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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 370982, 14 pages
doi:10.1155/2011/370982
Research Article

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

Josef Diblík,1,2Β Miroslava Růžičková,3Β Ewa Schmeidel,4Β and Małgorzata Zbąszyniak4

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: ElenaΒ Braverman

Copyright Β© 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 π‘₯ ( 𝑛 + 1 ) = π‘Ž ( 𝑛 ) + 𝑏 ( 𝑛 ) π‘₯ ( 𝑛 ) + βˆ‘ 𝑛 𝑖 = 0 𝐾 ( 𝑛 , 𝑖 ) π‘₯ ( 𝑖 ) , where π‘₯ ∢ β„• 0 β†’ ℝ , π‘Ž ∢ β„• 0 β†’ ℝ , 𝐾 ∢ β„• 0 Γ— β„• 0 β†’ ℝ and 𝑏 ∢ β„• 0 β†’ 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 ∏ πœ” βˆ’ 1 𝑗 = 0 𝑏 ( 𝑗 ) is assumed. The results generalize some of the recent results.

1. Introduction

In the paper, we study a linear Volterra difference equation π‘₯ ( 𝑛 + 1 ) = π‘Ž ( 𝑛 ) + 𝑏 ( 𝑛 ) π‘₯ ( 𝑛 ) + 𝑛  𝑖 = 0 𝐾 ( 𝑛 , 𝑖 ) π‘₯ ( 𝑖 ) , ( 1 . 1 ) where 𝑛 ∈ β„• 0 ∢ = { 0 , 1 , 2 , … } , π‘Ž ∢ β„• 0 β†’ ℝ , 𝐾 ∢ β„• 0 Γ— β„• 0 β†’ ℝ , and 𝑏 ∢ β„• 0 β†’ ℝ ⧡ { 0 } is πœ” -periodic, πœ” ∈ β„• ∢ = { 1 , 2 , … } . We will also adopt the customary notations π‘˜  𝑖 = π‘˜ + 𝑠 π’ͺ ( 𝑖 ) = 0 , π‘˜  𝑖 = π‘˜ + 𝑠 π’ͺ ( 𝑖 ) = 1 , ( 1 . 2 ) where π‘˜ is an integer, 𝑠 is a positive integer, and “ π’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 assumption πœ” βˆ’ 1  𝑗 = 0 𝑏 ( 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 𝑦 ∢ β„• 0 β†’ ℝ is called πœ” -periodic if 𝑦 ( 𝑛 + πœ” ) = 𝑦 ( 𝑛 ) for all 𝑛 ∈ β„• 0 . The sequence 𝑦 is called asymptotically πœ” -periodic if there exist two sequences 𝑒 , 𝑣 ∢ β„• 0 β†’ ℝ such that 𝑒 is πœ” -periodic, l i m 𝑛 β†’ ∞ 𝑣 ( 𝑛 ) = 0 , and 𝑦 ( 𝑛 ) = 𝑒 ( 𝑛 ) + 𝑣 ( 𝑛 ) ( 1 . 4 ) for all 𝑛 ∈ β„• 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 𝑦 ∢ β„• 0 β†’ ℝ is called weighted asymptotically πœ” -periodic if there exist two sequences 𝑒 , 𝑣 ∢ β„• 0 β†’ ℝ such that 𝑒 is πœ” -periodic and l i m 𝑛 β†’ ∞ 𝑣 ( 𝑛 ) = 0 , and, moreover, if there exists a sequence 𝑀 ∢ β„• 0 β†’ ℝ ⧡ { 0 } such that 𝑦 ( 𝑛 ) 𝑀 ( 𝑛 ) = 𝑒 ( 𝑛 ) + 𝑣 ( 𝑛 ) , ( 1 . 5 ) for all 𝑛 ∈ β„• 0 .

