On the Global Asymptotic Stability of a Second-Order System of Difference Equations

Abstract

A sufficient condition is obtained for the global asymptotic stability of the following system of difference equations 𝑧_𝑛+1=(𝑡_𝑛𝑧_𝑛−1+𝑎)/(𝑡_𝑛+𝑧_𝑛−1),𝑡_𝑛+1=(𝑧_𝑛𝑡_𝑛−1+𝑎)/(𝑧_𝑛+𝑡_𝑛−1),𝑛=0,1,2,…, where the parameter 𝑎𝜖(0,∞) and the initial values (𝑧_𝑘,𝑡_𝑘)𝜖(0,∞) (for 𝑘=−1,0).

1. Introduction

Recently, there has been an increasing interest in the study of qualitative analyses of rational difference equations and systems of difference equations. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so forth. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the global behaviors of their solutions (see [1–15] and the references cited therein).

In [9, 10] Papaschinopoluos and Schinas studied the behavior of the positive solutions of the system of two Lyness difference equations𝑥_𝑛+1=𝑏𝑦_𝑛+𝑐𝑥_𝑛−1,𝑦_𝑛+1=𝑑𝑥_𝑛+𝑒𝑦_𝑛−1,𝑛=0,1,2,…, (1.1)where 𝑏,𝑐,𝑑,𝑒 are positive constants and initial values 𝑥₋₁,𝑥₀,𝑦₋₁,𝑦₀ are positive.

Iričanin and Stević [6] studied, among others, the following system:𝑥⁽¹⁾_𝑛+1=1+𝑥_𝑛^(𝑘)𝑥^(𝑘−1)_𝑛−1,𝑥⁽²⁾_𝑛+1=1+𝑥_𝑛⁽¹⁾𝑥^(𝑘)_𝑛−1,…,𝑥^(𝑘)_𝑛+1=1+𝑥_𝑛^(𝑘−1)𝑥^(𝑘−2)_𝑛−1, (1.2)where 𝑘∈𝑁.

In [2], Amleh et al. proved that all positive solutions of the difference equations𝑥_𝑛+1=𝑥_𝑛+𝑥_𝑛−1𝑥_𝑛−2𝑥_𝑛𝑥_𝑛−1+𝑥_𝑛−2,𝑥_𝑛+1=𝑥_𝑛−1+𝑥_𝑛𝑥_𝑛−2𝑥_𝑛𝑥_𝑛−1+𝑥_𝑛−2,𝑥_𝑛+1=𝑥_𝑛+𝑥_𝑛−1𝑥_𝑛−2𝑥_𝑛𝑥_𝑛−2+𝑥_𝑛−1, (1.3)where initial values 𝑥₋₂,𝑥₋₁,𝑥₀ are positive, converge to 1 as 𝑛→∞.

Moreover, Xianyi and Deming [8] proved that the unique positive equilibrium of the difference equation𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−1+𝑎𝑥_𝑛+𝑥_𝑛−1,𝑛=0,1,2,…, (1.4)where 𝑎∈[0,∞) and 𝑥₋₁,𝑥₀ are positive, is globally asymptotically stable.

In [1], we extended the results obtained in [8] to the following difference equation:𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−𝑘+𝑎𝑥_𝑛+𝑥_𝑛−𝑘,𝑛=0,1,2,…, (1.5)where 𝑘 is nonnegative integer, 𝑎∈[0,∞) and 𝑥_−𝑘,…,𝑥₀ are positive and are globally asymptotically stable.

Also in [14], we extended the results obtained in [8] to the following system of difference equations:𝑧_𝑛+1=𝑧_𝑛𝑡_𝑛−1+𝑎𝑧_𝑛+𝑡_𝑛−1,𝑡_𝑛+1=𝑡_𝑛𝑧_𝑛−1+𝑎𝑡_𝑛+𝑧_𝑛−1,𝑛=0,1,2,…, (1.6)where 𝑎∈(0,∞) and the initial values (𝑧_𝑘,𝑡_𝑘)∈(0,∞) (for 𝑘=−1,0) are globally asymptotically stable.

In this paper, we consider the following system of difference equations:𝑧_𝑛+1=𝑡_𝑛𝑧_𝑛−1+𝑎𝑡_𝑛+𝑧_𝑛−1,𝑡_𝑛+1=𝑧_𝑛𝑡_𝑛−1+𝑎𝑧_𝑛+𝑡_𝑛−1,𝑛=0,1,2,…, (1.7)where 𝑎∈(0,∞) and the initial values (𝑧_𝑘,𝑡_𝑘)∈(0,∞) (for 𝑘=−1,0). Our main aim is to investigate the global asymptotic behavior of its solutions.

It is clear that the change of variables(𝑧_𝑛,𝑡_𝑛)=(√𝑎𝑥_𝑛,√𝑎𝑦_𝑛) (1.8)reduces the system (1.7) to the system𝑥_𝑛+1=𝑦_𝑛𝑥_𝑛−1+1𝑦_𝑛+𝑥_𝑛−1,𝑦_𝑛+1=𝑥_𝑛𝑦_𝑛−1+1𝑥_𝑛+𝑦_𝑛−1,𝑛=0,1,2,…, (1.9)where the initial values (𝑥_𝑘,𝑦_𝑘)∈(0,∞) for (𝑘=−1,0).

