On the Behavior of a System of Rational

(1)

Volume 2012, Article ID 105496,9pages doi:10.1155/2012/105496

Research Article

On the Behavior of a System of Rational

Difference Equations x

_n1

x

_n−1

/y

n

x

_n−1

− 1 , y

n1

y

n−1

/x

n

y

n−1

− 1 , z

n1

1/x

n

z

n−1

Liu Keying,

¹

Wei Zhiqiang,

¹

Li Peng,

^{1, 2}

and Zhong Weizhou

²

1School of Mathematics, North China University of Water Resources and Electric Power, Zhengzhou 450045, China

2School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710061, China

Correspondence should be addressed to Zhong Weizhou,[email protected] Received 28 June 2012; Accepted 24 August 2012

Academic Editor: Cengiz C¸ inar

Copyrightq2012 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.

We are concerned with a three-dimensional system of rational diﬀerence equations with nonzero initial values. We present solutions of the system in an explicit way and obtain the asymptotical behavior of solutions.

1. Introduction

Difference equations, also referred to recursive sequence, is a hot topic. There has been an increasing interest in the study of qualitative analysis of difference equations and systems of difference equations. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economics, physics, computer sciences, and so on. Especially, Gu and Ding 1have considered the state space models described by difference equations.

Particularly, there is a class of nonlinear difference equations, known as rational difference equations or fractional difference equations. A lot of work has been concentrated on it 2–12. There is one way to study rational difference equations—giving the exact expression of solutions 4, 5. Another way is studying the qualitative behavior such as asymptotical stability using the linearized method, semicycle analysis, and so on2.

At the same time, more and more attention is paid to systems of rational diﬀerence equations composed by two or three rational diﬀerence equations 3, 6–12. The single equation is simple, but the coupled ways of systems are various and thus such systems have no fixed ways to follow to investigate their behavior.

(2)

In4,5, C¸ inar has obtained the solutions of the following diﬀerence equations:

xn1 xn−1

1xnxn−1 , xn1 axn−1

1bxnxn−1.

1.1

In 6, C¸ inar has proved the periodicity of positive solutions of the following diﬀerence equation system:

xn1 1

yn , yn1 yn

xn−1yn−1. 1.2

In7, Stevic has investigated the following system of diﬀerence equations:

xn1 axn−1

bynxn−1c, yn1 αyn−1

βxnyn−1γ. 1.3

In fact, such a general system has no explicit solutions and the author has classified the parameters to give explicit solutions for 14 special cases.

In8, Kurbanli et al. have studied the behavior of positive solutions of the system of the following rational diﬀerence equations:

xn1 xn−1

ynx_n−1−1, yn1 yn−1

xny_n−1−1. 1.4

Based on it, other three-dimensional systems have been investigated in9,10, and 11, respectively,

xn1 xn−1

ynxn−1−1, yn1 yn−1

xnyn−1−1, zn1 zn−1

ynzn−1−1; 1.5 x_n1 x_n−1

ynxn−1−1, y_n1 y_n−1

xnyn−1−1, z_n1 1 ynzn

; 1.6

xn1 xn−1

ynx_n−1−1, yn1 y_n−1

xny_n−1−1, zn1 xn

ynz_n−1. 1.7

In 12, we improved the results on 1.5 of those in 9 and also investigated the system

xn1 xn−1

ynx_n−1−1, yn1 yn−1

xny_n−1−1, zn1 zn−1

xnz_n−1−1. 1.8

Some other results would be presented in3.

(3)

In this paper, motivated by the above references and the references cited therein, we consider the following system:

x_n1 x_n−1

ynxn−1−1, y_n1 yn−1

xnyn−1−1, z_n1 1

xnzn−1, 1.9

where the initial conditions are nonzero real numbers.

In next section, we express solutions of the system 1.9 and try to describe the behavior of solutions.

2. Main Results

Through the paper, we suppose the initial values to be

y0 a, x0c, y−1b, x−1d, z0e, z−1 f. 2.1

Here,a,b,c,d,e, andfare real numbers such thatad−1cb−1/0,cdef /0. We call this to be the hypothesisH.

Theorem 2.1. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. Then all solutions of1.9are

xn

⎧⎪

⎨

⎪⎩ d

ad−1^k , n2k−1, ccb−1^k, n2k,

k1,2, . . . , 2.2

yn

⎧⎪

⎨

⎪⎩ b

cb−1^k , n2k−1, aad−1^k, n2k,

k1,2, . . . , 2.3

zn

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1

cfcb−1^k−1 , n4k−1 1, ad−1^k

de , n4k−1 2,

f

cb−1^k , n4k−1 3, ead−1^k, n4k−1 4.

k1,2, . . . . 2.4

Proof. It is obvious to obtain2.2and2.3and referred to8. Here, we only focus on2.4.

