1. Introduction
Max-type difference equations and max-type difference systems have been wisely applied in biology, computer science and automatic control systems and so on. There has been great interest in studying these equations in recent years.
For example, Briden et al. [1] investigated the periodicity character of the solution of the max-type difference equation
Xiao Qian et al. [2] showed that the solution of the max-type difference equation
is periodic with period two.
W. Q. Ji et al. [3] showed that the solution of the max-type difference system
is periodic with period two.
In addition, E. M. Elasyed, Stevo Stević and others investigated some periodic max-type difference equations and periodic max-type difference systems in [4] - [7] .
In this paper we show that every solution of the following max-type difference system
(1)
where the initial conditions are arbitrary non-zero real numbers and, is periodic with period four.
Remark 1. Note that if, then System (1) becomes, , from which it follows that, and every solution is periodic with period four.
2. Some Lemmas
Lemma 1 Assume that is a solution of System (1) and there exists a such that
(2)
Then every solution is periodic with period four.
Proof Frist, we will prove that
(3)
where, from which the lemma follows.
Now, we use the method of induction. For, Equation (3) becomes the following equations
(4)
By System (1) and Equation (2), we obtain that
From which, Equation (4) holds.
Assume Equation (3) holds for, and by using System (1) and Equation (2), we obtain that
So we complete the proof.
Lemma 2 Assume that. Then every solution of System (1) is positive if initial conditions satisfy one of the following conditions or or or.
Proof Without loss of generality, we assume that and from System (1) we have
By using the method of induction, we have
Similarly, when or or, there exists an such that
The proof is completed.
Lemma 3 Assume that. Then every solution of System (1) with positive initial conditions is periodic with period four.
Proof By System (1), we obtain that
Let, , and there are four cases which need to be discussed.
Case 1.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four. Moreover, we have
where, and the solution has the following form
Case 2.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four. Moreover, we have
where, and the solution has the following form
Case 3.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four. Moreover, we have
where, and the solution has the following form
Case 4.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four. Moreover, we have
where, and the solution has the following form
So we complete the proof.
Lemma 4 Assume that. Then every solution of System (1) with negative initial conditions is periodic with period four.
Proof Since and, by induction we have. If we use the change and System (1) can be rewritten as follows
(5)
where.
Now, we will prove that every solution of System (5) with positive initial conditions is periodic with period four.
Let, ,. Similar to the proof of Lemma (3), there are four cases which need to be discussed.
Case 1.. We obtain that
Case 2.. We obtain that
Case 3.. We obtain that
Case 4.. We obtain that
So we complete the proof.
Lemma 5 Assume that. Then every solution of System (1) is periodic with period four if initial conditions satisfy one of the following conditions
Proof If, by using System (1), we know that there is only one case which needs to be discussed. That is
Then we have
Hence,. And by Lemma 1, we have that the solution is periodic with period four.
The proof of case is similar to the proof of case, so we omit it. Then, the proof is completed.
Lemma 6 Assume that. Then every solution of System (1) is periodic with period four if initial conditions satisfy one of the following conditions
Proof If, by using System (1), we know that so there are two cases which need to be discussed. That is
Case 1.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four.
Case 2.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four.
The proof of case is similar to the proof of case, so we omit it. Then, the proof is completed.
Lemma 7 Assume that. Then every solution of System (1) is periodic with period four if initial conditions satisfy one of the following conditions
Proof If, by using System (1), we know that there are four cases which need to be discussed. That is
Case 1.. We have
Hence,. And by Lemma 1, we have that the solution is periodic with period four.
The proof of case is similar to the proof of case in Lemma 3, so we omit it and case is completed.
Similarly, the proof of case is similar to the proof of case, so we omit it.
3. Main Results
By using Lemma 2 and Lemma 3, we obtain the following result.
Theorem 1 Assume that. Then every solution of System (1) is periodic with period four if initial conditions satisfies one of the following conditions or or or.
By using Theorem 1 and Lemma 4, we obtain the following result.
Theorem 2 Assume that. Then every well-defined solution of System (1) is periodic with period four.
By using Lemma 5, Lemma 6 and Lemma 7, we obtain the following result.
Theorem 3 Assume that. Then every well-defined solution of System (1) is periodic with period four.
By using Theorem 2 and Theorem 3, we obtain the following result.
Theorem 4 Assume that. Then every well-defined solution of System (1) is periodic with period four.
Acknowledgements
We thank the Editor and the referee for their comments. Research supported by Distinguished Expert Foundation and Youth Science Foundation of Naval Aeronautical and Astronautical University.