# Eventually Periodic Solutions of a Max-Type System of Difference Equations of Higher Order

Authors: Guangwang Su; Taixiang Sun; Bin Qin
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We study in this paper the following max-type system of difference equations of higher order:xn=max{A,yn-k/xn-1} and yn=max{B,xn-k/yn-1}, n∈{0,1,2,…}, where A≥B>0, k≥1, and the initial conditions x-k,y-k,x-k+1,y-k+1,…,x-1,y-1∈(0,+∞). We show that (1) if AB>1, then every solution of the above system is periodic with period 2 eventually. (2) If AB=1>B, then every solution of the above system is periodic with period 2k or 2 eventually. (3) If A=B=1 or AB<1, then the above system has a solution which is not periodic eventually.


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## Body

## 1. Introduction

Our purpose in this paper is to study eventual periodicity of the following max-type system of difference equations of higher order:(1)xn=max⁡A,yn-kxn-1,yn=max⁡B,xn-kyn-1,n∈N0≡0,1,…with (2)A≥B>0,where k∈N≡{1,2,…} and the initial conditions x-k,y-k,x-k+1,y-k+1,…,x-1,y-1∈R+≡(0,+∞).Recently, there has been a great interest in studying max-type systems of difference equations. For example, Fotiades and Papaschinopoulos [1] studied the following max-type system of difference equations: (3)xn=max⁡A,yn-1xn-2,yn=max⁡B,xn-1yn-2,n∈N0,and they showed that every positive solution of (3) is eventually periodic.In 2012, Stević [2] obtained in an elegant way the general solution to the following max-type system of difference equations: (4)xn+1=max⁡Axn,ynxn,yn+1=max⁡Ayn,xnyn,n∈N0,for the case x0,y0≥A>0 and y0/x0≥max⁡{A,1/A}.In [3], we have recently studied the following max-type system of difference equations: (5)xn=max⁡1xn-m,min⁡1,Ayn-r,yn=max⁡1yn-m,min⁡1,Bxn-t,n∈N0,where A,B∈R+, m,r,t∈N, and the initial conditions x-d,y-d,x-d+1,y-d+1,…,x-1,y-1∈R+ with d=max⁡{m,r,t}, and we showed that every positive solution of (5) is eventually periodic with period 2m.Whenm=r=t=1 and A=B, (5) reduces to the max-type system of difference equations (6)xn=max⁡1xn-1,min1,Ayn-1,yn=max⁡1yn-1,min⁡1,Axn-1,n∈N0.Yazlik et al. [4] in 2015 obtained in an elegant way the general solution of (6).In 2014, Stević et al. [5] investigated the behavior of positive solutions of the following max-type system of difference equations: (7)xn=maxB,yn-1pzn-2p,yn=maxB,zn-1pxn-2p,zn=maxB,xn-1pyn-2p,n∈N0,where B,p∈R+. It is proved that system (7) is permanent when p∈(0,4).For some other results on solutions of many max-type difference equations and systems, such as eventual periodicity, the boundedness character, and attractivity, see, for example, [6–20] and the related references therein.


