On the Behavior of a System of Rational Difference Equations 𝑥_𝑛+1=𝑥_𝑛−1/(𝑦_𝑛𝑥_𝑛−1−1),𝑦_𝑛+1 = 𝑦_𝑛−1/(𝑥_𝑛𝑦_𝑛−1−1),𝑧_𝑛+1=1/𝑥_𝑛𝑧_𝑛−1

Abstract

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 [1] have 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 on [2].

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.

In [4, 5], Çinar has obtained the solutions of the following difference equations: 𝑥_𝑛+1=𝑥_𝑛−11+𝑥_𝑛𝑥_𝑛−1,𝑥_𝑛+1=𝑎𝑥_𝑛−11+𝑏𝑥_𝑛𝑥_𝑛−1. (1.1) In [6], Çinar has proved the periodicity of positive solutions of the following difference equation system: 𝑥_𝑛+1=1𝑦_𝑛,𝑦_𝑛+1=𝑦_𝑛𝑥_𝑛−1𝑦_𝑛−1. (1.2)

In [7], Stevic has investigated the following system of difference equations: 𝑥_𝑛+1=𝑎𝑥_𝑛−1𝑏𝑦_𝑛𝑥_𝑛−1+𝑐,𝑦_𝑛+1=𝛼𝑦_𝑛−1𝛽𝑥_𝑛𝑦_𝑛−1.+𝛾 (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.

In [8], Kurbanli et al. have studied the behavior of positive solutions of the system of the following rational difference equations: 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1.−1 (1.4)

Based on it, other three-dimensional systems have been investigated in [9], [10], and [11], respectively, 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=𝑧_𝑛−1𝑦_𝑛𝑧_𝑛−1;−1𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=1𝑦_𝑛𝑧_𝑛;𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=𝑥_𝑛𝑦_𝑛𝑧_𝑛−1. (1.5)(1.6)(1.7)

In [12], we improved the results on (1.5) of those in [9] and also investigated the system 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=𝑧_𝑛−1𝑥_𝑛𝑧_𝑛−1.−1 (1.8) Some other results would be presented in [3].

In this paper, motivated by the above references and the references cited therein, we consider the following system: 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=1𝑥_𝑛𝑧_𝑛−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,𝑥₀=𝑐,𝑦₋₁=𝑏,𝑥₋₁=𝑑,𝑧₀=𝑒,𝑧₋₁=𝑓. (2.1) Here, 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, and 𝑓 are real numbers such that (𝑎𝑑−1)(𝑐𝑏−1)≠0, 𝑐𝑑𝑒𝑓≠0. We call this to be the hypothesis 𝐻.

Theorem 2.1. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). Then all solutions of (1.9) are 𝑥_𝑛=𝑑(𝑎𝑑−1)^𝑘𝑐(),𝑛=2𝑘−1,𝑐𝑏−1^𝑘,𝑛=2𝑘,𝑘=1,2,…,𝑦_𝑛=𝑏(𝑐𝑏−1)^𝑘𝑎(),𝑛=2𝑘−1,𝑎𝑑−1^𝑘,𝑛=2𝑘,𝑘=1,2,…,𝑧_𝑛=1𝑐𝑓(𝑐𝑏−1)^𝑘−1()(),𝑛=4𝑘−1+1,𝑎𝑑−1^𝑘()𝑓𝑑𝑒,𝑛=4𝑘−1+2,(𝑐𝑏−1)^𝑘()𝑒(),𝑛=4𝑘−1+3,𝑎𝑑−1^𝑘,𝑛=4(𝑘−1)+4.𝑘=1,2,…. (2.2)(2.3)(2.4)

Proof. It is obvious to obtain (2.2) and (2.3) and referred to [8]. Here, we only focus on (2.4).
First, for 𝑘=1, from (1.9) and (2.2), we easily check that 𝑧₁=_𝑥01_𝑧−1=1,𝑧𝑐𝑓₂=1𝑥₁𝑧₀=1((𝑒=𝑑/𝑎𝑑−1))𝑎𝑑−1,𝑧𝑑𝑒₃=1𝑥₂𝑧₁=𝑓,𝑧𝑐𝑏−1₄=1𝑥₃𝑧₂=𝑒(𝑎𝑑−1). (2.5)
Next, we assume the conclusion is true for 𝑘, that is, (2.4) holds.
Then, for 𝑘+1, we confirm it. In fact, from (1.9), (2.2), and (2.4), we have the following: 𝑧_4𝑘+1=1𝑥_4𝑘𝑧_{4(𝑘−1)+3}=1𝑐(𝑐𝑏−1)^2𝑘×𝑓/(𝑐𝑏−1)^𝑘=1𝑐𝑓(𝑐𝑏−1)^𝑘,𝑧_4𝑘+2=1𝑥_4𝑘+1𝑧_4𝑘=1𝑑/(𝑎𝑑−1)^2𝑘+1×𝑒(𝑎𝑑−1)^𝑘=(𝑎𝑑−1)^𝑘+1,𝑧𝑑𝑒_4𝑘+3=1𝑥_4𝑘+2𝑧_4𝑘+1=1𝑐(𝑐𝑏−1)^2𝑘+1×1/𝑐𝑓(𝑐𝑏−1)^𝑘=𝑓(𝑐𝑏−1)^𝑘+1,𝑧_4𝑘+4=1𝑥_4𝑘+3𝑧_4𝑘+2=1𝑑/(𝑎𝑑−1)^2𝑘+2×(𝑎𝑑−1)^𝑘+1()/𝑑𝑒=𝑒𝑎𝑑−1^𝑘+1, (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 of (2.4).