We need the following definitions and theorem.

Let 𝐼 be some interval of real numbers and let𝑓,𝑔∶𝐼×𝐼→𝐼 (1.10)be continuously differentiable functions. Then, for all initial values (𝑥_𝑘,𝑦_𝑘)𝜖𝐼,𝑘=−1,0, the system of difference equations𝑥_𝑛+1=𝑓(𝑥_𝑛,𝑦_𝑛−1),𝑦_𝑛+1=𝑔(𝑦_𝑛,𝑥_𝑛−1),𝑛=0,1,2,…, (1.11)has a unique solution {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1.Definition 1.1. A point (𝑥,𝑦) is called an equilibrium point of the system (1.11) if𝑥=𝑓(𝑥,𝑦),𝑦=𝑔(𝑥,𝑦). (1.12)

It is easy to see that the system (1.9) has the unique positive equilibrium (𝑥,𝑦)=(1,1) [7].Definition 1.2. Let (𝑥,𝑦) be an equilibrium point of the system (1.11).
(a) An equilibrium point (𝑥,𝑦) is said to be stable if for any 𝜀>0 there is 𝛿>0 such that for every initial points (𝑥₋₁,𝑦₋₁) and (𝑥₀,𝑦₀) for which ‖(𝑥₋₁,𝑦₋₁)−(𝑥,𝑦)‖+‖(𝑥₀,𝑦₀)−(𝑥,𝑦)‖<𝛿, the iterates (𝑥_𝑛,𝑦_𝑛) of (𝑥₋₁,𝑦₋₁) and (𝑥₀,𝑦₀) satisfy ‖(𝑥_𝑛,𝑦_𝑛)−(𝑥,𝑦)‖<𝜀 for all 𝑛>0. An equilibrium point (𝑥,𝑦) is said to be unstable if it is not stable (the Euclidean norm in ℝ² given by ‖(𝑥,𝑦)‖=𝑥²+𝑦²) is denoted by ‖⋅‖.
(b) An equilibrium point (𝑥,𝑦) is said to be asymptotically stable if there exists 𝑟>0 such that (𝑥_𝑛,𝑦_𝑛)→(𝑥,𝑦) as 𝑛→∞ for all (𝑥₋₁,𝑦₋₁) and (𝑥₀,𝑦₀) that satisfy ‖(𝑥₋₁,𝑦₋₁)−(𝑥,𝑦)‖+‖(𝑥₀,𝑦₀)−(𝑥,𝑦)‖<𝑟 [7].Definition 1.3. Let (𝑥,𝑦) be an equilibrium point of a map 𝐹=(𝑓,𝑔), where 𝑓 and 𝑔 are continuously differentiable functions at (𝑥,𝑦). The Jacobian matrix of 𝐹 at (𝑥,𝑦) is the matrix𝐽_𝐹(𝑥,𝑦)=𝜕𝑓(𝜕𝑥𝑥,𝑦)𝜕𝑓(𝜕𝑦𝑥,𝑦)𝜕𝑔(𝜕𝑥𝑥,𝑦)𝜕𝑔(𝜕𝑦𝑥,𝑦). (1.13)
The linear map 𝐽_𝐹(𝑥,𝑦)∶ℝ²→ℝ² given by𝐽_𝐹(𝑝,𝑞)(𝑥,𝑦)=𝜕𝑓(𝜕𝑥𝑥,𝑦)𝑥+𝜕𝑓(𝜕𝑦𝑥,𝑦)𝑦𝜕𝑔(𝜕𝑥𝑥,𝑦)𝑥+𝜕𝑔(𝜕𝑦𝑥,𝑦)𝑦 (1.14)is called the linearization of the map𝐹 at (𝑥,𝑦) [7].Theorem 1.4 (linearized stability theorem [7]). Let 𝐹=(𝑓,𝑔) be a continuously differentiable function defined on an open set 𝐼 in ℝ², and let (𝑥,𝑦) in 𝐼 be an equilibrium point of the map 𝐹=(𝑓,𝑔).
(a)If all the eigenvalues of the Jacobian matrix 𝐽_𝐹(𝑥,𝑦) have modulus less than one, then the equilibrium point (𝑥,𝑦) is asymptotically stable.(b)If at least one of the eigenvalues of the Jacobian matrix 𝐽_𝐹(𝑥,𝑦) has modulus greater than one, then the equilibrium point (𝑥,𝑦) is unstable.(c)An equilibrium point (𝑥,𝑦) of the map 𝐹=(𝑓,𝑔) is locally asymptotically stable if and only if every solution of the characteristic equation𝜆²−tr𝐽_𝐹(𝑥,𝑦)𝜆+det𝐽_𝐹(𝑥,𝑦)=0 (1.15)lies inside the unit circle, that is, if and only if|tr𝐽_𝐹(𝑥,𝑦)|<1+det𝐽_𝐹(𝑥,𝑦)<2. (1.16)Definition 1.5. Let (𝑥,𝑦) be positive equilibrium point of the system (1.11).
A “string” of consecutive terms {𝑥_𝑠,…,𝑥_𝑚} (resp., {𝑦_𝑠,…,𝑦_𝑚}),𝑠≥−1,𝑚≤∞ is said to be a positive semicycle if 𝑥_𝑖⩾𝑥 (resp., 𝑦_𝑖⩾𝑦), 𝑖𝜖{𝑠,…,𝑚},𝑥_𝑠−1<𝑥 (resp., 𝑦_𝑠−1<𝑦), and 𝑥_𝑚+1<𝑥 (resp., 𝑦_𝑚+1<𝑦).
A “string” of consecutive terms {𝑥_𝑠,…,𝑥_𝑚} (resp., {𝑦_𝑠,…,𝑦_𝑚}),𝑠≥−1,𝑚≤∞ is said to be a negative semicycle if 𝑥_𝑖<𝑥 (resp., 𝑦_𝑖<𝑦), 𝑖𝜖{𝑠,…,𝑚},𝑥_𝑠−1⩾𝑥 (resp., 𝑦_𝑠−1⩾𝑦), and 𝑥_𝑚+1⩾𝑥 (resp., 𝑦_𝑚+1⩾𝑦).
A “string” of consecutive terms {(𝑥_𝑠,𝑦_𝑠),…,(𝑥_𝑚,𝑦_𝑚)} is said to be a positive (resp., negative) semicycle if {𝑥_𝑠,…,𝑥_𝑚},{𝑦_𝑠,…,𝑦_𝑚} are positive (resp., negative) semicycles. Finally, a “string” of consecutive terms {(𝑥_𝑠,𝑦_𝑠),…,(𝑥_𝑚,𝑦_𝑚)} is said to be a semicycle positive (resp., negative) with respect to 𝑥_𝑛 and negative (resp., positive) with respect to 𝑦_𝑛 if {𝑥_𝑠,…,𝑥_𝑚} is a positive (resp., negative) semicycle and {𝑦_𝑠,…,𝑦_𝑚} is a negative (resp., positive) semicycle [9].

