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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 178483, 9 pages
http://dx.doi.org/10.1155/2011/178483
Research Article

More on Three-Dimensional Systems of Rational Difference Equations

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Henan, Zhengzhou 450045, China

Received 6 June 2011; Accepted 24 September 2011

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2011 Liu Keying 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 are concerned with a kind of three-dimensional system of rational difference equations, given by Kurbanli (2011). A new expression of solution of the system is presented, and the asymptotical behavior is described. At the same time, we also consider a different system and obtain some results, which expand the study of such a kind of difference equations and the method can be applied to other systems.

1. Introduction

Difference equations is a hot topic in that they are widely used to investigate equations arising in mathematical models describing real-life situations such as population biology, probability theory, and genetics. Recently, rational difference equations have appealed more and more scholars for their wide application. For details, see [1]. However, there are few literatures on the system of two or three rational difference equations [28].

In [2], Kurbanli studied a three-dimensional system of rational difference equations𝑥𝑛+1=𝑥𝑛1𝑦𝑛𝑥𝑛11,𝑦𝑛+1=𝑦𝑛1𝑥𝑛𝑦𝑛11,𝑧𝑛+1=𝑧𝑛1𝑦𝑛𝑧𝑛1,1(1.1) where the initial conditions are arbitrary real numbers. He expressed the solution of (1.1) and investigated the behavior and computed for some initial values.

The following theorem is cited from [2].

Theorem 1.1. Let 𝑦0, 𝑦1, 𝑥0, 𝑥1, 𝑧0, 𝑧1 be arbitrary real numbers and 𝑦0=𝑎, 𝑦1=𝑏, 𝑥0=𝑐, 𝑥1=𝑑, 𝑧0=𝑒, 𝑧1=𝑓, and let {𝑥𝑛,𝑦𝑛,𝑧𝑛} be a solution of the system (1.1). Also, assume that 𝑎𝑑1 and 𝑏𝑐1; then all solutions of (1.1) are 𝑥𝑛=𝑑(𝑎𝑑1)𝑛,𝑛isodd,𝑐(𝑏𝑐1)𝑛𝑦,𝑛iseven,(1.2)𝑛=𝑏(𝑏𝑐1)𝑛,𝑛isodd,𝑎(a𝑑1)𝑛𝑧,𝑛iseven,(1.3)𝑛=𝑓(1)0(𝑛/0)𝑎𝑛𝑓𝑑𝑛1+(1)1(𝑛/1)𝑎𝑛1𝑓𝑑𝑛2++(1)𝑛1(𝑛/(𝑛1))𝑎1𝑓𝑑0+(1)𝑛,(𝑛/𝑛)𝑛isodd,(1)𝑛(𝑏𝑐1)𝑛𝑒(1)𝑛(𝑛/1)𝑏1𝑐0𝑒++(1)1(𝑛/𝑛)𝑏𝑛𝑐𝑛1𝑒+(1)0(𝑛/0)𝑏𝑛𝑐𝑛++(1)𝑛(𝑛/𝑛)𝑏0𝑐0,𝑛iseven.(1.4)

From (1.4), the expression of 𝑧𝑛 is so tedious. Although the solution is given, we are so tired to compute for large 𝑛.

In [2], Kurbanli only considered the asymptotical behavior of 𝑥𝑛 and 𝑦𝑛, and he has no way to consider that of 𝑧𝑛 since its expression (1.4) is too difficult to deal with.

In this paper, first, we give more results of the solution of (1.1) including a new and simple expression of 𝑧𝑛 and the asymptotical behavior of the solution. Then, we consider a system similar to (1.1) and obtain some conclusions.

2. More Results on the System (1.1)

First, we give another form of the expression of 𝑧𝑛.

In fact, (1.4) could be rewritten as𝑧𝑛=𝑑𝑓(𝑎𝑑1)𝑘𝑓+(1)𝑘(𝑑𝑓),𝑛=2𝑘1,(1)𝑘𝑐𝑒𝑐𝑒+𝑒/(1𝑏𝑐)𝑘,𝑛=2𝑘,𝑘=1,2,.(2.1)

From (2.1), it is easy to check the following: 𝑧1=𝑑𝑓=𝑓(𝑎𝑑1)𝑓+(1)(𝑑𝑓),𝑧𝑎𝑓12=(1)𝑐𝑒=𝑐𝑒+(𝑒/(1𝑏𝑐))𝑒(𝑏𝑐1),𝑧𝑏𝑒𝑏𝑐+13=𝑑𝑓(𝑎𝑑1)2𝑓+(1)2=𝑓(𝑑𝑓)𝑎2,𝑑𝑓2𝑎𝑓+1(2.2) which are consistent with (1.17) in [2]. The proof is omitted here for the limited space and one could see a similar proof in the next section.

