Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 903593, 10 pages

http://dx.doi.org/10.1155/2013/903593

## Solutions Form for Some Rational Systems of Difference Equations

^{1}Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 13 February 2013; Accepted 8 April 2013

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2013 H. El-Metwally. 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 form of the solutions for the following systems of rational difference equations , , with nonzero real numbers initial conditions. Also we investigate some properties of the obtained solutions and present some numerical examples.

#### 1. Introduction

Our aim in this paper is to find the solutions form for the following systems of rational difference equations: with nonzero real numbers initial conditions and then investigate the obtained solutions.

Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on. So, recently there has been an increasing interest in the study of qualitative analysis of scalar rational difference equations and systems of rational difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. See [1–7] and the references cited therein.

The periodicity of the positive solutions for the following system of rational difference equations was studied by Cinar et al. [8].

Özban [9] has studied the positive solutions for the following system: The behavior of the positive solutions for the following system has been studied by Kurbanlı et al. [10].

Touafek and Elsayed [11] studied the periodicity and gave the form of the solutions for the following systems: Yalcinkaya [12] investigated the sufficient condition for the global asymptotic stability for the following system of difference equations: Yang [13] investigated the positive solutions for the system Clark et al. [14, 15] investigate the global asymptotic stability of the following difference equations: Camouzis and Papaschinopoulos [16] studied the global asymptotic behavior of the positive solutions of the system of rational difference equations as follows:

#### 2. On the System:,

In this section, we study the existence of analytical forms of the solutions for the following system of difference equations: with nonzero real initials conditions , , , , , and .

In the sequel we assume that , for any real numbers and .

Theorem 1. * Suppose that is a solution for system (10), then for , one obtains
**
where , , , , , and .*

*Proof. *For the result holds. Now suppose that and that our assumption holds for . That is,
Now, it follows from system (10) that
Also, from system (10), we see that
The proof is complete.

Lemma 2. *Every positive solution for system (10) is bounded, and .*

*Proof. *It follows from system (10) that
for large, we see that

Then the subsequences , , , and are decreasing and so are bounded from above by , , , and , respectively, where .

The proofs of the following theorems are similar to that of Theorem 1 and will be omitted.

Theorem 3. * Assume that is a solution for the system
**
Then for ,
*

Theorem 4. *The solutions form for the following system:
**
are given by the following formulas:
*

Theorem 5. *Let be a solution for the following system of difference equations
**
Then for ,
*

*Example 6. *We consider an interesting numerical example for system (10) with the initial conditions , , , , , and . See Figure 1.

#### 3. On the System: ,

In this section, we obtain the solutions form for the following system of two difference equations: with nonzero real numbers initial conditions , , , , , and provided that and .

Theorem 7. * Suppose that is a solution for system (23). Then for ,
*

*Proof. *For the result holds. Now suppose that and that our assumption holds for . That is,
Now, it follows from system (23) that
Similarly one can prove the other relations. The proof is complete.

Lemma 8. *Every positive solution of the equation is bounded, and .*

The following theorems deal with the solutions form for the following systems, and their proofs will be omitted:

Theorem 9. *Assume that is a solution for system (27) with and . Then for ,
*

Theorem 10. *Assume that is a solution for system (28) with and . Then for ,
*

Theorem 11. *The solution form for system (29) is given by
**
where and . *

*Example 12. * Consider system (23) with the initial conditions , , , , , and . See Figure 2.

#### 4. On the System: ,

In this section, we present the solutions form for the following system: with nonzero real numbers initial conditions where and .

The following theorems can be proved similarly to those in Sections 2 and 3.

Theorem 13. * Suppose that is a solution for system (33). Assume that , , , , , and are arbitrary nonzero real numbers. Then
*

Lemma 14. *Every positive solution of the equation is bounded and .*

Theorem 15. * Let be a solution for the system
**
with and . Then for ,
*

Theorem 16. * The solution form for the following system
**
with and is given by
*

Theorem 17. *The following system
**
has a solution form given by the following relations:
**
where and . *

*Example 18. * Consider system (33) with the initial values , , , , , and . See Figure 3.

#### 5. Other Systems

In this section, we give the solutions form for the following systems of difference equations: with nonzero real numbers initial conditions.

Theorem 19. * Let be a solution for system (41) with and . Then
*

Theorem 20. * Suppose that is a solution for system (42) with , , , and . Then
*

Theorem 21. *The solution for system (43) is given by the following formula; for ;
**
where , , , and . *

Theorem 22. * If is a solution for system (44) with , , , and , then
*

*Example 23. * Figure 4 shows the behavior of the solution for system (41) with the initial conditions , , , , , and .

#### References

- R. P. Agarwal,
*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. - R. P. Agarwal and E. M. Elsayed, “On the solution of fourth-order rational recursive sequence,”
*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 4, pp. 525–545, 2010. View at Google Scholar · View at MathSciNet - 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 · View at MathSciNet - H. El-Metwally, “Qualitative properties of some higher order difference equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 4, pp. 686–692, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - 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 · View at MathSciNet - C. Wang, S. Wang, and X. Yan, “Global asymptotic stability of 3-species mutualism models with diffusion and delay effects,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 317298, 20 pages, 2009. View at Publisher · View at Google Scholar - E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequence ${x}_{n+1}=a{x}_{n}-b{x}_{n}/(c{x}_{n}-d{x}_{n-k})$,”
*Communications on Applied Nonlinear Analysis*, vol. 15, no. 2, pp. 47–57, 2008. View at Google Scholar · View at MathSciNet - C. Cinar, I. Yalçinkaya, and R. Karatas, “On the positive solutions of the difference equation system ${x}_{n+1}=m/{y}_{n},{y}_{n+1}=p{y}_{n}/{x}_{n-1}{y}_{n-1}$,”
*Journal of Institute of Mathematics and Computer Sciences*, vol. 18, pp. 135–136, 2005. 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 · View at MathSciNet - A. S. Kurbanlı, C. Çinar, and I. Yalçinkaya, “On the behavior of positive solutions of the system of rational difference equations,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 1261–1267, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - N. Touafek and E. M. Elsayed, “On the periodicity of some systems of nonlinear difference equations,”
*Bulletin Mathématiques de la Société des Sciences*, vol. 55(103), no. 2, pp. 217––224, 2012. View at Google Scholar · View at MathSciNet - 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 · View at MathSciNet - X. Yang, “On the system of rational difference equations ${x}_{n}=A+{y}_{n-1}/{x}_{n-p}{y}_{n-q}$, $A+{x}_{n-1}/{x}_{n-r}{y}_{n-s}$,”
*Journal of Mathematical Analysis and Applications*, vol. 307, no. 1, pp. 305–311, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - 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 · View at MathSciNet - D. Clark, M. R. S. Kulenović, and J. F. Selgrade, “Global asymptotic behavior of a two-dimensional difference equation modelling competition,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 52, no. 7, pp. 1765–1776, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - E. Camouzis and G. Papaschinopoulos, “Global asymptotic behavior of positive solutions on the system of rational difference equations,”
*Applied Mathematics Letters*, vol. 17, no. 6, pp. 733–737, 2004. View at Publisher · View at Google Scholar · View at MathSciNet