# 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

Authors: Abdullah Selçuk Kurbanli
Journal: Discrete Dynamics in Nature and Society
 (2011)

---

## Abstract

We investigate the solutions of the system of difference equationsxn+1=xn-1/(ynxn-1-1), yn+1=yn-1/(xnyn-1-1), zn+1=zn-1/(ynzn-1-1), where y0,y-1,x0,x-1,z-1,z0∈ℝ.


---

## Body

## 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], Çinar studied the solutions of the systems of the difference equations (1.1)xn+1=1yn,yn+1=ynxn-1yn-1.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 (1.2)xn+1=A+ynxn-p,yn+1=A+xnyn-q,n=0,1,…,p,q.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 (1.3)xn+1=∑i=0kAiyn-ipi,yn+1=∑i=0kBixn-iqi.In [4, 5] Özban studied the positive solutions of the system of rational difference equations (1.4)xn=ayn-3,yn=byn-3xn-qyn-q,xn+1=1yn-k,yn+1=ynxn-myn-m-k.In [6, 7] Clark and Kulenović investigate the global asymptotic stability (1.5)xn+1=xna+cyn,yn+1=ynb+dxn.In [8] Camouzis and Papaschinopoulos studied the global asymptotic behavior of positive solutions of the system of rational difference equations(1.6)xn+1=1+xnyn-m,yn+1=1+ynxn-m.In [9] Yang et al. considered the behavior of the positive solutions of the system of the difference equations (1.7)xn=ayn-p,yn=byn-pxn-qyn-q.In [10] Kulenović and Nurkanović studied the global asymptotic behavior of solutions of the system of difference equations (1.8)xn+1=a+xnb+yn,yn+1=c+ynd+zn,zn+1=e+znf+xn.In [11] Zhang et al. investigated the behavior of the positive solutions of the system of difference equations (1.9)xn=A+1yn-p,yn=A+yn-1xn-ryn-s.In [12] Zhang et al. studied the boundedness, the persistence, and global asymptotic stability of the positive solutions of the system of difference equations(1.10)xn+1=A+yn-mxn,yn+1=A+xn-myn.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(1.11)xn+1(1)=xn(2)xn(2)-1,xn+1(2)=xn(3)xn(3)-1,…,xn+1(k)=xn(1)xn(1)-1.In [15] Yalcinkaya studied the global asymptotic stability of the system of difference equations(1.12)zn+1=tnzn-1+atn+zn-1,tn+1=zntn-1+azn+tn-1.In [16] Irićanin and Stević studied the positive solutions of the system of difference equations  (1.13)xn+1(1)=1+xn(2)xn-1(3),xn+1(2)=1+xn(3)xn-1(4),…,xn+1(k)=1+xn(1)xn-1(2),xn+1(1)=1+xn(2)+xn-1(3)xn-2(4),xn+1(2)=1+xn(3)+xn-1(4)xn-2(5),…,xn+1(k)=1+xn(1)+xn-1(2)xn-2(3).In [17] Kurbanli et al. studied the behavaior of positive solutions of the system of rational difference equations (1.14)xn+1=xn-1ynxn-1+1,yn+1=yn-1xnyn-1+1.Also see references.In this paper, we investigate the behavior of the solutions of the difference equations system(1.15)xn+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.Theorem 1.1.