Comparing (2.1) with (1.4), we find that it is not only simple in the form, but also giving more obvious results on the asymptotical behavior of solution of (1.1).

Next, we give the following corollaries.

Corollary 2.1. Suppose that the initial values satisfy 𝑑=𝑓 and one of the following: (i)0<𝑎𝑑<1, 0<𝑏𝑐<1,(ii)1<𝑎𝑑<2, 1<𝑏𝑐<2.Then lim𝑘𝑥2𝑘1,𝑦2𝑘1,𝑧2𝑘1=(,,),lim𝑘𝑥2𝑘,𝑦2𝑘,z2𝑘=(0,0,0).(2.3)

Corollary 2.2. Suppose that the initial values satisfy 𝑐=𝑒 and one of the following: (i)2<𝑎𝑑<+, 2<𝑏𝑐<+,(ii)<𝑎𝑑<0, <𝑏𝑐<0.Then lim𝑘𝑥2𝑘1,𝑦2𝑘1,𝑧2𝑘1=(0,0,0),lim𝑘𝑥2𝑘,𝑦2𝑘,𝑧2𝑘=(,,).(2.4)

Corollary 2.3. Suppose that the initial values satisfy 𝑎=𝑒0, 𝑏=𝑓0, and 𝑎𝑑=𝑏𝑐=2. Then lim𝑘𝑥2𝑘1,𝑦2𝑘1,𝑧2𝑘1=(𝑑,𝑏,𝑏),lim𝑘𝑥2𝑘,𝑦2𝑘,𝑧2𝑘=(𝑐,𝑎,𝑎).(2.5)

Such results expand those in [2], where the behavior of 𝑧𝑛 could not be obtained from its expression. The proof is similar to that in the next section and we omit it here.

3. Main Results

Motivated by [2] and other references, such as [38] and the references cited therein, we consider the following system:𝑥𝑛+1=𝑥𝑛1𝑦𝑛𝑥𝑛11,𝑦𝑛+1=𝑦𝑛1𝑥𝑛𝑦𝑛11,𝑧𝑛+1=𝑧𝑛1𝑥𝑛𝑧𝑛1.1(3.1) Here, the last equation is different from that of (1.1).

Through the paper, we suppose the initial values to be 𝑦0=𝑎,𝑥0=𝑐,𝑧0=𝑒,𝑦1=𝑏,𝑥1=𝑑,𝑧1=𝑓.(3.2) Here, 𝑎,𝑏,𝑐,d,𝑒, and 𝑓 are nonzero real numbers such that 𝑎𝑑1 and 𝑏𝑐1. We call this hypothesis 𝐻).

Is the solution of (3.1) similar to that of (1.1)? The following theorem confirms this.

Theorem 3.1. Suppose that hypothesis (𝐻) holds, and let (𝑥𝑛,𝑦𝑛,𝑧𝑛) be a solution of the system (3.1). Then all solutions of (3.1) are 𝑥𝑛=𝑑(𝑎𝑑1)𝑛,𝑛=2𝑘1,𝑐(𝑏𝑐1)𝑛𝑦,𝑛=2𝑘,𝑛=𝑏(𝑏𝑐1)𝑛,𝑛=2𝑘1,𝑎(𝑎𝑑1)𝑛𝑧,𝑛=2𝑘,𝑛=𝑏𝑓(𝑏𝑐1)𝑘𝑓+(1)𝑘(𝑏𝑓),𝑛=2𝑘1,(1)𝑘𝑎𝑒𝑎𝑒+𝑒/(1𝑎𝑑)𝑘,𝑛=2𝑘(3.3) for 𝑘=1,2,.