We now make new definitions. These definitions can be used for different subsequences of {𝑥_𝑛} (resp., {𝑦_𝑛}).Definition 1.6. Let (𝑥,𝑦) be positive equilibrium point of the system (1.11).
A “string” of consecutive terms {𝑥_2𝑠,𝑥_2𝑠+2,…,𝑥_2𝑚} ( resp., {𝑦_2𝑠,…,𝑦_2𝑚}),𝑠≥1,𝑚≤∞ is said to be a positive sub-semicycle associated with {𝑥_2𝑛} (resp., {𝑦_2𝑛}) if 𝑥_𝑖⩾𝑥 (resp., 𝑦_𝑖⩾𝑦), 𝑖𝜖{2𝑠,2𝑠+2,…,2𝑚},𝑥_2𝑠−2<𝑥 (resp., 𝑦_2𝑠−2<𝑦), and 𝑥_2𝑚+2<𝑥 (resp., 𝑦_2𝑚+2<𝑦).
A “string” of consecutive terms {𝑥_2𝑠,𝑥_2𝑠+2,…,𝑥_2𝑚} ( resp., {𝑦_2𝑠,…,𝑦_2𝑚}),𝑠≥1,𝑚≤∞ is said to be a negative sub-semicycle associated with {𝑥_2𝑛} (resp., {𝑦_2𝑛}) if 𝑥_𝑖<𝑥 (resp., 𝑦_𝑖<𝑦), 𝑖𝜖{2𝑠,2𝑠+2,…,2𝑚},𝑥_2𝑠−2⩾𝑥 (resp., 𝑦_2𝑠−2⩾𝑦), and 𝑥_2𝑚+2⩾𝑥 (resp., 𝑦_2𝑚+2⩾𝑦).
A “string” of consecutive terms {(𝑥_2𝑠,𝑦_2𝑠),(𝑥_2𝑠+2,𝑦_2𝑠+2),…,(𝑥_2𝑚,𝑦_2𝑚)} is said to be a positive (resp., negative) sub-semicycle if {𝑥_2𝑠,𝑥_2𝑠+2,…,𝑥_2𝑚},{𝑦_2𝑠,…,𝑦_2𝑚} are positive (resp., negative) sub-semicycles. Finally, a “string” of consecutive terms {(𝑥_2𝑠,𝑦_2𝑠),(𝑥_2𝑠+2,𝑦_2𝑠+2),…,(𝑥_2𝑚,𝑦_2𝑚)} is said to be a sub-semicycle positive (resp., negative) with respect to 𝑥_2𝑛 and negative (resp., positive) with respect to 𝑦_2𝑛 if {𝑥_2𝑠,𝑥_2𝑠+2,…,𝑥_2𝑚} is a positive (resp., negative) sub-semicycle and {𝑦_2𝑠,…,𝑦_2𝑚} is a negative (resp., positive) sub-semicycle.Definition 1.7. Let (𝑥,𝑦) be positive equilibrium point of the system (1.11).
A “string” of consecutive terms {𝑥_2𝑠−1,𝑥_2𝑠+1,…,𝑥_2𝑚+1} (resp., {𝑦_2𝑠−1,…,𝑦_2𝑚+1}),𝑠≥1,𝑚≤∞, is said to be a positive sub-semicycle associated with {𝑥_2𝑛−1} (resp., {𝑦_2𝑛−1}) if 𝑥_𝑖⩾𝑥 (resp., 𝑦_𝑖⩾𝑦), 𝑖𝜖{2𝑠+1,2𝑠+1,…,2𝑚+1},𝑥_2𝑠−3<𝑥 (resp., 𝑦_2𝑠−3<𝑦), and 𝑥_2𝑚+3<𝑥 (resp., 𝑦_2𝑚+3<𝑦).
A “string” of consecutive terms {𝑥_2𝑠−1,𝑥_2𝑠+1,…,𝑥_2𝑚+1} (resp., {𝑦_2𝑠−1,…,𝑦_2𝑚+1}),𝑠≥1,𝑚≤∞, is said to be a negative sub-semicycle associated with {𝑥_2𝑛−1} (resp., {𝑦_2𝑛−1}) if 𝑥_𝑖<𝑥 (resp., 𝑦_𝑖<𝑦), 𝑖𝜖{2𝑠−1,2𝑠+1,…,2𝑚+1},𝑥_2𝑠−3⩾𝑥 (resp., 𝑦_2𝑠−3⩾𝑦), and 𝑥_2𝑚+3⩾𝑥 (resp., 𝑦_2𝑚+3⩾𝑦).
A “string” of consecutive terms {(𝑥_2𝑠−1,𝑦_2𝑠−1),(𝑥_2𝑠+1,𝑦_2𝑠+1),…,(𝑥_2𝑚+1,𝑦_2𝑚+1)} is said to be a positive (resp., negative) sub-semicycle if {𝑥_2𝑠−1,𝑥_2𝑠+1,…,𝑥_2𝑚+1},{𝑦_2𝑠−1,…,𝑦_2𝑚+1} are positive (resp., negative) sub-semicycles. Finally, a “string” of consecutive terms {(𝑥_2𝑠−1,𝑦_2𝑠−1),(𝑥_2𝑠+1,𝑦_2𝑠+1),…,(𝑥_2𝑚+1,𝑦_2𝑚+1)} is said to be a sub-semicycle positive (resp., negative) with respect to 𝑥_2𝑛−1 and negative (resp., positive) with respect to 𝑦_2𝑛−1, if {𝑥_2𝑠−1,𝑥_2𝑠+1,…,𝑥_2𝑚+1} is a positive (resp., negative) sub-semicycle and {𝑦_2𝑠−1,𝑦_2𝑠+1,…,𝑦_2𝑚+1} is a negative (resp., positive) sub-semicycle.

