# On a System of Difference Equations

Authors: Ozan Özkan; Abdullah Selçuk Kurbanli
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We have investigated the periodical solutions of the system of rational difference equationsxn+1=yn-2/(-1±yn-2xn-1yn),yn+1=xn-2/(-1±xn-2yn-1xn), and zn+1=(xn-2+yn-2)/(-1±xn-2yn-1xn), where y0,y-1,y-2,x0,x-1,x-2,z0,z-1,z-2∈ℝ.


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## Body

## 1. Introduction

Recently, a great interest has arisen on studying difference equation systems. One of the reasons for that is the necessity for some techniques which can be used in investigating equations which originate in mathematical models to describe real-life situations such as population biology, economics, probability theory, genetics, and psychology. There are many papers related to the difference equations system.In [1], Kurbanli et al. studied the periodicity of solutions of the system of rational difference equations
(1)xn+1=xn-1+ynynxn-1-1,yn+1=yn-1+xnxnyn-1-1.In [2], Çinar studied the solutions of the systems of difference equations
(2)xn+1=1yn,yn+1=ynxn-1yn-1.In [3, 4], Özban studied the positive solutions of the system of rational difference equations
(3)xn=ayn-3,yn=byn-3xn-qyn-q,xn+1=1yn-k,yn+1=ynxn-myn-m-k.In [5–16], Elsayed studied a variety of systems of rational difference equations; for more, see references.In this paper, we have investigated the periodical solutions of the system of difference equations(4)xn+1=yn-2-1±yn-2xn-1yn,yn+1=xn-2-1±xn-2yn-1xn,zn+1=xn-2+yn-2-1±xn-2yn-1xn,n∈ℕ0,
where the initial conditions are arbitrary real numbers.


