Research Article | Open Access

Stevo Stević, Josef Diblík, Bratislav Iričanin, Zdenĕk Šmarda, "On the Difference Equation ", *Abstract and Applied Analysis*, vol. 2012, Article ID 108047, 9 pages, 2012. https://doi.org/10.1155/2012/108047

# On the Difference Equation

**Academic Editor:**Norio Yoshida

#### 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.

#### References

- A. Andruch-Sobiło and M. Migda, “Further properties of the rational recursive sequence ${x}_{n+1}=a{x}_{n-1}/\left(b+c{x}_{n}{x}_{n-1}\right)$,”
*Opuscula Mathematica*, vol. 26, no. 3, pp. 387–394, 2006. View at: Google Scholar | Zentralblatt MATH - A. Andruch-Sobiło and M. Migda, “On the rational recursive sequence ${x}_{n+1}=a{x}_{n-1}/\left(b+c{x}_{n}{x}_{n-1}\right)$,”
*Tatra Mountains Mathematical Publications*, vol. 43, pp. 1–9, 2009. View at: Google Scholar | Zentralblatt MATH - I. Bajo and E. Liz, “Global behaviour of a second-order nonlinear difference equation,”
*Journal of Difference Equations and Applications*, vol. 17, no. 10, pp. 1471–1486, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Berg and S. Stević, “On difference equations with powers as solutions and their connection with invariant curves,”
*Applied Mathematics and Computation*, vol. 217, no. 17, pp. 7191–7196, 2011. View at: Publisher Site | Google Scholar - L. Berg and S. Stević, “On some systems of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 1713–1718, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Iričanin and S. Stević, “On some rational difference equations,”
*Ars Combinatoria*, vol. 92, pp. 67–72, 2009. View at: Google Scholar | Zentralblatt MATH - S. Stević, “More on a rational recurrence relation,”
*Applied Mathematics E-Notes*, vol. 4, pp. 80–85, 2004. View at: Google Scholar - S. Stević, “A short proof of the Cushing-Henson conjecture,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 37264, 5 pages, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “Existence of nontrivial solutions of a rational difference equation,”
*Applied Mathematics Letters*, vol. 20, no. 1, pp. 28–31, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “Global stability of a max-type difference equation,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 354–356, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “On a system of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3372–3378, 2011. View at: Publisher Site | Google Scholar - S. Stević, “On the difference equation ${x}_{n}={x}_{n-2}/\left({b}_{n}+{c}_{n}{x}_{n-1}{x}_{n-2}\right)$,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4507–4513, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “On a third-order system of difference equations,”
*Applied Mathematics and Computation*, vol. 218, pp. 7649–7654, 2012. View at: Publisher Site | Google Scholar - S. Stević, “On some solvable systems of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 9, pp. 5010–5018, 2012. View at: Publisher Site | Google Scholar - S. Stević, “On the difference equation ${x}_{n}={x}_{n-k}/\left(b+c{x}_{n-1}\cdots {x}_{n-k}\right)$,”
*Applied Mathematics and Computation*, vol. 218, no. 11, pp. 6291–6296, 2012. View at: Publisher Site | Google Scholar - S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, “On a third-order system of difference equations with variable coefficients,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 508523, 22 pages, 2012. View at: Publisher Site | Google Scholar - H. Levy and F. Lessman,
*Finite Difference Equations*, The Macmillan Company, New York, NY, USA, 1961. - H. El-Metwally and E. M. Elsayed, “Qualitative study of solutions of some difference equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 248291, 16 pages, 2012. View at: Publisher Site | Google Scholar - E. A. Grove and G. Ladas,
*Periodicities in Nonlinear Difference Equations*, vol. 4, 2005. - B. D. Iričanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,”
*Dynamics of Continuous, Discrete & Impulsive Systems A*, vol. 13, no. 3-4, pp. 499–507, 2006. View at: Google Scholar | Zentralblatt MATH - R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,”
*Canadian Journal of Mathematics. Journal Canadien de Mathématiques*, vol. 26, pp. 1356–1371, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Papaschinopoulos and C. J. Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations,”
*Communications on Applied Nonlinear Analysis*, vol. 5, no. 2, pp. 47–59, 1998. View at: Google Scholar | Zentralblatt MATH - G. Papaschinopoulos and C. J. Schinas, “Invariants for systems of two nonlinear difference equations,”
*Differential Equations and Dynamical Systems*, vol. 7, no. 2, pp. 181–196, 1999. View at: Google Scholar | Zentralblatt MATH - G. Papaschinopoulos and C. J. Schinas, “Invariants and oscillation for systems of two nonlinear difference equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 46, no. 7, pp. 967–978, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Stević, “Periodicity of max difference equations,”
*Utilitas Mathematica*, vol. 83, pp. 69–71, 2010. View at: Google Scholar | Zentralblatt MATH - S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,”
*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9562–9566, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

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.