## 2. Main Results and Proofs

In this section, we study the eventual periodicity of solutions of system (1). Let {(xn,yn)}n≥-k be a solution of (1) with the initial conditions x-k,y-k,x-k+1,y-k+1,…,x-1,y-1∈R+. From (1), it immediately follows that, for any n∈N0, (8)xn≥A,(9)yn≥B.Lemma 1.
Ifxn=A eventually, then yn is a periodic sequence with period 2 eventually. If yn=B eventually, then xn is a periodic sequence with period 2 eventually.Proof.
Assume thatxn=A eventually. By (1), we see (10)yn=max⁡B,Ayn-1eventually,which implies (11)ynyn-1≥Aeventually.We have from (9), (10), and (11) that (12)B≤yn=max⁡B,Ayn-2yn-1yn-2≤max⁡B,yn-2=yn-2eventually.Then, for any i∈0.1, y2n+i is eventually nonincreasing.
We claim that, for everyi∈{0.1}, y2n+i is a constant sequence eventually. Assume on the contrary that, for some i∈{0.1}, y2n+i is not a constant sequence eventually. Then, by (10) and (12), we see that there exists a sequence of positive integers k1<k2<⋯ such that, for any n∈N, (13)B<y2kn+1+i=Ay2kn+1+i-1<y2kn+i=Ay2kn+i-1,which implies y2kn+1+i-1>y2kn+i-1 for any n∈N. This is a contradiction. Thus, yn is a periodic sequence with period 2 eventually. The other case is treated similarly/dually, so we omit the detail. The proof is complete.Lemma 2.
IfAB≥1, then x2(n+1)k+i≤x2nk+i for any n≥k+1 and i∈N0.Proof.
Assume thatAB≥1. Then, for 2(n+1)k+i≥k+1, by (1) and (8), we have (14)A≤x2n+1k+i=max⁡A,Bx2n+1k+i-1,x2nk+ix2n+1k+i-1y2n+1k+i-k-1and by (2), (8), and (9), we have (15)Bx2n+1k+i-1≤BA≤1≤A,x2n+1k+i-1y2n+1k+i-k-1≥AB≥1.Thus, (16)x2n+1k+i≤max⁡A,x2nk+i=x2nk+ieventually.The proof is complete.Theorem 3.
LetAB>1. Then, xn=A eventually and yn is a periodic sequence with period 2 eventually.Proof.
Assume thatAB>1. For any i∈{0.1,…,2k-1} and n∈N0, let limn→∞x2nk+i=Ai since x2nk+i is eventually nonincreasing. Thus, by using (15) and by letting n→∞ in (14), it is obtained that xn=A eventually. By Lemma 1, we see that yn is a periodic sequence with period 2 eventually. The proof is complete.Now, we assumeAB=1, and by Lemma 2, we can assume that, for any i∈N0, (17)limn→∞x2nk+i=Ai.Then, Ai≥A and A2k+i=Ai for any i∈N0.Lemma 4.
IfAB=1 and Ai>A for some i∈N0, then, for any s∈N, x2nk+s+i and y2nk-k+s+i are constant sequences eventually.Proof.
Assume thatAB=1 and Ai>A for some i∈N0. Since Ai>A, it follows from (14), (15), and Lemma 2 that (18)x2nk+i=max⁡A,x2n-1k+ix2nk-1+iy2nk-k-1+i=x2n-1k+ix2nk-1+iy2nk-k-1+ieventually.By (9), (17), and (18), we have (19)B≤limn→∞y2nk-k-1+i=limn→∞x2n-1k+ix2nk+ix2nk-1+i=1A-1+i+2k≤1A=B.This implies (20)limn→∞x2nk-1+i=A,limn→∞y2nk-k-1+i=B.Note that x2nk+i=max⁡{A,y2nk-k+i/x2nk-1+i}=y2nk-k+i/x2nk-1+i>A eventually; it follows that (21)limn→∞y2nk-k+i=limn→∞x2nk+ix2nk-1+i=AiA.Since (22)x2nk+1+i=max⁡A,x2n-1k+1+ix2nk+iy2nk-k+i,limn→∞⁡x2n-1k+1+ix2nk+iy2nk-k+i=A1+iAi2A<A1+i,we see that x2nk+1+i=A eventually. This with (21) implies (23)limn→∞⁡x2n-1k+1+iy2nk-k+i=AAAi<B.From (23), it follows that (24)y2nk-k+1+i=maxB,x2n-1k+1+iy2nk-k+i=Beventually.By x2(n-1)k+2+i/y2nk-k+1+i≥A/B≥1≥B eventually and (24), it follows that (25)y2nk-k+2+i=maxB,x2n-1k+2+iy2nk-k+1+i=x2n-1k+2+iBeventually,x2nk+2+i=max⁡A,y2nk-k+2+ix2nk+1+i=max⁡A,x2n-1k+2+i=x2n-1k+2+ieventually.