2. Some Auxiliary Results

In this section, we give the following lemmas which show us the behavior of semicycles of positive solutions of system (1.9). The proof of Lemma 2.1 is clear from (1.9). So, it will be omitted.Lemma 2.1. Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 is a solution of the system (1.9) and consider the following cases:
(Case a) 𝑥₀=𝑥₋₁=1;
(Case b) 𝑦₀=𝑦₋₁=1;
(Case c) 𝑥₀=𝑦₀=1;
(Case d) 𝑥₋₁=𝑦₋₁=1.
If one of the above cases occurs, then every positive solution of system (1.9) is equal to (1,1).Lemma 2.2. Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 is a positive solution of the system (1.9) which is not eventually equal to (1,1). Then the following statements are true:
(i)(𝑥_𝑛+1−𝑥_𝑛−1)(𝑥_𝑛−1−1)<0𝑎𝑛𝑑(𝑦_𝑛+1−𝑦_𝑛−1)(𝑦_𝑛−1−1)<0𝑓𝑜𝑟𝑎𝑙𝑙𝑛≥0;(ii)(𝑥_𝑛+1−1)(𝑥_𝑛−1−1)(𝑦_𝑛−1)>0𝑎𝑛𝑑(𝑦_𝑛+1−1)(𝑦_𝑛−1−1)(𝑥_𝑛−1)>0𝑓𝑜𝑟𝑎𝑙𝑙𝑛≥0.Proof. In view of system (1.9), we obtain𝑥_𝑛+1−𝑥_𝑛−1=(1−𝑥_𝑛−1)(1+𝑥_𝑛−1)𝑦_𝑛+𝑥_𝑛−1,𝑦_𝑛+1−𝑦_𝑛−1=(1−𝑦_𝑛−1)(1+𝑦_𝑛−1)𝑥_𝑛+𝑦_𝑛−1,𝑥_𝑛+1−1=(𝑥_𝑛−1−1)(𝑦_𝑛−1)𝑦_𝑛+𝑥_𝑛−1,𝑦_𝑛+1−1=(𝑦_𝑛−1−1)(𝑥_𝑛−1)𝑥_𝑛+𝑦_𝑛−1 (2.1) for 𝑛=0,1,2,…, from which the inequalities in (i) and (ii) follow.Lemma 2.3. Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 is a solution of system (1.9) and suppose that the case,
(Case 1) 𝑥_𝑘,𝑦_𝑘>1 ( for 𝑘=−1,0), holds.
Then, (𝑥_2𝑛−1,𝑦_2𝑛−1) and (𝑥_2𝑛,𝑦_2𝑛) are positive sub-semicycles of system (1.9) with an infinite number of terms and they monotonically tend to the positive equilibrium (𝑥,𝑦)=(1,1).Proof. If 𝑥_𝑘,𝑦_𝑘>1 (for 𝑘=−1,0), then by Lemma 2.2(ii), it follows that𝑥_2𝑛−1,𝑥_2𝑛,𝑦_2𝑛−1,𝑦_2𝑛>1∀𝑛≥0, (2.2)that is, these positive sub-semicycles have an infinite number of terms. Furthermore, according to Lemma 2.2(i), we know that (𝑥_2𝑛−1,𝑦_2𝑛−1) and (𝑥_2𝑛,𝑦_2𝑛) are strictly decreasing for all 𝑛≥0. So, the limitslim_𝑛→∞𝑥_2𝑛−1=𝑙₁,lim_𝑛→∞𝑥_2𝑛=𝑙₂,lim_𝑛→∞𝑦_2𝑛−1=𝑙₃,lim_𝑛→∞𝑦_2𝑛=𝑙₄. (2.3) exist and are finite. From (1.9), we can write𝑥_2𝑛+1=𝑦_2𝑛𝑥_2𝑛−1+1𝑦_2𝑛+𝑥_2𝑛−1,𝑦_2𝑛+1=𝑥_2𝑛𝑦_2𝑛−1+1𝑥_2𝑛+𝑦_2𝑛−1,𝑛=0,1,2,…, (2.4) taking limits on both sides of (2.4), we have𝑙₁=𝑙₄𝑙₁+1𝑙₄+𝑙₁,𝑙₃=𝑙₂𝑙₃+1𝑙₂+𝑙₃, (2.5) and thus 𝑙₁=𝑙₃=1. Similarly, one can see that 𝑙₂=𝑙₄=1. Therefore, the proof is complete.Lemma 2.4. Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 is a solution of system (1.9), and consider the following cases:
(Case 2) 𝑥₋₁,𝑦₋₁>1 and 𝑥₀,𝑦₀<1;
(Case 3) 𝑥₋₁,𝑦₋₁<1 and 𝑥₀,𝑦₀>1;
(Case 4) 𝑥₋₁,𝑥₀<1 and 𝑦₋₁,𝑦₀>1;
(Case 5) 𝑥₋₁,𝑥₀>1 and 𝑦₋₁,𝑦₀<1;
(Case 6) 𝑥₋₁,𝑦₋₁,𝑦₀<1 and 𝑥₀>1;
(Case 7) 𝑥₀,𝑦₋₁,𝑦₀<1 and 𝑥₋₁>1;
(Case 8) 𝑥₋₁,𝑥₀,𝑦₋₁<1 and 𝑦₀>1;
(Case 9) 𝑥₋₁,𝑥₀,𝑦₀<1 and 𝑦₋₁>1;
(Case 10) 𝑥₋₁,𝑥₀,𝑦₋₁,𝑦₀<1.
If one of the above cases occurs, then the following hold.
(i)Every positive sub-semicycle associated with {𝑥_2𝑛−1} and {𝑥_2𝑛} (resp., {𝑦_2𝑛−1} and {𝑦_2𝑛}) of system (1.9) consists of one term.(ii)Every negative sub-semicycle associated with {𝑥_2𝑛−1} and {𝑥_2𝑛} (resp., {𝑦_2𝑛−1} and {𝑦_2𝑛}) of system (1.9) consists of two terms.(iii)Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.(iv)Every negative sub-semicycle of length two is followed by a positive sub-semicycle of length one.Proof. If Case 2 occurs, then in view of inequality (ii) of Lemma 2.2 we have: 𝑥₁,𝑦₁<1;𝑥₃,𝑦₃<1 and𝑥_6𝑛+5,𝑦_6𝑛+5>1;𝑥_6𝑛+7,𝑦_6𝑛+7<1,𝑥_6𝑛+9,𝑦_6𝑛+9<1∀𝑛≥0, (2.6)which imply that every positive sub-semicycle associated with {𝑥_2𝑛−1} and {𝑦_2𝑛−1} of system (1.9) of length one is followed by a negative sub-semicycle of length two, which in turn is followed by a positive sub-semicycle of length one.
Similarly, if Case 2 occurs, then in view of inequality (ii) of Lemma 2.2 we have𝑥_6𝑛+2,𝑦_6𝑛+2>1;𝑥_6𝑛+4,𝑦_6𝑛+4<1,𝑥_6𝑛+6,𝑦_6𝑛+6<1∀𝑛≥0, (2.7)which imply that every positive sub-semicycle associated with {𝑥_2𝑛} and {𝑦_2𝑛} of system (1.9) of length two is followed by a negative sub-semicycle of length four, which in turn is followed by a positive sub-semicycle of length two.
Proofs of the other cases are similar, so they will be omitted. Therefore, the proof is complete.

