Discrete Dynamics in Nature and Society

VolumeΒ 2011, Article IDΒ 932362, 12 pages

http://dx.doi.org/10.1155/2011/932362

## On the Behavior of Solutions of the System of Rational Difference Equations: , , and

Department of Mathematics, Faculty of Education, Selcuk University, 42090 Konya, Turkey

Received 23 December 2010; Accepted 26 January 2011

Academic Editor: IbrahimΒ Yalcinkaya

Copyright Β© 2011 Abdullah SelΓ§uk Kurbanli. 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 solutions of the system of difference equations , , , where .

#### 1. Introduction

Recently, there has been great interest in studying difference equation systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economic, probability theory, genetics, psychology, and so forth. There are many papers related to the difference equations system, for example, the following papers.

In [1], Γinar studied the solutions of the systems of the difference equations

In [2] Papaschinopoulos and Schinas studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of nonlinear difference equations

In [3] Papaschinopoulos and Schinas proved the boundedness, persistence, the oscillatory behavior, and the asymptotic behavior of the positive solutions of the system of difference equations

In [4, 5] Γzban studied the positive solutions of the system of rational difference equations

In [6, 7] Clark and KulenoviΔ investigate the global asymptotic stability

In [8] Camouzis and Papaschinopoulos studied the global asymptotic behavior of positive solutions of the system of rational difference equations

In [9] Yang et al. considered the behavior of the positive solutions of the system of the difference equations

In [10] KulenoviΔ and NurkanoviΔ studied the global asymptotic behavior of solutions of the system of difference equations

In [11] Zhang et al. investigated the behavior of the positive solutions of the system of difference equations

In [12] Zhang et al. studied the boundedness, the persistence, and global asymptotic stability of the positive solutions of the system of difference equations

In [13] Yalcinkaya studied the global asymptotic behavior of a system of two nonlinear difference equations.

In [14] Yalcinkaya et al. investigated the solutions of the system of difference equations

In [15] Yalcinkaya studied the global asymptotic stability of the system of difference equations

In [16] IriΔanin and SteviΔ studied the positive solutions of the system of difference equations

In [17] Kurbanli et al. studied the behavaior of positive solutions of the system of rational difference equations

Also see references.

In this paper, we investigate the behavior of the solutions of the difference equations system where the initial conditions are arbitrary real numbers.

Theorem 1.1. *Let be arbitrary real numbers and and let be a solution of the system (1.15). Also, assume that and then all solutions of (1.15) are
*

*Proof. *For , we have
for assume that
are true. Then for we will show that (1.16) is true. From (1.15), we have

Also, similarly from (1.15), we have
from properties of Binomial coefficients,
written and accurate.

Also, we have
wheredenotes.

Corollary 1.2. *Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15). If and then one has
*

*Proof. *From and we have and . Hence, we have

Then,

Also,

*Example 1.3. *If , then the solutions of (1.15) can be represented by Table 1.

Corollary 1.4. *Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15) If and then one has
*

*Proof. *
From and we have and . Hence, we have

Then,

Also,

*Example 1.5. *
If then the solutions of (1.15) can be represented by Table 2.

Corollary 1.6. *Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15) If and then one has
*

*Proof. *From and we have and . Hence, we have

Then,

Also,

*Example 1.7. *
If then the solutions of (1.15) can be represented by Table 3.

Corollary 1.8. *Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15). If and then one has
*

*Proof. *From and we have and . Hence, we have

Then,

Also,

*Example 1.9. *
If then the solutions of (1.15) can be represented by Table 4.

