Research Article | Open Access

# On Two Systems of Difference Equations

**Academic Editor:**Leonid Berezansky

#### 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.

#### References

- B. D. Iričanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 13, no. 3-4, pp. 499–507, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - İ. Yalçinkaya, C. Çinar, and M. Atalay, “On the solutions of systems of difference equations,”
*Advances in Difference Equations*, vol. 2008, Article ID 143943, 9 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On a $k$-order system of Lyness-type difference equations,”
*Advances in Difference Equations*, vol. 2007, Article ID 31272, 13 pages, 2007. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2010 Bratislav Iričanin and Stevo Stević. 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.