We omit the proofs of the following two results since they can easily be obtained in a way similar to the proof of Lemma 2.4.Lemma 2.5. Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 is a solution of system (1.9) and consider the following cases:
(Case 11) 𝑥₋₁,𝑦₋₁,𝑦₀>1 and 𝑥₀<1;
(Case 12) 𝑥₋₁,𝑥₀,𝑦₀>1 and 𝑦₋₁<1;
(Case 13) 𝑥₋₁,𝑦₀>1 and 𝑥₀,𝑦₋₁<1.
If one of the above cases occurs, then the following hold.
(i)(𝑥_2𝑛−1,𝑦_2𝑛) is a positive sub-semicycle of system (1.9) with an infinite number of terms (monotonically tend to the positive equilibrium (𝑥,𝑦)=(1,1)).(ii)Every positive sub-semicycle associated with {𝑥_2𝑛} and {𝑦_2𝑛−1} of system (1.9) consists of one term. (iii) Every negative sub-semicycle associated with {𝑥_2𝑛} and {𝑦_2𝑛−1} of system (1.9) consists of two terms.(iv)Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.(v)Every negative sub-semicycle of length two is followed by a positive sub-semicycle of length one.Lemma 2.6. Assume that {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 is a solution of system (1.9) and consider the following cases:
(Case 14) 𝑥₋₁,𝑥₀,𝑦₋₁>1 and 𝑦₀<1;
(Case 15) 𝑥₀,𝑦₋₁,𝑦₀>1 and 𝑥₋₁<1;
(Case 16) 𝑥₀,𝑦₋₁>1 and 𝑥₋₁,𝑦₀<1.
If one of the above cases occurs, then the following hold.
(i)(𝑥_2𝑛,𝑦_2𝑛−1) is a positive sub-semicycle of system (1.9) with an infinite number of terms (monotonically tend to the positive equilibrium (𝑥,𝑦)=(1,1)).(ii)Every positive sub-semicycle associated with {𝑥_2𝑛−1} and {𝑦_2𝑛} of system (1.9) consists of one term.(iii)Every negative sub-semicycle associated with {𝑥_2𝑛−1} and {𝑦_2𝑛} of system (1.9) consists of two terms.(iv)Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.(v)Every negative sub-semicycle of length two is followed by a positive sub-semicycle of length one.

