# On Somek-Dimensional Cyclic Systems of Difference Equations

Authors: Wanping Liu; Xiaofan Yang; Bratislav D. Iričanin
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Motivated by Iričanin and Stević's paper (2006) in which for the first time were considered some cyclic systems of difference equations, here we study the global attractivity of some nonlineark-dimensional cyclic systems of higher-order difference equations. To do this, we use the transformation method from Berenhaut et al. (2007) and Berenhaut and Stević (2007). The main results in this paper also extend our recent results in the work of (Liu and Yang 2010, in press).


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## Body

## 1. Introduction

Motivated by papers [1, 2], in [3], we proved that the unique positive equilibrium points of the following difference equations:(1.1)yn=yn-kr+yn-lr1+yn-kryn-lr,yn=yn-kryn-lryn-mr+yn-kr+yn-lr+yn-mryn-kryn-lr+yn-kryn-mr+yn-lryn-mr+1,n∈ℕ0,
where k,l,m∈ℕ,1≤k<l<m, and r∈(0,1] are globally asymptotically stable, respectively.Motivated by paper [4] by Iričanin and Stević, in which for the first time were considered some cyclic systems of difference equations, here we mainly investigate the global attractivity of the k-dimensional system of higher-order difference equations(1.2)zn(i)=1+(zn-l(i))r(zn-m(θ(i+1)))r(zn-l(i))r+(zn-m(θ(i+1)))r,i=1,2,…,k,n∈ℕ0,
where r∈(0,1],k,l,m∈ℕ,k≥2,θ(s)=s(modk), and θ(k)=k, as well as the following counterpart difference equation system:(1.3)zn(i)=(zn-p(i))r+(zn-q(θ(i+1)))r1+(zn-p(i))r(zn-q(θ(i+1)))r,i=1,2,…,k,n∈ℕ0,
where r∈(0,1],k,p,q∈ℕ,k≥2.Furthermore, we also present some similar results regarding thek-dimensional cyclic difference equation system:(1.4)zn(i)=(zn-u(i))r(zn-l(θ(i+1)))r(zn-m(θ(i+2)))r+(zn-u(i))r+(zn-l(θ(i+1)))r+(zn-m(θ(i+2)))r(zn-u(i))r(zn-l(θ(i+1)))r+(zn-u(i))r(zn-m(θ(i+2)))r+(zn-l(θ(i+1)))r(zn-m(θ(i+2)))r+1,n∈ℕ0,
where i=1,2,…,k and r∈(0,1],k,u,l,m∈ℕ,k≥3, as well as the following counterpart difference equation system(1.5)zn(i)=(zn-v(i))r(zn-p(θ(i+1)))r+(zn-v(i))r(zn-q(θ(i+2)))r+(zn-p(θ(i+1)))r(zn-q(θ(i+2)))r+1(zn-v(i))r(zn-p(θ(i+1)))r(zn-q(θ(i+2)))r+(zn-v(i))r+(zn-p(θ(i+1)))r+(zn-q(θ(i+2)))r,n∈ℕ0,
where i=1,2,…,k and r∈(0,1],k,v,p,q∈ℕ,k≥3.For some recent papers on systems of difference equations, see, for example, [4–10] and the related references therein. Some related scalar equations have been studied mainly by the semicycle structure analysis which was unnecessarily complicated and only useful for lower-order equations, as it was shown by Berg and Stević in [11] (see also papers [12–22]). Thus in this paper we will investigate systems (1.2)–(1.5) by the transformation method from [1, 2] which makes the proofs more concise and elegant, and moreover, it is also effective for higher-order ones.