(4)

First, fork1, from1.9and2.2, we easily check that

z1 1 x0z−1 1

cf,

z2 1

x1z0 1

d/ad−1e ad−1

de ,

z3 1

x2z1 f cb−1, z4 1

x3z2 ead−1.

2.5

Next, we assume the conclusion is true fork, that is,2.4holds.

Then, fork1, we confirm it. In fact, from1.9,2.2, and2.4, we have the following:

z_4k1 1

x4kz_4k−13 1

ccb−1^2k×

f/cb−1^k 1

cfcb−1^k,

z4k2 1

x4k1z4k 1

d/ad−1^2k1

×ead−1^k ad−1^k1

de ,

z4k3 1

x_4k2z_4k1 1

ccb−1^2k1×

1/cfcb−1^k f cb−1^k1,

z_4k4 1

x4k3z4k2 1

d/ad−1^2k2

×

ad−1^k1/de ead−1^k1,

2.6

and complete the proof.

By Theorem 2.1, the expressions of 2.2, 2.3, and 2.4 will greatly help us to investigate the asymptotical behavior of solutions of2.4.

Corollary 2.2. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. Also, ifadcb2, then all solutions of 1.9are four periodic.

Proof. In this case, from2.2,2.3, and2.4, we have the following:

xn

d, n2k−1,

c, n2k, k1,2, . . .

(5)

yn

b, n2k−1,

a, n2k, k1,2, . . .

zn

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1

cf, n4k−1 1, 1

de, n4k−1 2, f, n4k−1 3, e, n4k−1 4,

k1,2, . . . ,

2.7

Corollary 2.3. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. Also, ifad, cb∈1,2, andc >0, then all solutions of 1.9satisfy

nlim→ ∞ x2n−1, y2n−1, z2n−1

∞,∞,∞,

nlim→ ∞ x2n, y2n, z2n

0,0,0. 2.8

Proof. From the hypothesis andad, cb ∈ 1,2, andd > c, we obtain that 0 < ad−1 < 1, 0< cb−1<1 and thus,ad−1ⁿandcb−1ⁿtend to zero asntends to∞.

First, from2.2, we have

nlim→ ∞x2n−1 lim

n→ ∞

d

ad−1ⁿ d· ∞

−∞, d <0,

∞, d >0. 2.9

Similarly, from2.3, we have

nlim→ ∞y_2n−1 lim

n→ ∞

b

cb−1ⁿ b· ∞

−∞, b <0,

∞, b >0. 2.10

As far as z2n−1 is concerned, from 2.4 we could consider z4k1 and z4k3 for n k 1, respectively,

nlim→ ∞z4k1 lim

n→ ∞

1

cfcb−1^k 1 cf · ∞

−∞, f <0, c >0

∞, f >0,

n→ ∞limz4k3 lim

n→ ∞

f

cb−1^k1 f· ∞

−∞, f <0,

∞, f >0.

2.11

(6)

Thus,

nlim→ ∞z2n−1

−∞, f <0,

∞, f >0. 2.12

Therefore,

nlim→ ∞ x2n−1, y2n−1, z2n−1

∞,∞,∞. 2.13

Next, from2.2and2.3, we have

nlim→ ∞x2n lim

n→ ∞ccb−1ⁿ 0,

nlim→ ∞y2n lim

n→ ∞aad−1ⁿ 0. 2.14

At last, forz2n, we have

n→ ∞limz4k2 lim

n→ ∞

ad−1^k1

de 0,

n→ ∞limz_4k4 lim

n→ ∞ead−1^k10.

2.15

Thus,

nlim→ ∞z2n0 2.16

Corollary 2.4. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. Also, ifa,b,c,d∈0,1, then all solutions of 1.9satisfy

nlim→ ∞ x_2n−1, y_2n−1, z_2n−1

∞,∞,∞,

0,0,0. 2.17

Proof. Froma,b,c,d ∈ 0,1, we have−1 < ad−1 < 0,−1 < cb−1 < 0. The remainder is similar to that ofCorollary 2.3and we omit here.

Corollary 2.5. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. Also, ifad, cb∈2,∞, andd >0, then all solutions of 1.9satisfy

nlim→ ∞ x2n−1, y2n−1, z2n−1

0,0,0,

∞,∞,∞. 2.18

(7)

Corollary 2.6. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. Also, ifad, cb∈−∞,0, andd >0, then all solutions of 1.9satisfy

nlim→ ∞ x2n−1, y2n−1, z2n−1

0,0,0,

∞,∞,∞. 2.19

The above theorems describe the asymptotical behavior of solutions in case of the initial values lying in diﬀerent intervals. At last, we describe the behavior in another way.

