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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 820629, 11 pages
http://dx.doi.org/10.1155/2008/820629
Research Article

On the Periodicity of a Difference Equation with Maximum

Department of Mathematics, Education Faculty, Selcuk University, 42099 Konya, Turkey

Received 20 January 2008; Revised 8 April 2008; Accepted 18 May 2008

Academic Editor: Yong Zhou

Copyright © 2008 Ali Gelisken et al. 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.

Abstract

We investigate the periodic nature of solutions of the max difference equation , , where is a positive real parameter, and the initial conditions and such that and are positive rational numbers. The results in this paper answer the Open Problem 6.2 posed by Grove and Ladas (2005).

1. Introduction

Recently, there has been a great interest in studying the periodic nature of nonlinear difference equations. Although difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the periodic behavior of their solutions. The periodic nature of nonlinear difference equations of the max type has been investigated by many authors. See, for example, [120] (see also the references therein).

In [7], the following open problems were posed.

Problem 1.1 (Open Problem [7, Page 217, Open Problem 6.1]). Assume that and that and are positive rational numbers.
Investigate the periodic nature of the solution of the difference equation where the initial conditions and .

Problem 1.2 (Open Problem [7, Page 217, Open Problem 6.2]). Assume that and that and are positive rational numbers. Investigate the periodic nature of the solution of the difference equation where the initial conditions and .

Problem 1.3 (Open Problem [7, Page 217, Open Problem 6.3]). Assume that and that and are positive rational numbers. Investigate the periodic nature of the solution of the difference equation where the initial conditions and .

Problem 1.4 (Open Problem [7, Page 218, Open Problem 6.4]). Assume that and that and are positive rational numbers. Investigate the periodic nature of the solution of the difference equation where the initial conditions and .

Problem 1.5 (Open Problem [7, Page 218, Open Problem 6.5]). Assume that , and are natural numbers, and that and are positive rational numbers. Investigate the periodic nature of the solution of the difference equation where the initial conditions and .

In [6], we solved the open problem 6.1. And, in [20] we solved the open problem 6.4. Now, in this paper we give answer to the open problem 6.2.

2. The Case

We consider (1.2) where . It is clear that the change of variablesreduces (1.2) to the difference equationwhere the initial conditions are positive rational numbers.

In this section, we consider the behavior of the solutions of (2.2) (or equivalently of (1.2)) where . We give the following lemmas which give us explicit solutions of (2.2) for some consecutive terms and show us the pattern of the behavior of solutions of (2.2) (or equivalently of (1.2)). The proofs of some lemmas and theorems in this section are similar. So, some will be proved and the proofs of the others will be omitted.

Lemma 2.1. Suppose that is a solution of (2.2). If at least one of the initial conditions of (2.2) is less than or equal to one and then the following statements are true for some positive integer (a)If then (b)If then for all (c)If then or (d)If and then(e)If and then

Proof.(a) Let From (2.2), if we have and Thus, we get and If then we get , , and If then we get , , , and
Similarly, if we get that , , and then , , , and If this proceeds, we have for and
Now, let If from (2.2), we have and So, we obtain
If we have So, we obtain If this proceeds, we have if and So, the proof of (a) is complete.
(b) Let Then, we have and . So, the proof of (b) follows directly from (2.2).
(c) Let and From (2.2), we get
If then , and , from (a).
If then , , , and from (a). So, the proof of (c) is complete.
(d) Let and for some positive integer Suppose that From (2.2), we get that , , and From we have So, the proof of (d) is complete.
(e) Let and for some positive integer Suppose that From (2.2), we get that , , , and From we have So, the proof of (e) is complete.

The proof of the following lemma is similar and will be omitted.

Lemma 2.2. Suppose that is a solution of (2.2), where If then the following statements are true for some positive integer (a)If then (b)If then or (c)If and then(d)If and then

Remark 2.3. In view of Lemma 2.1 for it is clear that and for or But, it is not clear or If then or If then or So, we have or if for
In view of Lemma 2.1, for from Lemma 2.1(c), we obtain Also, from Lemma 2.1(d), we haveNow, applying Lemma 2.1(d) and then we get
Moreover, from Lemma 2.1(c), we have
It shows that the last corresponding two terms are the same in each two cases. So, we can apply Lemma 2.1(c) or Lemma 2.1(d) for getting the last two terms we need. Furthermore, it is reality there are infinite number of integers satisfying Lemma 2.1(b). If we determine exactly the number of integers , we can apply Lemmas 2.1(c) and 2.1(d), consecutively. Also, in view of Lemma 2.1,

Remark 2.4. In view of Lemma 2.2, we can get similar results as Remark 2.3. So, we have the following:Also, we can apply Lemma 2.2(c) or Lemma 2.2(d) firstly for getting the last two terms we need.

Clearly, there are infinite number of integers satisfying Lemma 2.1 or Lemma 2.2. We give the following two lemmas about the number of integers .

