Abstract
We study the periodicities of a system of difference equations , where initial values . We show that if are two periodic sequences, then every solution of the above system is eventually periodic with period 2. If is even, there must be one in and converges to period two solution.
1. Introduction
Difference equations are powerful tool that describe the law of nature. Recently, the theory of difference equations has received extensive attention because it can be applied to many areas of science and technology, such as the fields of information and e-science. Moreover, people pay more attention to the dynamics of max-type difference equations; they are concerned about the research of the period character of solutions and the convergence of positive solutions. Please refer to [1–7].
Szalkai [8] studied the max-type difference equation where , and at least one of them is nonzero. The initial values are nonzero real numbers, so every solution of this equation is periodic solution. Chen [9] studied the dynamics of max-type difference equationwhere is a two-cycle sequence, . The initial values , are nonzero real numbers; then every positive solution of this equation is bounded and persists, and for all . Furthermore, every solution of this equation is periodic with period 4. Yang [10] studied the max-type difference equationwhere , ; then every solution of this equation tends to be or is eventually periodic with period 4. Sun et al. [11] studied the dynamics of max-type difference equation where is a -cycle sequence with . The initial values , are positive real numbers. Let ; if there exist infinitely many such that and , then is eventually equal to 1. every positive solution of this equation is eventually periodic with period 2 provided .
Motivated by the work of above max-type difference equations, the aim of this paper is to investigate the periodicities of a system of max-type difference equationswhere initial values , , and are positive sequences. More precisely, we will prove that if are two periodic sequences, then every solution of (5) is eventually periodic with period 2. if is even, then one of and converges to a period two solution.
2. Main Results
Before proceeding with the proof of our main results in this section, we shall need the following lemmas.
Lemma 1. Let be a solution of system (5); then , , and , for all .
Proof. By system (5), we havefor all . Hence , , and , . This completes the proof of the lemma.
Lemma 2. Let be a solution of system (5). If , then
Proof. By Lemma 1 and , we obtain that, for any ,Similarly, we also obtain that, for any ,This completes the proof of the lemma.
Theorem 3. Let be a solution of system (5) and . Then(1) is eventually monotonically decreasing. Moreover, if , then ;(2).
Proof. By Lemma 2, if , we haveSimilarly, we obtain . Henceso is eventually monotonically decreasing. On the other hand, by Lemma 1, we know , , so . Hence .
It is easy to know that there are infinitely many such that or , so , or . On the other hand, and . Hence and . ThusThis completes the proof of the theorem.
Lemma 4. Let and . Then there exists such thatfor all .
Proof. Let . By Theorem 3, there exists such that :Then . By Lemma 2, it follows thatSimilarly, we obtain . This completes the proof of the lemma.
Lemma 5. There exists a positive integer such that, for all , (if ), or (if ).
Proof. Let and be as Lemma 4. By Theorem 3, if , then there exists a monotonically increasing sequence such that . On the other hand, there exists a subsequence of such that . Without loss of generality, we may assume that , (for all ). By Lemma 4, we getWrite . Then for all .
Similarly, we obtain that if , then . This completes the proof of the Lemma.
Lemma 6. (1) If , then there exists an integer such that both the number sequences and are monotonically decreasing, and , for all .
(2) If , then there exists an integer such that both the number sequences and are monotonically decreasing, and , for all .
Proof. Here, we only prove . By Lemma 5, if and , then there exists such that for all . So is monotonically decreasing for all . There exists a constant such that . Hence,This implies . Without loss of generality, we assume , soHence, , and is monotonically decreasing for all . This completes the proof of the Lemma.
Theorem 7. Let be a solution of system (5). If is even, then one of the and is eventually periodic with period 2.
Proof. By Lemma 6, if , then there exists such that both and are monotonically decreasing. We can easily get and for all .
We will prove that is eventually periodic with period 2. Otherwise, there will be infinitely many such thatWe take a subsequence and assume , . Since is even, . Take the limit on both sides of the inequality ; hence . Since , , we have , so . This is a contradiction. Therefore is eventually periodic with period 2.
Similarly, we obtain that if , then is eventually periodic with period 2. This completes the proof of the theorem.
Theorem 8. If both and are periodic sequences, then the solution of system (5) converges to a period two solution.
Proof. We will only prove the case of . By Lemma 6, we know that if , then there exists such that both and are monotonically decreasing. We can get and for all easily. So both and are monotonically increasing.
Now, we prove that is eventually periodic with period 2. Assume that is not eventually periodic with period 2; then is not an eventual constant, and there exist infinitely many such thatThereforeBy Lemma 6, we know that is monotonically increasing, not an eventual constant, so is monotonically increasing, not an eventual constant. This is a contradiction. So is eventually periodic with period 2.
Similarly, we obtain that is eventually periodic with period 2. So converges to a period two solution. This completes the proof of the theorem.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors read and approved the final manuscript.