## 2. Main Results

Theorem 1.
Lety0=a, y-1=b, y-2=c, x0=d, x-1=e, x-2=f, z0=k, z-1=p, and z-2=q  be arbitrary real numbers, and let {xn,yn,zn} be a solution of the system
(5)xn+1=yn-2-1+yn-2xn-1yn,yn+1=xn-2-1+xn-2yn-1xn,zn+1=xn-2+yn-2-1+xn-2yn-1xn,n∈ℕ0.
Also, assume that b≠0, e≠0, fbd≠1, and cea≠1. Then, all six-period solutions of (5) are as follows:
(6)x6n+1=c1-cea,y6n+1=ffbd-1,z6n+1=-f+cfbd-1,x6n+1=b(fbd-1),y6n+2=e(cea-1),z6n+2=-(e+b)(cea+1),x6n+1=a1-cea,y6n+3=dfbd-1,z6n+3=-d+afbd+1,x6n+1=f,y6n+4=c,z6n+4=c(fbd+1)+f(cea+1)fbd+1,x6n+1=e,y6n+5=b,z6n+5=b(fbd+1)+e(cea+1)fbd+1,x6n+1=d,y6n+6=a,z6n+6=a(fbd+1)+d(cea+1)fbd+1,z6n+2=-(e+b)(cea+1),n∈ℕ0.Proof.
Forn=0,1,2,3,4,5, we have
(7)x1=y-2-1+y-2x-1y0=c-1+cea,y1=x-2-1+x-2y-1x0=f-1+fbd,z1=x-2+y-2-1+y-2x-1y0=f+c-1+fbd,x2=y-1-1+y-1x0y1=b-1+bd(f/(fbd-1))=b1/(fbd-1)=b(fbd-1),y2=x-1-1+x-1y0x1=e-1+ea(c/(cea-1))=e1/(cea-1)=e(cea-1),z2=x-1+y-1-1+x-1y0x1=e+b-1+ea(c/(cea-1))=e+b1/(cea-1)=(e+b)(cea-1),x3=y0-1+y0x1y2=a-1+a(c/(cea-1))e(cea-1)=acea-1,y3=x0-1+x0y1x2=d-1+d(f/(fbd-1))b(fbd-1)=dfbd-1,z3=x0+y0-1+x0y1x2=d+a-1+d(f/(fbd-1))b(fbd-1)=d+afbd-1,x4=y1-1+y1x2y3=f/(fbd-1)-1+(f/(fbd-1))b(fbd-1)(d/(fbd-1))=f/(fbd-1)-1+(fbd/(fbd-1))=f/(fbd-1)1/(fbd-1)=f,y4=x1-1+x1y2x3=c/(cea-1)-1+(c/(cea-1))e(cea-1)(a/(cea-1))=c/(cea-1)-1+(cea/(cea-1))=c/(cea-1)1/(cea-1)=c,z4=x1+y1-1+x1y2x3=(c/(cea-1))+(f/(fbd-1))-1+(c/(cea-1))e(cea-1)(a/(cea-1))=c(fbd-1)+f(cea-1)fbd-1,x5=y2-1+y2x3y4=e(cea-1)-1+e(cea-1)(a/(cea-1))c=e,y5=x2-1+x2y3x4=b(fbd-1)-1+b(fbd-1)(d/(fbd-1))f=b,z5=x2+y2-1+x2y3x4=b(fbd-1)+e(cea-1)-1+b(fbd-1)(d/(fbd-1))f=b(fbd-1)+e(cea-1)fbd-1,x6=y3-1+y3x4y5=d/(fbd-1)-1+(d/(fbd-1))fb=d,y6=x3-1+x3y4x5=a/(cea-1)-1+(a/(cea-1))ce=a,z6=x3+y3-1+x3y4x5=(a/(cea-1))+(d/(fbd-1))-1+(a/(cea-1))ce=a(fbd-1)+d(cea-1)fbd-1.
For n=6,7,8,9,10,11, assume that
(8)x7=y4-1+y4x5y6=c-1+cea=x1,y7=x4-1+x4y5x6=f-1+fbd=y1,z7=x4+y4-1+x4y5x6=f+c-1+fbd=z1,x8=y5-1+y5x6y7=b-1+bd(f/(-1+fbd))=b(fbd-1)=x2,y8=x5-1+x5y6x7=e-1+ea(c/(-1+cea))=e(cea-1)=y2,z8=x5+y5-1+x5y6x7=e+b-1+ea(c/(-1+cea))=(e+b)(cea-1)=z2,x9=y6-1+y6x7y8=a-1+a(c/(-1+cea))e(cea-1)=acea-1=x3,y9=x6-1+x6y7x8=d-1+d(f/(-1+fbd))b(fbd-1)=dfbd-1=y3,z9=x6+y6-1+x6y7x8=d+a-1+d(f/(-1+fbd))b(fbd-1)=d+afbd-1=z3,x10=y7-1+y7x8y9=f/(-1+fbd)-1+(f/(-1+fbd))b(fbd-1)(d/(fbd-1))=f/(-1+fbd)-1+(fbd/(-1+fbd))=f=x4,y10=x7-1+x7y8x9=c/(-1+cea)-1+(c/(-1+cea))e(cea-1)(a/(cea-1))=c/(-1+cea)-1+(cea/(-1+cea))=c=y4,z10=x7+y7-1+x7y8x9=(c/(-1+cea))+(f/(-1+fbd))-1+(c/(-1+cea))e(cea-1)(a/(cea-1))=c(fbd-1)+f(cea-1)fbd-1=z4,x11=y8-1+y8x9y10=e(cea-1)-1+e(cea-1)(a/(cea-1))c=e=x5,y11=x8-1+x8y9x10=b(fbd-1)-1+b(fbd-1)(d/(fbd-1))f=b=y5,z11=x8+y8-1+x8y9x10=b(fbd-1)+e(cea-1)-1+b(fbd-1)(d/(fbd-1))f=b(fbd-1)+e(cea-1)fbd-1=z5,x12=y9-1+y9x10y11=d/(fbd-1)-1+(d/(fbd-1))fb=d=x6,y12=x9-1+x9y10x11=a/(cea-1)-1+(a/(cea-1))ce=a=y6,z12=x9+y9-1+x9y10x11=(a/(cea-1))+(d/(fbd-1))-1+(a/(cea-1))ce=a(fbd-1)+d(cea-1)fbd-1=z6