Lemma 2.5. Suppose that is a solution of (2.2) satisfying Lemma 2.1. If and then the following statements are true. (a)If is an even integer, then the number of integers satisfying Lemma 2.1(c) is and the number of integers satisfying Lemma 2.1(d) is for .(b)If is an odd integer, then the number of integers satisfying Lemma 2.1(c) is and the number of integers satisfying Lemma 2.1(d) is for

Proof.(a) Assume that is an even integer. So, both of and are even or odd integers. From and are odd integers. We have
Let Then, we have , , and If we get from (2.2), then we get from Lemma 2.1(d) and Remark 2.3. Similarly, If we get from (2.2), then we get from Lemma 2.1(d) and Remark 2.3. So, the number of integers satisfying Lemma 2.1(d) is for , such that or There are not any integers satisfying Lemma 2.1(c). So, the claim is true for
Let and the number of integers satisfying Lemma 2.1(c) is for From Remark 2.3, observe that is the smallest integer of the integers satisfying Lemma 2.1(b), such that or This assumption and Remark 2.3 allow us that Lemma 2.1(c) can be applied consecutively for iterated times such that
So, we have
Thus,
This shows that But, it contradicts Lemma 2.1(a). This means Lemma 2.1(c) cannot be applied consecutively for iterated times. So, the number of integers satisfying Lemma 2.1(c) is not more than for
Similarly, we assume that the number of integers satisfying Lemma 2.1(d) is for So, we can apply Lemma 2.1(d) consecutively for iterated times such that
So, we have
Thus, we have
This means that But, it contradicts Lemma 2.1(a) So, the number of integers satisfying Lemma 2.1(c) is not more than
We assume that the number of integers satisfying Lemma 2.1(c) is for We have just had the number of integers satisfying Lemma 2.1(d) is less than . We apply Lemma 2.1(d) for iterated times, and then we get that the first integer satisfying Lemma 2.1(c) is such that . So, we apply Lemma 2.1(c) for iterated times, and then we get that the biggest integer satisfying Lemma 2.1(b) for is such that From Lemma 2.1(b), or But, is not possible because the number of integers satisfying Lemma 2.1(d) is not So, it must be and it contradicts our assumption. Thus, the number of integers satisfying Lemma 2.1(c) is exactly for
Similarly, the number of integers satisfying Lemma 2.1(d) is not and can be showed for . So, the number of integers satisfying Lemma 2.1(d) is exactly for The proof is complete.
(b) The proof of (b) is similar and will be omitted.

The proof of the following lemma is similar and will be omitted.

Lemma 2.6. Suppose that is a solution of (2.2) satisfying Lemma 2.2. If and then the following statements are true. (a)If is an even integer, then the number of integers satisfying Lemma 2.2(a) is and the number of integers satisfying Lemma 2.2(b) is for (b)If is an odd integer, then the number of integers satisfying Lemma 2.2(a) is and the number of integers satisfying Lemma 2.2(b) is for

We give the following two lemmas which are generalized from Lemmas 2.1 and 2.5, and Remark 2.3. It allows us to more quickly calculate terms in the solution of (2.2).

Lemma 2.7. Suppose that is a solution of (2.2) satisfying Lemmas 2.1 and 2.5. If is an even integer, then the following statements are true. (a)If thenfor (b)If thenfor

Lemma 2.8. Suppose that is a solution of (2.2) satisfying Lemma 2.1 and 2.5. If is an odd integer, then the following statements are true. (a)If thenfor (b)If thenfor

The following two lemmas are generalized from Lemmas 2.2, 2.6, and Remark 2.4.

Lemma 2.9. Suppose that is a solution of (2.2) satisfying Lemmas 2.2 and 2.6. If is an even integer, then the following statements are true. (a)If thenfor (b)If thenfor

Lemma 2.10. Suppose that is a solution of (2.2) satisfying Lemmas 2.2 and 2.6. If is an odd integer, then the following statements are true. (a)If thenfor (b)If thenfor

Theorem 2.11. Suppose that is a solution of (1.2) with the initial conditions and , such that , and are positive rational numbers. Let at least one of and is less than or equal to one. If and then is periodic

Proof.Let is a solution of (2.2) satisfying Lemmas 2.1 and 2.5. We assume that is an odd integer. We must show that
From Lemma 2.8(a), we get thatThen, from Lemma 2.8(b), we get thatSo, at the end of this process, we have and . From we get immediately for all Also, it is easy to see that and for and It shows that is prime period. So, the proof is complete. The proof of the case that is an even integer is similar and will be omitted.

The proof of the following theorem is similar and follows directly from Lemmas 2.9 and 2.10.

Theorem 2.12. Suppose that is a solution of (1.2) with the initial conditions and such that , and are positive rational numbers. Let , If and then is periodic

3. The Case

We consider (1.2) where . It is clear that the change of variablesreduces (1.2) to the difference equationwhere the initial conditions are positive rational numbers.

In this section, we consider the behavior of the solutions of (3.2) (or equivalently of (1.2)) where We omit the proofs of the following results since they can easily be obtained in a way similar to the proofs of the lemmas and theorems in the previous section.

Lemma 3.1. Suppose that is a solution of (3.2). If then the following statements are true for some integer (a)If then (b)If then or (c)If and then(d)If and then

Lemma 3.2. Suppose that is a solution of (3.2) satisfying Lemma 3.1. If and then the following statements are true. (a)If is an even integer, then the number of integers satisfying Lemma 3.1(a) is and the number of integers satisfying Lemma 3.1(b) is for (b)If is an odd integer, then the number of integers satisfying Lemma 3.1(a) is and the number of integers satisfying Lemma 3.1(b) is for

The following two lemmas are generalized from Lemmas 3.1 and 3.2.

Lemma 3.3. Suppose that is a solution of (3.2) satisfying Lemmas 3.1 and 3.2. If is an even integer, then the following statements are true. (a)If thenfor (b)If thenfor

Lemma 3.4. Suppose that is a solution of (3.2) satisfying Lemmas 3.1 and 3.2. If is an odd integer, then the following statements are true. (a)If thenfor (b)If thenfor

Theorem 3.5. Suppose that is a solution of (1.2) with the initial conditions and such that , , and are positive rational numbers. If and then is periodic

4. Conclusion

In this paper, we have solved the open problem 6.2 which was proposed in [7]. We think that the method used in this work may solve the open problems 6.3 and 6.5 which were proposed in [7].

Acknowledgment

The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.

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