Apart from this, when we assume πœ” βˆ’ 1  π‘˜ = 0 𝑏 ( π‘˜ ) = βˆ’ 1 , ( 1 . 6 ) then, as a consequence of our main result (Theorem 2.2), the existence of an asymptotically 2 π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 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 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 set 𝛽 ( 𝑛 ) ∢ = 𝑛 βˆ’ 1  𝑗 = 0 𝑏 ( 𝑗 ) , 𝑛 ∈ β„• 0 , ( 2 . 1 ) ℬ ∢ = 𝛽 ( πœ” ) . ( 2 . 2 ) Moreover, we define 𝑛 βˆ— ∢ = 𝑛 βˆ’ 1 βˆ’ πœ” ξ‚ž 𝑛 βˆ’ 1 πœ” ξ‚Ÿ , ( 2 . 3 ) where ⌊ β‹… βŒ‹ is the floor function (the greatest-integer function) and 𝑛 βˆ— is the “remainder” of dividing 𝑛 βˆ’ 1 by πœ” . Obviously, { 𝛽 ( 𝑛 βˆ— ) } , 𝑛 ∈ β„• 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, 𝑏 ∢ β„• 0 β†’ ℝ ⧡ { 0 } be πœ” -periodic, π‘Ž ∢ β„• 0 β†’ ℝ , and 𝐾 ∢ β„• 0 Γ— β„• 0 β†’ ℝ . Assume that ∞  𝑖 = 0 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | < ∞ , ∞  𝑗 = 0 𝑗  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‘₯ ∢ β„• 0 β†’ ℝ of (1.1) with 𝑒 , 𝑣 ∢ β„• 0 β†’ ℝ and 𝑀 ∢ β„• 0 β†’ ℝ ⧡ { 0 } in representation (1.5) such that 𝑀 ( 𝑛 ) = ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ , 𝑒 ( 𝑛 ) ∢ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ , l i m 𝑛 β†’ ∞ 𝑣 ( 𝑛 ) = 0 , ( 2 . 5 ) that is, π‘₯ ( 𝑛 ) ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ + 𝑣 ( 𝑛 ) , 𝑛 ∈ β„• 0 . ( 2 . 6 )

Proof. We will use a notation 𝑀 ∢ = ∞  𝑗 = 0 𝑗  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 ∈ β„• 0 . We set 𝛼 ( 0 ) ∢ = βˆ‘ ∞ 𝑖 = 0 | | π‘Ž ( 𝑖 ) / ( 𝛽 ( 𝑖 + 1 ) ) | | + 𝑐 βˆ‘ ∞ 𝑗 = 0 βˆ‘ 𝑗 𝑖 = 0 | | ( 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) ) / ( 𝛽 ( 𝑗 + 1 ) ) | | 1 βˆ’ βˆ‘ ∞ 𝑗 = 0 βˆ‘ 𝑗 𝑖 = 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 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | + ( 𝑐 + 𝛼 ( 0 ) ) ∞  𝑗 = 𝑛 𝑗  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | , ( 2 . 9 ) for 𝑛 β‰₯ 1 . It is easy to see that l i m 𝑛 β†’ ∞ 𝛼 ( 𝑛 ) = 0 . ( 2 . 1 0 ) We show, moreover, that 𝛼 ( 𝑛 ) ≀ 𝛼 ( 0 ) , ( 2 . 1 1 ) for any 𝑛 ∈ β„• . Let us first remark that 𝛼 ( 0 ) = ∞  𝑖 = 0 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | + ( 𝑐 + 𝛼 ( 0 ) ) ∞  𝑗 = 0 𝑗  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | . ( 2 . 1 2 ) Then, due to the convergence of both series (see (2.4)), the inequality 𝛼 ( 0 ) = ∞  𝑖 = 0 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | + ( 𝑐 + 𝛼 ( 0 ) ) ∞  𝑗 = 0 𝑗  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | β‰₯ ∞  𝑖 = 𝑛 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | + ( 𝑐 + 𝛼 ( 0 ) ) ∞  𝑗 = 𝑛 𝑗  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | = 𝛼 ( 𝑛 ) ( 2 . 1 3 ) obviously holds for every 𝑛 ∈ β„• and (2.11) is proved.