3. Main Result

Theorem 3.1. The positive equilibrium point (𝑥,𝑦)=(1,1) of the system (1.9) is globally asymptotically stable.Proof. We must show that the positive equilibrium point (𝑥,𝑦)=(1,1) of the system (1.9) is both locally asymptotically stable and (𝑥_𝑛,𝑦_𝑛)→(𝑥,𝑦) as 𝑛→∞ (or equivalently (𝑥_2𝑛−1,𝑦_2𝑛−1)→(𝑥,𝑦) and (𝑥_2𝑛,𝑦_2𝑛)→(𝑥,𝑦) as 𝑛→∞). The characteristic equation of the system (1.9) about the positive equilibrium point (𝑥,𝑦)=(1,1) is𝜆²−0.𝜆+0=0, (3.1)and so it is clear from Theorem 1.4 that positive equilibrium point (𝑥,𝑦)=(1,1) of the system (1.9) is locally asymptotically stable. It remains to verify that every positive solution {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 of the system (1.9) converges to (𝑥,𝑦)=(1,1) as 𝑛→∞. Namely, we want to provelim_𝑛→∞𝑥_2𝑛=lim_𝑛→∞𝑥_2𝑛−1=𝑥=1,lim_𝑛→∞𝑦_2𝑛=lim_𝑛→∞𝑦_2𝑛−1=𝑦=1. (3.2)
If the solution {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 of (1.9) is nonoscillatory about the positive equilibrium point (𝑥,𝑦)=(1,1) of the system (1.9), then according to Lemmas 2.1 and 2.3, respectively, we know that the solution is either eventually equal to (1,1) or an eventually positive one which has an infinite number of terms and monotonically tends the positive equilibrium point (𝑥,𝑦)=(1,1) of the system (1.9) and so (3.2) holds. Therefore, it suffices to prove that (3.2) holds for strictly oscillatory solutions. Now, let {(𝑥_𝑛,𝑦_𝑛)}^∞_𝑛=−1 be strictly oscillatory about the positive equilibrium point (𝑥,𝑦)=(1,1) of the system (1.9). By virtue of Lemmas 2.2(ii) and 2.4, one can see that every positive sub-semicycle associated with {𝑥_2𝑛−1} (resp. {𝑥_2𝑛},{𝑦_2𝑛−1},{𝑦_2𝑛}) of this solution has one term, and every negative sub-semicycle associated with {𝑥_2𝑛−1} (resp., {𝑥_2𝑛},{𝑦_2𝑛−1},{𝑦_2𝑛}) except perhaps for the first has exactly two terms. Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.
We consider the sub-semicycles associated with {𝑥_2𝑛} and {𝑦_2𝑛}.
