^{1}

^{*}

^{1}

^{*}

^{1}

^{1}

In this paper, we show that every well-defined solution of the max-type system of difference equations , , is eventually periodic with period four.

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. [

Xiao Qian et al. [

is periodic with period two.

W. Q. Ji et al. [

is periodic with period two.

In addition, E. M. Elasyed, Stevo SteviÄ‡ and others investigated some periodic max-type difference equations and periodic max-type difference systems in [

In this paper we show that every solution of the following max-type difference system

where the initial conditions

Remark 1. Note that if

Lemma 1 Assume that

Then every solution is periodic with period four.

Proof Frist, we will prove that

where

Now, we use the method of induction. For

By System (1) and Equation (2), we obtain that

From which, Equation (4) holds.

Assume Equation (3) holds for

So we complete the proof.

Lemma 2 Assume that

Proof Without loss of generality, we assume that

By using the method of induction, we have

Similarly, when

The proof is completed.

Lemma 3 Assume that

Proof By System (1), we obtain that

Let

Case 1.

Hence,

where

Case 2.

Hence,

where

Case 3.

Hence,

where

Case 4.

Hence,

where

So we complete the proof.

Lemma 4 Assume that

Proof Since

where

Now, we will prove that every solution of System (5) with positive initial conditions is periodic with period four.

Let

Case 1.

Case 2.

Case 3.

Case 4.

So we complete the proof.

Lemma 5 Assume that

Proof

Then we have

Hence,

The proof of case

Lemma 6 Assume that

Proof

Case 1.

Hence,

Case 2.

Hence,

The proof of case

Lemma 7 Assume that

Proof

Case 1.

Hence,

The proof of case

Similarly, the proof of case

By using Lemma 2 and Lemma 3, we obtain the following result.

Theorem 1 Assume that

By using Theorem 1 and Lemma 4, we obtain the following result.

Theorem 2 Assume that

By using Lemma 5, Lemma 6 and Lemma 7, we obtain the following result.

Theorem 3 Assume that

By using Theorem 2 and Theorem 3, we obtain the following result.

Theorem 4 Assume that

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.