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 [2–8].

In [2], Kurbanli studied a three-dimensional system of rational difference equationsxn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=zn-1ynzn-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, y-1, x0, x-1, z0, z-1∈ℝ be arbitrary real numbers and y0=a, y-1=b, x0=c, x-1=d, z0=e, z-1=f, and let {xn,yn,zn} be a solution of the system (1.1). Also, assume that ad≠1 and bc≠1; then all solutions of (1.1) are
xn={d(ad-1)n,nisodd,c(bc-1)n,niseven,yn={b(bc-1)n,nisodd,a(ad-1)n,niseven,zn={f(-1)0(n/0)anfdn-1+(-1)1(n/1)an-1fdn-2+⋯+(-1)n-1(n/(n-1))a1fd0+(-1)n(n/n),nisodd,(-1)n(bc-1)ne(-1)n(n/1)b1c0e+⋯+(-1)1(n/n)bncn-1e+(-1)0(n/0)bncn+⋯+(-1)n(n/n)b0c0,niseven.

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 (<xref ref-type="disp-formula" rid="EEq1.1">1.1</xref>)

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

In fact, (1.4) could be rewritten aszn={df(ad-1)kf+(-1)k(d-f),n=2k-1,(-1)kcec-e+(e/(1-bc)k),n=2k,k=1,2,….

From (2.1), it is easy to check the following: z1=df(ad-1)f+(-1)(d-f)=faf-1,z2=(-1)cec-e+(e/(1-bc))=e(bc-1)be-bc+1,z3=df(ad-1)2f+(-1)2(d-f)=fa2df-2af+1,
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:

0<ad<1, 0<bc<1,

1<ad<2, 1<bc<2.

Then
limk→∞(x2k-1,y2k-1,z2k-1)=(∞,∞,∞),limk→∞(x2k,y2k,z2k)=(0,0,0).Corollary 2.2.

Suppose that the initial values satisfy c=e and one of the following:

2<ad<+∞, 2<bc<+∞,

-∞<ad<0, -∞<bc<0.

Then
limk→∞(x2k-1,y2k-1,z2k-1)=(0,0,0),limk→∞(x2k,y2k,z2k)=(∞,∞,∞).Corollary 2.3.

Suppose that the initial values satisfy a=e≠0, b=f≠0, and ad=bc=2. Then
limk→∞(x2k-1,y2k-1,z2k-1)=(d,b,b),limk→∞(x2k,y2k,z2k)=(c,a,a).

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 [3–8] and the references cited therein, we consider the following system:xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=zn-1xnzn-1-1.
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,y-1=b,x-1=d,z-1=f.
Here, a,b,c,d,e, and f are nonzero real numbers such that ad≠1 and bc≠1. 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(ad-1)n,n=2k-1,c(bc-1)n,n=2k,yn={b(bc-1)n,n=2k-1,a(ad-1)n,n=2k,zn={bf(bc-1)kf+(-1)k(b-f),n=2k-1,(-1)kaea-e+(e/(1-ad)k),n=2k
for k=1,2,….

Proof.

First, for k=1,2, from (3.1), we easily check that
x1=dad-1,y1=bbc-1,z1=z-1x0z-1-1=fcf-1=bf(bc-1)f-(b-f),x2=c(bc-1),y2=a(ad-1),z2=z0x1z0-1=e(ad-1)de-ad+1=-1aea-e+(e/(1-ad)),x3=d(ad-1)2,y3=b(bc-1)2,z3=z1x2z1-1=fbc2f-2cf+1=bf(bc-1)f-(b-f),x4=c(bc-1)2,y4=a(ad-1)2,z4=z2x3z2-1=e(ad-1)2(2d-a2d)e+((ad-1)2=(-1)2aea-e+(e/(1-ad)2).
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(ad-1)k+1,y2(k+1)-1=b(bc-1)k+1,z2(k+1)-1=z2k-1x2kz2k-1-1=bf(bc-1)kf+(-1)k(b-f)×1bfc(bc-1)k/((bc-1)kf+(-1)k(b-f))=bf(bc-1)k+1f+(-1)k+1(b-f),x2(k+1)=c(bc-1)k+1,y2(k+1)=a(ad-1)k+1,z2(k+1)=z2kx2k+1z2k-1=(-1)kaea-e+(e/(1-ad)k)×1(d/(ad-1)k)((-1)kae/(a-e+(e/(1-ad)k)))-1=(-1)k+1ae/(a-e+(e/(1-ad)k+1)),
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:

0<ad<1,0<bc<1;

1<ad<2,1<bc<2.

Then
limk→∞(x2k-1,y2k-1,z2k-1)=(∞,∞,∞),limk→∞(x2k,y2k,z2k)=(0,0,0). Proof.

First, for 2k-1, we consider the following two cases.