Let 𝐡 be the Banach space of all real bounded sequences 𝑧 ∢ β„• 0 β†’ ℝ equipped with the usual supremum norm β€– 𝑧 β€– = s u p 𝑛 ∈ β„• 0 | 𝑧 ( 𝑛 ) | for 𝑧 ∈ 𝐡 . We define a subset 𝑆 βŠ‚ 𝐡 as 𝑆 ∢ = ξ€½ 𝑧 ∈ 𝐡 ∢ 𝑐 βˆ’ 𝛼 ( 0 ) ≀ 𝑧 ( 𝑛 ) ≀ 𝑐 + 𝛼 ( 0 ) , 𝑛 ∈ β„• 0 ξ€Ύ . ( 2 . 1 4 ) It is not difficult to prove that 𝑆 is a nonempty, bounded, convex, and closed subset of 𝐡 .
Let us define a mapping 𝑇 ∢ 𝑆 β†’ 𝐡 as follows: ( 𝑇 𝑧 ) ( 𝑛 ) = 𝑐 βˆ’ ∞  𝑖 = 𝑛 π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) βˆ’ ∞  𝑗 = 𝑛 𝑗  𝑖 = 0 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) 𝑧 ( 𝑖 ) , ( 2 . 1 5 ) for any 𝑛 ∈ β„• 0 .
We will prove that the mapping 𝑇 has a fixed point in 𝑆 .
We first show that 𝑇 ( 𝑆 ) βŠ‚ 𝑆 . Indeed, if 𝑧 ∈ 𝑆 , then | 𝑧 ( 𝑛 ) βˆ’ 𝑐 | ≀ 𝛼 ( 0 ) for 𝑛 ∈ β„• 0 and, by (2.11) and (2.15), we have | | ( 𝑇 𝑧 ) ( 𝑛 ) βˆ’ 𝑐 | | ≀ ∞  𝑖 = 𝑛 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | + ( 𝑐 + 𝛼 ( 0 ) ) ∞  𝑗 = 𝑛 𝑗  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | = 𝛼 ( 𝑛 ) ≀ 𝛼 ( 0 ) . ( 2 . 1 6 ) Next, we prove that 𝑇 is continuous. Let 𝑧 ( 𝑝 ) be a sequence in 𝑆 such that 𝑧 ( 𝑝 ) β†’ 𝑧 as 𝑝 β†’ ∞ . Because 𝑆 is closed, 𝑧 ∈ 𝑆 . Now, utilizing (2.15), we get | | ξ€· 𝑇 𝑧 ( 𝑝 ) ξ€Έ ( 𝑛 ) βˆ’ ( 𝑇 𝑧 ) ( 𝑛 ) | | = | | | | | ∞  𝑗 = 𝑛 𝑗  𝑖 = 0 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) ξ€· 𝑧 ( 𝑝 ) ( 𝑖 ) βˆ’ 𝑧 ( 𝑖 ) ξ€Έ | | | | | ≀ 𝑀 s u p 𝑖 β‰₯ 0 | | 𝑧 ( 𝑝 ) ( 𝑖 ) βˆ’ 𝑧 ( 𝑖 ) | | = 𝑀 β€– β€– 𝑧 ( 𝑝 ) βˆ’ 𝑧 β€– β€– , 𝑛 ∈ β„• 0 . ( 2 . 1 7 ) Therefore, β€– β€– 𝑇 𝑧 ( 𝑝 ) βˆ’ 𝑇 𝑧 β€– β€– ≀ 𝑀 β€– β€– 𝑧 ( 𝑝 ) βˆ’ 𝑧 β€– β€– , l i m 𝑝 β†’ ∞ β€– β€– 𝑇 𝑧 ( 𝑝 ) βˆ’ 𝑇 𝑧 β€– β€– = 0 . ( 2 . 1 8 ) 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’ž πœ€ of 𝑇 ( 𝑆 ) is given. Then, from (2.10), we conclude that there exists an 𝑛 πœ€ ∈ β„• such that 𝛼 ( 𝑛 ) < πœ€ / 4 for 𝑛 β‰₯ 𝑛 πœ€ .
