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.