Global Behavior of Two Families of Nonlinear Symmetric Difference Equations

Abstract

We mainly investigate the global asymptotic stability and exponential convergence of positive solutions to two families of higher-order difference equations, one of which was recently studied in Stević's paper (2010). A new concise proof is given to a quite recent result by Stević and analogous parallel result of the other inverse equation, which extend related results of Aloqeili (2009), Berenhaut and Stević (2007), and Liao et al. (2009).

1. Introduction

The interest in investigating rational difference equations has a long history; for instance, see [1–24] and the references cited therein. More generally, it is meaningful to study not only rational recursive equations but also those with powers of arbitrary positive degrees.

For instance, at many conferences, Stević proposed to study the behavior of positive solutions of the following generic difference equation (see also [25]): 𝑥_𝑛𝑥=𝐴+^{𝑝𝑛−𝑚}𝑥^𝑞_𝑛−𝑘,𝑛∈ℕ₀, (1.1) where 𝐴,𝑝,𝑞>0, and 𝑘,𝑚∈ℕ,𝑘≠𝑚. For some recent results in this area, see [26–32] and the references therein.

By a useful transformation method from [4], the authors of [3] confirmed that the unique positive equilibrium of the rational recursive equation 𝑦_𝑛=𝑦_𝑛−𝑘+𝑦_𝑛−𝑚1+𝑦_𝑛−𝑘𝑦_𝑛−𝑚,𝑛∈ℕ₀, (1.2) where 1≤𝑘<𝑚, is globally asymptotically stable for all solutions with positive initial values. In the meantime, they also remarked that the global asymptotic stability for the unique equilibrium of the difference equation 𝑦_𝑛=1+𝑦_𝑛−𝑘𝑦_𝑛−𝑚𝑦_𝑛−𝑘+𝑦_𝑛−𝑚,𝑛∈ℕ₀, (1.3) can be shown through analogous calculations. Some particular cases of (1.3) had already been considered in [12, 13].

In [3] were proposed the following two conjectures.

Conjecture 1.1. Suppose that 1≤𝑘<𝑙<𝑚 and that {𝑦_𝑛} satisfies 𝑦_𝑛=𝑦_𝑛−𝑘𝑦_𝑛−𝑙𝑦_𝑛−𝑚+𝑦_𝑛−𝑘+𝑦_𝑛−𝑙+𝑦_𝑛−𝑚𝑦_𝑛−𝑘𝑦_𝑛−𝑙+𝑦_𝑛−𝑘𝑦_𝑛−𝑚+𝑦_𝑛−𝑙𝑦_𝑛−𝑚+1,𝑛∈ℕ₀, (1.4) with positive initial values. Then, the sequence {𝑦_𝑛} converges to the unique positive equilibrium point 𝑦=1.

Some special cases of (1.4) had been studied by Li [9, 10] with a semicycle analysis method, which is useful for lower-order difference equations but tedious and complicated to some extent(see the explanation in [33]). Finally, Conjecture 1.1 was also confirmed in [2] with the similar transformation method used in [3, 4]. However, it is somewhat harder to prove the following conjecture in the same way.

Conjecture 1.2. Assume that 𝑞 is odd and 1≤𝑘₁<𝑘₂<⋯<𝑘_𝑞, and define 𝑆={1,2,…,𝑞}. If {𝑦_𝑛} satisfies 𝑦_𝑛=𝑓₁(𝑦_𝑛−𝑘1,𝑦_𝑛−𝑘2,…,𝑦_{𝑛−𝑘𝑞})𝑓₂(𝑦_𝑛−𝑘1,𝑦_𝑛−𝑘2,…,𝑦_{𝑛−𝑘𝑞}),𝑛∈ℕ₀, (1.5) with 𝑦_−𝑘𝑞,𝑦_{−𝑘𝑞+1},…,𝑦₋₁∈(0,+∞), where 𝑓₁(𝑥₁,𝑥₂,…,𝑥_𝑞)=^𝑞_{𝑖𝑖=1odd{𝑡1,𝑡2,…,𝑡𝑖𝑡}⊂𝑆1<𝑡2<⋯<𝑡𝑖}𝑥_𝑡1𝑥_𝑡2⋯𝑥_𝑡𝑖,𝑓₂(𝑥₁,𝑥₂,…,𝑥_𝑞)=1+^𝑞−1_{𝑖𝑖=2even{𝑡1,𝑡2,…,𝑡𝑖𝑡}⊂𝑆1<𝑡2<⋯<𝑡𝑖}𝑥_𝑡1𝑥_𝑡2⋯𝑥_𝑡𝑖. (1.6) Then the sequence {𝑦_𝑛} converges to the unique positive equilibrium point 𝑦=1.

