# On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn}, yn+1=max{A/yn,xn/yn}

Authors: Dağistan Simsek; Bilal Demir; Cengiz  Cinar
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We study the behavior of the solutions of the following system
of difference equationsxn+1=max{A/xn,yn/xn}, yn+1=max{A/yn,xn/yn} where the constant A and the initial conditions are positive real numbers.


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## Body

## 1. Introduction

Recently, there has been a great interest in studying the periodic nature of nonlinear difference equations. Although difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the periodic behavior of their solutions. The periodic nature of nonlinear difference equations of the max type has been investigated by many authors. See, for example [1–24].In this paper we study the behavior of the solutions of the following system of difference equations:(1.1)xn+1=max{Axn,ynxn},yn+1=max{Ayn,xnyn},
where the constant A and the initial conditions are positive real numbers.


## 2. Main Result

Definition 2.1.
Fibonacci sequence isf1=1, f2=1 and for n≥3,fn=fn-1+fn-2.Definition 2.2.
The symbol[[]] symbolizes the greatest integer function.Definition 2.3.
The sequence ofa(n)≡n(mod2).Definition 2.4.
The sequence of(2.1)k(n)={n,n=0,2,4,…,n+1,n=1,3,5,….Theorem 2.5.
Let(xn,yn) be the solution of the system of difference equations (1.1) for A<x0<y0 and y0/x0>A.
Ifn≥1, then
(2.2)xn=(Afk(n)-1-a(n)x0fk(n)y0fk(n))(-1)n,y1=x0/y0, and if n≥2(2.3)yn=(y0fk(n-1)+1Afk(n-1)+a(n)-1x0fk(n-1)+1)(-1)n.Proof.
Lety0>A, then
(2)x1=max{Ax0,y0x0}=y0x0>A,y1=max{Ay0,x0y0}=x0y0<A,x2=max{Ax1,y1x1}=Ax0y0<A,y2=max{Ay1,x1y1}=x1y1=y02x02>A,x3=max{Ax2,y2x2}=y2x2=y03Ax03>A,y3=max{Ay2,x2y2}=Ay2=Ax02y02<A,x4=max{Ax3,y3x3}=A2x03y03<A,y4=max{Ay3,x3y3}=x3y3=y05A2x05>A,x5=max{Ax4,y4x4}=y4x4=y08A3x08>A,y5=max{Ay4,x4y4}=Ay4=A3x05y05<A,⋮n≥1 then
(2.5)xn=(Afk(n)-1-a(n)x0fk(n)y0fk(n))(-1)n,y1=x0/y0 then n≥2,
(2.6)yn=(y0fk(n-1)+1Afk(n-1)+a(n)-1x0fk(n-1)+1)(-1)n.Theorem 2.6.
Let(xn,yn) be the solution of the system of difference equations (1.1) for A<y0<x0.
x1=x0/y0 and if n≥2,
(2.7)xn=(Afk(n-1)+a(n)-1x0fk(n-1)+1y0fk(n-1)+1)(-1)n.
Ifn≥1, then
(2.8)yn=(y0fk(n)Afk(n)-1-a(n)x0fk(n))(-1)n.Proof.
Similarly we can obtain the proof as the proof of Theorem2.5.Theorem 2.7.
Let(xn,yn) be the solution of the system of difference equations (1.1) for A<x0<y0 and (y0/x0)>A. 							(2.9)(a)limn→∞x2n=0,limn→∞y2n=∞.(b)limn→∞x2n+1=∞,limn→∞y2n+1=0.Proof.
(a) We obtain that(2.10)limn→∞x2n=limn→∞(Afk(2n)-1-a(2n)x0fk(2n)y0fk(2n))(-1)2n=limn→∞(Afk(2n)-1-a(2n)x0fk(2n)y0fk(2n))(-1)2k=limn→∞(Af2n-1-a(2n)x0f2ny0f2n)=0,limn→∞y2n=limn→∞(y0fk(2n-1)+1Afk(2n-1)+a(2n)-1x0fk(2n-1)+1)(-1)2n=limn→∞(y0fk(2n-1)+1Afk(2n-1)+a(2n)-1x0fk(2n-1)+1)(-1)2k=limk→∞(y0f2n+1Af2n-1x0f2n+1)=∞.
(b) Similarly we can obtain the proof of (b) as the proof of (a).Theorem 2.8.
Let(xn,yn) be the solution of the system of difference equations (1.1) for A<y0<x0 and (x0/y0)>A. 							(2.11)(a)limn→∞x2n=∞,limn→∞y2n=0.(b)limn→∞x2n+1=0,limn→∞y2n+1=∞.Proof.
Similarly we can obtain the proof as the proof of Theorem2.7.Theorem 2.9.
Let(xn,yn) be the solution of the system of difference equations (1.1) for 1<x0<y0<A.
Ifn≥1, then
(2.12)xn=(x0Aa(n))(-1)n.
Ifn≥1, then
(2.13)yn=(y0Aa(n))(-1)n.Proof.
Let(2)x1=max{Ax0,y0x0}=Ax0<A,y1=max{Ay0,x0y0}=Ay0<A,x2=max{Ax1,y1x1}=Ax0A=x0<A,y2=max{Ay1,x1y1}=Ay1=y0<A,x3=max{Ax2,y2x2}=Ax2=Ax0<A,y3=max{Ay2,x2y2}=Ay2=Ay0<A,x4=max{Ax3,y3x3}=Ax3=x0<A,y4=max{Ay3,x3y3}=Ay3=y0<A,x5=max{Ax4,y4x4}=Ax4=Ax0<A,y5=max{Ay4,x4y4}=Ay4=Ay0<A,⋮n≥1, then