## 2. Preliminary Lemmas

Before proving the main results in Section3, in this section we will first present some useful lemmas which are extensions of those ones in [1].Lemma 2.1.
Define a mappingℱ:ℝ+×ℝ+→ℝ+ by ℱ(x,y)=(1+xryr)/(xr+yr), where r∈(0,1]. Then,(1)
ℱ(x,y) is nondecreasing in x if y≥1 and strictly decreasing in x if 0<y<1;(2)
ℱ(x,y) is nondecreasing in y if x≥1 and strictly decreasing in y if 0<x<1.Proof.
The results follow directly from the following fact:(2.1)ℱ(x,y)=yr+1-y2rxr+yr=xr+1-x2rxr+yr.For simplicity, systems (1.2) and (1.3) can be respectively rewritten in the following forms:(2.2)zn(i)=ℱ(zn-l(i),zn-m(θ(i+1))),i=1,2,…,k,n∈ℕ0,(2.3)zn(i)=1ℱ(zn-p(i),zn-q(θ(i+1))),i=1,2,…,k,n∈ℕ0.Lemma 2.2.
Denote a transformation(2.4)ŷ={yify≥1,1yif0<y<1.
Then for the mapping ℱ(x,y) defined in Lemma 2.1, we have ℱ̂(x,y)=ℱ(x̂,ŷ).Proof.
It is easy to see thatℱ defined in Lemma 2.1 is invariant, if both arguments are replaced by the reciprocal ones, but it turns over into 1/ℱ if only one argument is replaced by its reciprocal value. Note that
(2.5)ℱ(x,y)-1=(xr-1)(yr-1)xr+yr,
from which the result directly follows by considering the next four cases (x≥1,y≥1), (x≥1,y<1), (x<1,y≥1) and (x<1,y<1). The proof is complete.The following lemma is a corollary of Lemma2.2.Lemma 2.3.
Letℬ:ℝ+×ℝ+→ℝ+ such that ℬ(x,y)=1/ℱ(x,y); then we have
(2.6)ℬ̂(x,y)=1ℬ(x̂,ŷ)=ℱ(x̂,ŷ).Let(Zn)n=-α∞ be a positive solution to system (2.2), where Zn=(zn(1),zn(2),…,zn(k))T,α=max{l,m}; then by the transformation (2.4), we can define a transformed sequence (Ẑn)n=-α∞, where Ẑn=(ẑn(1),ẑn(2),…,ẑn(k))T, and(2.7)ẑn(i)={zn(i)ifzn(i)≥1,1zn(i)if0<zn(i)<1,i=1,2,…,k.
Note that transformation (2.7) is a natural extension of the transformation in papers [1, 2]. Then by Lemma 2.2 and the transformation (2.4), we derive the following corollary.Corollary 2.4.
For system (2.2), we have
(2.8)ẑn(i)=ℱ(ẑn-l(i),ẑn-m(θ(i+1)))=1+(ẑn-l(i))r(ẑn-m(θ(i+1)))r(ẑn-l(i))r+(ẑn-m(θ(i+1)))r,i=1,2,…,k,n∈ℕ0.From Corollary2.4, we easily see that (Ẑn)n=-α∞ is also a positive solution to the system (2.2).Lemma 2.5.
For the functionℱ(x,y) defined in Lemma 2.1, there hold
(2.9)1≤ℱ̂(x,y)≤max{x̂,ŷ},x,y>0.Proof.
By (2.4), it immediately follows that ℱ̂(x,y)≥1.
Letλ=max{x̂,ŷ}≥1, which indicates that 1≤x̂≤λ and 1≤ŷ≤λ. Employing Lemma 2.1 two times and Lemma 2.2, we have(2.10)ℱ̂(x,y)=ℱ(x̂,ŷ)≤ℱ(x̂,λ)≤ℱ(λ,λ)=1+λrλrλr+λr≤λ.
The proof is complete.The following corollary follows directly from Corollary2.4 and Lemma 2.5.Corollary 2.6.
For the transformed sequence(Ẑn)n=-α∞, we get
(2.11)1≤ẑn(i)≤max{ẑn-l(i),ẑn-m(θ(i+1))},i=1,2,…,k,n∈ℕ0.Let(2.12)Hn=maxn-α≤j≤n-1{max1≤i≤k{ẑj(i)}},n∈ℕ0,
for a positive solution (Zn)n=-α∞ to system (2.2).Lemma 2.7.
The sequence(Hn)n=0∞ is monotonically nonincreasing.Proof.
The proof is a straightforward consequence of Corollary2.6 and (2.12), and hence is omitted.


