On the solutions of a max type system of difference equations with period two parameters

R E S E A R C H

Open Access

On the solutions of a max-type system of

difference equations with period-two

parameters

Guangwang Su

_{, Taixiang Sun}

_{and Bin Qin}

*_{Correspondence:} [email protected] 1_{College of Information and}

Statistics, Guangxi University of Finance and Economics, Nanning, China

Abstract

In this paper, we study the following max-type system of difference equations:

xn= max{An,y_xn_n–1_–2},

yn= max{Bn,x_yn_n–1_–2},

n∈ {0, 1, 2,. . .},

whereAn,Bn∈(0, +∞) are periodic sequences with period 2 and the initial values

x–1,y–1,x–2,y–2∈(0, +∞). We show that every solution of the above system is

eventually periodic.

Keywords: Max-type system of difference equations; Solution; Eventual periodicity

1 Introduction

Our purpose in this paper is to study eventual periodicity of the following max-type system of difference equations:

⎧ ⎨ ⎩

xn=max{An,yn_xn–1_–2},

yn=max{Bn,xn_yn–1_–2},

n∈N0≡ {0, 1, . . .}, (1.1)

whereAn,Bn∈R+≡(0, +∞) are periodic sequences with period 2 and the initial values

x–2,y–2,x–1,y–1∈R+.

In [1], Fotiades and Papaschinopoulos studied the following max-type system of differ-ence equations:

⎧ ⎨ ⎩

xn=max{A,yn_xn–1_–2},

yn=max{B,xn_yn–1_–2},

n∈N0 (1.2)

withA,B∈R+and showed that every positive solution of (1.2) is eventually periodic.

In [2], we studied the eventually periodic solutions of the following max-type system of difference equations:

⎧ ⎨ ⎩

xn=max{A,yn_xn–_–1k},

yn=max{B,x_ynn–_–1k},

n∈N₀, (1.3)

whereA,B∈R+,k∈N≡ {1, 2, . . .}and the initial valuesx–k,y–k,x–k+1,y–k+1, . . . ,x–1,y–1∈

R+.

Recently, there has been a great interest in studying max-type systems of difference equations. In 2012, Stević in [3] obtained in an elegant way the general solution to the following max-type system of difference equations:

⎧ ⎨ ⎩

xn+1=max{_xnA,yn_xn},

yn+1=max{_ynA,xn_yn},

n∈N0 (1.4)

for the casex0,y0≥A> 0 andy0/x0≥max{A, 1/A}. The solvability of various systems of difference equations has reattracted some recent interest, see, e.g., [4–6] and the refer-ences therein.

In 2016, we in [7] studied the following max-type system of difference equations: ⎧

⎨ ⎩

xn=max{_xn1_–m,min{1,_ynA_–r}},

yn=max{_yn1_–m,min{1,_xnB_–t}},

n∈N0, (1.5)

whereA,B∈R+,m,r,t∈Nand the initial valuesx–d,y–d,x–d+1,y–d+1, . . . ,x–1,y–1∈R+with

d=max{m,r,t}and showed that every positive solution of (1.5) is eventually periodic with period 2m.

Whenm=r=t= 1 andA=B, (1.5) reduces to the max-type system of difference equa-tions

⎧ ⎨ ⎩

xn=max{_xn1_–1,min{1,_ynA_–1}},

yn=max{_yn1_–1,min{1,_xnA_–1}},

n∈N0. (1.6)

In 2015, the authors of [8] obtained the general solution of system (1.6).