Assume that (i) holds; then -1<ad-1<0, -1<bc-1<0.

From (3.3), we have
limk→∞x2k-1=limk→∞d(ad-1)k={+∞,d>0,kiseven,+∞,d<0,kisodd,-∞,d>0,kisodd,-∞,d<0,kiseven,limk→∞y2k-1=limk→∞b(bc-1)k={+∞,b>0,kiseven,+∞,b<0,kisodd,-∞,b>0,kisodd,-∞,b<0,kiseven,limk→∞z2k-1=limk→∞bf(bc-1)kf+(-1)k(b-f)={+∞,b>0,kiseven,+∞,b<0,kisodd,-∞,b>0,kisodd,-∞,b<0,kiseven,
where the last equation is from b=f.

Assume that (ii) holds; then 0<ad-1<1, 0<bc-1<1. Similarly, we have

limk→∞x2k-1=limk→∞d(ad-1)k={+∞,d>0,-∞,d<0,limk→∞y2k-1=limk→∞b(bc-1)k={+∞,b>0,-∞,b<0,limk→∞z2k-1=limk→∞bf(bc-1)kf+(-1)k(b-f)={+∞,b>0,-∞,b<0.
Next, for 2k, we always have
limk→∞x2k=limk→∞c(bc-1)k=0,limk→∞y2k=limk→∞a(ad-1)k=0,limk→∞z2k=limk→∞(-1)kaea-e+(e/(1-ad)k)=0,
and complete the proof.Corollary 3.3.

Suppose that hypothesis (H),a=e, and one of the following hold:

2<ad<+∞,2<bc<+∞,

-∞<ad<0,-∞<bc<0.

Then
limk→∞(x2k-1,y2k-1,z2k-1)=(0,0,0),limk→∞(x2k,y2k,z2k)=(∞,∞,∞). Proof.

First, for 2k-1, in view of (i) or (ii), we have |ad-1|>1, |bc-1|>1 and thus
limk→∞x2k-1=limk→∞d(ad-1)k=0,limk→∞y2k-1=limk→∞b(bc-1)k=0,limk→∞z2k-1=limk→∞bf(bc-1)kf+(-1)k(b-f)=0.
Now, for 2k, we consider the following two cases.

Assume that (i) holds; then 1<ad-1<+∞, 1<bc-1<+∞.

From (3.3), we have
limk→∞x2k=limk→∞c(bc-1)k={+∞,c>0,-∞,c<0,limk→∞y2k=limk→∞a(ad-1)k={+∞,a>0,-∞,a<0,limk→∞z2k=limk→∞(-1)kaea-e+(e/(1-ad)k)={+∞,a>0,-∞,a<0,
where the last equation is from a=e.

Assume that (ii) holds; then -∞<ad-1<-1,-∞<bc-1<-1.

Similarly, we have
limk→∞x2k=limk→∞c(bc-1)k={+∞,c>0,k-even,+∞,c<0,k-odd,-∞,c>0,k-odd,-∞,c<0,k-even,limk→∞y2k=limk→∞a(ad-1)k={+∞,a>0,k-even,+∞,a<0,k-odd,-∞,a>0,k-odd,-∞,a<0,k-even,limk→∞z2k=limk→∞(-1)kaea-e+(e/(1-ad)k)={+∞,a>0,k-even,+∞,a<0,k-odd,-∞,a>0,k-odd,-∞,a<0,k-even,
and complete the proof.Corollary 3.4.

Suppose that hypothesis (H) holds and a=e≠0, b=f≠0, and ad=bc=2. Then
limk→∞(x2k-1,y2k-1,z2k-1)=(d,b,b),limk→∞(x2k,y2k,z2k)=(c,a,a).

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 [2–8] 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.

KulenovićM. R. S.LadasG.KurbanliA. S.On the behavior of solutions of the system of rational difference equations: xn+1=xn−1/(ynxn−1−1), yn+1=yn−1/(xnyn−1−1) and zn+1=zn−1/(ynzn−1−1)KurbanlıA. S.ÇinarC.YalçinkayaI.On the behavior of positive solutions of the system of rational difference equations xn+1=xn−1/(ynxn−1−1), yn+1=yn−1/(xnyn−1−1)YalcinkayaI.ÇinarC.AtalayM.On the solutions of systems of differennce equationsKurbanliA. S.ÇinarC.ŞimşsekD.On the periodicty of solutions of the system of rational difference equations xn+1=xn−1/(ynxn−1−1), yn+1=yn−1/(xnyn−1−1)ÖzbanA. Y.On the system of rational difference equations xn=a/yn−3, yn=byn−3/xn−qyn−q−1DasS. E.BayramM.On a system of rational difference equationsIričaninB. D.StevićS.Some systems of nonlinear difference equations of higher order with periodic solutions