Next, we present two definitions as defined in [1].

Definition 1.3. A function of 𝑛 variables is symmetric if it is invariant under any permutation of its variables. That is, a function 𝜑(𝑥₁,…,𝑥_𝑛) is called symmetric if 𝜑(𝑥₁,…,𝑥_𝑛)=𝜑(𝑥_𝜋(1),…,𝑥_𝜋(𝑛)), (1.7) where 𝜋(𝑖) is any permutation of the numbers {1,2,…,𝑛}.

Definition 1.4. The 𝑘th elementary symmetric function 𝜎_𝑘 of variables 𝑥₁,…,𝑥_𝑛, where 𝑘∈{1,2,…,𝑛} is defined by 𝜎_𝑘(𝑥₁,…,𝑥_𝑛)=_{𝑖1<𝑖2<⋯<𝑖𝑘}𝑥_𝑖1𝑥_𝑖2⋯𝑥_𝑖𝑘, (1.8) where the sum is taken over all 𝐶^𝑛_𝑘 choices of the indices 𝑖₁,…,𝑖_𝑘 from the set of integers {1,2,…,𝑛}.

Obviously, the functions 𝑓₁,𝑓₂ defined by (1.6) and (1.7) are symmetric and can be rewritten as 𝑓₁(𝑥₁,𝑥₂,…,𝑥_𝑞)=^𝑞_{𝑖𝑖=1odd}𝜎_𝑖,𝑓₂(𝑥₁,𝑥₂,…,𝑥_𝑞)=1+^𝑞−1_{𝑖𝑖=2even}𝜎_𝑖. (1.9)

In this paper, we give a new proof of a quite recent result by Stević in [34] where he, among others, studied the stability of one of the following two difference equations, which are dual: 𝑦_𝑛=𝑓₂(𝑦^𝑟_𝑛−𝑘1,𝑦^𝑟_𝑛−𝑘2,…,𝑦^𝑟_{𝑛−𝑘𝑞})𝑓₁(𝑦^𝑟_𝑛−𝑘1,𝑦^𝑟_𝑛−𝑘2,…,𝑦^𝑟_{𝑛−𝑘𝑞}),𝑛∈ℕ₀,𝑦_𝑛=𝑓₁(𝑦^𝑟_𝑛−𝑘1,𝑦^𝑟_𝑛−𝑘2,…,𝑦^𝑟_{𝑛−𝑘𝑞})𝑓₂(𝑦^𝑟_𝑛−𝑘1,𝑦^𝑟_𝑛−𝑘2,…,𝑦^𝑟_{𝑛−𝑘𝑞}),𝑛∈ℕ₀, (1.10)(1.11) where 3≤𝑞∈ℕ is odd, 𝑟∈(0,1] and 1≤𝑘₁<𝑘₂<⋯<𝑘_𝑞.

Apparently, Equation (1.11) is the generic form of (1.2), (1.4), and (1.5).

In [6, 18] the authors proved that the main results in some of papers [9–12] are direct consequences of a result confirmed by Kruse and Nesemann [35]. For example, in [6] was showed that the main result in [13] is also a consequence of Corollary 3 in [35]. On basis of these works, in 2008, Aloqeili [1] confirmed Conjecture 1.2 in the same way.

Later, Liao et al. [14] proved Conjecture 1.2 by using a new approach. They used a sort of “frame sequences” method(the notion suggested by Stević), which has been widely used in [5, 7, 18, 36–41]. Through careful analysis, we find that the method used in [14] can be further simplified and applied in proving Stević’s result in a more concise and interesting way. Namely, we give a new proof of the following result, which generalizes related results in [1, 2, 9, 10, 14].

Theorem 1.5. Assume that 𝑦_−𝑘𝑞,𝑦_{−𝑘𝑞+1},…,𝑦₋₁∈(0,+∞),𝑟∈(0,1],3≤𝑞∈ℕ is odd and positive integers 𝑘₁,𝑘₂,…,𝑘_𝑞 are satisfying 1≤𝑘₁<𝑘₂<⋯<𝑘_𝑞. Then(1)the unique positive equilibrium point 𝑦=1 of (1.10) is globally asymptotically stable;(2)the unique positive equilibrium point 𝑦=1 of (1.11) is globally asymptotically stable.

