Research Article | Open Access
On Two Systems of Difference Equations
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
where we regard that
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 . 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  is a trivial consequence of a result in  (see Remark 5 therein). Just note that the simple change of variables , transforms their system () satisfying conditions and , into system () in .
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.
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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.