For the convenience of statement, without loss of generality, we use the following notation. We denote by 𝑥_2𝑝 and 𝑦_2𝑝 the terms of a positive sub-semicycle of length one, followed by 𝑥_2𝑝+2,𝑥_2𝑝+4 and 𝑦_2𝑝+2,𝑦_2𝑝+4 which are the terms of a negative sub-semicycle of length two. Afterwards, there are the positive sub-semicycles 𝑥_2𝑝+6 and 𝑦_2𝑝+6 in turn followed by the negative sub-semicycles, and so on.
Therefore, we have the following sequences consisting of positive and negative sub-semicycles (for 𝑛=0,1,…):{𝑥_2𝑝+6𝑛}^∞_𝑛=0,{𝑥_{2𝑝+6𝑛+2},𝑥_𝑝+6𝑛+4}^∞_𝑛=0,{𝑦_2𝑝+6𝑛}^∞_𝑛=0,{𝑦_{2𝑝+6𝑛+2},𝑦_{2𝑝+6𝑛+4}}^∞_𝑛=0. (3.3)
We have the following assertions:
(i)𝑥_{2𝑝+6𝑛+2}<𝑥_{2𝑝+6𝑛+4} and 𝑦_{2𝑝+6𝑛+2}<𝑦_{2𝑝+6𝑛+4};(ii)𝑥_2𝑝+6𝑛𝑥_{2𝑝+6𝑛+2}>1 and 𝑦_2𝑝+6𝑛𝑦_{2𝑝+6𝑛+2}>1;(iii)𝑥_{2𝑝+6𝑛+4}𝑥_{2𝑝+6𝑛+6}<1 and 𝑦_{2𝑝+6𝑛+4}𝑦_{2𝑝+6𝑛+6}<1.
In fact, inequality (i) immediately follows from Lemma 2.2(i). From the observations that𝑥_{2𝑝+6𝑛+2}=𝑦_{2𝑝+6𝑛+1}𝑥_2𝑝+6𝑛+1𝑦_{2𝑝+6𝑛+1}+𝑥_2𝑝+6𝑛>𝑦_{2𝑝+6𝑛+1}𝑥_2𝑝+6𝑛+1𝑦_{2𝑝+6𝑛+1}𝑥²_2𝑝+6𝑛+𝑥_2𝑝+6𝑛=1𝑥_2𝑝+6𝑛,𝑦_{2𝑝+6𝑛+2}=𝑥_{2𝑝+6𝑛+1}𝑦_2𝑝+6𝑛+1𝑥_{2𝑝+6𝑛+1}+𝑦_2𝑝+6𝑛>𝑥_{2𝑝+6𝑛+1}𝑦_2𝑝+6𝑛+1𝑥_{2𝑝+6𝑛+1}𝑦²_2𝑝+6𝑛+𝑦_2𝑝+6𝑛=1𝑦_2𝑝+6𝑛, (3.4) one can see that (ii) is valid.
As for (iii), it is obtained from𝑥_{2𝑝+6𝑛+6}=𝑦_{2𝑝+6𝑛+5}𝑥_{2𝑝+6𝑛+4}+1𝑦_{2𝑝+6𝑛+5}+𝑥_{2𝑝+6𝑛+4}<𝑦_{2𝑝+6𝑛+5}𝑥_{2𝑝+6𝑛+4}+1𝑦_{2𝑝+6𝑛+5}𝑥²_{2𝑝+6𝑛+4}+𝑥_{2𝑝+6𝑛+4}=1𝑥_{2𝑝+6𝑛+4},𝑦_{2𝑝+6𝑛+6}=𝑥_{2𝑝+6𝑛+5}𝑦_{2𝑝+6𝑛+4}+1𝑥_{2𝑝+6𝑛+5}+𝑦_{2𝑝+6𝑛+4}<𝑥_{2𝑝+6𝑛+5}𝑦_{2𝑝+6𝑛+4}+1𝑥_{2𝑝+6𝑛+5}𝑦²_{2𝑝+6𝑛+4}+𝑦_{2𝑝+6𝑛+4}=1𝑦_{2𝑝+6𝑛+4}, (3.5) for 𝑛=0,1,2,….
Combining the above inequalities, we derive1𝑥_2𝑝+6𝑛<𝑥_{2𝑝+6𝑛+2}<𝑥_{2𝑝+6𝑛+4}<1𝑥_{2𝑝+6𝑛+6},1𝑦_2𝑝+6𝑛<𝑦_{2𝑝+6𝑛+2}<𝑦_{2𝑝+6𝑛+4}<1𝑦_{2𝑝+6𝑛+6}. (3.6)
From (3.6), one can see that {𝑥_{2𝑝+6𝑛+2}}^∞_𝑛=0 and {𝑦_{2𝑝+6𝑛+2}}^∞_𝑛=0 are increasing with upper bound 1. So the limitslim_𝑛→∞𝑥_{2𝑝+6𝑛+2}=𝐿₁,lim_𝑛→∞𝑦_{2𝑝+6𝑛+2}=𝐿₂ (3.7) exist and are finite. Accordingly, in view of (3.6), we obtainlim_𝑛→∞𝑥_{2𝑝+6𝑛+4}=𝐿₁,lim_𝑛→∞𝑥_{2𝑝+6𝑛+6}=1𝐿₁,lim_𝑛→∞𝑦_{2𝑝+6𝑛+4}=𝐿₂,lim_𝑛→∞𝑦_{2𝑝+6𝑛+6}=1𝐿₂. (3.8)