2. Auxiliary Results and Notation

In this section, we will introduce some useful notation and lemmas. Consider the following notation(for similar ones see [14]), which play an important role in the paper: 𝛼(𝑥₁,𝑥₂,…,𝑥_𝑞1)=2[^𝑞_𝑖=1(𝑥^𝑟_𝑖+1)−^𝑞_𝑖=1(𝑥^𝑟_𝑖−1)],𝛽(𝑥₁,𝑥₂,…,𝑥_𝑞1)=2[^𝑞_𝑖=1(𝑥^𝑟_𝑖+1)+^𝑞_𝑖=1(𝑥^𝑟_𝑖−1)]. (2.1) Employing 𝛼 and 𝛽, define a mapping Φ₁∶ℝ^𝑞₊→ℝ as follows: Φ₁(𝑥₁,𝑥₂,…,𝑥_𝑞)=𝛼(𝑥₁,…,𝑥_𝑞)𝛽(𝑥₁,…,𝑥_𝑞)=∏^𝑞_𝑖=1(𝑥^𝑟_𝑖+1)−∏^𝑞_𝑖=1(𝑥^𝑟_𝑖−1)∏^𝑞_𝑖=1(𝑥^𝑟_𝑖+1)+∏^𝑞_𝑖=1(𝑥^𝑟_𝑖.−1) (2.2) Then (1.10) can be rewritten as 𝑦_𝑛=∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}+1)−∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}−1)∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}+1)+∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}−1),𝑛∈ℕ₀, (2.3) or 𝑦_𝑛=Φ₁(𝑦_𝑛−𝑘1,𝑦_𝑛−𝑘2,…,𝑦_{𝑛−𝑘𝑞}),𝑛∈ℕ₀, (2.4) with 3≤𝑞∈ℕ being odd, and 𝑟∈ℝ₊.

By the notation defined by (2.1), define the other function Φ₂∶ℝ^𝑞₊→ℝ such that: Φ₂(𝑥₁,𝑥₂,…,𝑥_𝑞)=𝛽(𝑥₁,…,𝑥_𝑞)𝛼(𝑥₁,…,𝑥_𝑞)=∏^𝑞_𝑖=1(𝑥^𝑟_𝑖+1)+∏^𝑞_𝑖=1(𝑥^𝑟_𝑖−1)∏^𝑞_𝑖=1(𝑥^𝑟_𝑖+1)−∏^𝑞_𝑖=1(𝑥^𝑟_𝑖.−1) (2.5) Then (1.11) can be rewritten as 𝑦_𝑛=∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}+1)+∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}−1)∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}+1)−∏^𝑞_𝑖=1(𝑦^𝑟_{𝑛−𝑘𝑖}−1),𝑛∈ℕ₀, (2.6) or 𝑦_𝑛=Φ₂(𝑦_𝑛−𝑘1,𝑦_𝑛−𝑘2,…,𝑦_{𝑛−𝑘𝑞}),𝑛∈ℕ₀, (2.7) with 3≤𝑞∈ℕ being odd, and 𝑟∈ℝ₊.

Lemma 2.1. If 𝑟∈(0,1], then both (2.4) and (2.7) have the unique positive equilibrium point 𝑦=1.

Proof. Suppose that 𝜆₁>0 is an equilibrium of (2.4), then 𝜆₁=Φ₁(𝜆₁,…,𝜆₁)=(𝜆^𝑟₁+1)^𝑞−(𝜆^𝑟₁−1)^𝑞(𝜆^𝑟₁+1)^𝑞+(𝜆^𝑟₁−1)^𝑞, (2.8) which implies (1−𝜆₁)(𝜆^𝑟₁+1)^𝑞=(𝜆₁+1)(𝜆^𝑟₁−1)^𝑞. (2.9) Obviously 𝜆₁=1, due to the different signs of both sides of the last equality for the case 𝜆₁≠1. Likewise, let 𝜆₂>0 be an equilibrium of (2.7); then 𝜆₂=Φ₂(𝜆₂,…,𝜆₂)=(𝜆^𝑟₂+1)^𝑞+(𝜆^𝑟₂−1)^𝑞(𝜆^𝑟₂+1)^𝑞−(𝜆^𝑟₂−1)^𝑞, (2.10) which indicates (𝜆₂−1)(𝜆^𝑟₂+1)^𝑞=(𝜆₂+1)(𝜆^𝑟₂−1)^𝑞. (2.11) Assume that 𝜆₂≠1. If 𝜆₂>1, then by the monotonicity of the map ℎ(𝑥)=(𝑥+1)/(𝑥−1), 𝑥>1 we have that 𝜆₂+1𝜆₂≤𝜆−1^𝑟2+1𝜆^𝑟₂𝜆−1<(^𝑟2+1𝜆^𝑟₂)−1^𝑞, (2.12) which contradicts (2.11).Similarly, if 0<𝜆₂<1, then by the monotonicity of the function 𝑙(𝑥)=(1+𝑥)/(1−𝑥), 𝑥∈(0,1), we have that 1+𝜆₂1−𝜆₂≤1+𝜆^𝑟₂1−𝜆^𝑟₂<(1+𝜆^𝑟₂1−𝜆^𝑟₂)^𝑞, (2.13) which also contradicts (2.11). Thus 𝜆₂=1.
The proof is complete.

