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Discrete Dynamics in Nature and Society
VolumeΒ 2011Β (2011), Article IDΒ 932362, 12 pages
http://dx.doi.org/10.1155/2011/932362
Research Article

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 © 2011 Abdullah Selçuk 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], Γ‡inar 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] Γ–zban studied the positive solutions of the system of rational difference equations

In [6, 7] Clark and Kulenović 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] Kulenović and Nurkanović 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] Irićanin and Stević 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.

tab1
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.

tab2
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.

tab3
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.

tab4
Table 4

Acknowledgment

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

References

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