Corollary 2.7. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. If one of the following holds:

11< ad < cb;

2cb < ad <1;

3ad <1< cbandadcb >2;

4cb <1< adandadcb <2, then all solutions of 1.9satisfy

nlim→ ∞x2ny_2n−1cb,

nlim→ ∞x2n−1y2nad,

nlim→ ∞z2n−1z2n0.

2.20

Proof. In view of2.2,2.3, and2.4, we have

nlim→ ∞x2ny2n−1 lim

n→ ∞

ccb−1ⁿ × b cb−1ⁿ

cb,

nlim→ ∞x_2n−1y2n lim

n→ ∞

d

ad−1ⁿ ×aad−1ⁿ

ad.

2.21

As far asz_2n−1 and z2n are concerned, from 2.4we could consider z_4k1 andz_4k2, z4k3andz4k4fornk1, respectively. In fact, we have

z4k1z4k2 1

cfcb−1^k ×ad−1^k1

de ad−1

cdef

ad−1 cb−1

k

,

z4k3z4k4 f

cb−1^k1 ×ead−1^k1ef

ad−1 cb−1

_k1 .

2.22

If one of the four conditions holds, we obtain|ad−1/cb−1| < 1 and the conclusion is apparent.

(8)

Corollary 2.8. Suppose that the hypothesisHholds and let{xn, yn, zn}be a solution of the system 1.9. If one of the following holds:

11< cb < ad;

2ad < cb <1;

3ad <1< cbandadcb <2;

4cb <1< adandadcb >2

andad−1/cd >0, then all solutions of 1.9satisfy

nlim→ ∞x2ny_2n−1cb,

nlim→ ∞x_2n−1y2nad,

n→ ∞lim z2n−1z2n∞.

2.23

The proof is omitted here. In fact, we could obtain|ad−1/cb−1|>1 if one of the four conditions holds and the condition ofad−1/cd >0 is to keep the sign.

Funding

This paper is supported by the National Natural Science Foundation of ChinaNo. 71271086, 71172184and the foundation of Education Department of Henan ProvinceNo. 12A110014.

References

1 Y. Gu and R. Ding, “Observable state space realizations for multivariable systems,” Computers and Mathematics with Applications, vol. 63, no. 9, pp. 1389–1399, 2012.

2 M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Diﬀerence Equations, Chapman &

Hall/CRC, Boca Raton, Fla, USA, 2002.

3 K. Liu, P. Cheng, P. Li, and W. Zhong, “On a system of rational diﬀerence equations xn1 xn−1/ynxn−1−1, yn1yn−1/xnyn−1−1, zn11/ynzn−1,” Fasciculi Mathematici. In press.

4 C. C¸ inar, “On the positive solutions of the diﬀerence equationxn1 xn−1/1bxnxn−1,” Applied Mathematics and Computation, vol. 150, no. 1, pp. 21–24, 2004.

5 C. C¸ inar, “On the positive solutions of the diﬀerence equationxn1 axn−1/1bxnxn−1,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 587–590, 2004.

6 C. C¸ inar, “On the positive solutions of the diﬀerence equation system xn1 1/yn, yn1 yn/xn−1yn−1,” Applied Mathematics and Computation, vol. 158, no. 2, pp. 303–305, 2004.

7 S. Stevi´c, “On a system of diﬀerence equationsxn1axn−1/bynxn−1c, yn1αyn−1/βxnyn−1γ,”

Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372–3378, 2011.

8 A. S. Kurbanli, C. Ç inar, and I. Yalçinkaya, “On the behavior of solutions of the system of rational difference equationsxn1xn−1/ynxn−1−1, yn1yn−1/xnyn−1−1,” World Applied Sciences Journal, vol. 10, no. 11, pp. 1344–1350, 2010.

9 A. S. Kurbanli, “On the behavior of solutions of the system of rational diﬀerence equationsxn1 xn−1/ynxn−1−1, yn1yn−1/xnyn−1−1, zn1zn−1/ynzn−1−1,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 932632, 12 pages, 2011.

10 A. S. Kurbanli, “On the behavior of solutions of the system of rational diﬀerence equationsxn1 xn−1/ynxn−1−1, yn1yn−1/xnyn−1−1, zn11/ynzn,” Advances in Diﬀerence Equations, vol. 40, 2011.

(9)

11 A. S. Kurbanli, C. Ç inar, and M. E. Erdo ˘gan, “On the behavior of solutions of the system of rational difference equations of rational difference equationsxn1xn−1/ynxn−1−1, yn1yn−1/xnyn−1− 1, zn1xn/ynzn−1,” Applied Methematics, vol. 2, pp. 1031–1038, 2011.

12 K. Liu, Z. Zhao, X. Li, and P. Li, “More on three-dimensional systems of rational diﬀerence equations,”

Discrete Dynamics in Nature and Society, vol. 2011, Article ID 178483, 9 pages, 2011.

(10)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

On the Behavior of a System of Rational

Research Article