Ifx2nk+2+i>A eventually, then, in a similar fashion, we can obtain the following:(1)
x2nk+3+i=A eventually and y2nk-k+3+i=B eventually.(2)
x2nk+4+i and y2nk-k+4+i are constant sequences eventually.
Ifx2nk+2+i=A eventually, then y2nk-k+2+i=A/B eventually and (26)y2nk-k+3+i=max⁡B,x2n-1k+3+iy2nk-k+2+i=max⁡B,x2n-1k+3+iBA=x2n-1k+3+iBAeventually,(27)x2nk+3+i=maxA,y2nk-k+3+ix2nk+2+i=maxA,x2n-1k+3+iBA2eventually.From (27), we see that if A=B=1, then by x2(n-1)k+3+i≥A we have (28)x2nk+3+i=x2n-1k+3+ieventually,and if A>B, then (29)x2nk+3+i=Aeventuallysince (30)limn→∞x2n-1k+3+iBA2=A3+iBA2<A3+i.Using arguments similar to ones developed in the above given proof, we can show that, for any s∈N, x2nk+s+i and y2nk-k+s+i are constant sequences eventually. The proof is complete.Theorem 5.
AssumeAB=1>B. Then, one of the following statements holds:(i)
xn and yn are periodic sequences with period 2k eventually.(ii)
xn=A eventually and yn is a periodic sequence with period 2 eventually.Proof.
Assume thatA=1/B>B. If Ai>A for some i∈N0, then by Lemma 4 we see that, for any s∈N, x2nk+s+i and y2nk-k+s+i are constant sequences eventually, which implies that xn and yn are periodic sequences with period 2k eventually. In the following, we assume that Ai=A for any i∈N0.
We claim thatxn=A eventually. Indeed, if, for some i∈N0, x2nk+i>A, then by (1) we have (31)x2nk+i=max⁡A,y2nk-k+ix2nk-1+i=y2nk-k+ix2nk-1+ieventually,(32)x2nk+1+i=max⁡A,x2n-1k+1+ix2nk+iy2nk-k+i,(33)limn→∞⁡x2n-1k+1+ix2nk+iy2nk-k+i=limn→∞⁡x2n-1k+1+ix2nk+i2x2nk-1+i≤A1+iA3<A1+i.Then, we see that (34)x2nk+1+i=Aeventually.From (31), we have y2nk-k+i≥A2 eventually, and with (34) and B=1/A, it follows that (35)y2nk-k+1+i=max⁡B,x2n-1k+1+iy2nk-k+i=max⁡B,Ay2nk-k+i=Beventually,y2nk-k+2+i=maxB,x2n-1k+2+iy2nk-k+1+i=x2n-1k+2+iBeventually,x2nk+2+i=max⁡A,y2nk-k+2+ix2nk+1+i=max⁡A,x2n-1k+2+i=x2n-1k+2+ieventually.Thus, x2nk+2+i and y2nk-k+2+i are constant sequences eventually. Using arguments similar to ones developed in the proof of Lemma 4, we can show that, for any s∈N, x2nk+s+i and y2nk-k+s+i are constant sequences eventually. Thus, x2nk+i=Ai=A eventually; this is a contradiction.
By Lemma1, we see that xn=A eventually and yn is a periodic sequence with period 2 eventually. Theorem 5 is proven.Theorem 6.
IfA≤1 and B≤1, then (1) has a solution which is not periodic eventually.Proof.
We claim that the equation(36)zn=zn-kzn-1has a nonincreasing solution. Indeed, write (37)M=x1,…,xk:x1≥⋯≥xk≥x1>0,E=x1,…,xk:x1≥⋯≥xk≥x1xk-1>0.It is easy to see that M⊂E since, for any (x1,…,xk)∈M, we have x1≥⋯≥xk≥x1 and xk≥xk2/xk-1≥x1/xk-1. Define F:M→E by (38)Fx1,…,xk=u1,…,uk≡x2,…,xk,x1xkfor all (x1,…,xk)∈M.