Suppose that π‘₯ 1 𝑇 ∈ 𝑇 ( 𝑆 ) is one of the elements generating the πœ€ -cover π’ž πœ€ of 𝑇 ( 𝑆 ) . Then (as follows from (2.16)), for an arbitrary π‘₯ 𝑇 ∈ 𝑇 ( S ) , | | π‘₯ 1 𝑇 ( 𝑛 ) βˆ’ π‘₯ 𝑇 ( 𝑛 ) | | < πœ€ ( 2 . 1 9 ) if 𝑛 β‰₯ 𝑛 πœ€ . In other words, the πœ€ -neighborhood of π‘₯ 1 𝑇 βˆ’ 𝑐 βˆ— : β€– β€– π‘₯ 1 𝑇 βˆ’ 𝑐 βˆ— β€– β€– < πœ€ , ( 2 . 2 0 ) where 𝑐 βˆ— = { 𝑐 , 𝑐 , … } ∈ 𝑆 covers the set 𝑇 ( 𝑆 ) on an infinite interval 𝑛 β‰₯ 𝑛 πœ€ . It remains to cover the rest of 𝑇 ( 𝑆 ) on a finite interval for 𝑛 ∈ { 0 , 1 … , 𝑛 πœ€ βˆ’ 1 } by a finite number of πœ€ -neighborhoods of elements generating πœ€ -cover π’ž πœ€ . Supposing that π‘₯ 1 𝑇 itself is not able to generate such cover, we fix 𝑛 ∈ { 0 , 1 , … , 𝑛 πœ€ βˆ’ 1 } and split the interval [ 𝑐 βˆ’ 𝛼 ( 𝑛 ) , 𝑐 + 𝛼 ( 𝑛 ) ] ( 2 . 2 1 ) into a finite number β„Ž ( πœ€ , 𝑛 ) of closed subintervals 𝐼 1 ( 𝑛 ) , 𝐼 2 ( 𝑛 ) , … , 𝐼 β„Ž ( πœ€ , 𝑛 ) ( 𝑛 ) ( 2 . 2 2 ) each with a length not greater then πœ€ / 2 such that β„Ž ( πœ€ , 𝑛 )  𝑖 = 1 𝐼 𝑖 ( 𝑛 ) = [ 𝑐 βˆ’ 𝛼 ( 𝑛 ) , 𝑐 + 𝛼 ( 𝑛 ) ] , i n t 𝐼 𝑖 ( 𝑛 ) ∩ i n t 𝐼 𝑗 ( 𝑛 ) = βˆ… , 𝑖 , 𝑗 = 1 , 2 , … , β„Ž ( πœ€ , 𝑛 ) , 𝑖 β‰  𝑗 . ( 2 . 2 3 ) Finally, the set 𝑛 πœ€ βˆ’ 1  𝑛 = 0 [ 𝑐 βˆ’ 𝛼 ( 𝑛 ) , 𝑐 + 𝛼 ( 𝑛 ) ] ( 2 . 2 4 ) equals 𝑛 πœ€ βˆ’ 1  𝑛 = 0 β„Ž ( πœ€ , 𝑛 )  𝑖 = 1 𝐼 𝑖 ( 𝑛 ) ( 2 . 2 5 ) and can be divided into a finite number 𝑀 πœ€ ∢ = 𝑛 πœ€ βˆ’ 1  𝑛 = 0 β„Ž ( πœ€ , 𝑛 ) ( 2 . 2 6 ) of different subintervals (2.22). This means that, at most, 𝑀 πœ€ of elements generating the cover π’ž πœ€ are sufficient to generate a finite πœ€ -subcover of 𝑇 ( 𝑆 ) for 𝑛 ∈ { 0 , 1 , … , 𝑛 πœ€ βˆ’ 1 } . We remark that each of such elements simultaneously plays the same role as π‘₯ 1 𝑇 ( 𝑛 ) for 𝑛 β‰₯ 𝑛 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 𝑧 ∈ 𝑆 such that 𝑧 ( 𝑛 ) = ( 𝑇 𝑧 ) ( 𝑛 ) for 𝑛 ∈ β„• 0 . Thus, 𝑧 ( 𝑛 ) = 𝑐 βˆ’ ∞  𝑖 = 𝑛 π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) βˆ’ ∞  𝑗 = 𝑛 𝑗  𝑖 = 0 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) 𝐾 ( 𝑗 , 𝑖 ) 𝑧 ( 𝑖 ) , ( 2 . 2 7 ) for any 𝑛 ∈ β„• 0 .