are true. Also, we have
(9)x1=ccea-1=x7=x13=⋯=x6n+1,n∈ℕ0,x2=b(fbd-1)=x8=x14=⋯=x6n+2,n∈ℕ0,x3=acea-1=x9=x15=⋯=x6n+3,n∈ℕ0,x4=f=x10=x16=⋯=x6n+4,n∈ℕ0,x5=e=x11=x17=⋯=x6n+5,n∈ℕ0,x6=d=x12=x18=⋯=x6n+6,n∈ℕ0,y1=ffbd-1=y7=y13=⋯=y6n+1,n∈ℕ0,y2=e(cea-1)=y8=y14=⋯=y6n+2,n∈ℕ0,y3=dfbd-1=y9=y15=⋯=y6n+3,n∈ℕ0,y4=c=y10=y16=⋯=y6n+4,n∈ℕ0,y5=b=y11=y17=⋯=y6n+5,n∈ℕ0,y6=a=y12=y18=⋯=y6n+6,n∈ℕ0,z1=f+cfbd-1=z7=z13=⋯=z6n+1,n∈ℕ0,z2=(e+b)(cea-1)=z8=z14=⋯=z6n+2,n∈ℕ0,z3=d+afbd-1=z9=z15=⋯=z6n+3,n∈ℕ0,z4=c(fbd-1)+f(cea-1)fbd-1=z10=z16=⋯=z6n+4,n∈ℕ0,z5=b(fbd-1)+e(cea-1)fbd-1=z11=z17=⋯=z6n+5,n∈ℕ0,z6=a(fbd-1)+d(cea-1)fbd-1=z12=z18=⋯=z6n+6,n∈ℕ0.Theorem 2.
Lety0=a, y-1=b, y-2=c, x0=d, x-1=e, x-2=f, z0=k, z-1=p, and z-2=q  be arbitrary real numbers, and let {xn,yn,zn} be a solution of the system
(10)xn+1=yn-2-1-yn-2xn-1yn,yn+1=xn-2-1-xn-2yn-1xn,zn+1=xn-2+yn-2-1-xn-2yn-1xn,n∈ℕ0.
Also, assume that b≠0, e≠0, fbd≠-1, and cea≠-1. Then, all six-period solutions of (10) are as follows:
(11)x6n+1=-c1+cea,y6n+1=-ffbd+1,z6n+1=--f(1+fbd)+cfbd+1,x6n+1=-b(fbd+1),y6n+2=-e(cea+1),z6n+2=e-b(1+cea),x6n+1=-a1+cea,y6n+3=-dfbd+1,z6n+3=--d(fbd+1)+afbd+1,x6n+1=f,y6n+4=c,z6n+4=-c(fbd+1)+f(1+cea+c2e2a2)(fbd+1)(cea+1),x6n+1=e,y6n+5=b,z6n+5=-b(1+2fbd+f2b2d2)+e(1+cea)fbd+1,x6n+1=d,y6n+6=a,z6n+6=-a(fbd+1)+d(1+cea+c2e2a2)(1+cea)(fbd+1),n∈ℕ0.Proof.
Forn=0,1,2,3,4,5, we have
(12)x1=y-2-1-y-2x-1y0=c-1-cea=-c1+cea,y1=x-2-1-x-2y-1x0=f-1-fbd=-f1+fbd,z1=x-2+y-2-1-y-2x-1y0=f+c-1-fbd=-f+c1+fbd,x2=y-1-1-y-1x0y1=b-1-bd(-(f/(fbd+1)))=b-(1/(fbd+1))=-b(1+fbd),y2=x-1-1-x-1y0x1=e-1-ea(-(c/(1+cea)))=e-(1/(1+cea))=-e(1+cea),z2=x-1+y-1-1-x-1y0x1=e+b-1-ea(-(c/(1+cea)))=e+b-(1/(1+cea))=-(e+b)(1+cea),x3=y0-1-y0x1y2=a-1-a(-(c/(1+cea)))(-e(1+cea))=-a1+cea,y3=x0-1-x0y1x2=d-1-d(-(f/(1+fbd)))(-b(1+fbd))=-d1+fbd,z3=x0+y0-1-x0y1x2=d+a-1-d(-(f/(1+fbd)))(-b(1+fbd))=-d+a1+fbd,x4=y1-1-y1x2y3=-(f/(1+fbd))-1+(f/(1+fbd))b(1+fbd)(d/(1+fbd))=-(f/(1+fbd))-1+(fbd/(1+fbd))=-(f/(1+fbd))-(1/(1+fbd))=f,y4=x1-1-x1y2x3=-(c/(1+cea))-1+(c/(1+cea))e(1+cea)(a/(1+cea))=-(c/(1+cea))-1+(cea/(1+cea))=-(c/(1+cea))-(1/(1+cea))=c,z4=x1+y1-1-x1y2x3=-(c/(1+cea))-(f/(1+fbd))-1+(c/(1+cea))e(1+cea)(a/(1+cea))=c(1+fbd)+f(1+cea)1+fbd,x5=y2-1-y2x3y4=-e(1+cea)-1-e(1+cea)(a/(1+cea))c=e,y5=x2-1-x2y3x4=-b(1+fbd)-1-b(1+fbd)(d/(1+fbd))f=b,z5=x2+y2-1-x2y3x4=-b(1+fbd)-e(1+cea)-1-b(1+fbd)(d/(1+fbd))f=b(1+fbd)+e(1+cea)1+fbd,x6=y3-1-y3x4y5=-(d/(1+fbd))-1+(d/(1+fbd))fb=d,y6=x3-1-x3y4x5=-(a/(1+cea))-1+(a/(1+cea))ce=a,z6=x3+y3-1-x3y4x5=-(a/(1+cea))-(d/(1+fbd))-1+(a/(1+cea))ce=a(1+fbd)+d(1+cea)1+fbd.
