Research Article | Open Access

Ibrahim Yalcinkaya, Cengiz Cinar, Ali Gelisken, "On the Recursive Sequence ", *Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 583230, 13 pages, 2010. https://doi.org/10.1155/2010/583230

# On the Recursive Sequence

**Academic Editor:**Guang Y. Zhang

#### Abstract

We investigate the periodic nature of the solution of the max-type difference equation , , where the initial conditions are and for , and that and are positive rational numbers. The results in this paper solve the Open Problem proposed by Grove and Ladas (2005).

#### 1. Introduction

Max-type difference equations stem from, for example, certain models in automatic control theory (see [1, 2]). Although max-type difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the behavior of their solutions, see, for example, [1–41] and the relevant references cited therein. Furthermore, difference equation appear naturally as a discrete analogues, and as a numerical solution of differential and delay differential equations having applications various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology. For some papers on periodicity of difference equation see, for example, [8, 9, 11, 12, 15] and the relevant references cited therein.

In [22], the following open problem was posed.

*Open Problem 1 (see [22, page 218, Open Problem ]). *Assume that and that and are positive rational numbers. Investigate the periodic nature of the solution of the difference equation:
where the initial conditions are and .

Now, in this paper we give answer to the open problem

#### 2. Main Results

##### 2.1. The Case

We consider (1.1) where . It is clear that the change of variables reduces (1.1) to the difference equation: where the initial conditions and are positive rational numbers.

In this section we consider the behavior of the solutions of (2.2) (or equivalently of (1.1)) in this case . We give the following lemmas which give us explicit solutions for some consecutive terms and show us the behavior of the solutions of (2.2) (or equivalently of (1.1)).

Lemma 2.1. *Assume that is a solution of (2.2). If then the following statements are true for some integer such that *(i)*If then and .*(ii)*If then or .*(iii)*If and then and .*(iv)*If and then and *

*Proof. *(i) Assume that If then from (2.2) we get and If then we get If then and If then and

Working inductively we have and for So, the proof of (i) is complete.

(ii) Assume that From (i) and (2.2) we get that

If then

If then and So, the proof of (ii) is complete.

As for (iii) and (iv) they are immediately obtained from (ii) and (2.2)

Lemma 2.2. *Assume that is a solution of (2.2). If then the following statements are true.*(i)*If for then the number of integers for which Lemma 2.1(iii) holds is and the number of integers for which Lemma 2.1(iv) holds is *(ii)*If for then the number of integers for which Lemma 2.1(iii) holds is and the number of integers for which Lemma 2.1(iv) holds is *

*Proof. *(i) Assume that and

Assume that number of integers satisfying Lemma 2.1(iii) is This assumption made Lemma 2.1(iii) be applied consecutively for times such that;

Thus, But, from Lemma 2.1(i) we have and This means that Lemma 2.1(iii) cannot be applied consecutively for times. So, the number of integers satisfying Lemma 2.1(iii) is not more than

Similarly, assume that the number of integers satisfying Lemma 2.1(iv) is So, we can apply Lemma 2.1(iv) consecutively for times such that

Thus, we have and But, it contradicts Lemma 2.1(i) So, the number of integers satisfying Lemma 2.1(iv) is not more than

Now, assume that the number of integers satisfying Lemma 2.1(iii) is We have just had the number of integers satisfying Lemma 2.1(iii) is less than From the proof of Lemma 2.1(i) if is the smallest integer satisfying Lemma 2.1(i), then we have for We apply Lemma 2.1(iii) for times such that
and from Lemma 2.1(ii)

Thus, the number of integers satisfying Lemma 2.1(iv) is But it is not possible. So, the number of integers satisfying Lemma 2.1(iii) is

Similarly, assume that number of integers satisfying Lemma 2.1(iv) is From Lemma 2.1(ii)–(iv) we have

Thus, the number of integers satisfying Lemma 2.1(iii) is But it is not possible. So, the number of integers satisfying Lemma 2.1(iv) is So, the proof of (i) is completed.

(ii) Proof of (ii) is similar to the proof of (i). So, it is omitted.

We omit the proofs of Lemmas 2.3 and 2.4 since they can easily be obtained in a way similar to the proofs of Lemmas 2.1 and 2.2.

Lemma 2.3. *Assume that is a solution of (2.2). If then the following statements are true for some integer such that *(i)*If then and .*(ii)*If then or .*(iii)*If and then and .*(iv)*If and then and *

Lemma 2.4. *Assume that is a solution of (2.2). If then the following statements are true.*(i)*If for then number of integers for which Lemma 2.3(iii) holds is and the number of integers for which Lemma 2.3(iv) holds is *(ii)*If for then the number of integers for which Lemma 2.3(iii) holds is and the number of integers for which Lemma 2.3(iv) holds is *

Lemma 2.5. *Assume that is a solution of (2.2). If and then the following statements are true for some integer such that *(i)*Assume that , and If or if , then . *(ii)*If and then and *(iii)*If such that and then *

*Proof. *(i) Assume that From (2.2), we get that and If then If then , and or If then and If then , and or

Working inductively, we have for and

Assume that From (2.2), we get that and

If then and If then and If then and

Working inductively, we have for and So, the proof of (i) is completed.