Motivated by papers [9,10], in 2014, Stević et al. in [11] investigated the following max-type system of difference equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y(1)n =max1≤i1≤m1{f1i1(y (1)

n–k_i(1)

1,1 ,y(2)

n–k_i(1)

1,2 , . . . ,y(l)

n–k(1)_i

1,l

,n),y(_nσ_–(1))_t1s},

y(2)n =max1≤i2≤m2{f2i2(y (1)

n–k_i(2)

2,1 ,y(2)

n–k_i(2)

2,2 , . . . ,y(l)

n–k(2)_i

2,l

,n),y(nσ–(2))t2s},

· · ·

y(nl)=max1≤il≤ml{flil(y(1) n–k(_ill_,1),y

(2)

n–k(_ill)_,2, . . . ,y

(l)

n–k(l) il,l

,n),y(_nσ_–(_tlsl))},

n∈N0, (1.7)

wheres,l,mj,tj,k(ijj),h∈N(j,h∈ {1, 2, . . . ,l}), (σ(1), . . . ,σ(l)) is a permutation of (1, . . . ,l) and

solution of (1.7) is eventually periodic with periodsTfor someT∈Niffjij satisfy some

conditions.

For some results of some properties of many max-type difference equations and systems, such as eventual periodicity, the boundedness character, and attractivity, see, e.g., [12–30] and the related references therein.

2 Main results and proofs

In this section, we study the eventual periodicity of positive solutions of system (1.1). Write

x2n=pn,x2n+1=qn,y2n=sn,y2n+1=tn for anyn∈N0. Then system (1.1) reduces to the system

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

pn=max{A0,_pntn–1_–1},

tn=max{B1,_tnpn_–1},

qn=max{A1,_qnsn_–1},

sn=max{B0,qn_sn–1–1},

n∈N0, (2.1)

whereA0,A1,B0,B1∈R+and the initial valuess–1,t–1,p–1,q–1∈R+. The following lemma will be used in the proofs of our main results.

Lemma 2.1 Let{xn}n≥–1be a solution of the following equation:

xn=max

A, B

xn–1

, n∈N0 (2.2)

with A,B∈R+and the initial value x–1∈R+.Then xnis eventually periodic with period2.

Proof By (2.2) we seexnxn–1≥Bandxn≥Aforn∈N0and for anyn≥2,

A≤xn=max

A, B

xn–1

=max

A, Bxn–2

xn–1xn–2

≤max{A,xn–2}=xn–2. (2.3)

Then, for everyi∈ {0, 1},x2n+iis eventually nonincreasing.

We claim that, for everyi∈ {0, 1},x2n+iis an eventually constant sequence. Assume on

the contrary that for somei∈ {0, 1},x2n+iis not an eventually constant sequence. Then

there exists a sequence of positive integersk1<k2<· · · such that, for anyn∈N, we have

A<x2kn+1+i=

B x2kn+1+i–1

<x2kn+i=

B x2kn+i–1

,

which impliesx2kn+1+i–1>x2kn+i–1for anyn∈N. This is a contradiction. Thusxnis

From (2.1) we see that it suffices to consider the eventual periodicity of positive solutions of the following system:

⎧ ⎨ ⎩

un=max{A,vn_un–1_–1},

vn=max{B,_vnun_–1},

n∈N0, (2.4)

whereA,B∈R+and the initial valuesu–1,v–1∈R+. Let{(un,vn)}n≥–1be a solution of (2.4). From (2.4) it immediately follows that, for anyn∈N0,

un≥A (2.5)

and

vn≥B (2.6)

and ⎧ ⎨ ⎩

un=max{A,_unB_–1,_vn1_–2},

vn=max{B,_vnA_–1,_un1_–1}.

n∈N, (2.7)

Lemma 2.2 If there exist k,p∈Nsuch that up+k=uk and vp+k=vk,then un+p=unand

vn+p=vnfor any n≥k.

Proof It is easy to see that

uk+p+1=max

A,vk+p

uk+p

=max

A,vk

uk

=uk+1

and

vk+p+1=max

B,uk+p+1

vk+p

=max

B,uk+1

vk

=vk+1.

Assume that, for someN∈N, we haveuk+p+N=uk+N andvk+p+N =vk+N. Then

uk+p+N+1=max

A,vk+p+N

uk+p+N

=max

A,vk+N

uk+N

=uk+N+1

and

vk+p+N+1=max

B,uk+p+N+1

vk+p+N

=max

B,uk+N+1

vk+N

=vk+N+1.

