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.