On Some 𝑘-Dimensional Cyclic Systems of Difference Equations

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 nonlinear k-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).

1. Introduction

Motivated by papers [1, 2], in [3], we proved that the unique positive equilibrium points of the following difference equations:𝑦_𝑛=𝑦^𝑟_𝑛−𝑘+𝑦^𝑟_𝑛−𝑙1+𝑦^𝑟_𝑛−𝑘𝑦^𝑟_𝑛−𝑙,𝑦_𝑛=𝑦^𝑟_𝑛−𝑘𝑦^𝑟_𝑛−𝑙𝑦^𝑟_𝑛−𝑚+𝑦^𝑟_𝑛−𝑘+𝑦^𝑟_𝑛−𝑙+𝑦^𝑟_𝑛−𝑚𝑦^𝑟_𝑛−𝑘𝑦^𝑟_𝑛−𝑙+𝑦^𝑟_𝑛−𝑘𝑦^𝑟_𝑛−𝑚+𝑦^𝑟_𝑛−𝑙𝑦^𝑟_𝑛−𝑚+1,𝑛∈ℕ₀, (1.1) 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 equations𝑧_𝑛^(𝑖)=1+(𝑧^(𝑖)_𝑛−𝑙)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚)^𝑟(𝑧^(𝑖)_𝑛−𝑙)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚)^𝑟,𝑖=1,2,…,𝑘,𝑛∈ℕ₀, (1.2) 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:𝑧_𝑛^(𝑖)=(𝑧^(𝑖)_𝑛−𝑝)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑞)^𝑟1+(𝑧^(𝑖)_𝑛−𝑝)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑞)^𝑟,𝑖=1,2,…,𝑘,𝑛∈ℕ₀, (1.3) where 𝑟∈(0,1],𝑘,𝑝,𝑞∈ℕ,𝑘≥2.

Furthermore, we also present some similar results regarding the 𝑘-dimensional cyclic difference equation system:𝑧_𝑛^(𝑖)=(𝑧^(𝑖)_𝑛−𝑢)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑙)^𝑟(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑚)^𝑟+(𝑧^(𝑖)_𝑛−𝑢)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑙)^𝑟+(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑚)^𝑟(𝑧^(𝑖)_𝑛−𝑢)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑙)^𝑟+(𝑧^(𝑖)_𝑛−𝑢)^𝑟(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑚)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑙)^𝑟(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑚)^𝑟+1,𝑛∈ℕ₀, (1.4) where 𝑖=1,2,…,𝑘 and 𝑟∈(0,1],𝑘,𝑢,𝑙,𝑚∈ℕ,𝑘≥3, as well as the following counterpart difference equation system𝑧_𝑛^(𝑖)=(𝑧^(𝑖)_𝑛−𝑣)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑝)^𝑟+(𝑧^(𝑖)_𝑛−𝑣)^𝑟(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑞)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑝)^𝑟(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑞)^𝑟+1(𝑧^(𝑖)_𝑛−𝑣)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑝)^𝑟(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑞)^𝑟+(𝑧^(𝑖)_𝑛−𝑣)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑝)^𝑟+(𝑧^{(𝜃(𝑖+2))}_𝑛−𝑞)^𝑟,𝑛∈ℕ₀, (1.5) 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 Section 3, in this section we will first present some useful lemmas which are extensions of those ones in [1].

Lemma 2.1. Define a mapping ℱ∶ℝ⁺×ℝ⁺→ℝ⁺ by ℱ(𝑥,𝑦)=(1+𝑥^𝑟𝑦^𝑟)/(𝑥^𝑟+𝑦^𝑟), 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: ℱ(𝑥,𝑦)=𝑦^𝑟+1−𝑦^2𝑟𝑥^𝑟+𝑦^𝑟=𝑥^𝑟+1−𝑥^2𝑟𝑥^𝑟+𝑦^𝑟. (2.1)

For simplicity, systems (1.2) and (1.3) can be respectively rewritten in the following forms:𝑧_𝑛^(𝑖)=ℱ(𝑧^(𝑖)_𝑛−𝑙,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚),𝑖=1,2,…,𝑘,𝑛∈ℕ₀,𝑧_𝑛^(𝑖)=1ℱ(𝑧^(𝑖)_𝑛−𝑝,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑞),𝑖=1,2,…,𝑘,𝑛∈ℕ₀. (2.2)(2.3)

Lemma 2.2. Denote a transformation 𝑦𝑦=if1𝑦≥1,𝑦if0<𝑦<1. (2.4) Then for the mapping ℱ(𝑥,𝑦) defined in Lemma 2.1, we have ℱ(𝑥,𝑦)=ℱ(𝑥,𝑦).