By mathematical induction, we see thatun+p=unandvn+p=vnfor anyn≥k. The proof is

complete.

Proposition 2.1 If A>B≥1,then un=A eventually and vnis eventually periodic with

Proof IfA>B≥1, then by (2.5)–(2.7) we see that, forn≥2,

A≤un=max

A, B

un–1 , 1

vn–2

≤max

A,B

A,

1

B

=A.

Thus, forn≥2, we haveun=Aand

vn=max

B, A

vn–1

.

By Lemma2.1we see thatvnis eventually periodic with period 2.

IfB≥A≥1, then by (2.5)–(2.7) we see that, forn≥2,

B≤vn=max

B, A

vn–1 , 1

un–1

≤max

B,A

B,

1

A

=B.

Thus, forn≥2, we havevn=Band

un=max

A, B

un–1

.

By Lemma2.1we see thatunis eventually periodic with period 2. The proof is complete.

Proposition 2.2 If B≥1 >A≥1/B,then vn=B eventually and unis eventually periodic

with period2.If1/A>B≥1 >A,then vn=B eventually and unis eventually periodic with

period2or un,vnare eventually periodic with period3.

Proof Assume thatB≥1 >A≥1/B. By (2.5)–(2.7) we see that, forn≥1,

vn=max

B, A

vn–1 , 1

un–1

=B

sinceA/vn–1≤1 and 1/un–1≤B. By Lemma2.1we see thatunis eventually periodic with

period 2.

Assume that 1/A>B≥1 >A. Then by (2.5)–(2.7) we obtain

vn=max

B, A

vn–1 , 1

un–1

=max

B, 1

un–1

(n≥1) (2.8)

sinceA/vn–1≤1.

Ifvn= 1/un–1eventually, thenvnun–1= 1 eventually and by (2.4) we have

un=max

A,vn–1

un–1

=vnvn–1 eventually (2.9)

sincevnvn–1≥B2>A. Thus from (2.9) it follows that

un+3=vn+3vn+2=

vn+3vn+2vn+1

vn+1

=vn+3un+2vn+2vn+1

un+2vn+1un

un

=1×un+2

un+2×1

un=un eventually,

which implies thatun,vnare eventually periodic with period 3.

Ifvn=Beventually, then by (2.4) we have

un=max

A, B

un–1

eventually.

By Lemma2.1we see thatunis eventually periodic with period 2.

If there exists somek∈Nsuch that

vk=B≥

1

uk–1

and vk+1= 1

uk

>B, (2.10)

then by (2.4), (2.6), (2.8), and (2.10) it follows

uk+1=max

A,vk

uk

=max{A,vk+1vk}=vk+1vk,

vk+2=max

B, 1

uk+1

=B,

uk+2=max

A,vk+1

uk+1

=max

A, vk+1

vk+1vk

=max

A, 1

vk

=1

B,

vk+3=max

B, 1

uk+2

=B,

uk+3=max

A,vk+2

uk+2

=B2,

vk+4=max

B, 1

uk+3

=B,

uk+4=max

A,vk+3

uk+3

=1

B,

vk+5=max

B, 1

uk+4

=B,

uk+5=max

A,vk+4

uk+4

=1

By Lemma2.2we see thatvn=B(n≥k+ 2) anduk+2n= 1/B(n≥1) anduk+2n+1=B2(n≥ 1), which implies thatvn=Beventually andunis eventually periodic with period 2. The

proof is complete.

Proposition 2.3 If A≥1 >B,then un=A eventually and vnis eventually periodic with

period2.

Proof IfA≥1 >B≥1/A, then by (2.5)–(2.7) we see that, forn≥2,

un=max

A, B

un–1 , 1

vn–2

=A

since 1/vn–2≤AandB<un–1. Thus from (2.4) it follows

vn=max

B, A

vn–1

eventually.

By Lemma2.1we see thatvnis eventually periodic with period 2.

