ℬ|>1, then, for every solution π‘₯=π‘₯(𝑛) of (1.1) described by formula (2.6), one has liminfπ‘›β†’βˆžπ‘₯(𝑛)=βˆ’βˆž(3.2) or/and limsupπ‘›β†’βˆžπ‘₯(𝑛)=∞.(3.3) Finally, if ℬ>1, then, for every solution π‘₯=π‘₯(𝑛) of (1.1) described by formula (2.6), one has limπ‘›β†’βˆžπ‘₯(𝑛)=∞,(3.4) and if ℬ<βˆ’1, then, for every solution π‘₯=π‘₯(𝑛) of (1.1) described by formula (2.6), one has limπ‘›β†’βˆžπ‘₯(𝑛)=βˆ’βˆž.(3.5)

Now, let us discuss the case when (1.6) holds, that is, when ℬ=πœ”βˆ’1𝑗=0𝑏(𝑗)=βˆ’1.(3.6)

Corollary 3.2. Let Theorem 2.2 be valid. Assume that ℬ=βˆ’1. Then, for any nonzero constant 𝑐, there exists an asymptotically 2πœ”-periodic solution π‘₯=π‘₯(𝑛), π‘›βˆˆβ„•0 of (1.1) such that π‘₯(𝑛)=(βˆ’1)⌊(π‘›βˆ’1)/πœ”βŒ‹π‘’(𝑛)+𝑧(𝑛),π‘›βˆˆβ„•0,(3.7) with 𝑒(𝑛)∢=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ,limπ‘›β†’βˆžπ‘§(𝑛)=0.(3.8)

Proof. Putting ℬ=βˆ’1 in Theorem 2.2, we get π‘₯(𝑛)=(βˆ’1)⌊(π‘›βˆ’1)/πœ”βŒ‹π‘’(𝑛)+(βˆ’1)⌊(π‘›βˆ’1)/πœ”βŒ‹π‘£(𝑛),(3.9) with 𝑒(𝑛)∢=π‘π›½ξ€·π‘›βˆ—+1ξ€Έ,limπ‘›β†’βˆžπ‘£(𝑛)=0.(3.10)
Due to the definition of π‘›βˆ—, we see that the sequence ξ€½π›½ξ€·π‘›βˆ—+1ξ€Έξ€Ύ={𝛽(πœ”),𝛽(1),𝛽(2),…,𝛽(πœ”),𝛽(1),𝛽(2),…,𝛽(πœ”),…},(3.11) is an πœ”-periodic sequence. Since ξ‚†ξ‚žπ‘›βˆ’1πœ”ξ‚Ÿξ‚‡=⎧βŽͺ⎨βŽͺβŽ©βˆ’1,0,…,0ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπœ”,1,…,1ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπœ”,2,β€¦βŽ«βŽͺ⎬βŽͺ⎭,(3.12) for π‘›βˆˆβ„•0, we have ξ€½(βˆ’1)⌊(π‘›βˆ’1)/πœ”βŒ‹ξ€Ύ=⎧βŽͺ⎨βŽͺβŽ©βˆ’1,1,…,1ξ„Ώξ…€ξ…€ξ…ƒξ…€ξ…€ξ…Œπœ”,βˆ’1,…,βˆ’1ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπœ”,1,β€¦βŽ«βŽͺ⎬βŽͺ⎭.(3.13) Therefore, the sequence ξ€½(βˆ’1)⌊(π‘›βˆ’1)/πœ”βŒ‹π‘’(𝑛)ξ€Ύ=𝑐{βˆ’π›½(πœ”),𝛽(1),𝛽(2),…,𝛽(πœ”),βˆ’π›½(1),βˆ’π›½(2),…,βˆ’π›½(πœ”),…}(3.14) is a 2πœ”-periodic sequence. Set 𝑧(𝑛)=(βˆ’1)⌊(π‘›βˆ’1)/πœ”βŒ‹π‘£(𝑛).(3.15) Then, limπ‘›β†’βˆžπ‘§(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 β€œasymptotically weighted πœ”-periodic solution” π‘₯=π‘₯(𝑛) of (1.1) as well. The formula (2.6) reduces to π‘₯(𝑛)=β„¬βŒŠ(π‘›βˆ’1)/πœ”βŒ‹π‘£(𝑛)=π‘œ(1),π‘›βˆˆβ„•0,(3.17) since 𝑒(𝑛)=0. In the light of Definition 1.2, we can treat this case as follows. We set (as a singular case) 𝑒≑0 with an arbitrary (possibly other than ξ…’πœ”'') period and with 𝑣=π‘œ(1), π‘›β†’βˆž.

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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