Research Article | Open Access

Volume 2009 |Article ID 325296 | https://doi.org/10.1155/2009/325296

Dağistan Simsek, Bilal Demir, Cengiz Cinar, "On the Solutions of the System of Difference Equations , ", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 325296, 11 pages, 2009. https://doi.org/10.1155/2009/325296

On the Solutions of the System of Difference Equations ,

Accepted13 Apr 2009
Published01 Jul 2009

Abstract

We study the behavior of the solutions of the following system of difference equations , where the constant A and the initial conditions are positive real numbers.

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 [124].

In this paper we study the behavior of the solutions of the following system of difference equations: where the constant and the initial conditions are positive real numbers.

2. Main Result

Definition 2.1. Fibonacci sequence is , and for .

Definition 2.2. The symbol symbolizes the greatest integer function.

Definition 2.3. The sequence of

Definition 2.4. The sequence of

Theorem 2.5. Let be the solution of the system of difference equations (1.1) for and
If , then , and if

Proof. Let then then then ,

Theorem 2.6. Let be the solution of the system of difference equations (1.1) for .
and if ,
If , then

Proof. Similarly we can obtain the proof as the proof of Theorem 2.5.

Theorem 2.7. Let be the solution of the system of difference equations (1.1) for and .

Proof. (a) We obtain that
Similarly we can obtain the proof of (b) as the proof of (a).

Theorem 2.8. Let be the solution of the system of difference equations (1.1) for and .

Proof. Similarly we can obtain the proof as the proof of Theorem 2.7.

Theorem 2.9. Let be the solution of the system of difference equations (1.1) for
If , then
If , then

Proof. Let , then

Theorem 2.10. Let be the solution of the system of difference equations (1.1) for .

Proof. (a) We obtain that
Similarly we can obtain the proof of (b) as the proof of (a).

Lemma 2.11. Let be the initial condition of (1.1) for ; there is at least an such that every for , .

Proof. We consider that hence and that proofs the existing of defined in hypothesis.

Theorem 2.12. Let be the solution of the system of difference equations (1.1) for , and is the number, defined by Lemma 2.11.

and when , the solutions will be different for every different constant

Proof. Let then ,

Lemma 2.13. Let be the initial condition of (1.1) for ; there is at least an such that every for , .

Proof. Similarly we can obtain the proof as the proof of Lemma 2.11.

Theorem 2.14. Let be the solution of the system of difference equations (1.1) for , and is the number, defined by Lemma 2.13.

and when , the solutions will be different for every different constant

Proof. Similarly we can obtain the proof of be as the proof of Theorem 2.12.

Lemma 2.15. Let be the initial condition of (1.1) for ; there is at least an such that every for , .

Proof. We consider that hence and that proofs the existing of defined in hypothesis.

Theorem 2.16. Let be the solution of the system of difference equations (1.1) for , , and is the number, defined by Lemma 2.15.
and if ,
, and if , and when , the solutions will be different for every different constant

Proof. Let then then then

Lemma 2.17. Let be the initial condition of (1.1) for ; there is at least an such that every for , .

Proof. Similarly we can obtain the proof as the proof of Lemma 2.15.

Theorem 2.18. Let be the solution of the system of difference equations (1.1) for , , and is the number, defined by Lemma 2.17.
, and if ,
, and if , and when , the solutions will be different for every different constant

Proof. Similarly we can obtain the proof as the proof of Theorem 2.16, which completes the proofs of theorems.

References

1. A. M. Amleh, Boundedness periodicity and stability of some difference equations, Ph.D. thesis, University of Rhode Island, Kingston, Rhode Island, USA, 1998.
2. C. Çinar, S. Stević, and İ. Yalçınkaya, “On positive solutions of a reciprocal difference equation with minimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307–314, 2005.
3. S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 1996. View at: Zentralblatt MATH | MathSciNet
4. J. Feuer, “Periodic solutions of the Lyness max equation,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 147–160, 2003.
5. A. Gelişken, C. Çinar, and R. Karataş, “A note on the periodicity of the Lyness max equation,” Advances in Difference Equations, vol. 2008, Article ID 651747, 5 pages, 2008.
6. A. Gelişken, C. Çinar, and İ. Yalçınkaya, “On the periodicity of a difference equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 820629, 11 pages, 2008.
7. E. Janowski, V. L. Kocic, G. Ladas, and G. Tzanetopoulos, “Global behavior of solutions of ${\text{x}}_{n+1}\text{=}\left[{\text{max{x}}_{\text{n}}^{\text{k}}\text{,A}\right\}\right]/{\text{x}}_{n-1}$,” Journal of Difference Equations and Applications, vol. 3, no. 3-4, pp. 297–310, 1998.
8. M. R. S. Kulenević and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at: Zentralblatt MATH
9. 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.
10. L. A. Moybé and A. S. Kapadia, Difference Equations with Public Health Applications, CRC Press, New York, NY, USA, 2000.
11. G. Papaschinopoulos and V. Hatzifilippidis, “On a max difference equation,” Journal of Mathematical Analysis andApplications, vol. 258, no. 1, pp. 258–268, 2001. View at: Publisher Site | Google Scholar | MathSciNet
12. G. Papaschinopoulos, J. Schinas, and V. Hatzifilippidis, “Global behavior of the solutions of a max-equation and of a system of two max-equations,” Journal of Computational Analysis and Applications, vol. 5, no. 2, pp. 237–254, 2003.
13. 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.
14. G. Stefanidou and G. Papaschinopoulos, “The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation,” Information Sciences, vol. 176, no. 24, pp. 3694–3710, 2006.
15. S. Stević, “On the recursive sequence ${\text{x}}_{n+1}=\text{max}\left\{c,{x}_{n}^{p}/{x}_{n-1}^{p}\right\}$,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791–796, 2008.
16. D. Şimsek, C. Çinar, and İ. Yalçınkaya, “On the solutions of the difference equation ${\text{x}}_{n+1}=\text{max}\left\{1/{x}_{n-1},{x}_{n-1}\right\}$,” International Journal of Contemporary Mathematical Sciences, vol. 1, no. 9–12, pp. 481–487, 2006. View at: Google Scholar | MathSciNet
17. D. Şimsek, B. Demir, and A. S. Kurbanlı, “${\text{x}}_{n+1}=\text{max}\left\{1{\text{/x}}_{n}{\text{,y}}_{n}{\text{/x}}_{n}\right\},{\text{y}}_{n+1}=\text{max}\left\{1{\text{/y}}_{n}{\text{,x}}_{n}{\text{/y}}_{n}\right\}$,” Denklem Sistemlerinin Çözümleri Üzerine. In press. View at: Google Scholar
18. C. T. Teixeria, Existence stability boundedness and periodicity of some difference equations, Ph.D. thesis, University of Rhode Island, Kingston, Rhode Island, USA, 2000.
19. S. Valicenti, Periodicity and global attractivity of some difference equations, Ph.D. thesis, University of Rhode Island, Kingston, Rhode Island, USA, 1999.
20. H. D. Voulov, “On the periodic character of some difference equations,” Journal of Difference Equations and Applications, vol. 8, no. 9, pp. 799–810, 2002.
21. 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.
22. İ. Yalçınkaya, 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.
23. İ. Yalçınkaya, C. Çinar, and M. Atalay, “On the solutions of systems of difference equations,” Advances in Difference Equations, vol. 2008, Article ID 143943, 9 pages, 2008.
24. X. Yan, X. Liao, and C. Li, “On a difference equation with maximum,” Applied Mathematics and Computation, vol. 181, pp. 1–5, 2006. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2009 Dağistan Simsek 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.