Lemma 2.2. (1) Let Φ₁ be defined by (2.2); then Φ₁ is monotonically increasing in 𝑥_𝑗 if and only if ∏^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)<0, and monotonically decreasing in 𝑥_𝑗 if and only if ∏^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)>0, for 𝑗=1,2,…,𝑞.
(2) Let Φ₂ be defined by (2.5), then Φ₂ is monotonically decreasing in 𝑥_𝑗 if and only if ∏^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)<0, and monotonically increasing in 𝑥_𝑗 if and only if ∏^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)>0, for 𝑗=1,2,…,𝑞.

Proof. The results follow directly from the facts below: 𝜕Φ₁𝜕𝑥_𝑗=−𝑟𝑥_𝑗^𝑟−1∏^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖^2𝑟−1)[𝛽(𝑥₁,𝑥₂,…,𝑥_𝑞)]²<0,^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)>0,>0,^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)<0,𝑗∈{1,…,𝑞},𝜕Φ₂𝜕𝑥_𝑗=𝑟𝑥_𝑗^𝑟−1∏^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖^2𝑟−1)[𝛼(𝑥₁,𝑥₂,…,𝑥_𝑞)]²>0,^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)>0,<0,^𝑞_{𝑖=1,𝑖≠𝑗}(𝑥_𝑖−1)<0,𝑗∈{1,…,𝑞}. (2.14)

Remark 2.3. The second statement (i.e., (2)) in Lemma 2.2 can also be found in Stević’s paper [34] (see Lemma 1 and Corollary 1).

For 𝑟∈ℝ⁺,3≤𝑞∈ℕ odd, define a map Ψ∶ℝ₊→ℝ such that Ψ(𝑥)=(1+𝑥^𝑟)^𝑞−(1−𝑥^𝑟)^𝑞(1+𝑥^𝑟)^𝑞+(1−𝑥^𝑟)^𝑞, (2.15) which has the following simple property: 1Ψ(𝑥1)=Ψ(𝑥),𝑥∈ℝ₊. (2.16)

Lemma 2.4. Suppose that 0<𝜉<1, and let Γ=Ψ(𝜉),Φ∈{Φ₁,Φ₂}. If 𝑥₁,𝑥₂,…,𝑥_𝑞∈[𝜉,1/𝜉], then Γ≤Φ(𝑥₁,𝑥₂,…,𝑥_𝑞1)≤Γ. (2.17)

Proof. Since Φ(𝑥₁,𝑥₂,…,𝑥_𝑞) is symmetric in 𝑥₁,𝑥₂,…,𝑥_𝑞, without loss of generality, we suppose that 𝜉≤𝑥₁≤𝑥₂≤⋯≤𝑥_𝑞≤1/𝜉. If there exists 𝑗∈{1,…,𝑞} such that 𝑥_𝑗=1, then by (2.2) and (2.5) we can easily get that Φ(𝑥₁,𝑥₂,…,𝑥_𝑞)=1. Thus, assume 𝑥_𝑗≠1 for all 𝑗∈{1,…,𝑞}.
Then we have the following 𝑞+1 cases to consider: (1)𝜉≤𝑥₁≤⋯≤𝑥_𝑞−1≤𝑥_𝑞1<1<𝜉;(2)𝜉≤𝑥₁≤⋯≤𝑥_𝑞−1<1<𝑥_𝑞≤1𝜉;(3)𝜉≤𝑥₁≤⋯<1<𝑥_𝑞−1≤𝑥_𝑞≤1𝜉;⋮(𝑞)𝜉≤𝑥₁<1<⋯≤𝑥_𝑞−1≤𝑥_𝑞≤1𝜉;(𝑞+1)𝜉<1<𝑥₁≤⋯≤𝑥_𝑞−1≤𝑥_𝑞≤1𝜉. (2.18) By Lemma 2.2, for the above cases, we have that (1)1<Φ₁(𝑥₁,…,𝑥_𝑞−1,𝑥_𝑞1)≤Γ;(2)Γ≤Φ₁(𝑥₁,…,𝑥_𝑞−1,𝑥_𝑞)<1;(3)1<Φ₁(𝑥₁,…,𝑥_𝑞−1,𝑥_𝑞1)≤Γ;⋮(𝑞)1<Φ₁(𝑥₁,…,𝑥_𝑞−1,𝑥_𝑞1)≤Γ;(𝑞+1)Γ≤Φ₁(𝑥₁,…,𝑥_𝑞−1,𝑥_𝑞)<1. (2.19) Obviously, Γ≤Φ₁(𝑥₁,𝑥₂,…,𝑥_𝑞)≤1/Γ follows directly from the above inequalities.
The proof of the case Φ=Φ₂ is analogous and hence omitted.

