# Eventually Periodic Solutions of a Max-Type Difference Equation

Authors: Taixiang Sun; Jing Liu; Qiuli He; Xin-He Liu; Chunyan Tao
Journal: The Scientific World Journal
 (2014)

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## Abstract

We study the following max-type difference equationxn=max⁡{An/xn-r,xn-k}, n=1,2,…, where {An}n=1+∞ is a periodic sequence with period p and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max⁡{r,k}. We show that if p=1 (or p≥2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevic´ (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p≥2 and k being even which has a well-defined solution that is not eventually periodic.


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## Body

## 1. Introduction

The max operator arises naturally in certain models in automatic control theory (see [1]). In recent years, the discrete case involving difference equations with maximum has been receiving increasing attention, for some results in this area; see, for example, [2–4].In this paper, we consider the following max-type equation:(1)xn=max⁡⁡{Anxn-r,xn-k},n=1,2,…,
where {An}n=1+∞ is a periodic sequence with period p and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max⁡⁡{r,k}.In [5], Iricˇanin and Elsayed showed that every well-defined solution of (1) is eventually periodic with period 4 when k=4, r=1, and p=1. Elsayed and Stevic´ [6] showed that every well-defined solution of (1) is eventually periodic with period 3 when k=3, r=1, and p=1. In [7], Xiao and Shi showed that if k=2, r=1, and p=1, then every well-defined solution of (1) is eventually periodic with period 2. Qin et al. [8] showed that every well-defined solution of (1) is eventually periodic with period k when r=1 and p=1.In this paper, we will generalize the results of [5–8] to the general case.


