Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 405121 | 4 pages | https://doi.org/10.1155/2010/405121

On Two Systems of Difference Equations

Academic Editor: Leonid Berezansky
Received11 Jan 2010
Accepted09 Mar 2010
Published24 Mar 2010

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

  1. 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
  2. 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
  3. 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 © 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.


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