- Research
- Open Access
- Published:
On the behavior of solutions of the system of rational difference equations
Advances in Difference Equations volume 2011, Article number: 40 (2011)
Abstract
In this article, we investigate the solutions of the system of difference equations , , where x 0, x -1, y 0, y -1, z 0, z -1 real numbers such that y 0 x -1 ≠ 1, x 0 y -1 ≠ 1 and y 0 z 0 ≠ 0.
1. Introduction
In [1], Kurbanli et al. studied the behavior of positive solutions of the system of rational difference equations
In [2], Cinar studied the solutions of the systems of difference equations
In [3], Kurbanli, studied the behavior of solutions of the system of rational difference equations
In [4], Papaschinnopoulos and Schinas proved the boundedness, persistence, the oscillatory behavior, and the asymptotic behavior of the positive solutions of the system of difference equations
In [5], Clark and Kulenović investigate the global stability properties and asymptotic behavior of solutions of the system of difference equations
In [6], Camouzis and Papaschinnopoulos studied the global asymptotic behavior of positive solutions of the system of rational difference equations
In [7], Kulenović and Nurkanović studied the global asymptotic behavior of solutions of the system of difference equations
In [8], Özban studied the positive solutions of the system of rational difference equations
In [9], Zhang et al. investigated the behavior of the positive solutions of the system of the difference equations
In [10], Yalcinkaya studied the global asymptotic stability of the system of difference equations
In [11], Irićanin and Stević studied the positive solutions of the system of difference equations
Although difference equations are very simple in form, it is extremely difficult to understand throughly the global behavior of their solutions, for example, see Refs. [12–34].
In this article, we investigate the behavior of the solutions of the difference equation system
where x 0, x -1, y 0, y -1, z 0, z -1 real numbers such that y 0 x -1 ≠ 1, x 0 y -1 ≠ 1 and y 0 z 0 ≠ 0.
2. Main results
Theorem 1. Let y 0 = a, y -1 = b, x 0 = c, x -1 = d, z 0 = e, z -1 = f be real numbers such that y 0 x -1 ≠ 1, x 0 y -1 ≠ 1 and y 0 z 0 ≠ 0. Let {x n , y n , z n } be a solution of the system (1.1). Then all solutions of (1.1) are
Proof. For n = 0, 1, 2, 3, we have
for n = k, assume that
and
are true. Then, for n = k + 1 we will show that (1.2), (1.3), and (1.4) are true. From (1.1), we have
Also, similarly from (1.1), we have
Also, we have
and
□
Corollary 1. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0 and b ≠ 0. Also, if ad, cb ∈ (1, 2) and b > a then we have
and
Proof. From ad, cb ∈ (1, 2) and b > a we have 0 < ad -1 < 1 and 0 < cb - 1 < 1.
Hence, we obtain
and
Similarly, from ad, cb ∈ (1, 2) and b > a, we have 0 < ad - 1 < 1 and 0 < cb - 1 < 1.
Hence, we obtain
and
□
Corollary 2. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0 and b ≠ 0. If a = b and cb = ad = 2 then we have
and
Proof. From a = b and cb = ad = 2 then we have, cb - 1 = ad - 1 = 1. Hence, we have
and
Also, we have
and
Similarly, we have
and
□
Corollary 3. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0 and b ≠ 0. Also, if 0 < a, b, c, d, e, f < 1 then we have
and
Proof. From 0 < a, b, c, d, e, f < 1 we have -1 < ad - 1 < 0 and - 1 < cb - 1 < 0. Hence, we obtain
and
Similarly, we have
and
□
Corollary 4. Let {x n , y n , z n } be a solution of the system (1.1). Let a, b, c, d, e, f be real numbers such that ad ≠ 1, cb ≠ 1, ae ≠ 0, and b ≠ 0. Also, if 0 < a, b, c, d, e, f < 1 then we have
and
Proof. The proof is clear from Theorem 1. □
References
- 1.
Kurbanli AS, Çinar C, Yalcinkaya I:On the behavior of positive solutions of the system of rational difference equations , . Math Comput Model 2011,53(5-6):1261-1267. 10.1016/j.mcm.2010.12.009
- 2.
Çinar C:On the positive solutions of the difference equation system , . Appl Math Comput 2004, 158: 303-305. 10.1016/j.amc.2003.08.073
- 3.
Kurbanli AS:On the behavior of solutions of the system of rational difference equations , , . Discrete Dynamics Natural and Society 2011, 2011: 12. Article ID 932362
- 4.
Papaschinopoulos G, Schinas CJ: On the system of two difference equations. J Math Anal Appl 2002, 273: 294-309. 10.1016/S0022-247X(02)00223-8
- 5.
Clark D, Kulenović MRS: A coupled system of rational difference equations. Comput Math Appl 2002, 43: 849-867. 10.1016/S0898-1221(01)00326-1
- 6.
Camouzis E, Papaschinopoulos G:Global asymptotic behavior of positive solutions on the system of rational difference equations , . Appl Math Lett 2004, 17: 733-737. 10.1016/S0893-9659(04)90113-9
- 7.
