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, [6–20] 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.