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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 8467682, 6 pages
https://doi.org/10.1155/2018/8467682
Research Article

Eventually Periodic Solutions of a Max-Type System of Difference Equations of Higher Order

College of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, China

Correspondence should be addressed to Guangwang Su; moc.361@3w6g1s

Received 24 January 2018; Accepted 21 March 2018; Published 24 April 2018

Academic Editor: Douglas R. Anderson

Copyright © 2018 Guangwang Su 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 study in this paper the following max-type system of difference equations of higher order: and , , where , , and the initial conditions . We show that if , then every solution of the above system is periodic with period eventually. If , then every solution of the above system is periodic with period or eventually. If or , then the above system has a solution which is not periodic eventually.

1. Introduction

Our purpose in this paper is to study eventual periodicity of the following max-type system of difference equations of higher order: with where and the initial conditions .

Recently, there has been a great interest in studying max-type systems of difference equations. For example, Fotiades and Papaschinopoulos [1] studied the following max-type system of difference equations: and they showed that every positive solution of (3) is eventually periodic.

In 2012, Stević [2] obtained in an elegant way the general solution to the following max-type system of difference equations: for the case and .

In [3], we have recently studied the following max-type system of difference equations: where , , and the initial conditions with , and we showed that every positive solution of (5) is eventually periodic with period .

When and , (5) reduces to the max-type system of difference equations Yazlik et al. [4] in 2015 obtained in an elegant way the general solution of (6).

In 2014, Stević et al. [5] investigated the behavior of positive solutions of the following max-type system of difference equations: where . It is proved that system (7) is permanent when .

For some other results on solutions of many max-type difference equations and systems, such as eventual periodicity, the boundedness character, and attractivity, see, for example, [620] and the related references therein.

2. Main Results and Proofs

In this section, we study the eventual periodicity of solutions of system (1). Let be a solution of (1) with the initial conditions . From (1), it immediately follows that, for any ,

Lemma 1. If eventually, then is a periodic sequence with period eventually. If eventually, then is a periodic sequence with period eventually.

Proof. Assume that eventually. By (1), we see which implies We have from (9), (10), and (11) that Then, for any , is eventually nonincreasing.
We claim that, for every , is a constant sequence eventually. Assume on the contrary that, for some , is not a constant sequence eventually. Then, by (10) and (12), we see that there exists a sequence of positive integers such that, for any , which implies for any . This is a contradiction. Thus, is a periodic sequence with period eventually. The other case is treated similarly/dually, so we omit the detail. The proof is complete.

Lemma 2. If , then for any and .

Proof. Assume that . Then, for , by (1) and (8), we have and by (2), (8), and (9), we have Thus, The proof is complete.

Theorem 3. Let . Then, eventually and is a periodic sequence with period eventually.

Proof. Assume that . For any and , let since is eventually nonincreasing. Thus, by using (15) and by letting in (14), it is obtained that eventually. By Lemma 1, we see that is a periodic sequence with period eventually. The proof is complete.

Now, we assume , and by Lemma 2, we can assume that, for any , Then, and for any .

Lemma 4. If and for some , then, for any , and are constant sequences eventually.

Proof. Assume that and for some . Since , it follows from (14), (15), and Lemma 2 that By (9), (17), and (18), we have This implies Note that eventually; it follows that Since we see that eventually. This with (21) implies From (23), it follows that By eventually and (24), it follows that If eventually, then, in a similar fashion, we can obtain the following:(1) eventually and eventually.(2) and are constant sequences eventually.If eventually, then eventually and From (27), we see that if , then by we have and if , then since Using arguments similar to ones developed in the above given proof, we can show that, for any , and are constant sequences eventually. The proof is complete.

Theorem 5. Assume . Then, one of the following statements holds:(i) and are periodic sequences with period eventually.(ii) eventually and is a periodic sequence with period eventually.

Proof. Assume that . If for some , then by Lemma 4 we see that, for any , and are constant sequences eventually, which implies that and are periodic sequences with period eventually. In the following, we assume that for any .
We claim that eventually. Indeed, if, for some , , then by (1) we have Then, we see that From (31), we have eventually, and with (34) and , it follows that Thus, and are constant sequences eventually. Using arguments similar to ones developed in the proof of Lemma 4, we can show that, for any , and are constant sequences eventually. Thus, eventually; this is a contradiction.
By Lemma 1, we see that eventually and is a periodic sequence with period eventually. Theorem 5 is proven.

Theorem 6. If and , then (1) has a solution which is not periodic eventually.

Proof. We claim that the equation has a nonincreasing solution. Indeed, write It is easy to see that since, for any , we have and . Define by for all .
We claim that is well defined. Indeed, from (38) and the definition of , we have Thus, .
We also claim that is a bijection from to . Indeed, let , with ; we have . On the other hand, for any , we have Choose It follows from (40) and (41) that which implies , and by (38) we obtain .
Furthermore, since is continuous, is a homeomorphism from to .
Since and is a homeomorphism from onto , it follows that By induction, we have for every . Because is an unbounded connected closed set, we know that is an unbounded connected closed set for every . Let Then, and is also an unbounded connected set.
Now, let be a solution of (36) with the initial conditions ; we can show that, for all , Thus, for any and . It is easy to see that is also a solution of (1) which is not periodic eventually. Theorem 6 is proven.

Theorem 7. If , then (1) has an unbounded solution.

Proof. Let be a solution of (1) with the initial values and for .
If for some , then it is easy to see that, for , we have , . Thus, By induction, we can obtain that, for any and , If for some , then we also can obtain that, for any , It is easy to see that is an unbounded solution of (1). Theorem 7 is proven.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was supported by the NNSF of China (Grant no. 11761011), NSF of Guangxi (Grants nos. 2016GXNSFBA380235 and 2016GXNSFAA380286), and YMTBAPP of Guangxi Colleges (Grant no. 2017KY0598).

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