Lemma 2.5. Suppose that 0<𝑟≤1,𝜉∈(0,1) is fixed and let Γ=Ψ(𝜉). Then we have Γ>𝜉. (2.20)

Proof. By the monotonicity of the function ℎ(𝑥)=(1+𝑥)/(1−𝑥),𝑥∈(0,1), we have that 1+𝜉≤1−𝜉1+𝜉^𝑟1−𝜉^𝑟<(1+𝜉^𝑟1−𝜉^𝑟)^𝑞, (2.21) which implies (1−𝜉)(1+𝜉^𝑟)^𝑞>(1+𝜉)(1−𝜉^𝑟)^𝑞. (2.22) Therefore, ((1+𝜉^𝑟)^𝑞−(1−𝜉^𝑟)^𝑞)/((1+𝜉^𝑟)^𝑞+(1−𝜉^𝑟)^𝑞)>𝜉, that is, Γ>𝜉.
The proof is complete.

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

Corollary 2.6. Assume that 0<𝜉<1. If any positive solution (𝑦_𝑛)^+∞_{𝑛=−𝑘𝑞} to (2.4) or (2.7) has the initial values 𝑦_−𝑘𝑞,𝑦_{−𝑘𝑞+1},…,𝑦₋₁1∈[𝜉,𝜉], (2.23) then we have 𝑦_𝑛∈[𝜉,1/𝜉], for 𝑛∈ℕ₀.

Define two sequences (𝜉_𝑖)^+∞_𝑖=0 and (𝜂_𝑖)^+∞_𝑖=0 as follows: 𝜉_𝑖+1=Ψ(𝜉_𝑖),𝜂_𝑖+1=Ψ(𝜂_𝑖),𝑖∈ℕ₀, (2.24) with initial values 𝜉₀,𝜂₀>0.

Lemma 2.7. For the sequences (𝜉_𝑖)^+∞_𝑖=0and(𝜂_𝑖)^+∞_𝑖=0 defined by (2.24), if 0<𝜉₀<1, and 𝜉₀𝜂₀=1, then lim_𝑖→∞𝜉_𝑖=lim_𝑖→∞𝜂_𝑖=1. (2.25)

Proof. Inductively, we can simply obtain that 0<𝜉_𝑖<1<𝜂_𝑖<+∞, 𝑖∈ℕ₀. Through simple calculations, by (2.16), we have that 𝜉_𝑖𝜂_𝑖≡1,∀𝑖∈ℕ₀. (2.26) Therefore by Lemma 2.4 and Lemma 2.5, we get that 0<𝜉_𝑖≤𝜉_𝑖+1<1<𝜂_𝑖+1≤𝜂_𝑖,𝑖∈ℕ₀, (2.27) which implies that the sequences (𝜉_𝑖)^+∞_𝑖=0and(𝜂_𝑖)^+∞_𝑖=0 converge to some limits (denoted by 𝜉^∗ and 𝜂^∗, resp.), that is, 𝜉^∗=lim_𝑖→+∞𝜉_𝑖∈[𝜉₀,1],𝜂^∗=lim_𝑖→+∞𝜂_𝑖∈[1,𝜂₀]. (2.28) By taking limits on both sides of the first identity of (2.24), we get 𝜉^∗=(1+(𝜉^∗)^𝑟)^𝑞−(1−(𝜉^∗)^𝑟)^𝑞(1+(𝜉^∗)^𝑟)^𝑞+(1−(𝜉^∗)^𝑟)^𝑞, (2.29) which implies (1−𝜉^∗)(1+(𝜉^∗)^𝑟)^𝑞=(1+𝜉^∗)(1−(𝜉^∗)^𝑟)^𝑞. (2.30) Suppose that 𝜉^∗≠1; then by the monotonicity of the function 𝑓(𝑥)=(1+𝑥)/(1−𝑥), 𝑥∈(0,1), we have that 1+𝜉^∗1−𝜉^∗≤1+(𝜉^∗)^𝑟1−(𝜉^∗)^𝑟<(1+(𝜉^∗)^𝑟1−(𝜉^∗)^𝑟)^𝑞, (2.31) which contradicts (2.29). Hence, we have that 𝜉^∗=1 and then obviously it follows by (2.26) and (2.28) that 𝜉^∗=𝜂^∗=1.
The proof is complete.