Kulenović MRS, Nurkanović Z: Global behavior of a three-dimensional linear fractional system of difference equations. J Math Anal Appl 2005, 310: 673-689.
- 8.
Özban AY:On the positive solutions of the system of rational difference equations , J Math Anal Appl 2006, 323: 26-32. 10.1016/j.jmaa.2005.10.031
- 9.
Zhang Y, Yang X, Megson GM, Evans DJ:On the system of rational difference equations , . Appl Math Comput 2006, 176: 403-408. 10.1016/j.amc.2005.09.039
- 10.
Yalcinkaya I: On the global asymptotic stability of a second-order system of difference equations. Discrete Dyn Nat Soc 2008, 2008: 12. (Article ID 860152)
- 11.
Irićanin B, Stević S: Some systems of nonlinear difference equations of higher order with periodic solutions. Dyn Contin Discrete Impuls Syst Ser A Math Anal 2006, 13: 499-507.
- 12.
Agarwal RP, Li WT, Pang PYH: Asymptotic behavior of a class of nonlinear delay difference equations. J Difference Equat Appl 2002, 8: 719-728. 10.1080/1023619021000000735
- 13.
Agarwal RP: Difference Equations and Inequalities. 2nd edition. Marcel Dekker, New York; 2000.
- 14.
Papaschinopoulos G, Schinas CJ: On a system of two nonlinear difference equations. J Math Anal Appl 1998, 219: 415-426. 10.1006/jmaa.1997.5829
- 15.
Özban AY:On the system of rational difference equations , Appl Math Comput 2007, 188: 833-837. 10.1016/j.amc.2006.10.034
- 16.
Clark D, Kulenovic MRS, Selgrade JF: Global asymptotic behavior of a two-dimensional difference equation modelling competition. Nonlinear Anal 2003, 52: 1765-1776. 10.1016/S0362-546X(02)00294-8
- 17.
Yang X, Liu Y, Bai S:On the system of high order rational difference equations , . Appl Math Comput 2005, 171: 853-856. 10.1016/j.amc.2005.01.092
- 18.
Yang X:On the system of rational difference equations , . J Math Anal Appl 2005, 307: 305-311. 10.1016/j.jmaa.2004.10.045
- 19.
Zhang Y, Yang X, Evans DJ, Zhu C:On the nonlinear difference equation system , Comput Math Appl 2007, 53: 1561-1566. 10.1016/j.camwa.2006.04.030
- 20.
Yalcinkaya I, Cinar C:Global asymptotic stability of two nonlinear difference equations , . Fasciculi Mathematici 2010, 43: 171-180.
- 21.
Yalcinkaya I, Çinar C, Simsek D: Global asymptotic stability of a system of difference equations. Appl Anal 2008,87(6):689-699. 10.1080/00036810802163279
- 22.
Yalcinkaya I, Cinar C: On the solutions of a systems of difference equations. Int J Math Stat Autumn 2011.,9(A11):
- 23.
Cinar C:On the positive solutions of the difference equation Appl Math Comput 2004, 150: 21-24. 10.1016/S0096-3003(03)00194-2
- 24.
Cinar C:On the positive solutions of the difference equation Appl Math Comput 2004, 156: 587-590. 10.1016/j.amc.2003.08.010
- 25.
Cinar C:On the positive solutions of the difference equation Appl Math Comput 2004, 158: 809-812. 10.1016/j.amc.2003.08.140
- 26.
Cinar C:On the periodic cycle of Appl Math Comput 2004, 150: 1-4. 10.1016/S0096-3003(03)00182-6
- 27.
Abu-Saris R, Çinar C, Yalcinkaya I:On the asymptotic stability of Comput Math Appl 2008,56(5):1172-1175. 10.1016/j.camwa.2008.02.028
- 28.
Çinar C:On the difference equation Appl Math Comput 2004, 158: 813-816. 10.1016/j.amc.2003.08.122
- 29.
Çinar C:On the solutions of the difference equation Appl Math Comput 2004, 158: 793-797. 10.1016/j.amc.2003.08.139
- 30.
Kurbanli AS:On the behavior of solutions of the system of rational difference equations , . World Appl Sci J 2010, in press.
- 31.
Elabbasy EM, El-Metwally H, Elsayed EM: On the solutions of a class of difference equations systems. Demonstratio Mathematica 2008,41(1):109-122.
- 32.
Elsayed EM: On the solutions of a rational system of difference equations. Fasciculi Mathematici 2010, 45: 25-36.
- 33.
Elsayed EM: Dynamics of a recursive sequence of higher order. Commun Appl Nonlinear Anal 2009,16(2):37-50.
- 34.
Elsayed EM: On the solutions of higher order rational system of recursive sequences. Mathematica Balkanica 2008,21(3-4):287-296.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kurbanli, A.S. On the behavior of solutions of the system of rational difference equations . Adv Differ Equ 2011, 40 (2011). https://doi.org/10.1186/1687-1847-2011-40
Received:
Accepted:
Published:
Keywords
- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis