Abstract
We study the following max-type difference equation , , where is a periodic sequence with period and with and , and the initial conditions are real numbers with . We show that if (or and is odd), then every well-defined solution of this equation is eventually periodic with period , which generalizes the results of (Elsayed and Stevi (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with and being even which has a well-defined solution that is not eventually periodic.
1. Introduction
The max operator arises naturally in certain models in automatic control theory (see [1]). In recent years, the discrete case involving difference equations with maximum has been receiving increasing attention, for some results in this area; see, for example, [2–4].
In this paper, we consider the following max-type equation: where is a periodic sequence with period and with and , and the initial conditions are real numbers with .
In [5], Irianin and Elsayed showed that every well-defined solution of (1) is eventually periodic with period 4 when , , and . Elsayed and Stevi [6] showed that every well-defined solution of (1) is eventually periodic with period 3 when , , and . In [7], Xiao and Shi showed that if , , and , then every well-defined solution of (1) is eventually periodic with period 2. Qin et al. [8] showed that every well-defined solution of (1) is eventually periodic with period when and .
In this paper, we will generalize the results of [5–8] to the general case.
2. Main Results and Example
In this section, we are ready to state and prove the main results.
Theorem 1. Let be a periodic sequence with period , and with and .(1)If and is odd, then every well-defined solution of (1) is eventually periodic with period .(2)If , then every well-defined solution of (1) is eventually periodic with period .
Proof. Let be a well-defined solution of (1). It follows from (1) that, for any and any ,
Then, for every , is increasing, and for all or there exists such that for all .
We claim that is a constant sequence eventually. Indeed, if is not constant sequence eventually, then there exist such that and is a constant sequence for all since is a periodic sequence. Thus we have
From this we obtain that, for all ,
It follows that, for all ,
Therefore we have eventually. By induction, we can show that eventually for all and every is not constant sequence eventually.
If and is odd, then we have eventually. This is a contradiction.
If , then we write for all and choose such that and for any . Thus
which is a contradiction. This completes the proof of the claim.
By the above claim we may choose an such that for all . Since is a periodic sequence, we can choose an such that . Then, for all ,
Thus for all (otherwise, if for some , then we have , which is a contradiction). By induction, we can show that is a constant sequence eventually for every . Note since . Then is a constant sequence eventually for every , which implies that is eventually periodic with period .
From the proof of Theorem 1 we obtain the following corollary.
Corollary 2. Let with . If is a periodic sequence, then every positive (or negative) solution of (1) is eventually periodic with period .
Theorem 3. Let be a periodic sequence with period , and with and . If for some , then every well-defined solution of (1) is eventually periodic with period .
Proof. Let be a well-defined solution of (1). Using arguments similar to the ones developed in the proof of Theorem 1, we know that, for every , is increasing, and for all or there exists such that for all .
We may assume without loss of generality that . We claim that is a constant sequence eventually. Indeed, if is not constant sequence eventually, then there exist such that with being a constant sequence for all . Thus we have
From this we obtain that, for all ,
Thus and
It follows that, for all ,
This is a contradiction.
Using arguments similar to the ones developed in the proof of Theorem 1, we can show that is a constant sequence eventually for every . Note since . Then is a constant sequence eventually for every , which implies that is eventually periodic with period .
Now we construct an example with and being even which has a well-defined solution that is not eventually periodic.
Example 4. Consider the max-type equation where and is even with and and is a periodic sequence with for all . Choose the initial conditions for odd and for even with ; we can obtain a solution of (12) such that (1) If , then It is easy to verify that and . (2) If , then It is easy to verify that and .
Remark 5. Consider the max-type equation where is a periodic sequence with period and with and , and (or ), and the initial conditions are real numbers with . Write for every and . Then (12) reduces to the equation (1)If (or and is odd), then it follows from Theorem 1 that, for every , every well-defined solution of equation is eventually periodic with period . Thus every well-defined solution of (15) is eventually periodic with period .(2)If and, for every , there exists some such that , then it follows from Theorem 3 that for every , every well-defined solution of equation is eventually periodic with period . Thus every well-defined solution of (15) is eventually periodic with period .(3)If and is even, then it follows from Example 4 that, for every , we can construct an equation such that it has a well-defined solution which is not eventually periodic. Thus we can construct an equation such that it has a well-defined solution which is not eventually periodic.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The project is supported by NNSF of China (11261005) and NSF of Guangxi (2012GXNSFDA276040).