Due to (2.10) and (2.16), for fixed point 𝑧 ∈ 𝑆 of 𝑇 , we have l i m 𝑛 β†’ ∞ | | 𝑧 ( 𝑛 ) βˆ’ 𝑐 | | = l i m 𝑛 β†’ ∞ | | ( 𝑇 𝑧 ) ( 𝑛 ) βˆ’ 𝑐 | | ≀ l i m 𝑛 β†’ ∞ 𝛼 ( 𝑛 ) = 0 , ( 2 . 2 8 ) or, equivalently, l i m 𝑛 β†’ ∞ 𝑧 ( 𝑛 ) = 𝑐 . ( 2 . 2 9 ) 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 βˆ’ 1 ) / πœ” βŒ‹ provides an asymptotically πœ” -periodic sequence. Considering (2.27) for 𝑧 ( 𝑛 + 1 ) and 𝑧 ( 𝑛 ) , we get Ξ” 𝑧 ( 𝑛 ) = π‘Ž ( 𝑛 ) 𝛽 ( 𝑛 + 1 ) + 𝑛  𝑖 = 0 𝛽 ( 𝑖 ) 𝛽 ( 𝑛 + 1 ) 𝐾 ( 𝑛 , 𝑖 ) 𝑧 ( 𝑖 ) , ( 2 . 3 0 ) where 𝑛 ∈ β„• 0 . Hence, we have 𝑧 ( 𝑛 + 1 ) βˆ’ 𝑧 ( 𝑛 ) = π‘Ž ( 𝑛 ) 𝛽 ( 𝑛 + 1 ) + 1 𝛽 ( 𝑛 + 1 ) 𝑛  𝑖 = 0 𝛽 ( 𝑖 ) 𝐾 ( 𝑛 , 𝑖 ) 𝑧 ( 𝑖 ) , 𝑛 ∈ β„• 0 . ( 2 . 3 1 ) Putting 𝑧 ( 𝑛 ) = π‘₯ ( 𝑛 ) 𝛽 ( 𝑛 ) , 𝑛 ∈ β„• 0 ( 2 . 3 2 ) 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 ) + 1 𝛽 ( 𝑛 + 1 ) 𝑛  𝑖 = 0 𝐾 ( 𝑛 , 𝑖 ) π‘₯ ( 𝑖 ) , 𝑛 ∈ β„• 0 ( 2 . 3 3 ) yields π‘₯ ( 𝑛 + 1 ) = π‘Ž ( 𝑛 ) + 𝑏 ( 𝑛 ) π‘₯ ( 𝑛 ) + 𝑛  𝑖 = 0 𝐾 ( 𝑛 , 𝑖 ) π‘₯ ( 𝑖 ) , 𝑛 ∈ β„• 0 . ( 2 . 3 4 ) 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 . 3 5 ) for 𝑛 β†’ ∞ (where π‘œ ( 1 ) is the Landau order symbol). Hence, π‘₯ ( 𝑛 ) = 𝛽 ( 𝑛 ) ( 𝑐 + π‘œ ( 1 ) ) , 𝑛 ⟢ ∞ . ( 2 . 3 6 ) It is easy to show that the function 𝛽 defined by (2.1) can be expressed in the form 𝛽 ( 𝑛 ) = 𝑛 βˆ’ 1  𝑗 = 0 𝑏 ( 𝑗 ) = ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ β‹… 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ , ( 2 . 3 7 ) for 𝑛 ∈ β„• 0 . Then, as follows from (2.36), π‘₯ ( 𝑛 ) = ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ β‹… 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ ( 𝑐 + π‘œ ( 1 ) ) , 𝑛 ⟢ ∞ , ( 2 . 3 8 ) or π‘₯ ( 𝑛 ) ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ + 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ π‘œ ( 1 ) , 𝑛 ⟢ ∞ . ( 2 . 3 9 )