Corollary 2.2. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). Also, if 𝑎𝑑=𝑐𝑏=2, then all solutions of (1.9) are four periodic.

Proof. In this case, from (2.2), (2.3), and (2.4), we have the following: 𝑥_𝑛=𝑦𝑑,𝑛=2𝑘−1,𝑐,𝑛=2𝑘,𝑘=1,2,…_𝑛=𝑧𝑏,𝑛=2𝑘−1,𝑎,𝑛=2𝑘,𝑘=1,2,…_𝑛=1()1𝑐𝑓,𝑛=4𝑘−1+1,()()()𝑑𝑒,𝑛=4𝑘−1+2,𝑓,𝑛=4𝑘−1+3,𝑒,𝑛=4𝑘−1+4,𝑘=1,2,…, (2.7) and complete the proof.

Corollary 2.3. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). Also, if 𝑎𝑑,𝑐𝑏∈(1,2), and 𝑐>0, then all solutions of (1.9) satisfy lim_𝑛→∞𝑥_2𝑛−1,𝑦_2𝑛−1,𝑧_2𝑛−1=(∞,∞,∞),lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛,𝑧_2𝑛=(0,0,0). (2.8)

Proof. From the hypothesis and 𝑎𝑑,𝑐𝑏∈(1,2), and 𝑑>𝑐, we obtain that 0<𝑎𝑑−1<1, 0<𝑐𝑏−1<1 and thus, (𝑎𝑑−1)^𝑛 and (𝑐𝑏−1)^𝑛 tend to zero as 𝑛 tends to ∞.
First, from (2.2), we have lim_𝑛→∞𝑥_2𝑛−1=lim_𝑛→∞𝑑(𝑎𝑑−1)^𝑛=𝑑⋅∞=−∞,𝑑<0,+∞,𝑑>0. (2.9) Similarly, from (2.3), we have lim_𝑛→∞𝑦_2𝑛−1=lim_𝑛→∞𝑏(𝑐𝑏−1)^𝑛=𝑏⋅∞=−∞,𝑏<0,+∞,𝑏>0. (2.10) As far as 𝑧_2𝑛−1 is concerned, from (2.4) we could consider 𝑧_4𝑘+1 and 𝑧_4𝑘+3 for 𝑛=𝑘+1, respectively, lim_𝑛→∞𝑧_4𝑘+1=lim_𝑛→∞1𝑐𝑓(𝑐𝑏−1)^𝑘=1𝑐𝑓⋅∞=−∞,𝑓<0,𝑐>0+∞,𝑓>0,lim_𝑛→∞𝑧_4𝑘+3=lim_𝑛→∞𝑓(𝑐𝑏−1)^𝑘+1=𝑓⋅∞=−∞,𝑓<0,+∞,𝑓>0. (2.11) Thus, lim_𝑛→∞𝑧_2𝑛−1=−∞,𝑓<0,+∞,𝑓>0. (2.12) Therefore, lim_𝑛→∞𝑥_2𝑛−1,𝑦_2𝑛−1,𝑧_2𝑛−1=(∞,∞,∞). (2.13)
Next, from (2.2) and (2.3), we have lim_𝑛→∞𝑥_2𝑛=lim_𝑛→∞𝑐(𝑐𝑏−1)^𝑛=0,lim_𝑛→∞𝑦_2𝑛=lim_𝑛→∞𝑎(𝑎𝑑−1)^𝑛=0. (2.14)
At last, for 𝑧_2𝑛, we have lim_𝑛→∞𝑧_4𝑘+2=lim_𝑛→∞(𝑎𝑑−1)^𝑘+1𝑑𝑒=0,lim_𝑛→∞𝑧_4𝑘+4=lim_𝑛→∞𝑒(𝑎𝑑−1)^𝑘+1=0. (2.15) Thus, lim_𝑛→∞𝑧_2𝑛=0 (2.16) and complete the proof.