For n=6,7,8,9,10,11, assume that
(13)x7=y4-1-y4x5y6=c-1-cea=-c1+cea=x1,y7=x4-1-x4y5x6=f-1-fbd=-f1+fbd=y1,z7=x4+y4-1-x4y5x6=f+c-1-fbd=-f+c1+fbd=z1,x8=y5-1-y5x6y7=b-1+bd(f/(1+fbd))=-b(1+fbd)=x2,y8=x5-1-x5y6x7=e-1+ea(c/(1+cea))=-e(1+cea)=y2,z8=x5+y5-1-x5y6x7=e+b-1+ea(c/(1+cea))=-(e+b)(1+cea)=z2,x9=y6-1-y6x7y8=a-1-a(c/(1+cea))e(1+cea)=-a1+cea=x3,y9=x6-1-x6y7x8=d-1-d(f/(1+fbd))b(1+fbd)=-d1+fbd=y3,z9=x6+y6-1-x6y7x8=d+a-1-d(f/(1+fbd))b(1+fbd)=-d+a1+fbd=z3,x10=y7-1-y7x8y9=-(f/(1+fbd))-1+(f/(1+fbd))b(1+fbd)(d/(1+fbd))=-(f/(1+fbd))-1/(1+fbd)=f=x4,y10=x7-1-x7y8x9=-(c/(1+cea))-1+(c/(1+cea))e(1+cea)(a/(1+cea))=-(c/(1+cea))-1/(1+cea)=c=y4,z10=x7+y7-1-x7y8x9=-(c/(1+cea))-(f/(1+fbd))-1+(c/(1+cea))e(1+cea)(a/(1+cea))=c(1+fbd)+f(1+cea)1+fbd=z4,x11=y8-1-y8x9y10=-e(1+cea)-1-e(1+cea)(a/(1+cea))c=e=x5,y11=x8-1-x8y9x10=-b(1+fbd)-1-b(1+fbd)(d/(1+fbd))f=b=y5,z11=x8+y8-1-x8y9x10=-b(1+fbd)-e(1+cea)-1-b(1+fbd)(d/(1+fbd))f=b(1+fbd)+e(1+cea)1+fbd=z5,x12=y9-1-y9x10y11=-(d/(1+fbd))-1+(d/(1+fbd))fb=d=x6,y12=x9-1-x9y10x11=-(a/(1+cea))-1+(a/(1+cea))ce=a=y6,z12=x9+y9-1-x9y10x11=-(a/(1+cea))-(d/(1+fbd))-1+(a/(1+cea))ce=a(1+fbd)+d(1+cea)1+fbd=z6
are true. Also, we have
(14)x1=-c1+cea=x7=x13=⋯=x6n+1,n=0,1,2,3,…,x2=-b(1+fbd)=x8=x14=⋯=x6n+2,n=0,1,2,3,…,x3=-a1+cea=x9=x15=⋯=x6n+3,n=0,1,2,3,…,x4=f=x10=x16=⋯=x6n+4,n=0,1,2,3,…,x5=e=x11=x17=⋯=x6n+5,n=0,1,2,3,…,x6=d=x12=x18=⋯=x6n+6,n=0,1,2,3,…,y1=-f1+fbd=y7=y13=⋯=y6n+1,n∈ℕ0,y2=-e(1+cea)=y8=y14=⋯=y6n+2,n∈ℕ0,y3=-d1+fbd=y9=y15=⋯=y6n+3,n∈ℕ0,y4=c=y10=y16=⋯=y6n+4,n∈ℕ0,y5=b=y11=y17=⋯=y6n+5,n∈ℕ0,y6=a=y12=y18=⋯=y6n+6,n∈ℕ0,z1=-f+c1+fbd=z7=z13=⋯=z6n+1,n∈ℕ0,z3=-d+a1+fbd=z9=z15=⋯=z6n+3,n∈ℕ0,z4=c(1+fbd)+f(1+cea)1+fbd=z10=z16=⋯=z6n+4,n∈ℕ0,z5=b(1+fbd)+e(1+cea)1+fbd=z11=z17=⋯=z6n+5,n∈ℕ0,z6=a(1+fbd)+d(1+cea)1+fbd=z12=z18=⋯=z6n+6,n∈ℕ0.The following corollary follows from Theorem1.Corollary 3.
The following conclusions are valid forn∈ℕ:(i)
x6n+2y6n+3=x6n+6y6n+5,(ii)
x6n+1y6n+2=x6n+5y6n+6,(iii)
x6n+1y6n+6=x6n+3y6n+4,(iv)
x6n+4y6n+3=x6n+6y6n+1.The following corollary follows from Theorem2.Corollary 4.
The following conclusions are valid forn∈ℕ:(i)
x6n+1y6n+6=x6n+3y6n+4,(ii)
x6n+6y6n+1=x6n+4y6n+3,(iii)
x6n+3y6n+2=x6n+5y6n+6,(iv)
x6n+1y6n+2=x6n+5y6n+4.



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*Source: 970316-2013-03-31.xml*