The proof is completed since the sequence { 𝛽 ( 𝑛 βˆ— + 1 ) } is πœ” -periodic, hence bounded and, due to the properties of Landau order symbols, we have 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ π‘œ ( 1 ) = π‘œ ( 1 ) , 𝑛 ⟢ ∞ , ( 2 . 4 0 ) and it is easy to see that the choice 𝑒 ( 𝑛 ) ∢ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ , 𝑀 ( 𝑛 ) ∢ = ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ , 𝑛 ∈ β„• 0 , ( 2 . 4 1 ) and an appropriate function 𝑣 ∢ β„• 0 β†’ ℝ such that l i m 𝑛 β†’ ∞ 𝑣 ( 𝑛 ) = 0 ( 2 . 4 2 ) 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 = 0 𝐾 ( 𝑛 , 𝑖 ) 𝑦 ( 𝑖 ) ( 2 . 4 3 ) 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 ) ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ + 𝑣 ( 𝑛 ) , 𝑛 ∈ β„• 0 . ( 2 . 4 4 ) Or, in other words, (1.1) has a solution that can be represented by formula (2.44) as π‘₯ ( 𝑛 ) ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ = 𝑐 0 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ + 𝑣 βˆ— ( 𝑛 ) , 𝑛 ∈ β„• 0 , ( 2 . 4 5 ) with 𝑐 0 = βˆ’ 𝑐 and 𝑣 βˆ— ( 𝑛 ) = βˆ’ 𝑣 ( 𝑛 ) . In (2.45), 𝑐 0 < 0 and the function 𝑣 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 + 1 ξ‚€ 1 βˆ’ 1 3 𝑛 + 1  , 𝑏 ( 𝑛 ) = 3 ( βˆ’ 1 ) 𝑛 , 𝐾 ( 𝑛 , 𝑖 ) = ( βˆ’ 1 ) 𝑛 + ( 𝑖 ( 𝑖 βˆ’ 1 ) ) / 2 1 3 2 𝑖 , ( 2 . 4 6 ) that is, the equation π‘₯ ( 𝑛 + 1 ) = ( βˆ’ 1 ) 𝑛 + 1 ξ‚€ 1 βˆ’ 1 3 𝑛 + 1  + 3 ( βˆ’ 1 ) 𝑛 π‘₯ ( 𝑛 ) + 𝑛  𝑖 = 0 ( βˆ’ 1 ) 𝑛 + ( 𝑖 ( 𝑖 βˆ’ 1 ) ) / 2 1 3 2 𝑖 π‘₯ ( 𝑖 ) . ( 2 . 4 7 ) The sequence 𝑏 ( 𝑛 ) is 2-periodic and 𝛽 ( 𝑛 ) = 𝑛 βˆ’ 1  𝑗 = 0 𝑏 ( 𝑗 ) = ( βˆ’ 1 ) 𝑛 ( 𝑛 βˆ’ 1 ) / 2 3 𝑛 , ℬ = 𝛽 ( πœ” ) = 𝛽 ( 2 ) = βˆ’ 9 , 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ = βˆ’ 3 + 6 ( βˆ’ 1 ) 𝑛 + 1 , π‘Ž ( 𝑛 ) 𝛽 ( 𝑛 + 1 ) = ( βˆ’ 1 ) ( βˆ’ 𝑛 2 + 𝑛 + 2 ) / 2 ξ‚€ 1 3 𝑛 + 1 βˆ’ 1 3 2 ( 𝑛 + 1 )  , ∞  𝑖 = 0 | | | | π‘Ž ( 𝑖 ) 𝛽 ( 𝑖 + 1 ) | | | | < ∞ , ∞  𝑗 = 0 𝑗  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | < ∞  𝑗 = 0 ∞  𝑖 = 0 | | | | 𝐾 ( 𝑗 , 𝑖 ) 𝛽 ( 𝑖 ) 𝛽 ( 𝑗 + 1 ) | | | | = ∞  𝑗 = 0 ∞  𝑖 = 0 1 3 𝑖 + 𝑗 + 1 = 1 3 βŽ› ⎜ ⎝ ∞  𝑗 = 0 1 3 𝑗 ⎞ ⎟ ⎠ βŽ› ⎜ ⎝ ∞  𝑖 = 0 1 3 𝑖 ⎞ ⎟ ⎠ = 1 3 β‹… 1 1 βˆ’ 1 / 3 β‹… 1 1 βˆ’ 1 / 3 = 1 3 β‹… 3 2 β‹… 3 2 = 3 4 < 1 . ( 2 . 