## 3. Attractivity of (1.2) and (1.3)

In this section, we will formulate and prove the main results of this paper developing the methods and ideas from [1].Lemma 3.1.
Both systems (2.2) and (2.3) have the unique positive equilibrium point Z¯=(1,1,…,1︸k)T.Proof.
LetZ¯=(e1,e2,…,ek)T be a positive equilibrium point of system (2.2); then we have
(3.1)e1=ℱ(e1,e2),e2=ℱ(e2,e3),…,ek=ℱ(ek,e1).
Through certain calculations, we obtain
(3.2)e1-1=(e1r-1)(e2r-1)e1r+e2r,e2-1=(e2r-1)(e3r-1)e2r+e3r,…,ek-1=(ekr-1)(e1r-1)ekr+e1r.
If there exists some s∈{1,2,…,k} such that es=1, then from the previous system easily follows that ej=1 for all j=1,2,…,k. Hence, suppose that ej≠1 for all j=1,2,…,k; then by comparing the signs in the last system it is easy to get that it must be ej>1,j=1,2,…,k. However, since r∈(0,1], then
(3.3)e1-1≥e1r-1>(e1r-1)(e2r-1)e1r+e2r,
which contradicts e1=ℱ(e1,e2). Therefore, ej=1 for all j∈{1,2,…,k}.
The uniqueness of equilibrium of system (2.3) can be analogously proved and thus is omitted.Theorem 3.2.
The unique equilibrium point of system (1.2) is a global attractor.Proof.
Let(Zn)n=-α∞ be an arbitrary positive solution to system (1.2), where Zn=(zn(1),zn(2),…,zn(k))T; then we need to prove that
(3.4)limn→∞Zn=Z¯.
Define a transformed sequence (Ẑn)n=-α∞ by (2.4) and (2.7), where Ẑn=(ẑn(1),ẑn(2),…,ẑn(k))T; then it suffices to confirm that
(3.5)limn→∞Ẑn=Z¯.
Let the sequence (Hn)n=0∞ be defined by (2.12); then using Lemma 2.7 we know that there is a finite limit of Hn as n→∞, say H. Note that H≥1. By Corollary 2.6, we have ẑj(i)∈[1,Hj], for all i=1,2,…,k and j=0,1,2,…. It suffices to show that H=1.
Assume thatH>1; then by Lemma 2.7 and (2.12), for arbitrary ɛ>0, there exists a sufficiently large N∈ℕ such that ẑN(β)∈[H,H+ɛ], for some β∈{1,2,…,k} and(3.6)ẑn(i)∈[1,H+ɛ],n≥N-α,i=1,2,…,k.
Employing Lemma 2.1 two times and (3.6), we get
(3.7)H≤ẑN(β)=ℱ(ẑN-l(β),ẑN-m(θ(β+1)))≤ℱ(ẑN-l(β),H+ɛ)≤ℱ(H+ɛ,H+ɛ)=1+(H+ɛ)r(H+ɛ)r(H+ɛ)r+(H+ɛ)r.
Since ɛ>0 is arbitrary, we get
(3.8)H≤limɛ→0+ℱ(H+ɛ,H+ɛ)=ℱ(H,H)=1+HrHrHr+Hr,
which implies H2r-2Hr+1+1≥0 for all H>1.
On the other hand, letJ(x)=x2r-2xr+1+1,x>1; then the derivative of J(x) is(3.9)J′(x)=2rx2r-1-2(r+1)xr=2xr(rxr-1-r-1)<0.
Since J(1)=0, we get H2r-2Hr+1+1<0 which contradicts H2r-2Hr+1+1≥0, for all H>1.
ThereforeH=1. The proof is complete.By Theorem3.2, the following corollary easily follows.Corollary 3.3.
The unique positive equilibrium pointX¯=(A,A,…,A︸k)T of the following difference equation system:
(3.10)xn(i)=A+xn-l(i)xn-m(θ(i+1))xn-l(i)+xn-m(θ(i+1)),i=1,2,…,k,n∈ℕ0,
where the parameter A∈ℝ+,r∈(0,1],k,l,m∈ℕ,k≥2,l≠m, and α=max{l,m}, is a global attractor.Theorem 3.4.
The unique positive equilibrium point of system (1.3) is a global attractor.Proof.
Let(Zn)n=-μ∞ be any positive solution to system (1.3), where Zn=(zn(1),zn(2),…,zn(k))T, μ=max{p,q}. By using Lemma 2.3 and (1.3), we obtain
(3.11)ẑn(i)=ℱ(ẑn-p(i),ẑn-q(θ(i+1)))=1+(ẑn-p(i))r(ẑn-q(θ(i+1)))r(ẑn-p(i))r+(ẑn-q(θ(i+1)))r,i=1,2,…,k,n∈ℕ0,
which indicates that (Ẑn)n=-μ∞ is a positive solution to system (1.2), where Ẑn=(ẑn(1),ẑn(2),…,ẑn(k))T. Hence by Theorem 3.2, we have
(3.12)limn→∞Ẑn=Z¯
and then by the transformation (2.4) we easily get
(3.13)limn→∞Zn=Z¯.
The proof is complete.


