DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation32529610.1155/2009/325296325296Research ArticleOn the Solutions of the System of Difference Equations xn+1=max{A/xn,yn/xn}, yn+1=max{A/yn,xn/yn}SimsekDağistan1DemirBilal2CinarCengiz 2ZhangGuang1Department of Industrial EngineeringEngineering-Architecture FacultySelcuk UniversityKampüs 42075 KonyaTurkeyselcuk.edu.tr2Mathematics DepartmentEducation FacultySelcuk UniversityMeram Yeni Yol42090 KonyaTurkeyselcuk.edu.tr20092852009200917012009130420092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the behavior of the solutions of the following system of difference equations xn+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.

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 .

In this paper we study the behavior of the solutions of the following system of difference equations: xn+1=max{Axn,ynxn},yn+1=max{Ayn,xnyn}, where the constant A and the initial conditions are positive real numbers.

2. Main ResultDefinition 2.1.

Fibonacci sequence is f1=1, f2=1 and for n3,fn=fn-1+fn-2.

Definition 2.2.

The symbol [[]] symbolizes the greatest integer function.

Definition 2.3.

The sequence of a(n)n(mod2).

Definition 2.4.

The sequence of 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.

If n1, then xn=(Afk(n)-1-a(n)x0fk(n)y0fk(n))(-1)n,y1=x0/y0, and if n2yn=(y0fk(n-1)+1Afk(n-1)+a(n)-1x0fk(n-1)+1)(-1)n.

Proof.

Let y0>A, then 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,n1 then xn=(Afk(n)-1-a(n)x0fk(n)y0fk(n))(-1)n,y1=x0/y0 then n2, 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 n2, xn=(Afk(n-1)+a(n)-1x0fk(n-1)+1y0fk(n-1)+1)(-1)n.

If n1, then yn=(y0fk(n)Afk(n)-1-a(n)x0fk(n))(-1)n.

Proof.

Similarly we can obtain the proof as the proof of Theorem 2.5.

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. (a)limnx2n=0,limny2n=.(b)limnx2n+1=,limny2n+1=0.

Proof.

(a) We obtain that limnx2n=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,limny2n=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. (a)limnx2n=,limny2n=0.(b)limnx2n+1=0,limny2n+1=.

Proof.

Similarly we can obtain the proof as the proof of Theorem 2.7.

Theorem 2.9.

Let (xn,yn) be the solution of the system of difference equations (1.1) for 1<x0<y0<A.

If n1, then xn=(x0Aa(n))(-1)n.

If n1, then yn=(y0Aa(n))(-1)n.

Proof.

Let 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,n1, then 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.   (a)limnx2n=x0,limny2n=y0.(b)limnx2n+1=Ax0,limny2n+1=Ay0.

Proof.

(a) We obtain that limnx2n=limn(x0Aa(2n))(-1)2n=limn(x0A0)(-1)2n=limn(x0A0)=x0,limny2n=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 i0N such that every nN for n>i0, y0/x0n>A.

Proof.

We consider that x0<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.

1ni0xn=Aa(n)(x0)(-1)n,

1ni0yn=Aa(n)(y0x0[[n/2]])(-1)n, and when n>i0, the solutions will be different for every different constant A.

Proof.

Let y0<A, then 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=Ax02y0,1ni0, 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 i0N such that every nN for n>i0, x0/y0n>A.

Proof.

Similarly we can obtain the proof as the proof of Lemma 2.11.

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.

1ni0xn=Aa(n)(x0y0[[n/2]])(-1)n,

1ni0yn=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 Theorem 2.12.

Lemma 2.15.

Let (x0,y0) be the initial condition of (1.1) for 0<y0<x0<1; there is at least an i0N such that every nN for n>i0, (x0/y0)n>A.

Proof.

We consider that y0<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<ni0,   xn=Aa(n)(x0y0)[[n/2]](-1)n,

y1=A/y0, and if 1<ni0, yn=Aa(n)(y0x0)(-1)n, and when n>i0, the solutions will be different for every different constant A.

Proof.

Let y0<A, then 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 xn=Aa(n)(x0y0)[[n/2]](-1)n,n1 then 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 i0N such that every nN for n>i0, (y0/x0)n>A.

Proof.

Similarly we can obtain the proof as the proof of Lemma 2.15.

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<ni0, xn=Aa(n)(x0y0)(-1)n,

y1=A/y0, and if 1<ni0, 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 Theorem 2.16, which completes the proofs of theorems.

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