Proof. It is easy to see that ℱ 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 ℱ(𝑥,𝑦)−1=(𝑥^𝑟−1)(𝑦^𝑟−1)𝑥^𝑟+𝑦^𝑟, (2.5) 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 Lemma 2.2.

Lemma 2.3. Let ℬ∶ℝ⁺×ℝ⁺→ℝ⁺ such that ℬ(𝑥,𝑦)=1/ℱ(𝑥,𝑦); then we have 1ℬ(𝑥,𝑦)=ℬ(𝑥,𝑦)=ℱ(𝑥,𝑦). (2.6)

Let (𝐙_𝑛)^∞_𝑛=−𝛼 be a positive solution to system (2.2), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇,𝛼=max{𝑙,𝑚}; then by the transformation (2.4), we can define a transformed sequence (𝐙_𝑛)^∞_𝑛=−𝛼, where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇, and𝑧_𝑛^(𝑖)=𝑧_𝑛^(𝑖)if𝑧_𝑛^(𝑖)1≥1,𝑧_𝑛^(𝑖)if0<𝑧_𝑛^(𝑖)<1,𝑖=1,2,…,𝑘. (2.7) 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 𝑧_𝑛^(𝑖)=ℱ(𝑧^(𝑖)_𝑛−𝑙,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚𝑧)=1+(^(𝑖)_𝑛−𝑙)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚)^𝑟(𝑧^(𝑖)_𝑛−𝑙)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚)^𝑟,𝑖=1,2,…,𝑘,𝑛∈ℕ₀. (2.8)

From Corollary 2.4, we easily see that (𝐙_𝑛)^∞_𝑛=−𝛼 is also a positive solution to the system (2.2).

Lemma 2.5. For the function ℱ(𝑥,𝑦) defined in Lemma 2.1, there hold 1≤ℱ(𝑥,𝑦)≤max{𝑥,𝑦},𝑥,𝑦>0. (2.9)

Proof. By (2.4), it immediately follows that ℱ(𝑥,𝑦)≥1.
Let 𝜆=max{𝑥,𝑦}≥1, which indicates that 1≤𝑥≤𝜆 and 1≤𝑦≤𝜆. Employing Lemma 2.1 two times and Lemma 2.2, we haveℱ(𝑥,𝑦)=ℱ(𝑥,𝑦)≤ℱ(𝑥,𝜆)≤ℱ(𝜆,𝜆)=1+𝜆^𝑟𝜆^𝑟𝜆^𝑟+𝜆^𝑟≤𝜆. (2.10) The proof is complete.

The following corollary follows directly from Corollary 2.4 and Lemma 2.5.

Corollary 2.6. For the transformed sequence (𝐙_𝑛)^∞_𝑛=−𝛼, we get 1≤𝑧_𝑛^(𝑖)≤max{𝑧^(𝑖)_𝑛−𝑙,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑚},𝑖=1,2,…,𝑘,𝑛∈ℕ₀. (2.11)

Let 𝐻_𝑛=max_{𝑛−𝛼≤𝑗≤𝑛−1}{max_{1≤𝑖≤𝑘}{𝑧_𝑗^(𝑖)}},𝑛∈ℕ₀, (2.12) for a positive solution (𝐙_𝑛)^∞_𝑛=−𝛼 to system (2.2).

Lemma 2.7. The sequence (𝐻_𝑛)^∞_𝑛=0 is monotonically nonincreasing.