## 4. Attractivity of (1.4) and (1.5)

Similar to the proofs of Lemmas2.1–2.7, we can get the following lemmas. We omit their proofs.Lemma 4.1.
Define a mappingℳ:(ℝ+)3→ℝ by ℳ(x0,x1,x2)=(1+x0rx1r+x0rx2r+x1rx2r)/(x0r+x1r+x2r+x0rx1rx2r), where r∈(0,1]. Then for j=0,1,2 we have (1)
ℳ(x0,x1,x2) is nonincreasing in xj if 0<xω(j+1), xω(j+2)≤1 or xω(j+1),xω(j+2)≥1;(2)
ℳ(x0,x1,x2) is nondecreasing in xj if xω(j+1)≥1, 0<xω(j+2)≤1 or xω(j+2)≥1, 0<xω(j+1)≤1, where ω(s)=s(mod3).Lemma 4.2.
For the mappingℳ(x0,x1,x2) defined in Lemma 4.1, we have
(4.1)ℳ̂(x0,x1,x2)=1ℳ(x0̂,x1̂,x2̂).Lemma 4.3.
Let𝒩:(ℝ+)3→ℝ+ such that 𝒩(x0,x1,x2)=1/ℳ(x0,x1,x2), then we have
(4.2)𝒩̂(x0,x1,x2)=𝒩(x0̂,x1̂,x2̂).Let(Zn)n=-δ∞ be a positive solution to the system (1.5), where Zn=(zn(1),zn(2),…,zn(k))T,δ=max{v,p,q}; then by Lemma 4.2 and the transformation (2.7), we get the following corollary.Corollary 4.4.
For system (1.5), we have
(4.3)ẑn(i)=1ℳ(ẑn-v(i),ẑn-p(θ(i+1)),ẑn-q(θ(i+2))),i=1,2,…,k,n∈ℕ0.Lemma 4.5.
For the functionℳ(x0,x1,x2) defined in Lemma 4.1, there hold
(4.4)1≤ℳ̂(x0,x1,x2)≤max{x0̂,x1̂,x2̂},x0,x1,x2>0.The following corollary follows directly from Corollary4.4 and Lemma 4.5.Corollary 4.6.
For the transformed sequence(Ẑn)n=-δ∞, we get
(4.5)1≤ẑn(i)≤max{ẑn-v(i),ẑn-p(θ(i+1)),ẑn-q(θ(i+2))},i=1,2,…,k,n∈ℕ0.Let(4.6)Qn=maxn-δ≤j≤n-1{max1≤i≤k{ẑj(i)}},n∈ℕ0,
for a positive solution (Zn)n=-δ∞ to the system (1.5), where δ=max{v,p,q}.Lemma 4.7.
The sequence(Qn)n=0∞ is monotonically nonincreasing.Note that, by the proof of Lemma3.1 we can similarly confirm that both system (1.4) and (1.5) have the same unique positive equilibrium Z¯=(1,1,…,1︸k)T.Theorem 4.8.
The unique equilibrium point of system (1.5) is a global attractor.Proof.
Let(Zn)n=-δ∞ be an arbitrary positive solution to system (1.5), where Zn=(zn(1),zn(2),…,zn(k))T, then we need to prove that
(4.7)limn→∞Zn=Z¯.
