Discrete Dynamics in Nature and Society

Volume 2012, Article ID 892571, 12 pages

http://dx.doi.org/10.1155/2012/892571

## On the Solutions of a General System of Difference Equations

^{1}Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 4 December 2011; Revised 14 February 2012; Accepted 16 February 2012

Academic Editor: Elena Braverman

Copyright © 2012 E. M. Elsayed 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.

#### Abstract

We deal with the solutions of the systems of the difference equations , , and , , , with a nonzero real numbers initial conditions. Also, the periodicity of the general system of variables will be considered.

#### 1. Introduction

In this paper, we deal with the solutions of the systems of the difference equations with a nonzero real numbers initial conditions. Also, the periodicity of the general system of variables will be considered.

Recently, there has been a great interest in studying nonlinear difference equations and systems (cf. [1–14] and the references therein). 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, economy, probability theory, genetics, psychology, sociology, and so forth. Such equations also appear naturally as discrete analogues of differential equations which model various biological and economical systems.

Cinar [3] has obtained the positive solution of the difference equation system: Also, Çinar and Yalçinkaya [4] have obtained the positive solution of the difference equation system: Clark and Kulenović [5] have investigated the global stability properties and asymptotic behavior of solutions of the system Elabbasy et al. [6] have obtained the solution of particular cases of the following general system of difference equations: Elsayed [10] has obtained the solution of systems of difference equations of rational form.

Also, the behavior of the solutions of the following systems: has been studied by Elsayed [15].

Özban [16] has investigated the positive solutions of the system of rational difference equations: Özban [17] has investigated the solutions of the following system: Yang et al. [18] have investigated the positive solutions of the systems: Similar nonlinear systems of rational difference equations were investigated see [15–32].

*Definition 1.1 (Periodicity). *A sequence is said to be periodic with period if for all .

#### 2. Main Results

##### 2.1. The First System

In this section, we deal with the solutions of the system of the difference equations with a nonzero real numbers initial conditions and .

Theorem 2.1. *Suppose that are solutions of system (2.1). Also, assume that the initial conditionsare arbitrary nonzero real numbers. Then all solutions of equation system (2.1) are eventually periodic with period .*

*Proof. *From (2.1), we see that
Hence, the proof is completed.

*Numerical Example*

In order to illustrate the results of this section and to support our theoretical discussions, we consider the following numerical example.

*Example 2.2. *Consider the difference system (2.1) with , , and the initial conditions , , , , , and . (See Figure 1).

##### 2.2. The Second System

In this section, we deal with the solutions of the system of the difference equations with a nonzero real numbers initial conditions and .

Theorem 2.3. *Suppose that are solutions of system (2.3). Also, assume that the initial conditionsare arbitrary nonzero real numbers. Then all solutions of equation system (2.3) are eventually periodic with period .*

*Proof. *From (2.3), we see that
Hence, the proof is completed.

*Example 2.4. *Consider the difference system (2.3) with , , , and the initial conditions , , ,, , , , , and (See Figure 2).

Proposition 2.5. *It is easy to see by induction that the following general system is periodic with period :
*

#### Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

#### References

- N. Battaloglu, C. Cinar, and I. Yalçinkaya, “The dynamics of the difference equation,”
*Ars Combinatoria*, vol. 97, pp. 281–288, 2010. View at Google Scholar - E. Camouzis and G. Ladas,
*Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures*, vol. 5 of*Advances in Discrete Mathematics and Applications*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008. - C. Çinar, “On the positive solutions of the difference equation system ${x}_{n+1}=1/{y}_{n},{y}_{n+1}={y}_{n}/{x}_{n-1}{y}_{n-1}$,”
*Applied Mathematics and Computation*, vol. 158, no. 2, pp. 303–305, 2004. View at Publisher · View at Google Scholar - C. Çinar and I. Yalçinkaya, “On the positive solutions of the difference equation system ${x}_{n+1}=1/{z}_{n}$, ${y}_{n+1}={x}_{n}/{x}_{n-1}$, ${z}_{n+1}=1/{x}_{n-1}$,”
*International Mathematical Journal*, vol. 5, pp. 525–527, 2004. View at Google Scholar - D. Clark and M. R. S. Kulenović, “A coupled system of rational difference equations,”
*Computers & Mathematics with Applications*, vol. 43, no. 6-7, pp. 849–867, 2002. View at Publisher · View at Google Scholar - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the solutions of a class of difference equations systems,”
*Demonstratio Mathematica*, vol. 41, no. 1, pp. 109–122, 2008. View at Google Scholar - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Global behavior of the solutions of difference equation,”
*Advances in Difference Equations*, vol. 2011, article 28, 2011. View at Publisher · View at Google Scholar - E. M. Elabbasy and E. M. Elsayed, “Global attractivity and periodic
nature of a difference equation,”
*World Applied Sciences Journal*, vol. 12, no. 1, pp. 39–47, 2011. View at Google Scholar - S. N. Elaydi,
*An Introduction to Difference Equations*, Springer, New York, NY, USA, 2nd edition, 1999. - 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. View at Google Scholar - E. M. Elsayed, “On the solutions of a rational system of difference equations,”
*Fasciculi Mathematici*, no. 45, pp. 25–36, 2010. View at Google Scholar - E. M. Elsayed, “Dynamics of recursive sequence of order two,”
*Kyungpook Mathematical Journal*, vol. 50, no. 4, pp. 483–497, 2010. View at Google Scholar - E. M. M. Elsayed, “Behavior of a rational recursive sequences,”
*Studia. Universitatis Babeş-Bolyai Mathematica*, vol. 56, no. 1, pp. 27–42, 2011. View at Google Scholar - E. M. Elsayed, “Solution of a recursive sequence of order ten,”
*General Mathematics*, vol. 19, no. 1, pp. 145–162, 2011. View at Google Scholar - E. M. Elsayed, “Solutions of rational difference systems of order two,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 378–384, 2012. View at Publisher · View at Google Scholar - A. Y. Özban, “On the positive solutions of the system of rational difference equations, ${x}_{n+1}=1/{y}_{n-k},{y}_{n+1}={y}_{n}/{x}_{n-m}{y}_{n-m-k}$,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 1, pp. 26–32, 2006. View at Publisher · View at Google Scholar - A. Y. Özban, “On the system of rational difference equations ${x}_{n+1}=a/{y}_{n-3},{y}_{n+1}=b{y}_{n-3}/{x}_{n-q}{y}_{n-q}$,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 833–837, 2007. View at Publisher · View at Google Scholar - X. Yang, Y. Liu, and S. Bai, “On the system of high order rational difference equations ${x}_{n}=a/{y}_{n-p}$, ${y}_{n}=b{y}_{n-p}/{x}_{n-q}{y}_{n-q}$,”
*Applied Mathematics and Computation*, vol. 171, no. 2, pp. 853–856, 2005. View at Publisher · View at Google Scholar - E. M. Elsayed, “Solution and attractivity for a rational recursive sequence,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 982309, 17 pages, 2011. View at Publisher · View at Google Scholar - E. M. Elsayed, “On the solution of some difference equations,”
*European Journal of Pure and Applied Mathematics*, vol. 4, no. 3, pp. 287–303, 2011. View at Google Scholar - E. M. Elsayed, “On the dynamics of a higher-order rational recursive sequence,”
*Communications in Mathematical Analysis*, vol. 12, no. 1, pp. 117–133, 2012. View at Google Scholar - E. A. Grove and G. Ladas,
*Periodicities in Nonlinear Difference Equations*, vol. 4, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. - E. A. Grove, G. Ladas, L. C. McGrath, and C. T. Teixeira, “Existence and behavior of solutions of a rational system,”
*Communications on Applied Nonlinear Analysis*, vol. 8, no. 1, pp. 1–25, 2001. View at Google Scholar - A. S. Kurbanlı, C. Çinar, and I. Yalçinkaya, “On the behavior of positive solutions of the system of rational difference equations ${x}_{n+1}={x}_{n-1}/{y}_{n}{x}_{n-1}+1$, ${y}_{n+1}={y}_{n-1}/{x}_{n}{y}_{n-1}+1$,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 1261–1267, 2011. View at Publisher · View at Google Scholar - N. Touafek and E. M. Elsayed, “On the solutions of systems of rational
difference equations,”
*Mathematical and Computer Modelling*, vol. 55, no. 7-8, pp. 1987–1997, 2012. View at Google Scholar - I. Yalcinkaya, “On the global asymptotic stability of a second-order system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 860152, 12 pages, 2008. View at Publisher · View at Google Scholar - I. Yalçinkaya, “On the difference equation ${x}_{n+1}=\alpha +{x}_{n-m}/{x}_{n}^{k}$,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 805460, 8 pages, 2008. View at Publisher · View at Google Scholar - I. Yalcinkaya, “On the global asymptotic stability of a second-order system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 860152, 12 pages, 2008. View at Publisher · View at Google Scholar - I. Yalcinkaya, “On the global asymptotic behavior of a system of two nonlinear difference equations,”
*ARS Combinatoria*, vol. 95, pp. 151–159, 2010. View at Google Scholar - I. Yalçinkaya, 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. View at Publisher · View at Google Scholar - X. Yang, “On the system of rational difference equations ${x}_{n}=A+{y}_{n-1}/{x}_{n-p}{y}_{n-q}$
, ${y}_{n}=A+{x}_{n-1}/{x}_{n-r}{y}_{n-s}$,”
*Journal of Mathematical Analysis and Applications*, vol. 307, no. 1, pp. 305–311, 2005. View at Publisher · View at Google Scholar - Y. Zhang, X. Yang, G. M. Megson, and D. J. Evans, “On the system of rational difference equations ${x}_{n}=A+1/{y}_{n-p}$, ${y}_{n}=A+{y}_{n-1}/{x}_{n-t}{y}_{n-s}$,”
*Applied Mathematics and Computation*, vol. 176, no. 2, pp. 403–408, 2006. View at Publisher · View at Google Scholar