Proof. The proof is a straightforward consequence of Corollary 2.6 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. Let 𝐙=(𝑒₁,𝑒₂,…,𝑒_𝑘)^𝑇 be a positive equilibrium point of system (2.2); then we have 𝑒₁=ℱ(𝑒₁,𝑒₂),𝑒₂=ℱ(𝑒₂,𝑒₃),…,𝑒_𝑘=ℱ(𝑒_𝑘,𝑒₁). (3.1) Through certain calculations, we obtain 𝑒₁−1=(𝑒^𝑟₁−1)(𝑒^𝑟₂−1)𝑒^𝑟₁+𝑒^𝑟₂,𝑒₂−1=(𝑒^𝑟₂−1)(𝑒^𝑟₃−1)𝑒^𝑟₂+𝑒^𝑟₃,…,𝑒_𝑘−1=(𝑒^𝑟_𝑘−1)(𝑒^𝑟₁−1)𝑒^𝑟_𝑘+𝑒^𝑟₁. (3.2) If there exists some 𝑠∈{1,2,…,𝑘} such that 𝑒_𝑠=1, then from the previous system easily follows that 𝑒_𝑗=1 for all 𝑗=1,2,…,𝑘. Hence, suppose that 𝑒_𝑗≠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 𝑒_𝑗>1,𝑗=1,2,…,𝑘. However, since 𝑟∈(0,1], then 𝑒₁−1≥𝑒^𝑟₁−1>(𝑒^𝑟₁−1)(𝑒^𝑟₂−1)𝑒^𝑟₁+𝑒^𝑟₂, (3.3) which contradicts 𝑒₁=ℱ(𝑒₁,𝑒₂). Therefore, 𝑒_𝑗=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 (𝐙_𝑛)^∞_𝑛=−𝛼 be an arbitrary positive solution to system (1.2), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇; then we need to prove that lim_𝑛→∞𝐙_𝑛=𝐙. (3.4) Define a transformed sequence (𝐙_𝑛)^∞_𝑛=−𝛼 by (2.4) and (2.7), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇; then it suffices to confirm that lim_𝑛→∞𝐙_𝑛=𝐙. (3.5) Let the sequence (𝐻_𝑛)^∞_𝑛=0 be defined by (2.12); then using Lemma 2.7 we know that there is a finite limit of 𝐻_𝑛 as 𝑛→∞, say 𝐻. Note that 𝐻≥1. By Corollary 2.6, we have 𝑧_𝑗^(𝑖)∈[1,𝐻_𝑗], for all 𝑖=1,2,…,𝑘 and 𝑗=0,1,2,…. It suffices to show that 𝐻=1.
Assume that 𝐻>1; then by Lemma 2.7 and (2.12), for arbitrary 𝜀>0, there exists a sufficiently large 𝑁∈ℕ such that 𝑧_𝑁^(𝛽)∈[𝐻,𝐻+𝜀], for some 𝛽∈{1,2,…,𝑘} and𝑧_𝑛^(𝑖)∈[1,𝐻+𝜀],𝑛≥𝑁−𝛼,𝑖=1,2,…,𝑘. (3.6) Employing Lemma 2.1 two times and (3.6), we get 𝐻≤𝑧_𝑁^(𝛽)=ℱ(𝑧^(𝛽)_𝑁−𝑙,𝑧^{(𝜃(𝛽+1))}_𝑁−𝑚)≤ℱ(𝑧^(𝛽)_𝑁−𝑙,𝐻+𝜀)≤ℱ(𝐻+𝜀,𝐻+𝜀)=1+(𝐻+𝜀)^𝑟(𝐻+𝜀)^𝑟(𝐻+𝜀)^𝑟+(𝐻+𝜀)^𝑟. (3.7) Since 𝜀>0 is arbitrary, we get 𝐻≤lim_𝜀→0+ℱ(𝐻+𝜀,𝐻+𝜀)=ℱ(𝐻,𝐻)=1+𝐻^𝑟𝐻^𝑟𝐻^𝑟+𝐻^𝑟, (3.8) which implies 𝐻^2𝑟−2𝐻^𝑟+1+1≥0 for all 𝐻>1.
On the other hand, let 𝐽(𝑥)=𝑥^2𝑟−2𝑥^𝑟+1+1,𝑥>1; then the derivative of 𝐽(𝑥) is𝐽^′(𝑥)=2𝑟𝑥^2𝑟−1−2(𝑟+1)𝑥^𝑟=2𝑥^𝑟(𝑟𝑥^𝑟−1−𝑟−1)<0. (3.9) Since 𝐽(1)=0, we get 𝐻^2𝑟−2𝐻^𝑟+1+1<0 which contradicts 𝐻^2𝑟−2𝐻^𝑟+1+1≥0, for all 𝐻>1.
Therefore 𝐻=1. The proof is complete.

By Theorem 3.2, the following corollary easily follows.

Corollary 3.3. The unique positive equilibrium point 𝐗=(√𝐴,√𝐴,…,√𝐴_𝑘)^𝑇 of the following difference equation system: 𝑥_𝑛^(𝑖)=𝐴+𝑥^(𝑖)_𝑛−𝑙𝑥^{(𝜃(𝑖+1))}_𝑛−𝑚𝑥^(𝑖)_𝑛−𝑙+𝑥^{(𝜃(𝑖+1))}_𝑛−𝑚,𝑖=1,2,…,𝑘,𝑛∈ℕ₀, (3.10) 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 (𝐙_𝑛)^∞_𝑛=−𝜇 be any positive solution to system (1.3), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇, 𝜇=max{𝑝,𝑞}. By using Lemma 2.3 and (1.3), we obtain 𝑧_𝑛^(𝑖)=ℱ(𝑧^(𝑖)_𝑛−𝑝,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑞𝑧)=1+(^(𝑖)_𝑛−𝑝)^𝑟(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑞)^𝑟(𝑧^(𝑖)_𝑛−𝑝)^𝑟+(𝑧^{(𝜃(𝑖+1))}_𝑛−𝑞)^𝑟,𝑖=1,2,…,𝑘,𝑛∈ℕ₀, (3.11) which indicates that (𝐙_𝑛)^∞_𝑛=−𝜇 is a positive solution to system (1.2), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇. Hence by Theorem 3.2, we have lim_𝑛→∞𝐙_𝑛=𝐙 (3.12) and then by the transformation (2.4) we easily get lim_𝑛→∞𝐙_𝑛=𝐙. (3.13) The proof is complete.

