Research Article | Open Access

Volume 2010 |Article ID 583230 | https://doi.org/10.1155/2010/583230

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 𝑥 𝑛 + 𝟏 = m a x { 𝑥 𝑛 , 𝐴 } / 𝑥 𝟐 𝑛 𝑥 𝑛 − 𝟏

Accepted11 May 2010
Published12 Aug 2010

#### 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,  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 , 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)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

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  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 .

1. A. D. Myškis, “Some problems in the theory of differential equations with deviating argument,” Uspekhi Matematicheskikh Nauk, vol. 32, no. 2(194), pp. 173–202, 1977. View at: Google Scholar | MathSciNet
2. E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966.
3. R. Abu-Saris and F. Allan, “Periodic and nonperiodic solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{x}_{n}^{2},A\right\}/{x}_{n}{x}_{n-1}$,” in Advances in Difference Equations (Veszprém, 1995), pp. 9–17, Gordon and Breach, Amsterdam, The Netherlands, 1997.
4. R. M. Abu-Sarris and F. M. Allan, “Rational recursive sequences involving the maximum function,” Far East Journal of Mathematical Sciences, vol. 1, no. 3, pp. 335–342, 1999.
5. K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max difference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193–1199, 2006.
6. C. Çinar, S. Stević, and I. Yalçinkaya, “On positive solutions of a reciprocal difference equation with minimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307–314, 2005.
7. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the periodic nature of some max-type difference equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 14, pp. 2227–2239, 2005.
8. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Global attractivity and periodic character of a fractional difference equation of order three,” Yokohama Mathematical Journal, vol. 53, no. 2, pp. 89–100, 2007.
9. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of higher order difference equation,” Soochow Journal of Mathematics, vol. 33, no. 4, pp. 861–873, 2007.
10. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of some max-type difference equations,” Vietnam Journal of Mathematics, vol. 36, no. 1, pp. 47–61, 2008.
11. E. M. Elabbasy and E. M. Elsayed, “On the solution of recursive sequence ${x}_{n+1}=\mathrm{max} \left\{{x}_{n-2},1/{x}_{n-2}\right\}$,” Fasciculi Mathematici, vol. 41, pp. 55–63, 2009. View at: Google Scholar
12. E. M. Elsayed, “On the solutions of higher order rational system of recursive sequences,” Mathematica Balkanica, vol. 22, no. 3-4, pp. 287–296, 2008.
13. E. M. Elsayed and S. Stević, “On the max-type equation ${x}_{n+1}=\mathrm{max} \left\{A/{x}_{n},{x}_{n-2}\right\}$,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 910–922, 2009.
14. E. M. Elsayed and B. D. Iričanin, “On a max-type and a min-type difference equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 608–614, 2009.
15. E. M. Elsayed, B. D. Iricanin, and S. Stevic, “On the max-type equation ${x}_{n+1}=\mathrm{max} \left\{A/{x}_{n},{x}_{n-1}\right\}$,” ARS Combinatoria, vol. 95, 2010. View at: Google Scholar
16. J. Feuer, “On the eventual periodicity of ${x}_{n+1}=\mathrm{max} \left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$ with a period-four parameter,” Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467–486, 2006.
17. J. Feuer and K. T. Donnell, “On the eventual periodicity of ${x}_{n+1}=\mathrm{max} \left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$ with a period-five parameter,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 883–889, 2008. View at: Google Scholar
18. J. Feuer, “Periodic solutions of the Lyness max equation,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 147–160, 2003.
19. A. Gelisken, C. Cinar, and R. Karatas, “A note on the periodicity of the Lyness max equation,” Advances in Difference Equations, vol. 2008, Article ID 651747, 5 pages, 2008.
20. A. Gelisken, C. Cinar, and I. Yalcinkaya, “On the periodicity of a difference equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 820629, 11 pages, 2008.
21. A. Gelişken and C. Çinar, “On the global attractivity of a max-type difference equation,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 812674, 5 pages, 2009.
22. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. View at: MathSciNet
23. C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n-1}\right\}$ with periodic parameters,” Dynamics of Continuous, Discrete and Impulsive Systems. Series B, vol. 2003, pp. 11–15, 2003. View at: Google Scholar
24. E. J. Janowski, V. L. Kocic, G. Ladas, and S. W. Schultz, “Global behavior of solutions of ${x}_{n+1}=\mathrm{max} \left\{{x}_{n},A\right\}/{x}_{n-1}$,” in Proceedings of the 1st International Conference on Difference Equations, pp. 273–282, Gordon and Breach, Basel, Switzerland, 1995.
25. G. Ladas, “Open problems and conjectures,” Journal of Difference Equations and Applications, vol. 22, pp. 339–341, 1996. View at: Google Scholar
26. G. Ladas, “Open problems and conjectures,” Journal of Difference Equations and Applications, vol. 4, no. 3, p. 312, 1998. View at: Google Scholar
27. X. Li, D. Zhu, and G. Xiao, “Behavior of solutions of certain recursions involving the maximum function,” Journal of Mathematics, vol. 23, no. 2, pp. 199–206, 2003.
28. D. P. Mishev, W. T. Patula, and H. D. Voulov, “A reciprocal difference equation with maximum,” Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 1021–1026, 2002.
29. D. P. Mishev, W. T. Patula, and H. D. Voulov, “Periodic coefficients in a reciprocal difference equation with maximum,” Panamerican Mathematical Journal, vol. 13, no. 3, pp. 43–57, 2003.
30. W. T. Patula and H. D. Voulov, “On a max type recurrence relation with periodic coefficients,” Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 329–338, 2004.
31. E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York, NY, USA, 1969. View at: MathSciNet
32. E. C. Pielou, Population and Community Ecology, Gordon and Breach, Yverdon, Switzerland, 1974.
33. D. Simsek, C. Cinar, and I. Yalcinkaya, “On the solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{x}_{n-1},1/{x}_{n-1}\right\}$,” International Journal of Contemporary Mathematical Sciences, vol. 1, no. 9–12, pp. 481–487, 2006.
34. D. Simsek, “On the solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{x}_{n-2},1/{x}_{n-2}\right\}$,” Selcuk University Journal of Education Faculty, vol. 23, pp. 367–377, 2007. View at: Google Scholar
35. S. Stević, “On the recursive sequence ${x}_{n+1}=\mathrm{max} \left\{c,{x}_{n}^{p}/{x}_{n-1}^{p}\right\}$,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791–796, 2008.
36. F. Sun, “On the asymptotic behavior of a difference equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 243291, 6 pages, 2008.
37. T. Sun, B. Qin, H. Xi, and C. Han, “Global behavior of the max-type difference equation ${x}_{n+1}=\mathrm{max} \left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$,” Abstract and Applied Analysis, vol. 2009, Article ID 152964, 10 pages, 2009.
38. I. Szalkai, “On the periodicity of the sequence ${x}_{n+1}=\mathrm{max} \left\{{A}_{\text{0}}/{x}_{n},{A}_{1}/{x}_{n-1},\dots ,{A}_{k}/{x}_{n-k}\right\}$,” Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25–29, 1999.
39. H. D. Voulov, “Periodic solutions to a difference equation with maximum,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2155–2160, 2003.
40. H. D. Voulov, “On the periodic nature of the solutions of the reciprocal difference equation with maximum,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 32–43, 2004.
41. I. Yalçinkaya, B. D. Iričanin, and C. Çinar, “On a max-type difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 47264, 10 pages, 2007.

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