(2.15)xn=(x0Aa(n))(-1)n,yn=(y0Aa(n))(-1)n.Theorem 2.10.
Let(xn,yn) be the solution of the system of difference equations (1.1) for 1<x0<y0<A. 							(2.16)(a)limn→∞x2n=x0,limn→∞y2n=y0.(b)limn→∞x2n+1=Ax0,limn→∞y2n+1=Ay0.Proof.
(a) We obtain that(2.17)limn→∞x2n=limn→∞(x0Aa(2n))(-1)2n=limn→∞(x0A0)(-1)2n=limn→∞(x0A0)=x0,limn→∞y2n=limn→∞(y0Aa(2n))(-1)2n=limn→∞(y0A0)(-1)2n=limn→∞(y0A0)=y0.
(b) Similarly we can obtain the proof of (b) as the proof of (a).Lemma 2.11.
Let(x0,y0) be the initial condition of (1.1) for 0<x0<1<y0<A; there is at least an i0∈N such that every n∈N for n>i0, y0/x0n>A.Proof.
We consider thatx0<1 hence limn→∞(y0/x0n)=∞ and that proofs the existing of i0 defined in hypothesis.Theorem 2.12.
Let(xn,yn) be the solution of the system of difference equations (1.1) for 0<x0<1<y0<A, and i0 is the number, defined by Lemma 2.11.
1≤n≤i0(2.18)xn=Aa(n)(x0)(-1)n,
1≤n≤i0(2.19)yn=Aa(n)(y0x0[[n/2]])(-1)n,
and when n>i0, the solutions will be different for every different constant A.Proof.
Lety0<A, then
(2.20)x1=max{Ax0,y0x0}=Ax0>1,y1=max{Ay0,x0y0}=Ay0>1,x2=max{Ax1,y1x1}=Ax1=x0<1,y2=max{Ay1,x1y1}=x1y1=y0x0>1,x3=max{Ax2,y2x2}=Ax2=Ax0>1,y3=max{Ay2,x2y2}=Ay2=Ax0y0,x4=max{Ax3,y3x3}=Ax3=x0<1,y4=max{Ay3,x3y3}=x3y3=y0x02,x5=max{Ax4,y4x4}=Ax4=Ax0>1,y5=max{Ay4,x4y4}=Ay4=Ax02y0,1≤n≤i0,
(2.21)xn=Aa(n)(x0)(-1)n,yn=Aa(n)(y0x0[[n/2]])(-1)n.Lemma 2.13.
Let(x0,y0) be the initial condition of (1.1) for 0<y0<1<x0<A; there is at least an i0∈N such that every n∈N for n>i0, x0/y0n>A.Proof.
Similarly we can obtain the proof as the proof of Lemma2.11.Theorem 2.14.
Let(xn,yn) be the solution of the system of difference equations (1.1) for 0<y0<1<x0<A, and i0 is the number, defined by Lemma 2.13.
1≤n≤i0(2.22)xn=Aa(n)(x0y0[[n/2]])(-1)n,
1≤n≤i0(2.23)yn=Aa(n)(y0)(-1)n,
and when n>i0, the solutions will be different for every different constant A.Proof.
Similarly we can obtain the proof of be as the proof of Theorem2.12.Lemma 2.15.
Let(x0,y0) be the initial condition of (1.1) for 0<y0<x0<1; there is at least an i0∈N such that every n∈N for n>i0, (x0/y0)n>A.Proof.
We consider thaty0<x0 hence limn→∞(x0/y0)n=∞ and that proofs the existing of i0 defined in hypothesis.Theorem 2.16.
Let(xn,yn) be the solution of the system of difference equations (1.1) for 0<y0<x0<1, A>1, and i0 is the number, defined by Lemma 2.15.
x1=A/x0 and if 1<n≤i0,
(2.24)xn=Aa(n)(x0y0)[[n/2]](-1)n,
y1=A/y0, and if 1<n≤i0,
(2.25)yn=Aa(n)(y0x0)(-1)n,
and when n>i0, the solutions will be different for every different constant A.Proof.
Lety0<A, then
(2.26)x1=max{Ax0,y0x0}=Ax0>1,y1=max{Ay0,x0y0}=Ay0>1,x2=max{Ax1,y1x1}=y1x1=x0y0>1,y2=max{Ay1,x1y1}=x1y1=y0x0<1,x3=max{Ax2,y2x2}=Ax2=Ay0x0,y3=max{Ay2,x2y2}=Ay2=Ax0y0>1,x4=max{Ax3,y3x3}=y3x3=x02y02>1,y4=max{Ay3,x3y3}=Ay3=y0x0<1,n>1 then
(2.27)xn=Aa(n)(x0y0)[[n/2]](-1)n,n≥1 then
(2.28)yn=Aa(n)(y0x0)(-1)n.Lemma 2.17.
Let(x0,y0) be the initial condition of (1.1) for 0<y0<x0<1; there is at least an i0∈N such that every n∈N for n>i0, (y0/x0)n>A.Proof.
Similarly we can obtain the proof as the proof of Lemma2.15.Theorem 2.18.
Let(xn,yn) be the solution of the system of difference equations (1.1) for 0<x0<y0<1, A>1, and i0 is the number, defined by Lemma 2.17.
x1=A/x0, and if 1<n≤i0,
(2.29)xn=Aa(n)(x0y0)(-1)n,
y1=A/y0, and if 1<n≤i0,
(2.30)yn=Aa(n)(y0x0)[[n/2]](-1)n,
and when n>i0, the solutions will be different for every different constant A.Proof.
Similarly we can obtain the proof as the proof of Theorem2.16, which completes the proofs of theorems.



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*Source: 325296-2009-07-01.xml*