Now assume that 1/B>A≥1 >B. We claim that there exists a sequence of positive integersn1<n2<· · · such thatunk=A. Indeed, ifun=vn–1/un–1>Aeventually, then

A2<unun–1=vn–1=max

B,un–1

vn–2

=un–1

vn–2 =max

A vn–2

, 1

un–2

= 1

un–2

eventually,

which implies 1≤A3_<u

nun–1un–2= 1, a contradiction.

Ifun=Aeventually, then by Lemma2.1we see thatvnis eventually periodic with

pe-riod 2.

If there exists somek∈Nsuch that

uk=A≥

vk–1

uk–1

and uk+1=

vk

uk

=vk

A >A, (2.11)

thenvk=uk+1uk>A2and by (2.4) and (2.11) it follows

vk+1=max

B,uk+1

vk

=max

B, 1

uk

= 1

A,

uk+2=max

A,vk+1

uk+1

=max

A, 1

vk

=A,

vk+2=max

B,uk+2

vk+1

=A2,

uk+3=max

A,vk+2

uk+2

=A,

vk+3=max

B,uk+3

vk+2

= 1

uk+4=max

A,vk+3

uk+3

=A,

vk+4=max

B,uk+4

vk+3

=A2.

By Lemma2.2we see thatun=A(n≥k+ 2) andvk+2n–1= 1/A(n≥1) andvk+2n=A2(n≥

1), which implies thatun=Aeventually andvnis eventually periodic with period 2. The

proof is complete.

Proposition 2.4 If A< 1and B< 1,then un=A eventually and vnis eventually periodic

with period2or vn=B eventually and unis eventually periodic with period2or un,vnare

eventually periodic with period3.

Proof Note

un=max

A,vn–1

un–1

.

There are three cases to consider.

Case1. Assume thatun=A. By Lemma2.1we see thatvnis eventually periodic with

period 2.

Case2. Assume that

un=vn–1/un–1>A eventually. (2.12)

Then by (2.7) it follows

vn=max

B, A

vn–1 , 1

un–1

=max

B, 1

un–1

eventually. (2.13)

Ifvn=Beventually, then by Lemma2.1we see thatunis eventually periodic with

pe-riod 2.

Ifvn= 1/un–1>Beventually, then by (2.12) we have

un+3=

vn+2

un+2

= 1

un+2un+1

= un

un+2un+1un

= un

vn+1un

=un eventually,

which implies thatun,vnare eventually periodic with period 3.

In the following, we assume that there exists somek∈Nsuch that, for everyn≥k,

un=

vn–1

un–1

, vk=B, vk+1= 1

uk

Thus by (2.13) and (2.14) it follows

eventually periodic with period 3.

Ifvk+2=uk/B>B, then by (2.13)–(2.16) we have

are eventually periodic with period 3.

Case3. Assume that there exists somek∈Nsuch that

Thenvk=uk+1uk>A2and by (2.7) and (2.17) we have

vk+7=max

eventually periodic with period 3. Ifuk+2=A≥1/vkand

eventually periodic with period 3.

Ifuk+2= 1/vk>A≥Bvk, then by (2.4), (2.18), and (2.19) it follows

eventually periodic with period 3.

uk+3=max

eventually periodic with period 3.

Ifuk+2= 1/vk>Aandvk> 1/B2, thenA<B2and by (2.4), (2.18), and (2.19) it follows

eventually periodic with period 3. The proof is complete.

Combining (2.1) with (2.3), from Propositions2.1–2.4we obtain the following theorem.

Theorem 2.1 Let{(xn,yn)}n≥–2be a positive solution of(1.1).Then xnand ynare eventually

periodic with periods Txand Ty,respectively,and Tx,Ty∈ {2, 4, 6, 12}.

Acknowledgements

Funding

The research was supported by NNSF of China (11761011) and NSF of Guangxi (2016GXNSFBA380235,

2016GXNSFAA380286) and YMTBAPP of Guangxi Colleges (2017KY0598) and PIT of Guangxi University of Finance and Economics.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors participated in every phase of research conducted for this paper. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 10 September 2018 Accepted: 30 September 2018

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