Abstract and Applied Analysis

Volume 2012 (2012), Article ID 108047, 9 pages

doi:10.1155/2012/108047

Research Article

On the Difference Equation

Stevo Stević,¹ Josef Diblík,2,3 Bratislav Iričanin,⁴ and Zdenĕk Šmarda³

¹Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Belgrade, Serbia

²Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic

³Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic

⁴Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia

Received 15 April 2012; Accepted 23 July 2012

Academic Editor: Norio Yoshida

Copyright © 2012 Stevo Stević 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.

We show that the difference equation , where , the parameters , and initial values , are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.

1. Introduction

Recently, there has been some reestablished interest in difference equations which can be solved, as well as in their applications, see, for example, [1–16]. For some old results in the topic see, for example, the classical book [17].

In recently accepted paper [18] are given formulae for the solutions of the following four difference equations:

and some of these formulae are proved by induction.

Here, we show that the formulae obtained in [18] follow from known results in a natural way. Related idea was exploited in paper [7].

Moreover, we will consider here the following more general equation:

where and the parameters as well as initial values are real numbers, and describe the behaviour of all well-defined solutions of the equation.

For a solution , of the difference equation

is said that it is eventually periodic with period , if there is an such that

If , then it is said that the solution is periodic with period . For some results in this area see, for example, [19–26] and the references therein.

2. Solutions of Equation (1.2)

By using the change of variables

equation (1.2) is transformed into the following linear first-order difference equation:

for which it is known (and easy to see) that

if , and

if .

From (2.1), we have

for , from which it follows that

for every and .

Using (2.3) and (2.4) in (2.6), we get

if , and

if , for every and .

By using formulae (2.7) and (2.8), the behavior of well-defined solutions of equation (1.2) can be obtained. This is done in the following section.

Remark 2.1. It is easy to check that the formulae in Theorems 2.1 and 4.1 from [18] are direct consequences of formula (2.8), whereas formulae in Theorem 3.1 and Theorem 5.1 from [18] are direct consequences of formula (2.7).

Remark 2.2. Note that from formula (2.8), it follows that in the case , , all well-defined solutions of equation (1.2) are periodic with period . This can be also obtained from (1.2), without knowing explicit formulae for its solutions. Namely, in this case, (1.2) can be written as follows:

since we assume , for all , from which it follows that the sequence is constant, that is, for some . Hence,

as claimed.

Remark 2.3. Note also that in the case , , from (2.7), we have

for every and , from which the behaviour of the solutions in the case easily follows.

3. Asymptotic Behavior of Well-Defined Solutions of Equation (1.2)

In this section, we derive some results on asymptotic behavior of well-defined solutions of equation (1.2). We will use well-known asymptotic formulae as follows:

for , where is the Landau “big-oh” symbol.

Theorem 3.1. Let and in (1.2). Then, every well-defined solution of equation (1.2) converges to zero.

Proof. By formula (2.8), we have

where is sufficiently large so that (3.1) can be applied below, and

Since

we conclude, using (3.1), that

Therefore,

for every , and consequently, , as claimed.

Before we formulate and prove our next result, we will prove an auxiliary result which is incorporated in the lemma that follows.

Lemma 3.2. If , then equation (1.2) has -periodic solutions.

Proof. Let

Then

Since , we see that equation (3.8) has equilibrium solution as follows:

For this solution of equation (3.8), we have

from which the lemma follows.

Remark 3.3. Note that -periodic solutions in the previous lemma could be prime -periodic. For this it is enough to choose initial conditions , such that the string

is not periodic with a period less then .

Theorem 3.4. Let and in (1.2). Then, every well-defined solution of equation (1.2) converges to a, not necessarily prime, -periodic solution of the equation.

Proof. First note that by Lemma 3.2, in this case, there are -periodic solutions of the equation. We know that in this case well-defined solutions of the equation are given by formula (2.7). From this and by using asymptotic formulae (3.1), we obtain that for sufficiently large

where

From (3.12) and since , it easily follows that the sequences are convergent for each , from which the theorem follows.

Theorem 3.5. Let and in (1.2). Then, every well-defined solution of equation (1.2) converges to zero.

Proof. In this case, well-defined solutions of equation (1.2) are also given by formula (2.7). Further note that for each holds

Now note that , due to the assumption . Using this fact and (3.14), it follows that for sufficiently large , say we have

From this, we have

as , where

from which the theorem follows.

Theorem 3.6. Let , , and be even in (1.2). Then, every well-defined solution of equation (1.2) is eventually periodic with, not necessarily prime, period .

Proof. From (2.2), in this case, we have

which means that the sequence is two-periodic, and consequently the sequence is two-periodic. Hence

from which it follows that

that is, the sequences and are -periodic from which the result easily follows.

Theorem 3.7. Let , , be odd in (1.2), and be a well-defined solution of equation (1.2). Then the following statements are true.

(a)If , then the solution is -periodic.(b)If , then and , as .(c)If , then and , as .

Proof. As in Theorem 3.6, we obtain (3.18) and consequently (3.19) holds. If for some , then from (1.2) and (3.19) we have

From (3.21) we obtain

From relation (3.22), the statements in this theorem easily follow.

Remark 3.8. Note that the case is not possible in Theorem 3.7. Namely, in this case , due to the assumption and, as we can see from (1.2), the solution is not well-defined.

Acknowledgments

The second author is supported by Grant no. P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government Grant no. MSM 00 216 30519. The fourth author is supported by Grant no. FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science Projects III 41025, III 44006, and OI 171007.

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