We claim thatF is well defined. Indeed, from (38) and the definition of M, we have (39)ui=xi+1,fori∈1,…,k-1,uk=x1xk,u1≥⋯≥uk-1=xk≥x1x1≥x1xk=uk≥u1uk-1.Thus, (u1,…,uk)∈E.
We also claim thatF is a bijection from M to E. Indeed, let z=(z1,…,zk), y=(y1,…,yk)∈M with z≠y; we have F(y)≠F(z). On the other hand, for any u=(u1,…,uk)∈E, we have (40)u1≥⋯≥uk≥u1uk-1.Choose (41)x=x1,…,xk≡ukuk-1,u1,…,uk-1.It follows from (40) and (41) that (42)x1=ukuk-1≥u1=x2≥⋯≥uk-1=xk=uk-1ukuk≥uk-1ukuk-1uk=x1,which implies x∈M, and by (38) we obtain F(x)=u.
Furthermore, sinceF-1(u1,…,uk)=(ukuk-1,u1,…,uk-1) is continuous, F is a homeomorphism from M to E.
SinceM⊂E and F is a homeomorphism from M onto E, it follows that F-1(M)⊂F-1(E)=M. By induction, we have (43)p=1,1,…,1∈F-nM⊂F-n+1Mfor every n∈N. Because M is an unbounded connected closed set, we know that F-n(M) is an unbounded connected closed set for every n∈N. Let (44)S=⋂i=0∞F-iM.Then, S≠∅ and S is also an unbounded connected set.
Now, let{zn}n=-k∞ be a solution of (36) with the initial conditions (z-k,…,z-1)∈S-{p}; we can show that, for all n∈N, (45)Fnz-k,…,z-1=zn-k,…,zn-1∈M-p.Thus, zn≥zn+1>1 for any n≥-k and limn→∞zn=1. It is easy to see that {(zn,zn)}n=-k∞ is also a solution of (1) which is not periodic eventually. Theorem 6 is proven.Theorem 7.
IfAB<1<A, then (1) has an unbounded solution.Proof.
Let{(xn,yn)}n=-k∞ be a solution of (1) with the initial values xi=A and yi=B for i∈{-k,-k+1,…,-1}.
Ifk=2m for some m∈N, then it is easy to see that, for i∈{0,1,…,m-1}, we have (x2i,y2i)=(A,A/B), (x2i+1,y2i+1)=(A,B). Thus, (46)xk,yk=max⁡A,y0xk-1,max⁡B,x0yk-1=max⁡A,1B,max⁡B,AB=1B,AB,xk+1,yk+1=max⁡A,y1xk,max⁡B,x1yk=max⁡A,B2,max⁡B,B=A,B,xk+2,yk+2=max⁡A,y2xk+1,max⁡B,x2yk+1=max⁡A,1B,max⁡B,AB=1B,AB,xk+3,yk+3=max⁡A,y3xk+3,max⁡B,x3yk+3=max⁡A,B2,max⁡B,B=A,B,⋮x2k-2,y2k-2=max⁡A,yk-2x2k-3,max⁡B,xk-2yk-3=max⁡A,1B,max⁡B,AB=1B,AB,x2k-1,y2k-1=max⁡A,yk-1x2k-2,max⁡B,xk-1y2k-2=max⁡A,B2,max⁡B,B=A,B,x2k,y2k=max⁡A,ykx2k-1,max⁡B,xky2k-1=max⁡A,1B,max⁡B,1B2=1B,1B2,x2k+1,y2k+1=max⁡A,yk+1x2k,max⁡B,xk+1y2k=max⁡A,B2,max⁡B,B2A=A,B,x2k+2,y2k+2=max⁡A,yk+2x2k+1,max⁡B,xk+2y2k+1=max⁡A,1B,max⁡B,1B2=1B,1B2,x2k+3,y2k+3=max⁡A,yk+3x2k+2,max⁡B,xk+3y2k+2=max⁡A,B2,max⁡B,B2A=A,B,⋮x3k-2,y3k-2=max⁡A,y2k-2x3k-3,max⁡B,x2k-2y3k-3=max⁡A,1B,max⁡B,1B2=1B,1B2,x3k-1,y3k-1=max⁡A,y2k-1x3k-2,max⁡B,x2k-1y3k-2=max⁡A,B2,max⁡B,B2A=A,B.By induction, we can obtain that, for any n∈N and i∈{0,1,…,m-1}, (47)x2n-1k+2i,y2n-1k+2i=1An-1Bn,1An-2Bn,x2nk+2i,y2nk+2i=1An-1Bn,1An-1Bn+1,xnk+2i+1,ynk+2i+1=A,B.
Ifk=2m+1 for some m∈N, then we also can obtain that, for any n∈N, (48)x2n-1k+2i,y2n-1k+2i=1An-1Bn,Bi∈0,1,…,m,x2n-1k+2i+1,y2n-1k+2i+1=A,1An-2Bni∈0,1,…,m-1,x2nk+2i,y2nk+2i=A,1An-1Bn+1i∈0,1,…,m,x2nk+2i+1,y2nk+2i+1=1An-1Bn,Bi∈0,1,…,m-1.It is easy to see that {(xn,yn)}n=-k∞ is an unbounded solution of (1). Theorem 7 is proven.



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