#### Abstract

We give very short and elegant proofs of the main results in the work of Yalcinkaya et al. (2008).

#### 1. Introduction and a Proof of Some Resent Results

Motivated by our paper [1], the authors of [2] studied the following two systems of difference equations:

where we regard that

Following line by line the proofs of the main results in [1] they proved the following result (see Theorems 2.1 and 2.4 in [2])

Theorem 1 A. *Assume then the following statements are true.*(a)*If then every (well-defined) solution of systems (1.1) and (1.2) is periodic with period *(b)*If , then every (well-defined) solution of systems (1.1) and (1.2) is periodic with period *

Here we give a very short and elegant proof of Theorem A.

*Proof of Theorem A. *By using the change , system (1.1) becomes
while system (1.2) becomes

From (1.3) and (1.4), for each and , we obtain correspondingly that
From (1.5), with it follows that
from which the statement in (a) easily follows.

If we have that
from which it follows that
, implying the statement in (b), as desired.

#### 2. An Extension on Theorem A

Here we extend Theorem A in a natural way. Let denote the greatest common divisor of the integers and , the least common multiple of and , and for let where

Theorem 2.1. *Assume that is a real function such that on its domain of definition, for some then all well-defined solutions of the system of difference equations
**
are periodic with period .*

*Proof. *We use our method of “prolongation" described in [1]. Note that for each system (2.1) is equivalent to a system of difference equations of the same form, where
for every , and

From (2.1) and since , for we have
for each and every

It is clear that
where are such that and .

From (2.3) we have
for each and , from which the result follows.

The following result is proved similarly. Hence we omit its proof.

Theorem 2.2. *Assume that is a real function such that on its domain of definition, for some then all well-defined solutions of the system of difference equations
**
are periodic with period . *

*Remark 2.3. *The proof of Theorem A follows from Theorems 2.1 and 2.2. Indeed, note that the function satisfies the condition on its domain of definition. By Theorems 2.1 and 2.2 we know that all well-defined solutions of systems (1.1) and (1.2) are periodic with period from which the result follows.

*Remark 2.4. *We also have to say that the main result in [3] is a trivial consequence of a result in [1] (see Remark 5 therein). Just note that the simple change of variables , transforms their system () satisfying conditions and , into system () in [1].

#### Acknowledgment

The results in this note were presented at the talk: S. Stević, on a class of max-type difference equations and some of our old results, *Progress on Difference Equations 2009*, Bedlewo, Poland, May 25–29, 2009.