Define a transformed sequence (Ẑn)n=-δ∞ by (2.4) and (2.7), where Ẑn=(ẑn(1),ẑn(2),…,ẑn(k))T, then it suffices to confirm that
(4.8)limn→∞Ẑn=Z¯.
Let the sequence (Qn)n=0∞ be defined by (4.6); then by using Lemma 4.7 we know that the limit of (Qn)n=0∞ exists, say Q. Note that Q≥1. By Corollary 4.6 we get ẑj(i)∈[1,Qj], for all i=1,2,…,k and j=0,1,2,…. Obviously, it suffices to show that Q=1.
Assume thatQ>1; then by Lemma 4.7 and (4.6), for arbitrary ɛ>0, there exists a sufficiently large N∈ℕ such that ẑN(β)∈[Q,Q+ɛ], for some β∈{1,2,…,k} and(4.9)ẑn(i)∈[1,Q+ɛ],n≥N-δ,i=1,2,…,k.
Employing Lemma 4.1 three times and (4.9), we get
(4.10)Q≤ẑN(β)=1ℳ(ẑN-v(β),ẑN-p(θ(β+1)),ẑN-q(θ(β+2)))≤1ℳ(Q+ɛ,ẑN-p(θ(β+1)),ẑN-q(θ(β+2)))≤1ℳ(Q+ɛ,Q+ɛ,ẑN-q(θ(β+2)))≤1ℳ(Q+ɛ,Q+ɛ,Q+ɛ)=3(Q+ɛ)r+(Q+ɛ)3r1+3(Q+ɛ)2r.
Since ɛ>0 is arbitrary, we get
(4.11)Q≤limɛ→0+1ℳ(Q+ɛ,Q+ɛ,Q+ɛ)=1ℳ(Q,Q,Q)=3Qr+Q3r1+3Q2r.
Since Q>1; then by Lemma 2.2 in [3], we have that
(4.12)3Qr+Q3r1+3Q2r<Q
which contradicts (4.11). Therefore Q=1. The proof is complete.Theorem 4.9.
The unique positive equilibrium point of system (1.4) is a global attractor.Proof.
The proof is similar to that of Theorem4.8 and hence is omitted.


## 5. Conclusions

Following the proofs in this paper line by line, we can also similarly confirm the following two results.Remark 5.1.
Every positive solution to the difference equation system(5.1)zn(i)=1+(zn-l(θ(i-1)))r(zn-m(i))r(zn-l(θ(i-1)))r+(zn-m(i))r,i=1,2,…,k,n∈ℕ0,
where r∈(0,1],k,l,m∈ℕ,k≥2 and θ(0)=k, converges to the unique positive equilibrium point Z¯=(1,1,…,1︸k)T.Remark 5.2.
Every positive solution to the difference equation system(5.2)zn(i)=(zn-p(θ(i-1)))r+(zn-q(i))r1+(zn-p(θ(i-1)))r(zn-q(i))r,i=1,2,…,k,n∈ℕ0,
where r∈(0,1],k,p,q∈ℕ and k≥2, converges to the unique positive equilibrium point Z¯=(1,1,…,1︸k)T.



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