#### Acknowledgment

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

#### References

- C. Çinar, βOn the positive solutions of the difference equation system ${x}_{n+1}=1/{y}_{n}$, ${y}_{n+1}={y}_{n}/{x}_{n-1}{y}_{n-1}$,β
*Applied Mathematics and Computation*, vol. 158, no. 2, pp. 303β305, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - G. Papaschinopoulos and C. J. Schinas, βOn a system of two nonlinear difference equations,β
*Journal of Mathematical Analysis and Applications*, vol. 219, no. 2, pp. 415β426, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - G. Papaschinopoulos and C. J. Schinas, βOn the system of two difference equations,β
*Journal of Mathematical Analysis and Applications*, vol. 273, no. 2, pp. 294β309, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. Y. Özban, βOn the system of rational difference equations ${x}_{n}=a/{y}_{n-3}$, ${y}_{n}=b{y}_{n-3}/{x}_{n-q}{y}_{n-q}$,β
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 833β837, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. Y. Özban, βOn the positive solutions of the system of rational difference equations ${x}_{n+1}=1/{y}_{n-k}$, ${y}_{n+1}={y}_{n}/{x}_{n-m}{y}_{n-m-k}$,β
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 1, pp. 26β32, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - D. Clark and M. R. S. Kulenović, βA coupled system of rational difference equations,β
*Computers & Mathematics with Applications*, vol. 43, no. 6-7, pp. 849β867, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - D. Clark, M. R. S. Kulenović, and J. F. Selgrade, βGlobal asymptotic behavior of a two-dimensional difference equation modelling competition,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 52, no. 7, pp. 1765β1776, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - E. Camouzis and G. Papaschinopoulos, βGlobal asymptotic behavior of positive solutions on the system of rational difference equations ${x}_{n+1}=1+{x}_{n}/{y}_{n-m}$
, ${y}_{n+1}=1+{y}_{n}/{x}_{n-m}$,β
*Applied Mathematics Letters*, vol. 17, no. 6, pp. 733β737, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - X. Yang, Y. Liu, and S. Bai, βOn the system of high order rational difference equations ${x}_{n}=a/{y}_{n-p},{y}_{n}=b{y}_{n-p}/{x}_{n-q}{y}_{n-q}$,β
*Applied Mathematics and Computation*, vol. 171, no. 2, pp. 853β856, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. R. S. Kulenović and Z. Nurkanović, βGlobal behavior of a three-dimensional linear fractional system of difference equations,β
*Journal of Mathematical Analysis and Applications*, vol. 310, no. 2, pp. 673β689, 2005. View at Google Scholar Β· View at Zentralblatt MATH - Y. Zhang, X. Yang, G. M. Megson, and D. J. Evans, βOn the system of rational difference equations ${x}_{n}=A+1/{y}_{n-p}$, ${y}_{n}=A+{y}_{n-1}/{x}_{n-r}{y}_{n-s}$,β
*Applied Mathematics and Computation*, vol. 176, no. 2, pp. 403β408, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - Y. Zhang, X. Yang, D. J. Evans, and C. Zhu, βOn the nonlinear difference equation system ${x}_{n+1}=A+{y}_{n-m}/{x}_{n}$, ${y}_{n+1}=A+{x}_{n-m}/{y}_{n}$,β
*Computers & Mathematics with Applications*, vol. 53, no. 10, pp. 1561β1566, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - I. Yalcinkaya, βOn the global asymptotic behavior of a system of two nonlinear difference equations,β
*Ars Combinatoria*, vol. 95, pp. 151β159, 2010. View at Google Scholar - I. Yalcinkaya, C. Çinar, and M. Atalay, βOn the solutions of systems of difference equations,β
*Advances in Difference Equations*, vol. 2008, Article ID 143943, 9 pages, 2008. View at Google Scholar Β· View at Zentralblatt MATH - I. Yalcinkaya, βOn the global asymptotic stability of a second-order system of difference equations,β
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 860152, 12 pages, 2008. View at Google Scholar Β· View at Zentralblatt MATH - B. D. Irićanin and S. Stević, βSome systems of nonlinear difference equations of higher order with periodic solutions,β
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis*, vol. 13, no. 3-4, pp. 499β507, 2006. View at Google Scholar Β· View at Zentralblatt MATH - A. S. Kurbanlı, C. Çinar, and I. Yalcinkaya, βOn the behavaior of positive solutions of the system of rational difference equations ${x}_{n+1}={x}_{n-1}/({y}_{n}{x}_{n-1})+1$, ${y}_{n+1}={y}_{n-1}/({x}_{n}{y}_{n-1})+1$,β
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 1261β1267, 2011. View at Google Scholar