4 8 ) By virtue of Theorem 2.2, for any nonzero constant 𝑐 , there exists a solution π‘₯ ∢ β„• 0 β†’ ℝ of (1.1) which is weighed asymptotically 2-periodic. Let, for example, 𝑐 = 2 / 3 . Then, 𝑀 ( 𝑛 ) = ( βˆ’ 9 ) ⌊ ( 𝑛 βˆ’ 1 ) / 2 βŒ‹ , 𝑒 ( 𝑛 ) = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ = 2 3 ξ€· βˆ’ 3 + 6 ( βˆ’ 1 ) 𝑛 + 1 ξ€Έ = βˆ’ 2 + 4 ( βˆ’ 1 ) 𝑛 + 1 , ( 2 . 4 9 ) and the sequence π‘₯ ( 𝑛 ) given by π‘₯ ( 𝑛 ) ( βˆ’ 9 ) ⌊ ( 𝑛 βˆ’ 1 ) / 2 βŒ‹ = βˆ’ 2 + 4 ( βˆ’ 1 ) 𝑛 + 1 + 𝑣 ( 𝑛 ) , 𝑛 ∈ β„• 0 , ( 2 . 5 0 ) or, equivalently, π‘₯ ( 𝑛 ) = ( βˆ’ 9 ) ⌊ ( 𝑛 βˆ’ 1 ) / 2 βŒ‹ ξ€· βˆ’ 2 + 4 ( βˆ’ 1 ) 𝑛 + 1 ξ€Έ + 𝑣 ( 𝑛 ) , 𝑛 ∈ β„• 0 ( 2 . 5 1 ) 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 , 𝑛 ∈ β„• 0 , that is, π‘₯ βˆ— ( 𝑛 ) = ( βˆ’ 9 ) ⌊ ( 𝑛 βˆ’ 1 ) / 2 βŒ‹ ξ€· βˆ’ 2 + 4 ( βˆ’ 1 ) 𝑛 + 1 ξ€Έ = 2 3 β‹… ( βˆ’ 1 ) 𝑛 ( 𝑛 βˆ’ 1 ) / 2 β‹… 3 𝑛 , 𝑛 ∈ β„• 0 ( 2 . 5 2 ) 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 l i m 𝑛 β†’ ∞ π‘₯ ( 𝑛 ) = 0 . ( 3 . 1 ) If | ℬ | > 1 , then, for every solution π‘₯ = π‘₯ ( 𝑛 ) of (1.1) described by formula (2.6), one has l i m i n f 𝑛 β†’ ∞ π‘₯ ( 𝑛 ) = βˆ’ ∞ ( 3 . 2 ) or/and l i m s u p 𝑛 β†’ ∞ π‘₯ ( 𝑛 ) = ∞ . ( 3 . 3 ) Finally, if ℬ > 1 , then, for every solution π‘₯ = π‘₯ ( 𝑛 ) of (1.1) described by formula (2.6), one has l i m 𝑛 β†’ ∞ π‘₯ ( 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 l i m 𝑛 β†’ ∞ π‘₯ ( 𝑛 ) = βˆ’ ∞ . ( 3 . 5 )

Now, let us discuss the case when (1.6) holds, that is, when ℬ = πœ” βˆ’ 1  𝑗 = 0 𝑏 ( 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 2 πœ” -periodic solution π‘₯ = π‘₯ ( 𝑛 ) , 𝑛 ∈ β„• 0 of (1.1) such that π‘₯ ( 𝑛 ) = ( βˆ’ 1 ) ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ 𝑒 ( 𝑛 ) + 𝑧 ( 𝑛 ) , 𝑛 ∈ β„• 0 , ( 3 . 7 ) with 𝑒 ( 𝑛 ) ∢ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ , l i m 𝑛 β†’ ∞ 𝑧 ( 𝑛 ) = 0 . ( 3 . 8 )