Corollary 2.4. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). Also, if 𝑎, 𝑏, 𝑐, 𝑑∈(0,1), then all solutions of (1.9) satisfy lim_𝑛→∞𝑥_2𝑛−1,𝑦_2𝑛−1,𝑧_2𝑛−1=(∞,∞,∞),lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛,𝑧_2𝑛=(0,0,0). (2.17)

Proof. From 𝑎, 𝑏, 𝑐, 𝑑∈(0,1), we have −1<𝑎𝑑−1<0, −1<𝑐𝑏−1<0. The remainder is similar to that of Corollary 2.3 and we omit here.

Corollary 2.5. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). Also, if 𝑎𝑑,𝑐𝑏∈(2,+∞), and 𝑑>0, then all solutions of (1.9) satisfy lim_𝑛→∞𝑥_2𝑛−1,𝑦_2𝑛−1,𝑧_2𝑛−1=(0,0,0),lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛,𝑧_2𝑛=(∞,∞,∞). (2.18)

Corollary 2.6. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). Also, if 𝑎𝑑,𝑐𝑏∈(−∞,0), and 𝑑>0, then all solutions of (1.9) satisfy lim_𝑛→∞𝑥_2𝑛−1,𝑦_2𝑛−1,𝑧_2𝑛−1=(0,0,0),lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛,𝑧_2𝑛=(∞,∞,∞). (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 hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). If one of the following holds:(1)1<𝑎𝑑<𝑐𝑏; (2)𝑐𝑏<𝑎𝑑<1; (3)𝑎𝑑<1<𝑐𝑏and𝑎𝑑+𝑐𝑏>2; (4)𝑐𝑏<1<𝑎𝑑and𝑎𝑑+𝑐𝑏<2, then all solutions of (1.9) satisfy lim_𝑛→∞𝑥_2𝑛𝑦_2𝑛−1=𝑐𝑏,lim_𝑛→∞𝑥_2𝑛−1𝑦_2𝑛=𝑎𝑑,lim_𝑛→∞𝑧_2𝑛−1𝑧_2𝑛=0. (2.20)

Proof. In view of (2.2), (2.3), and (2.4), we have lim_𝑛→∞𝑥_2𝑛𝑦_2𝑛−1=lim_𝑛→∞𝑐(𝑐𝑏−1)^𝑛×𝑏(𝑐𝑏−1)^𝑛=𝑐𝑏,lim_𝑛→∞𝑥_2𝑛−1𝑦_2𝑛=lim_𝑛→∞𝑑(𝑎𝑑−1)^𝑛×𝑎(𝑎𝑑−1)^𝑛=𝑎𝑑. (2.21)
As far as 𝑧_2𝑛−1 and 𝑧_2𝑛 are concerned, from (2.4) we could consider 𝑧_4𝑘+1 and 𝑧_4𝑘+2, 𝑧_4𝑘+3 and 𝑧_4𝑘+4 for 𝑛=𝑘+1, respectively. In fact, we have 𝑧_4𝑘+1𝑧_4𝑘+2=1𝑐𝑓(𝑐𝑏−1)^𝑘×(𝑎𝑑−1)^𝑘+1=𝑑𝑒𝑎𝑑−1𝑐𝑑𝑒𝑓𝑎𝑑−1𝑐𝑏−1^𝑘,𝑧_4𝑘+3𝑧_4𝑘+4=𝑓(𝑐𝑏−1)^𝑘+1×𝑒(𝑎𝑑−1)^𝑘+1=𝑒𝑓𝑎𝑑−1𝑐𝑏−1^𝑘+1. (2.22) If one of the four conditions holds, we obtain |(𝑎𝑑−1)/(𝑐𝑏−1)|<1 and the conclusion is apparent.

Corollary 2.8. Suppose that the hypothesis 𝐻 holds and let {𝑥_𝑛,𝑦_𝑛,𝑧_𝑛} be a solution of the system (1.9). If one of the following holds:(1)1<𝑐𝑏<𝑎𝑑; (2)𝑎𝑑<𝑐𝑏<1; (3)𝑎𝑑<1<𝑐𝑏and𝑎𝑑+𝑐𝑏<2; (4)𝑐𝑏<1<𝑎𝑑and𝑎𝑑+𝑐𝑏>2and (𝑎𝑑−1)/𝑐𝑑>0, then all solutions of (1.9) satisfy lim_𝑛→∞𝑥_2𝑛𝑦_2𝑛−1=𝑐𝑏,lim_𝑛→∞𝑥_2𝑛−1𝑦_2𝑛=𝑎𝑑,lim_𝑛→∞𝑧_2𝑛−1𝑧_2𝑛=∞. (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 of (𝑎𝑑−1)/𝑐𝑑>0 is to keep the sign.

Funding

This paper is supported by the National Natural Science Foundation of China (No. 71271086, 71172184) and the foundation of Education Department of Henan Province (No. 12A110014).