Lety0,y-1,x0,x-1,z-1,z0∈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 (1.15). Also, assume that af≠1 and bc≠1 then all solutions of (1.15) are
(1.16)xn={d(ad-1)n,n-odd,c(bc-1)n,n-even,yn={b(bc-1)n,n-odd,a(ad-1)n,n-even,zn={f(-1)0(n0)anfdn-1+(-1)1(n1)an-1fdn-2+⋯+(-1)n-1(nn-1)a1fd0+(-1)n(nn),n-odd(-1)n(bc-1)ne(-1)n(n1)b1c0e+⋯+(-1)1(nn)bncn-1e+(-1)0(n0)bncn+⋯+(-1)n(nn)b0c0,n-even.Proof.
Forn=0,1,2,3, we have
(1.17)x1=x-1y0x-1-1=dad-1,y1=y-1x0y-1-1=bbc-1,z1=z-1y0z-1-1=faf-1,x2=x0y1x0-1=c(b/(bc-1))c-1=c(bc-1)bc-bc+1=c(bc-1),y2=y0x1y0-1=a(d/(ad-1))a-1=a(ad-1)ad-ad+1=a(ad-1),z2=z0y1z0-1=e(b/(bc-1))e-1=e(bc-1)be-bc+1=-e(bc-1)-1⋅be+bc-1,x3=x1y2x1-1=d/(ad-1)a(ad-1)⋅d/(ad-1)-1=d(ad-1)2,y3=y1x2y1-1=b/(bc-1)c(bc-1)⋅b/(bc-1)-1=b/(bc-1)bc-1=b(bc-1)2,z3=z1y2z1-1=f/(af-1)a(ad-1)⋅f/(af-1)-1=f/(af-1)(a2df-af-af+1)/(af-1)=fa2df-2af+1
for n=k assume that
(1.18)xk={d(ad-1)k,k-odd,c(bc-1)k,k-even,yk={b(bc-1)k,k-odd,a(ad-1)k,k-even,z2k-1=f(-1)0(k0)akfdk-1+(-1)1(k1)ak-1fdk-2+⋯+(-1)k-1(kk-1)a1fd0+(-1)k(kk),k=1,2,…,z2k=(-1)k(bc-1)ke(-1)k(k1)b1c0e+⋯+(-1)1(kk)bkck-1e+(-1)0(k0)bkck+⋯+(-1)k(kk)b1c1,k=1,2,…
are true. Then for n=k+1 we will show that (1.16) is true. From (1.15), we have
(1.19)x2k+1=x2k-1y2kx2k-1-1=d/(ad-1)ka(ad-1)k⋅d/(ad-1)k-1=d(ad-1)k+1.
Also, similarly from (1.15), we have(1.20)y2k+1=y2k-1x2ky2k-1-1b/(bc-1)kc(bc-1)k⋅b/(bc-1)k-1=b(bc-1)k+1,(nk)=n!(n-k)!⋅k!,(nk)+(nk-1)=(n+1k),(n0)=(nn)=(n+1n+1)
from properties of Binomial coefficients,
(1.21)z2k+1=z2k-1y2kz2k-1-1=f/((-1)0(k0)akfdk-1+⋯+(-1)k-1(kk-1)a1fd0+(-1)k(kk))a(ad-1)kf/((-1)0(k0)akfdk-1+⋯+(-1)k-1(kk-1)a1fd0+(-1)k(kk))-1=f(-1)0(k+10)ak+1fdk+(-1)1(k+11)akfdk-1+⋯+(-1)k(k+1k)a1fd0+(-1)k+!(k+1k+1)
written and accurate.
Also, we have(1.22)x2k+2=x2ky2k+1x2k-1=c(bc-1)k(b/(bc-1)k+1)c(bc-1)k-1=c(bc-1)kbc/(bc-1)-1=c(bc-1)k+1,y2k+2=y2kx2k+1y2k-1=a(ad-1)k(d/(ad-1)k+1)⋅a(ad-1)k-1=a(ad-1)kad/(ad-1)-1=a(ad-1)k+!,z2k+2=z2ky2k+1z2k-1=(-1)k(bc-1)ke/(A+(-1)0(k0)bkck+⋯+(-1)k(kk)b0c0)(b/(bc-1)k+1)⋅(-1)k(bc-1)ke/(A+(-1)0(k0)bkck+⋯+(-1)k(kk)b0c0)-1=(-1)k+1×(bc-1)k+1e(-1)k+1(k+11)b1c0e+⋯+(-1)1(k+1k+1)bk+1cke+(-1)0(k+10)bk+1ck+1+⋯+(-1)k+1(k+1k+1)b1c1,