Now, we consider the sub-semicycles associated with {𝑥_2𝑛−1} and {𝑦_2𝑛−1}.
Similarly, for the convenience of statement, without loss of generality, we use the following notation. We denote by 𝑥_2𝑝+1 and 𝑦_2𝑝+1 the terms of a positive sub-semicycle of length one, followed by 𝑥_2𝑝+3,𝑥_2𝑝+5 and 𝑦_2𝑝+3,𝑦_2𝑝+5 which are the terms of a negative sub-semicycle of length two. Afterwards, there are the positive sub-semicycles 𝑥_2𝑝+7 and 𝑦_2𝑝+7 in turn followed by the negative sub-semicycles, and so on.
Therefore, we have the following sequences consisting of positive and negative sub-semicycles (for 𝑛=0,1,…):{𝑥_{2𝑝+6𝑛+1}}^∞_𝑛=0,{𝑥_{2𝑝+6𝑛+3},𝑥_𝑝+6𝑛+5}^∞_𝑛=0,{𝑦_{2𝑝+6𝑛+1}}^∞_𝑛=0,{𝑦_{2𝑝+6𝑛+3},𝑦_{2𝑝+6𝑛+5}}^∞_𝑛=0. (3.9)
We have the following assertions:
(i)𝑥_{2𝑝+6𝑛+3}<𝑥_{2𝑝+6𝑛+5} and 𝑦_{2𝑝+6𝑛+3}<𝑦_{2𝑝+6𝑛+5};(ii)𝑥_{2𝑝+6𝑛+1}𝑥_{2𝑝+6𝑛+3}>1 and 𝑦_{2𝑝+6𝑛+1}𝑦_{2𝑝+6𝑛+3}>1;(iii)𝑥_{2𝑝+6𝑛+5}𝑥_{2𝑝+6𝑛+7}<1 and 𝑦_{2𝑝+6𝑛+5}𝑦_{2𝑝+6𝑛+7}<1.
Combining the above inequalities, we derive1𝑥_{2𝑝+6𝑛+1}<𝑥_{2𝑝+6𝑛+3}<𝑥_{2𝑝+6𝑛+5}<1𝑥_{2𝑝+6𝑛+7},1𝑦_{2𝑝+6𝑛+1}<𝑦_{2𝑝+6𝑛+3}<𝑦_{2𝑝+6𝑛+5}<1𝑦_{2𝑝+6𝑛+7}. (3.10)
From (3.10), one can see that {𝑥_{2𝑝+6𝑛+3}}^∞_𝑛=0 and {𝑦_{2𝑝+6𝑛+3}}^∞_𝑛=0 are increasing with upper bound 1. So the limitslim_𝑛→∞𝑥_{2𝑝+6𝑛+3}=𝐿₃,lim_𝑛→∞𝑦_{2𝑝+6𝑛+3}=𝐿₄ (3.11) exist and are finite. Accordingly, in view of (3.10), we obtainlim_𝑛→∞𝑥_{2𝑝+6𝑛+5}=𝐿₃,lim_𝑛→∞𝑥_{2𝑝+6𝑛+7}=1𝐿₃,lim_𝑛→∞𝑦_{2𝑝+6𝑛+5}=𝐿₄,lim_𝑛→∞𝑦_{2𝑝+6𝑛+7}=1𝐿₄. (3.12)
It suffices to verify that𝐿₁=𝐿₂=𝐿₃=𝐿₄=1. (3.13)
To this end, note that𝑥_{2𝑝+6𝑛+6}=𝑦_{2𝑝+6𝑛+5}𝑥_{2𝑝+6𝑛+4}+1𝑦_{2𝑝+6𝑛+5}+𝑥_{2𝑝+6𝑛+4},𝑦_{2𝑝+6𝑛+6}=𝑥_{2𝑝+6𝑛+5}𝑦_{2𝑝+6𝑛+4}+1𝑥_{2𝑝+6𝑛+5}+𝑦_{2𝑝+6𝑛+4}. (3.14)
Take the limits on both sides of the above equality and obtain1𝐿₁=𝐿₄⋅𝐿₁+1𝐿₄+𝐿₁,1𝐿₂=𝐿₃⋅𝐿₂+1𝐿₃+𝐿₂, (3.15) which imply that 𝐿₁=𝐿₂=1. Similarly, one can see that 𝐿₃=𝐿₄=1.
Moreover, by virtue of Lemmas 2.2(ii) and 2.5 (resp., 2.6), one can see that (3.2) holds. Therefore, the proof is complete.