Proof. First, for 𝑘=1,2, from (3.1), we easily check that 𝑥1=𝑑,𝑦𝑎𝑑11=𝑏,𝑧𝑏𝑐11=𝑧1𝑥0𝑧1=𝑓1=𝑐𝑓1𝑏𝑓,𝑥(𝑏𝑐1)𝑓(𝑏𝑓)2𝑦=𝑐(𝑏𝑐1),2𝑧=𝑎(𝑎𝑑1),2=𝑧0𝑥1𝑧0=𝑒1(𝑎𝑑1)=𝑑𝑒𝑎𝑑+11𝑎𝑒,𝑥𝑎𝑒+(𝑒/(1𝑎𝑑))3=𝑑(𝑎𝑑1)2,𝑦3=𝑏(𝑏𝑐1)2,𝑧3=𝑧1𝑥2𝑧1=𝑓1𝑏𝑐2=𝑓2𝑐𝑓+1𝑏𝑓,𝑥(𝑏𝑐1)𝑓(𝑏𝑓)4=𝑐(𝑏𝑐1)2,𝑦4=𝑎(𝑎𝑑1)2,𝑧4=𝑧2𝑥3𝑧2=1𝑒(𝑎𝑑1)22𝑑𝑎2𝑑𝑒+(𝑎𝑑1)2=(1)2𝑎𝑒𝑎𝑒+𝑒/(1𝑎𝑑)2.(3.4) Next, we assume the conclusion is true for 𝑘, that is, (3.3) hold.
Then, for 𝑘+1, from (3.1) and (3.3), we have𝑥2(𝑘+1)1=𝑑(𝑎𝑑1)𝑘+1,𝑦2(𝑘+1)1=𝑏(𝑏𝑐1)𝑘+1,𝑧2(𝑘+1)1=𝑧2𝑘1𝑥2𝑘𝑧2𝑘1=1𝑏𝑓(𝑏𝑐1)𝑘𝑓+(1)𝑘×1(𝑏𝑓)𝑏𝑓𝑐(𝑏𝑐1)𝑘/(𝑏𝑐1)𝑘𝑓+(1)𝑘=(𝑏𝑓)𝑏𝑓(𝑏𝑐1)𝑘+1𝑓+(1)𝑘+1,𝑥(𝑏𝑓)2(𝑘+1)=𝑐(𝑏𝑐1)𝑘+1,𝑦2(𝑘+1)=𝑎(𝑎𝑑1)𝑘+1,𝑧2(𝑘+1)=𝑧2𝑘𝑥2𝑘+1𝑧2𝑘=1(1)𝑘𝑎𝑒𝑎𝑒+𝑒/(1𝑎𝑑)𝑘×1𝑑/(𝑎𝑑1)𝑘(1)𝑘𝑎𝑒/𝑎𝑒+𝑒/(1𝑎𝑑)𝑘1=(1)𝑘+1𝑎𝑒/𝑎𝑒+𝑒/(1𝑎𝑑)𝑘+1,(3.5) which complete the proof.

From the above theorem, such a simple expression of the solution of (3.1) will greatly help us to investigate the behavior of the solution.

Corollary 3.2. Suppose that hypothesis (𝐻),𝑏=𝑓, and one of the following hold: (i)0<𝑎𝑑<1,0<𝑏𝑐<1;(ii)1<𝑎𝑑<2,1<𝑏𝑐<2.Then lim𝑘𝑥2𝑘1,𝑦2𝑘1,𝑧2𝑘1=(,,),lim𝑘𝑥2𝑘,𝑦2𝑘,𝑧2𝑘=(0,0,0).(3.6)

Proof. First, for 2𝑘1, we consider the following two cases.(1)Assume that (i) holds; then 1<𝑎𝑑1<0, 1<𝑏𝑐1<0.From (3.3), we have lim𝑘𝑥2𝑘1=lim𝑘𝑑(𝑎𝑑1)𝑘=+,𝑑>0,𝑘iseven,+,𝑑<0,𝑘isodd,,𝑑>0,𝑘isodd,,𝑑<0,𝑘iseven,lim𝑘𝑦2𝑘1=lim𝑘𝑏(𝑏𝑐1)𝑘=+,𝑏>0,𝑘iseven,+,𝑏<0,𝑘isodd,,𝑏>0,𝑘isodd,,𝑏<0,𝑘iseven,lim𝑘𝑧2𝑘1=lim𝑘𝑏𝑓(𝑏𝑐1)𝑘𝑓+(1)𝑘(=𝑏f)+,𝑏>0,𝑘iseven,+,𝑏<0,𝑘isodd,,𝑏>0,𝑘isodd,,𝑏<0,𝑘iseven,(3.7) where the last equation is from 𝑏=𝑓.(2)Assume that (ii) holds; then 0<𝑎𝑑1<1, 0<𝑏𝑐1<1. Similarly, we havelim𝑘𝑥2𝑘1=lim𝑘𝑑(𝑎𝑑1)𝑘=+,𝑑>0,,𝑑<0,lim𝑘𝑦2𝑘1=lim𝑘𝑏(𝑏𝑐1)𝑘=+,𝑏>0,,𝑏<0,lim𝑘𝑧2𝑘1=lim𝑘𝑏𝑓(𝑏𝑐1)𝑘𝑓+(1)𝑘=(𝑏𝑓)+,𝑏>0,,𝑏<0.(3.8) Next, for 2𝑘, we always have lim𝑘𝑥2𝑘=lim𝑘𝑐(𝑏𝑐1)𝑘=0,lim𝑘𝑦2𝑘=lim𝑘𝑎(𝑎𝑑1)𝑘=0,lim𝑘𝑧2𝑘=lim𝑘(1)𝑘𝑎𝑒𝑎𝑒+𝑒/(1𝑎𝑑)𝑘=0,(3.9) and complete the proof.