4. Attractivity of (1.4) and (1.5)

Similar to the proofs of Lemmas 2.1–2.7, we can get the following lemmas. We omit their proofs.

Lemma 4.1. Define a mapping ℳ∶(ℝ⁺)³→ℝ by ℳ(𝑥₀,𝑥₁,𝑥₂)=(1+𝑥^𝑟₀𝑥^𝑟₁+𝑥^𝑟₀𝑥^𝑟₂+𝑥^𝑟₁𝑥^𝑟₂)/(𝑥^𝑟₀+𝑥^𝑟₁+𝑥^𝑟₂+𝑥^𝑟₀𝑥^𝑟₁𝑥^𝑟₂), where 𝑟∈(0,1]. Then for 𝑗=0,1,2 we have (1)ℳ(𝑥₀,𝑥₁,𝑥₂) is nonincreasing in 𝑥_𝑗 if 0<𝑥_𝜔(𝑗+1), 𝑥_𝜔(𝑗+2)≤1 or 𝑥_𝜔(𝑗+1),𝑥_𝜔(𝑗+2)≥1;(2)ℳ(𝑥₀,𝑥₁,𝑥₂) is nondecreasing in 𝑥_𝑗 if 𝑥_𝜔(𝑗+1)≥1, 0<𝑥_𝜔(𝑗+2)≤1 or 𝑥_𝜔(𝑗+2)≥1, 0<𝑥_𝜔(𝑗+1)≤1, where 𝜔(𝑠)=𝑠(mod3).

Lemma 4.2. For the mapping ℳ(𝑥₀,𝑥₁,𝑥₂) defined in Lemma 4.1, we have ℳ(𝑥₀,𝑥₁,𝑥₂1)=ℳ(𝑥₀,𝑥₁,𝑥₂). (4.1)

Lemma 4.3. Let 𝒩∶(ℝ⁺)³→ℝ⁺ such that 𝒩(𝑥₀,𝑥₁,𝑥₂)=1/ℳ(𝑥₀,𝑥₁,𝑥₂), then we have 𝒩(𝑥₀,𝑥₁,𝑥₂)=𝒩(𝑥₀,𝑥₁,𝑥₂). (4.2)

Let (𝐙_𝑛)^∞_𝑛=−𝛿 be a positive solution to the system (1.5), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(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 𝑧_𝑛^(𝑖)=1ℳ(𝑧^(𝑖)_𝑛−𝑣,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑝,𝑧^{(𝜃(𝑖+2))}_𝑛−𝑞),𝑖=1,2,…,𝑘,𝑛∈ℕ₀. (4.3)

Lemma 4.5. For the function ℳ(𝑥₀,𝑥₁,𝑥₂) defined in Lemma 4.1, there hold 1≤ℳ(𝑥₀,𝑥₁,𝑥₂)≤max{𝑥₀,𝑥₁,𝑥₂},𝑥₀,𝑥₁,𝑥₂>0. (4.4)

The following corollary follows directly from Corollary 4.4 and Lemma 4.5.

Corollary 4.6. For the transformed sequence (𝐙_𝑛)^∞_𝑛=−𝛿, we get 1≤𝑧_𝑛^(𝑖)≤max{𝑧^(𝑖)_𝑛−𝑣,𝑧^{(𝜃(𝑖+1))}_𝑛−𝑝,𝑧^{(𝜃(𝑖+2))}_𝑛−𝑞},𝑖=1,2,…,𝑘,𝑛∈ℕ₀. (4.5)

Let𝑄_𝑛=max_{𝑛−𝛿≤𝑗≤𝑛−1}{max_{1≤𝑖≤𝑘}{𝑧_𝑗^(𝑖)}},𝑛∈ℕ₀, (4.6) for a positive solution (𝐙_𝑛)^∞_𝑛=−𝛿 to the system (1.5), where 𝛿=max{𝑣,𝑝,𝑞}.