(ii)Assume that and that From (2.2) and (i), we get that
Then,

So, the proof of (ii) is completed.

(iii)Assume that the smallest integer of integers satisfying (i) is . From the proof of (i), we have for Also, from this assumption we have the subsequence such that and Then, from (ii) we get that and or If we get that It means that and are not the element previous and later subsequences satisfying (ii) If then we get that and that is a element of the subsequence such that and If this proceeds, we have for such that and .

Lemma 2.6. *Assume that is a solution of (2.2). If then the following statements are true.*(i)*If for then number of integers for which Lemma 2.5(ii) holds is and the number of integers for which Lemma 2.5(iii) holds is *(ii)*If for then number of integers for which Lemma 2.5(ii) holds is and the number of integers for which Lemma 2.5(iii) holds is *

Lemma 2.7. *Assume that is a solution of (2.2). If then the following statements are true.*(i)*If for then number of integers for which Lemma 2.5(ii) holds is and the number of integers for which Lemma 2.5(iii) holds is *(ii)

Theorem 2.8. *Consider (2.2). If and then the following statements are true.*(i)*If then is periodic with prime period *(ii)*If then is periodic with prime period *

*Proof. *(i) Assume that and It suffices to prove for We must show that and From Lemmas 2.5 and 2.6, for getting the last two terms we need, we can assume that
and that

From this assumption and (2.2) we get that for
and that

Then we get and that

So, we have for

(ii) Assume that From (2.2), we get immediately and So, we have for If and then the proof of (ii) is similar to the proof of (i). So, it is omitted.

Theorem 2.9. *Consider (2.2). If and then the following statements are true.*(i)*If then is periodic with prime period *(ii)*If then is periodic with prime period *

*Proof. *(i) If then from (2.2) we get that and So, we have for

In the case the proof is similar to the proof of Theorem 2.8(i) such that; from Lemmas 2.5 and 2.7, we can assume that
and that
for Thus, we get that
for and that

So, we have for

Assume that From Lemmas 2.3 and 2.4, for getting the last two terms, we can assume that
for and From this assumption and Lemma 2.4(iii)-(iv) we get that
and that

Thus, we get and From we have for Also, it is easy to see that and for and which imply that is the smallest period.

In the case the proof is similar. So, it is omitted.

(ii) Assume that If then from (2.2) we get that and If then and So, we have for The rest of proof is similar to the proof of (i) and is omitted.

##### 2.2. The Case

We consider (1.1) where . It is clear that the change of variables reduces (1.1) to the difference equation: where the initial conditions and are positive rational numbers.

In this section we consider the behavior of the solutions of (2.24) (or equivalently of (1.1)) in this case 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 2.10. *Assume that is a solution of (2.24). If and then the following statements are true for some integer *(i)*If then and .*(ii)*If then or .*(iii)*If and then and .*(iv)*If and then and *

Lemma 2.11. *Assume that is a solution of (2.24) for which Lemma 2.10 holds. If , , then the following statements are true.*(i)*If for then number of integers for which Lemma 2.10(iii) holds is and the number of integers for which Lemma 2.10(iv) holds is *(ii)*If for then number of integers for which Lemma 2.10(iii) holds is and the number of integers for which Lemma 2.10(iv) holds is *

Lemma 2.12. *Assume that is a solution of (2.24). If and one of the initial contitions is less than or equal to one then the following statements are true for some integer *(i)*If then and .*(ii)*If then or .*(iii)*If and then and .*(iv)*If and then and *

Lemma 2.13. *Assume that is a solution of (2.24) for which Lemma 2.12 holds. If , , then the following statements are true.*(i)*If for then number of integers for which Lemma 2.12(iii) holds is and the number of integers for which Lemma 2.12(iv) holds is *(ii)*If for then number of integers for which Lemma 2.12(iii) holds is and the number of integers for which Lemma 2.12(iv) holds is *

Theorem 2.14. *Consider (2.24). If and then the following statements are true.*(i)*If then is periodic with prime period *(ii)*If then is periodic with prime period *

Theorem 2.15. *Consider (2.24). If at least one of the initial conditions of (2.24) is less than or equal to one, and then the following statements are true.*(i)*If then is periodic with prime period *(ii)*If then is periodic with prime period *

#### 3. Conclusion

Difference equation appears naturally as a discrete analogue and as a numerical solutions of differential and delay differential equations having applications various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology. Specially, max-type difference equations stem from, for example, certain models in automatic control theory. In this paper, periodic nature of the solution of (1.1) which was open problem proposed by Grove and Ladas [22] was investigated. We describe a new method in investigating periodic character of max-type difference equations. It is expected that after some modifications our method will be applicable to Open Problem in [22].

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