Proof. Putting ℬ = βˆ’ 1 in Theorem 2.2, we get π‘₯ ( 𝑛 ) = ( βˆ’ 1 ) ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ 𝑒 ( 𝑛 ) + ( βˆ’ 1 ) ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ 𝑣 ( 𝑛 ) , ( 3 . 9 ) with 𝑒 ( 𝑛 ) ∢ = 𝑐 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ , l i m 𝑛 β†’ ∞ 𝑣 ( 𝑛 ) = 0 . ( 3 . 1 0 )
Due to the definition of 𝑛 βˆ— , we see that the sequence ξ€½ 𝛽 ξ€· 𝑛 βˆ— + 1 ξ€Έ ξ€Ύ = { 𝛽 ( πœ” ) , 𝛽 ( 1 ) , 𝛽 ( 2 ) , … , 𝛽 ( πœ” ) , 𝛽 ( 1 ) , 𝛽 ( 2 ) , … , 𝛽 ( πœ” ) , … } , ( 3 . 1 1 ) is an πœ” -periodic sequence. Since  ξ‚ž 𝑛 βˆ’ 1 πœ” ξ‚Ÿ  = ⎧ βŽͺ ⎨ βŽͺ ⎩ βˆ’ 1 , 0 , … , 0 ξ„Ώ ξ…€ ξ…€ ξ…ƒ ξ…€ ξ…€ ξ…Œ πœ” , 1 , … , 1 ξ„Ώ ξ…€ ξ…€ ξ…ƒ ξ…€ ξ…€ ξ…Œ πœ” , 2 , … ⎫ βŽͺ ⎬ βŽͺ ⎭ , ( 3 . 1 2 ) for 𝑛 ∈ β„• 0 , we have ξ€½ ( βˆ’ 1 ) ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ ξ€Ύ = ⎧ βŽͺ ⎨ βŽͺ ⎩ βˆ’ 1 , 1 , … , 1 ξ„Ώ ξ…€ ξ…€ ξ…ƒ ξ…€ ξ…€ ξ…Œ πœ” , βˆ’ 1 , … , βˆ’ 1 ξ„Ώ ξ…€ ξ…€ ξ…€ ξ…€ ξ…€ ξ…€ ξ…ƒ ξ…€ ξ…€ ξ…€ ξ…€ ξ…€ ξ…€ ξ…Œ πœ” , 1 , … ⎫ βŽͺ ⎬ βŽͺ ⎭ . ( 3 . 1 3 ) Therefore, the sequence ξ€½ ( βˆ’ 1 ) ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ 𝑒 ( 𝑛 ) ξ€Ύ = 𝑐 { βˆ’ 𝛽 ( πœ” ) , 𝛽 ( 1 ) , 𝛽 ( 2 ) , … , 𝛽 ( πœ” ) , βˆ’ 𝛽 ( 1 ) , βˆ’ 𝛽 ( 2 ) , … , βˆ’ 𝛽 ( πœ” ) , … } ( 3 . 1 4 ) is a 2 πœ” -periodic sequence. Set 𝑧 ( 𝑛 ) = ( βˆ’ 1 ) ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ 𝑣 ( 𝑛 ) . ( 3 . 1 5 ) Then, l i m 𝑛 β†’ ∞ 𝑧 ( 𝑛 ) = 0 . ( 3 . 1 6 ) 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 πœ” -periodic solution” π‘₯ = π‘₯ ( 𝑛 ) of (1.1) as well. The formula (2.6) reduces to π‘₯ ( 𝑛 ) = ℬ ⌊ ( 𝑛 βˆ’ 1 ) / πœ” βŒ‹ 𝑣 ( 𝑛 ) = π‘œ ( 1 ) , 𝑛 ∈ β„• 0 , ( 3 . 1 7 ) 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 ) , 𝑛 β†’ ∞ .

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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