## 2. Main Results and Example

In this section, we are ready to state and prove the main results.Theorem 1.
Let{An}n=1+∞ be a periodic sequence with period p, and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r.(1)
Ifp≥2 and k is odd, then every well-defined solution of (1) is eventually periodic with period k.(2)
Ifp=1, then every well-defined solution of (1) is eventually periodic with period k.Proof.
Let{xn}n=1-d+∞ be a well-defined solution of (1). It follows from (1) that, for any n≥0 and any i≥0,
(2)x(n+1)k+i=max⁡⁡{A(n+1)k+ix(n+1)k+i-r,xnk+i}≥xnk+i.
Then, for every i≥0, {xnk+i}n=0+∞ is increasing, and xnk+i<0 for all n≥0 or there exists Ni>0 such that xnk+i>0 for all n≥Ni.
We claim that{xnk}n=0+∞ is a constant sequence eventually. Indeed, if {xnk}n=0+∞ is not constant sequence eventually, then there exist kr<n1<n2<⋯ such that xnik>x(ni-1)k and Anik is a constant sequence for all i≥1 since {An}n=1+∞ is a periodic sequence. Thus we have
(3)xni+1k=max⁡⁡{Ani+1kxni+1k-r,x(ni+1-1)k}=Ani+1kxni+1k-r>x(ni+1-1)k≥xnik=max⁡⁡{Anikxnik-r,x(ni-1)k}=Anikxnik-r.
From this we obtain that, for all i≥1,
(4)Anik(xni+1k-r-xnik-r)xni+1k-rxnik-r<0.
It follows that, for all i≥1,
(5)xnikxnik-r=Anik<0,xni+1k-r>xnik-r.
Therefore we have xnkxnk-r<0 eventually. By induction, we can show that xnk-irxnk-(i+1)r<0 eventually for all 1≤i≤k-1 and every {xnk-ir}n=r+∞(1≤i≤k-1) is not constant sequence eventually.
Ifp≥2 and k is odd, then we have xnkxnk-kr∏i=1k-1xnk-irxnk-ir<0 eventually. This is a contradiction.
Ifp=1, then we write An=A for all n≥1 and choose m0≥m1≥⋯≥m2k such that xmjk-jr<x(mj+1)k-jr and x(mj+1)k-(j+1)r=x(mj+1+1)k-(j+1)r for any j∈{0,1,…,2k-1}. Thus
(6)x(m0+1)k=max⁡⁡{Ax(m0+1)k-r,xm0k}=Ax(m0+1)k-r=Ax(m1+1)k-r=Amax⁡{A/x(m1+1)k-2r,xm1k-r}=x(m1+1)k-2r=x(m2+1)k-2r⋮=x(m2k+1)k-2kr≤xm0k,
which is a contradiction. This completes the proof of the claim.
By the above claim we may choose anN>0 such that xnk=xNk for all n≥N. Since An is a periodic sequence, we can choose an N1>N such that AN1k+r/xNk=max⁡{Ank+r/xNk:n≥N}. Then, for all n≥N1,
(7)xnk+r=max⁡{Ank+rxNk,x(n-1)k+r}.
Thus xnk+r=x(n-1)k+r for all n≥N1+1 (otherwise, if xmk+r>x(m-1)k+r for some m≥N1+1, then we have xmk+r=Amk+r/xNk>x(m-1)k+r≥xN1k+r≥AN1k+r/xNk, which is a contradiction). By induction, we can show that xnk+jr is a constant sequence eventually for every 1≤j≤k. Note {jr:1≤j≤k}mod⁡k={0,1,2,…,k-1} since gcd(k,r)=1. Then xnk+i is a constant sequence eventually for every i∈{0,1,…,k-1}, which implies that {xn}n=1-d+∞ is eventually periodic with period k.From the proof of Theorem1 we obtain the following corollary.Corollary 2.
Letk,r∈{1,2,…} with k≠r. If {An}n=1+∞ is a periodic sequence, then every positive (or negative) solution of (1) is eventually periodic with period k.Theorem 3.
Let{An}n=1+∞ be a periodic sequence with period p≥2, and k,r∈{1,2,…} with gcd(k,r)=1 and k≠r. If As≥0 for some s∈{1,2,…,p}, then every well-defined solution of (1) is eventually periodic with period k.Proof.
Let{xn}n=1-d+∞ be a well-defined solution of (1). Using arguments similar to the ones developed in the proof of Theorem 1, we know that, for every i≥0, {xnk+i}n=0+∞ is increasing, and xnk+i<0 for all n≥0 or there exists Ni>0 such that xnk+i>0 for all n≥Ni.
We may assume without loss of generality thatA1≥0. We claim that {xnk}n=0+∞ is a constant sequence eventually. Indeed, if {xnk}n=0+∞ is not constant sequence eventually, then there exist kr<n1<n2<⋯ such that xnik>x(ni-1)k with Anik being a constant sequence for all i≥1. Thus we have
(8)xni+1k=max⁡⁡{Ani+1kxni+1k-r,x(ni+1-1)k}=Ani+1kxni+1k-r>x(ni+1-1)k≥xnik=max⁡⁡{Anikxnik-r,x(ni-1)k}=Anikxnik-r,xnpk=max⁡{Anpkxnpk-r,x(np-1)k}≥Anpkxnpk-r.
From this we obtain that, for all i≥1,
(9)xnikxnik-r=Anik,xnpkxnpk-r≥A1≥0.
Thus Anik≥0 and
(10)Anik(xni+1k-r-xnik-r)xni+1k-rxnik-r<0.
It follows that, for all i≥1,
(11)xni+1k-r<xnik-r.
This is a contradiction.Using arguments similar to the ones developed in the proof of Theorem1, we can show that xnk+jr is a constant sequence eventually for every 1≤j≤k. Note {jr:1≤j≤k}mod⁡k={0,1,2,…,k-1} since gcd(k,r)=1. Then xnk+i is a constant sequence eventually for every i∈{0,1,…,k-1}, which implies that {xn}n=1-d+∞ is eventually periodic with period k.Now we construct an example withp≥2 and k being even which has a well-defined solution that is not eventually periodic.Example 4.
Consider the max-type equation(12)xn=max⁡{Anxn-r,xn-k},n=1,2,…,
where k,r∈{1,2,…} and k is even with gcd(k,r)=1 and k≠r and An is a periodic sequence with A2i=A<A2i-1=B<0 for all i≥1. Choose the initial conditions x-i=B for odd i∈{0,1,…,d} and x-i=1 for even i∈{0,1,…,d} with d=max⁡{r,k}; we can obtain a solution {xn}n=-1∞ of (12) such that(1) Ifr<k, then
(13)xnk+i={{BA}nB,ifi∈{1,3,…,r},{BA}n+1B,ifi∈{r+2,…,k-1},{AB}n+1,ifi∈{2,…,k}. It is easy to verify that lim⁡n→∞x2n=∞ and lim⁡n→∞x2n-1=0.(2) Ifr>k, then
(14)xn2r+i={{BA}nB,ifi∈{1,3,…,r},{BA}n+1B,ifi∈{r+2,…,2r-1},{AB}n+1,ifi∈{2,…,2r}.It is easy to verify that lim⁡n→∞x2n=∞ and lim⁡i→∞x2n-1=0.Remark 5.
Consider the max-type equation(15)xn=max⁡⁡{Anxn-sr,xn-sk},n=1,2,…,
where {An}n=1+∞ is a periodic sequence with period ps and s,k,r∈{1,2,…} with gcd(k,r)=1 and k≠r, and p=1 (or p≥2), and the initial conditions x1-d,x2-d,…,x0 are real numbers with d=max⁡⁡{sr,sk}. Write yni=xns+i for every 1≤i≤s and n=0,1,2,…. Then (12) reduces to the equation
(16)yni=max⁡⁡{Ans+iyn-ri,yn-ki},1≤i≤s,n=0,1,2,….(1)
Ifp=1 (or p≥2 and k is odd), then it follows from Theorem 1 that, for every 1≤i≤s, every well-defined solution of equation yni=max⁡⁡{Ans+i/yn-ri,yn-ki} is eventually periodic with period k. Thus every well-defined solution of (15) is eventually periodic with period sk.(2)
Ifp≥2 and, for every 1≤i≤s, there exists some ji such that Ajis+i≥0, then it follows from Theorem 3 that for every 1≤i≤s, every well-defined solution of equation yni=max⁡⁡{Ans+i/yn-ri,yn-ki} is eventually periodic with period k. Thus every well-defined solution of (15) is eventually periodic with period sk.(3)
Ifp≥2 and k is even, then it follows from Example 4 that, for every 1≤i≤s, we can construct an equation yni=max⁡⁡{Ans+i/yn-ri,yn-ki} such that it has a well-defined solution which is not eventually periodic. Thus we can construct an equation
(17)xn=max⁡⁡{Anxn-sr,xn-sk},n=1,2,…such that it has a well-defined solution which is not eventually periodic.



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