Corollary 3.3. Suppose that hypothesis (𝐻),𝑎=𝑒, and one of the following hold: (i)2<𝑎𝑑<+,2<𝑏𝑐<+,(ii)<𝑎𝑑<0,<𝑏𝑐<0.Then lim𝑘𝑥2𝑘1,𝑦2𝑘1,𝑧2𝑘1=(0,0,0),lim𝑘𝑥2𝑘,𝑦2𝑘,𝑧2𝑘=(,,).(3.10)

Proof. First, for 2𝑘1, in view of (i) or (ii), we have |𝑎𝑑1|>1, |𝑏𝑐1|>1 and thus lim𝑘𝑥2𝑘1=lim𝑘𝑑(𝑎𝑑1)𝑘=0,lim𝑘𝑦2𝑘1=lim𝑘𝑏(𝑏𝑐1)𝑘=0,lim𝑘𝑧2𝑘1=lim𝑘𝑏𝑓(𝑏𝑐1)𝑘𝑓+(1)𝑘(𝑏𝑓)=0.(3.11) Now, for 2𝑘, we consider the following two cases.(1)Assume that (i) holds; then 1<𝑎𝑑1<+, 1<𝑏𝑐1<+.From (3.3), we have lim𝑘𝑥2𝑘=lim𝑘𝑐(𝑏𝑐1)𝑘=+,𝑐>0,,𝑐<0,lim𝑘𝑦2𝑘=lim𝑘𝑎(𝑎𝑑1)𝑘=+,𝑎>0,,𝑎<0,lim𝑘𝑧2𝑘=lim𝑘(1)𝑘𝑎𝑒𝑎𝑒+𝑒/(1𝑎𝑑)𝑘=+,𝑎>0,,𝑎<0,(3.12) where the last equation is from 𝑎=𝑒.(2)Assume that (ii) holds; then <𝑎𝑑1<1,<𝑏𝑐1<1.Similarly, we have lim𝑘𝑥2𝑘=lim𝑘𝑐(𝑏𝑐1)𝑘=+,𝑐>0,𝑘-even,+,𝑐<0,𝑘-odd,,𝑐>0,𝑘-odd,,𝑐<0,𝑘-even,lim𝑘𝑦2𝑘=lim𝑘𝑎(𝑎𝑑1)𝑘=+,𝑎>0,𝑘-even,+,𝑎<0,𝑘-odd,,𝑎>0,𝑘-odd,,𝑎<0,𝑘-even,lim𝑘𝑧2𝑘=lim𝑘(1)𝑘𝑎𝑒𝑎𝑒+𝑒/(1𝑎𝑑)𝑘=+,𝑎>0,𝑘-even,+,𝑎<0,𝑘-odd,,𝑎>0,𝑘-odd,,𝑎<0,𝑘-even,(3.13) and complete the proof.

Corollary 3.4. Suppose that hypothesis (𝐻) holds and 𝑎=𝑒0, 𝑏=𝑓0, and 𝑎𝑑=𝑏𝑐=2. Then lim𝑘𝑥2𝑘1,𝑦2𝑘1,𝑧2𝑘1=(𝑑,𝑏,𝑏),lim𝑘𝑥2𝑘,𝑦2𝑘,𝑧2𝑘=(𝑐,𝑎,𝑎).(3.14)

The proof is simple and we omit it here. From this theorem, we can see that 𝑦𝑛=𝑧𝑛 for such initial values.

4. Conclusion

It is popular to study kinds of difference equations. The results can be divided into two parts. On the one hand, by linear stability theorem, one could study the behavior of solutions. Such a method is widely used to deal with a single difference equation; See [1]. On the other hand, the exact expression of solutions with respect to some difference equations is given. Generally speaking, it is difficult to obtain such an expression and to apply to other systems.

On a system consisting of two or three rational difference equations, there are few literatures. For details, see [28] and the references cited therein. In these papers, the exact expressions of solution are given.

In this paper, we expand the results obtained by Kurbanli in [2] and also investigate the behavior of the solution. At the same time, we consider a similar system and give some related results. The method can be applied to other kinds of difference equations.

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