Research Article | Open Access

Volume 2010 |Article ID 891564 | https://doi.org/10.1155/2010/891564

Bratislav D. Iričanin, Wanping Liu, "On a Higher-Order Difference Equation", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 891564, 6 pages, 2010. https://doi.org/10.1155/2010/891564

# On a Higher-Order Difference Equation

Accepted23 Jun 2010
Published11 Aug 2010

#### Abstract

We describe in an elegant and short way the behaviour of positive solutions of the higher-order difference equation , , where and , extending some recent results in the literature.

#### 1. Introduction

Studying difference equations has attracted a considerable interest recently, see, for example, [139] and the references listed therein. The study of positive solutions of the following higher-order difference equations: and where are natural numbers such that , , and was proposed by Stević in several talks, see, for example, [21, 26]. For some results concerning equations related to (1.1) see, for example, [6, 7, 10, 29, 31, 32, 34, 38], while some results on equations related to (1.2) can be found, for example, in [3, 8, 9, 1114, 1820, 22, 25, 29, 32, 33, 35] (see also related references cited therein).

Case is of some less interest, since in this case positive solutions of (1.1) and (1.2), by using the change , become solutions of a linear difference equation with constant coefficients. However, some particular results for the case recently appeared in the literature, see [16, 17, 39].

Nevertheless, motivated by the above-mentioned papers, we will describe the behaviour of positive solutions of the higher-order difference equation where and in, let us say, an elegant and short way.

Let us introduce the following.

Definition 1.1. A solution of (1.3) is said to be eventually periodic with period if there is such that for all If , then we say that the sequence is periodic with period

For some results on equations all solutions of which are eventually periodic see, for example, [2, 4, 8, 15, 28, 37] and the references therein.

Definition 1.2. One says that a solution of a difference equation converges geometrically to if there exist and such that

First, note that if , then (1.3) becomes from which easily follow the following results:(a)if , then all positive solutions of (1.5) are periodic with period (b)if , then each positive solution of (1.5) converges to zero. Moreover, its subsequences converges decreasingly to zero as (c)if , then each positive solution of (1.5) tends to infinity as . Moreover, its subsequences tend increasingly to infinity as .

We may assume that and are relatively prime integers, that is, (the greatest common divisor of numbers and ). Namely, if , then by using the changes , (1.3) is reduced to copies of the following equation: where , and .

Further, note that from (1.3), we have that which implies that the sequence satisfies the following simple difference equation:

#### 2. Main Results

Here we formulate and prove our main results.

Theorem 2.1. Assume that , , and is odd, then all positive solutions of (1.3) are eventually periodic with period .

Proof. By using repeatedly relation (1.7) -times, we obtain Now, note that from (1.8), it follows that in this case is periodic with period . On the other hand, since for each , we have that Hence, the indices , and , belong to different subsequences. From this and the periodicity of , it follows that from which the theorem follows.

By taking the logarithm of (1.3) and using the change , we get The characteristic polynomial of the homogeneous part of (2.4) is from which it follows that all its roots are expressed by These roots are simple if and only if Clearly, if is odd, inequality (2.7) holds. If is even, that is, , for some , then, since must be odd. Then, for and , we will get that inequality (2.7) does not hold.

From the above consideration and Theorem 2.1, we get the next corollary.

Corollary 2.2. Assume that and . Then all positive solutions of (1.3) are eventually periodic if and only if is odd. Moreover, if is odd, then the period is .

Since the root of characteristic polynomial (2.5) is a simple one, a particular solution of nonhomogeneous (2.4) has the form from which, by a direct calculation, we easily get that .

Hence, if is odd, the general solution of (1.3) is Note that from (2.9), it follows that and that is a positive solution of (1.3) with .

From (2.9), (2.10), and Theorem 2.1 the following results directly follow.

Theorem 2.3. Assume that , , and is odd, then every positive solution of (1.3) converges geometrically to zero. Moreover, for each , the subsequence converges monotonically to zero as .

Theorem 2.4. Assume that , , and is odd, then every positive solution of (1.3) tends to infinity. Moreover, for each , the subsequence converges increasingly to infinity as .

Finally, there are two concluding interesting remarks.

Remark 2.5. Note that, since the functions and are periodic with period and the functions and are periodic with period , from the representation (2.9) we also obtain Theorem 2.1.

Remark 2.6. The results in papers [16, 17, 39], which are obtained in much complicated ways, are particular cases of our results. Namely, in [16] Özban studied a system which is transformed into (1.3) with and , in [17] he studied a system which is transformed into (1.3) with , and , while in [39] the authors considered a system which is transformed into (1.3) with , but they only considered the case when

#### Acknowledgments

The authors are indebted to the anonymous referees for their advice resulting in numerous improvements of the text. The research of the first author was partly supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, Project no. 144013.

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Copyright © 2010 Bratislav D. Iričanin and Wanping Liu. 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.