Lemma 4.7. The sequence (𝑄_𝑛)^∞_𝑛=0 is monotonically nonincreasing.

Note that, by the proof of Lemma 3.1 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 (𝐙_𝑛)^∞_𝑛=−𝛿 be an arbitrary positive solution to system (1.5), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇, then we need to prove that lim_𝑛→∞𝐙_𝑛=𝐙. (4.7) Define a transformed sequence (𝐙_𝑛)^∞_𝑛=−𝛿 by (2.4) and (2.7), where 𝐙_𝑛=(𝑧_𝑛⁽¹⁾,𝑧_𝑛⁽²⁾,…,𝑧_𝑛^(𝑘))^𝑇, then it suffices to confirm that lim_𝑛→∞𝐙_𝑛=𝐙. (4.8) Let the sequence (𝑄_𝑛)^∞_𝑛=0 be defined by (4.6); then by using Lemma 4.7 we know that the limit of (𝑄_𝑛)^∞_𝑛=0 exists, say 𝑄. Note that 𝑄≥1. By Corollary 4.6 we get 𝑧_𝑗^(𝑖)∈[1,𝑄_𝑗], for all 𝑖=1,2,…,𝑘 and 𝑗=0,1,2,…. Obviously, it suffices to show that 𝑄=1.
Assume that 𝑄>1; then by Lemma 4.7 and (4.6), for arbitrary 𝜀>0, there exists a sufficiently large 𝑁∈ℕ such that 𝑧_𝑁^(𝛽)∈[𝑄,𝑄+𝜀], for some 𝛽∈{1,2,…,𝑘} and𝑧_𝑛^(𝑖)∈[1,𝑄+𝜀],𝑛≥𝑁−𝛿,𝑖=1,2,…,𝑘. (4.9) Employing Lemma 4.1 three times and (4.9), we get 𝑄≤𝑧_𝑁^(𝛽)=1ℳ(𝑧^(𝛽)_𝑁−𝑣,𝑧^{(𝜃(𝛽+1))}_𝑁−𝑝,𝑧^{(𝜃(𝛽+2))}_𝑁−𝑞)≤1ℳ(𝑄+𝜀,𝑧^{(𝜃(𝛽+1))}_𝑁−𝑝,𝑧^{(𝜃(𝛽+2))}_𝑁−𝑞)≤1ℳ(𝑄+𝜀,𝑄+𝜀,𝑧^{(𝜃(𝛽+2))}_𝑁−𝑞)≤1=ℳ(𝑄+𝜀,𝑄+𝜀,𝑄+𝜀)3(𝑄+𝜀)^𝑟+(𝑄+𝜀)^3𝑟1+3(𝑄+𝜀)^2𝑟. (4.10) Since 𝜀>0 is arbitrary, we get 𝑄≤lim_𝜀→0+1=1ℳ(𝑄+𝜀,𝑄+𝜀,𝑄+𝜀)=ℳ(𝑄,𝑄,𝑄)3𝑄^𝑟+𝑄^3𝑟1+3𝑄^2𝑟. (4.11) Since 𝑄>1; then by Lemma 2.2 in [3], we have that 3𝑄^𝑟+𝑄^3𝑟1+3𝑄^2𝑟<𝑄 (4.12) 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 Theorem 4.8 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 𝑧_𝑛^(𝑖)=1+(𝑧^{(𝜃(𝑖−1))}_𝑛−𝑙)^𝑟(𝑧^(𝑖)_𝑛−𝑚)^𝑟(𝑧^{(𝜃(𝑖−1))}_𝑛−𝑙)^𝑟+(𝑧^(𝑖)_𝑛−𝑚)^𝑟,𝑖=1,2,…,𝑘,𝑛∈ℕ₀, (5.1) 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 𝑧_𝑛^(𝑖)=(𝑧^{(𝜃(𝑖−1))}_𝑛−𝑝)^𝑟+(𝑧^(𝑖)_𝑛−𝑞)^𝑟1+(𝑧^{(𝜃(𝑖−1))}_𝑛−𝑝)^𝑟(𝑧^(𝑖)_𝑛−𝑞)^𝑟,𝑖=1,2,…,𝑘,𝑛∈ℕ₀, (5.2) where 𝑟∈(0,1],𝑘,𝑝,𝑞∈ℕ and 𝑘≥2, converges to the unique positive equilibrium point 𝐙=(1,1,…,1_𝑘)^𝑇.

Acknowledgments

The authors are indebted to the anonymous referees for their much valuable advice resulting in numerous improvements of the text. This work was financially supported by National Natural Science Foundation of China (no. 10771227). The research of the third author was partially supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, project no. 144013.