3. Stability

In this section, we give a new, concise and clear proof of Stević’s Theorem 1.5, by the lemmas in Section 2.

Proof of Theorem 1.5. Employing Lemma 2.2, the linearized equations of (2.4) and (2.7) about the equilibrium 𝑦=1 are both 𝑧_𝑛=0⋅𝑧_𝑛−𝑘1+0⋅𝑧_𝑛−𝑘2+⋯+0⋅𝑧_{𝑛−𝑘𝑞}≡0,𝑛∈ℕ₀. (3.1) Then by the Linearized Stability Theorem, 𝑦=1 is locally stable.
Thus it suffices to confirm that 𝑦=1 is also a global attractor for all positive solutions of (2.4) and (2.7).
Let (𝑦_𝑛)^+∞_{𝑛=−𝑘𝑞} be a positive solution to (2.4) or (2.7) with initial values 𝑦_−𝑘𝑞,𝑦_{−𝑘𝑞+1},…,𝑦₋₁∈(0,+∞). (3.2) We need to prove that lim_𝑛→∞𝑦_𝑛=1.
Apparently, there exists 𝜉₀∈(0,1) such that 𝑦_𝑖∈[𝜉₀,𝜂₀],𝑖=−𝑘_𝑞,−𝑘_𝑞+1,…,−1, (3.3) where 𝜂₀=1/𝜉₀. Employing Corollary 2.6, we have 𝑦_𝑛∈[𝜉₀,𝜂₀],𝑛=−𝑘_𝑞,−𝑘_𝑞+1,…. (3.4) Let sequences (𝜉_𝑖)^+∞_𝑖=0 and (𝜂_𝑖)^+∞_𝑖=0 be defined by (2.24). Let Φ∈{Φ₁,Φ₂}; then in light of Lemma 2.4, (3.4), and (2.26), we get Ψ(𝜉₀)≤Φ(𝑦_𝑛−𝑘1,𝑦_𝑛−𝑘2,…,𝑦_{𝑛−𝑘𝑞}1)≤Ψ(𝜉₀)=Ψ(𝜂₀),𝑛∈ℕ₀. (3.5) That is, 𝑦_𝑛∈[𝜉₁,𝜂₁],𝑛∈ℕ₀. (3.6) In view of (3.6), (2.26) and Lemma 2.4, we have that Ψ(𝜉₁)≤Φ(𝑦_𝑛−𝑘1,𝑦_𝑛−𝑘2,…,𝑦_{𝑛−𝑘𝑞}1)≤Ψ(𝜉₁)=Ψ(𝜂₁),𝑛≥𝑘_𝑞. (3.7) That is, 𝑦_𝑛∈[𝜉₂,𝜂₂],𝑛≥𝑘_𝑞. (3.8) Reasoning inductively, we can get 𝑦_𝑛∈[𝜉_𝑖+1,𝜂_𝑖+1],𝑛≥𝑖𝑘_𝑞,𝑖∈ℕ₀. (3.9) By Lemma 2.7 and (3.9), we obtain 1=lim_𝑖→∞𝜉_𝑖+1≤lim_𝑛→∞𝑦_𝑛≤lim_𝑖→∞𝜂_𝑖+1=1, (3.10) which implies lim_𝑛→∞𝑦_𝑛=1. (3.11)
The proof is complete.

4. Exponential Convergence

In this section, we will prove that all positive solutions to (2.4) and (2.7) with 0<𝑟≤1 are exponentially convergent, by using an approach from paper [42].

Theorem 4.1. If 𝑟∈(0,1], then every positive solution to (2.4) and (2.7) exponentially converges to 1.