where𝒜denotes(-1)k(k1)b1c0e+⋯+(-1)1(kk)bkck-1e.Corollary 1.2.
Leta,b,c,d be only positive real numbers or negative real numbers and let e,f be arbitrary nonnegative real numbers, and let {xn,yn,zn} be a solution of the system (1.15). If ad≠1,bc≠1,0<ad<1 and 0<bc<1  then one has
(1.23)lim⁡n→∞x2n-1=lim⁡n→∞y2n-1=∞,lim⁡n→∞x2n=lim⁡n→∞y2n=0.Proof.
Fromad≠1,bc≠1,0<ad<1 and 0<bc<1 we have -1<ad-1<0 and -1<bc-1<0. Hence, we have
(1.24)x2n-1=d(ad-1)n,y2n-1=b(bc-1)n,x2n=c(bc-1)n,y2n=a(ad-1)n.
Then,(1.25)lim⁡n→∞x2n-1=lim⁡n→∞d(ad-1)n={+∞,d>0,n-even+∞,d<0,n-odd-∞,d>0,n-odd-∞,d<0,n-even,lim⁡n→∞y2n-1=lim⁡n→∞b(bc-1)n={+∞,b>0,n-even+∞,b<0,n-odd-∞,b>0,n-odd-∞,b<0,n-even.
Also,(1.26)lim⁡n→∞x2n=lim⁡n→∞c(bc-1)n=0,lim⁡n→∞y2n=lim⁡n→∞a(ad-1)n=0.Example 1.3.
Ify(0)=-1/2,y(-1)=-1/2,x(0)=-1/2,x(-1)=-1/2, then the solutions of (1.15) can be represented by Table 1.Table 1i12345678x2i-10.6667−0.88891.1852−1.58032.1070−2.80933.7458−4.9944x2i0.3750−0.28130.2109−0.15820.1187−0.08900.0667−0.0501y2i-10.6667−0.88891.1852−1.58032.1070−2.80933.7458−4.9944y2i0.3750−0.28130.2109−0.15820.1187−0.08900.0667−0.0501z2i-10.6667−0.88891.1852−1.58032.1070−2.80933.7458−4.9944z2i0.3750−0.28130.2109−0.15820.1187−0.08900.0667−0.0501Corollary 1.4.
Leta,b,c,d be only positive real numbers or negative real numbers and let e,f be arbitrary nonnegative real numbers, and let {xn,yn,zn} be a solution of the system (1.15) If ad≠1,bc≠1,1<ad<2 and 1<bc<2  then one has
(1.27)lim⁡n→∞x2n-1=lim⁡n→∞y2n-1=∞,lim⁡n→∞x2n=lim⁡n→∞y2n=0.Proof.
Fromad≠1,bc≠1,1<ad<2 and 1<bc<2 we have 0<ad-1<1 and 0<bc-1<1. Hence, we have
(1.28)x2n-1=d(ad-1)n,y2n-1=b(bc-1)n,x2n=c(bc-1)n,y2n=a(ad-1)n.
Then,(1.29)lim⁡n→∞x2n-1=lim⁡n→∞d(ad-1)n={+∞,d>0-∞,d<0,lim⁡n→∞y2n-1=lim⁡n→∞b(bc-1)n={+∞,b>0-∞,b<0.
Also,(1.30)lim⁡n→∞x2n=lim⁡n→∞c(bc-1)n=0,lim⁡n→∞y2n=lim⁡n→∞a(ad-1)n=0.Example 1.5.
Ify(0)=3/2,y(-1)=3/4,x(0)=3/2,x(-1)=3/4,z(0)=3/2,z(-1)=3/4 then the solutions of (1.15) can be represented by Table 2.Table 2i12345678x2i-16483843072245761.967·1051.573·1061.258·107x2i0.1880.0230.0030.00040.000040.000017.153·10-78.941·10-8y2i-16483843072245761.967·1051.573·1061.258·107y2i0.1880.0230.0030.00040.000040.000017.153·10-78.941·10-8z2i-16483843072245761.967·1051.573·1061.258·107z2i0.1880.0230.0030.00040.000040.000017.153·10-78.941·10-8Corollary 1.6.