Proof. Let (𝑦_𝑛)^∞_{𝑛=−𝑘𝑞} be a positive solution to (2.4) or (2.7); then by Theorem 1.5, there exists a sufficiently large natural number 𝑁 such that for arbitrary fixed 𝜀>0 we have |𝑦_𝑛−1|<𝜀 for all 𝑛≥𝑁.
Denote 𝑇_𝑛=|𝑦_𝑛−1|,𝑛≥−𝑘_𝑞; then 𝑇_𝑛<𝜀 for all 𝑛≥𝑁.
(1). For (2.4).
Let 0<𝜀≤1−^𝑟1/𝑞; then by (2.4), we have 𝑇_𝑛=|𝑦_𝑛−1|=2∏^𝑞_𝑗=1|𝑦^𝑟_{𝑛−𝑘𝑗}−1|∏^𝑞_𝑗=1(𝑦^𝑟_{𝑛−𝑘𝑗}+1)+∏^𝑞_𝑗=1(𝑦^𝑟_{𝑛−𝑘𝑗}≤−1)2∏^𝑞_𝑗=1|𝑦^𝑟_{𝑛−𝑘𝑗}−1|2∑^𝑞_𝑗=1𝑦^𝑟_{𝑛−𝑘𝑗}≤∏^𝑞_𝑗=1|𝑦^𝑟_{𝑛−𝑘𝑗}−1|𝑞(1−𝜀)^𝑟≤^𝑞_𝑗=1|𝑦^𝑟_{𝑛−𝑘𝑗}−1|≤^𝑞_𝑗=1|𝑦_{𝑛−𝑘𝑗}=−1|_𝑞𝑗=1𝑇_{𝑛−𝑘𝑗}≤𝜀^𝑞−1𝑇_{𝑛−𝑘𝑞},𝑛≥𝑁+𝑘_𝑞. (4.1)(2). For (2.7).
Let 0<𝜀<1 be fixed; then by (2.7), we get 𝑇_𝑛=|𝑦_𝑛−1|=2∏^𝑞_𝑗=1|𝑦^𝑟_{𝑛−𝑘𝑗}−1|∏^𝑞_𝑗=1(𝑦^𝑟_{𝑛−𝑘𝑗}+1)−∏^𝑞_𝑗=1(𝑦^𝑟_{𝑛−𝑘𝑗}≤−1)_𝑞𝑗=1|𝑦^𝑟_{𝑛−𝑘𝑗}−1|≤^𝑞_𝑗=1|𝑦_{𝑛−𝑘𝑗}=−1|_𝑞𝑗=1𝑇_{𝑛−𝑘𝑗}≤𝜀^𝑞−1𝑇_{𝑛−𝑘𝑞},𝑛≥𝑁+𝑟_𝑞. (4.2)
From this inequality and Lemma 1 in [43] (see also Corollary 1 therein), the result directly follows.

5. Other Simple Results

In this section, we will present some elementary results of (2.3) and (2.6) with 𝑟>1.

Proposition 5.1. If 𝑟>1, then there is no positive solution (𝑦_𝑛)^+∞_{𝑛=−𝑘𝑞} to (2.3) such that lim_𝑛→∞𝑦_𝑛=+∞.

Proof. Suppose (𝑦_𝑛)^+∞_{𝑛=−𝑘𝑞} is a positive solution to (2.3) such that lim_𝑛→∞𝑦_𝑛=+∞. (5.1) Then for some fixed 𝑀>1, there exists 𝑁∈ℕ such that ∀𝑛≥𝑁,𝑦_𝑛>𝑀. (5.2) Employing (2.3) and (5.2), we can simply get that 𝑦_{𝑁+𝑘𝑞}=∏^𝑞_𝑗=1(𝑦^𝑟_{𝑁+𝑘𝑞−𝑘𝑗}+1)−∏^𝑞_𝑗=1(𝑦^𝑟_{𝑁+𝑘𝑞−𝑘𝑗}−1)∏^𝑞_𝑗=1(𝑦^𝑟_{𝑁+𝑘𝑞−𝑘𝑗}+1)+∏^𝑞_𝑗=1(𝑦^𝑟_{𝑁+𝑘𝑞−𝑘𝑗}−1)<1 (5.3) which contradicts (5.2). The proof is complete.