Leta,b,c,d be only positive real numbers or negative real numbers and let e,f be arbitrary nonnegative real numbers, and let {xn,yn,zn} be a solution of the system (1.15) If -∞<ad<0  and -∞<bc<0 then one has
(1.31)lim⁡n→∞x2n-1=lim⁡n→∞y2n-1=0,lim⁡n→∞x2n=lim⁡n→∞y2n=∞.Proof.
From-∞<ad<0 and -∞<bc<0 we have -∞<ad-1<-1 and -∞<bc-1<-1. Hence, we have
(1.32)x2n-1=d(ad-1)n,y2n-1=b(bc-1)n,x2n=c(bc-1)n,y2n=a(ad-1)n.
Then,(1.33)lim⁡n→∞x2n-1=lim⁡n→∞d(ad-1)n=0,lim⁡n→∞y2n-1=lim⁡n→∞b(bc-1)n=0.
Also,(1.34)lim⁡n→∞x2n=lim⁡n→∞c(bc-1)n={+∞,c>0,n-even+∞,c<0,n-odd-∞,c>0,n-odd-∞,c<0,n-even,lim⁡n→∞y2n=lim⁡n→∞a(ad-1)n={+∞,a>0,n-even+∞,a<0,n-odd-∞,a>0,n-odd-∞,a<0,n-even.Example 1.7.
Ify(0)=-3/2,y(-1)=-3/4,x(0)=3/2,x(-1)=3/4,z(0)=3/2,z(-1)=3/4 then the solutions of (1.15) can be represented by Table 3.Table 3i12345678x2i-1−0.35290.1661−0.07820.0368−0.01730.0081−0.00380.0018x2i−3.18756.7734−14.393630.5863−64.9959138.1163−293.4971623.6813y2i-10.3529−0.16610.0782−0.03680.0173−0.00810.0038−0.0018y2i3.1875−6.773414.3936−30.586364.9959−138.1163293.4971−623.6813z2i-1−0.35290.1661−0.07820.0368−0.01730.0081−0.00380.0018z2i−3.18756.7734−14.393630.5863−64.9959138.1163−293.4971623.6813Corollary 1.8.
Leta,b,c,d be only positive real numbers or negative real numbers and let e,f be arbitrary nonnegative real numbers, and let {xn,yn,zn} be a solution of the system (1.15). If2<ad<+∞  and 2<bc<+∞  then one has
(1.35)lim⁡n→∞x2n-1=lim⁡n→∞y2n-1=0,lim⁡n→∞x2n=lim⁡n→∞y2n=∞.Proof.
From2<ad<+∞ and 2<bc<+∞ we have 1<ad-1<+∞ and 1<bc-1<+∞. Hence, we have
(1.36)x2n-1=d(ad-1)n,y2n-1=b(bc-1)n,x2n=c(bc-1)n,y2n=a(ad-1)n.
Then,(1.37)lim⁡n→∞x2n-1=lim⁡n→∞d(ad-1)n=0,lim⁡n→∞y2n-1=lim⁡n→∞b(bc-1)n=0.
Also,(1.38)lim⁡n→∞x2n=lim⁡n→∞c(bc-1)n={+∞,c>0-∞,c<0,lim⁡n→∞y2n=lim⁡n→∞a(ad-1)n={+∞,a>0-∞,a<0.Example 1.9.
Ify(0)=5/2,y(-1)=5/2,x(0)=3/2,x(-1)=3/2,z(0)=3/2,z(-1)=3/2 then the solutions of (1.15) can be represented by Table 4.Table 4i12345678x2i-10.5460.1980.0720.0260.0100.0040.0010.0005x2i4.12511.34431.19585.787235.915648.7651784.1044906.285y2i-10.9090.3310.1200.0440.0160.0060.0020.001y2i6.87518.90651.992142.979393.1911081.2752973.5068177.142z2i-10.5460.1980.0720.0260.0100.0040.0010.0005z2i4.12511.34431.19585.787235.915648.7651784.1044906.285



---

*Source: 932362-2011-03-28.xml*