Proposition 5.2. We have the following simple statements:(1)if 𝑟>1, then (2.6) has nonoscillatory positive solutions with all initial values 𝑦_𝑖≥1, −𝑘_𝑞≤𝑖≤−1, or 0<𝑦_𝑖<1,−𝑘_𝑞≤𝑖≤−1;(2)let 𝐻={𝑦_𝑖∣𝑦_𝑖≠1,−𝑘_𝑞≤𝑖≤−1} and denote by ‖𝐻‖ the cardinality of the set 𝐻. If ‖𝐻‖<𝑞, then for any positive solution (𝑦_𝑛)^+∞_{𝑛=−𝑘𝑞} to (2.3) or (2.6), we get 𝑦_𝑛≡1, for all 𝑛≥0.

6. Conclusions

In the following, let 𝑎^∗=max{𝑎,1/𝑎} for any 𝑎∈ℝ₊ as defined in [20] and firstly we present [20, Theorem 1].

Theorem 6.1 1 (see [20]). Let 𝑓,𝑔 satisfy the following two conditions: (H1)[𝑓(𝑢₁,𝑢₂,…,𝑢_𝑘)]^∗=𝑓(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑘) and [𝑔(𝑢₁,…,𝑢_𝑙)]^∗=𝑔(𝑢^∗₁,…,𝑢^∗_𝑙);(H2)𝑓(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑘)≤𝑢^∗₁.Then 𝑥=1 is the unique positive equilibrium for equation (1) which is globally asymptotically stable.
The Equation (1) mentioned in Theorem 6.1 is the following difference equation: 𝑥_𝑛+1=𝑓(𝑥_𝑛−𝑟1,…,𝑥_{𝑛−𝑟𝑘})𝑔(𝑥_𝑛−𝑚1,…,𝑥_{𝑛−𝑚𝑙})+1𝑓(𝑥_𝑛−𝑟1,…,𝑥_{𝑛−𝑟𝑘})+𝑔(𝑥_𝑛−𝑚1,…,𝑥_{𝑛−𝑚𝑙}),𝑛∈ℕ, (6.1) where 𝑓∈𝐶(𝑅^𝑘₊,𝑅₊) and 𝑔∈𝐶(𝑅^𝑙₊,𝑅₊) with 𝑘,𝑙∈{1,2,…},0≤𝑟₁<⋯<𝑟_𝑘, and 0≤𝑚₁<⋯<𝑚_𝑙, and the initial values are positive real numbers.

Remark 6.2. Equation (1.10) is a special case of equation (1) in [20].

Proof. Let 𝑘≥3,𝑓₁(𝑢)=𝑢^𝑟(𝑢>0), 𝑟∈ℝ₊, and define a recursive equation 𝑓_𝑗(𝑢₁,𝑢₂,…,𝑢_𝑗𝑓)=^𝑗−1(𝑢₁,𝑢₂,…,𝑢_𝑗−1)𝑢^𝑟_𝑗+1𝑓_𝑗−1(𝑢₁,𝑢₂,…,𝑢_𝑗−1)+𝑢^𝑟_𝑗, (6.2) for all 2≤𝑗≤𝑞. Then the following difference equation: 𝑦_𝑛=𝑓_𝑞(𝑦_𝑛−𝑟1,…,𝑦_{𝑛−𝑟𝑞}),𝑛∈ℕ₀, (6.3) where 1≤𝑟₁<𝑟₂<⋯<𝑟_𝑞 and the initial values 𝑦_−𝑟𝑞,𝑦_{−𝑟𝑞+1},…,𝑦₋₁∈(0,+∞), is the very Equation (1.10) in this paper.

Remark 6.3. Let 𝑓(𝑢)=𝑢^𝑟, 𝑟∈(0,1], and 𝑔(𝑢₁,𝑢₂,…,𝑢_𝑘)=𝑓_𝑘(𝑢₁,𝑢₂,…,𝑢_𝑘). Then through simple calculations, we have(H1)[𝑓(𝑢)]^∗=𝑓(𝑢^∗)and[𝑔(𝑢₁,𝑢₂,…,𝑢_𝑘)]^∗=𝑔(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑘);(H2)(𝐻₂)𝑓(𝑢^∗)≤𝑢^∗.Thus the conditions (H1) and (H2) of [20, Theorem 1] hold. By [20, Theorem 1], we know that the unique positive equilibrium 𝑦=1 of (1.10) (also (6.3)) is globally asymptotically stable.

Remark 6.4. Although the stability of (1.10) can be also obtained as a corollary from Theorem 1 of the paper by Sun and Xi [20], the method of proof of Theorem 1.5 in this paper is distinct.

Acknowledgments

The authors are grateful to the referees for their huge number of valuable suggestions, which considerably improved the presentation of this paper. This work was financially supported by National Natural Science Foundation of China (no. 10771227).