Global Stability of a Rational Difference Equation

Abstract

We consider the higher-order nonlinear difference equation 𝑥_𝑛+1=(𝑝+𝑞𝑥_𝑛−𝑘)/(1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘),𝑛=0,1,… with the parameters, and the initial conditions 𝑥_−𝑘,…,𝑥₀ are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).

1. Introduction and Preliminaries

Our aim in this paper is to investigate the global behavior of solutions of the following nonlinear difference equation:𝑥_𝑛+1=𝑝+𝑞𝑥_𝑛−𝑘1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘,𝑛=0,1,…, (1.1) where the parameters 𝑝, 𝑞, 𝑟 and the initial conditions 𝑥_−𝑘,…,𝑥₀ are nonnegative real numbers, 𝑘∈{1,2,…}.

In 2002, Kulenović and Ladas [1] proposed the following open problem.

Open Problem 1. Assume that 𝑝,𝑞,𝑟∈[0,∞) and 𝑘∈{2,3,…}. Investigate the global behavior of all positive solutions of (1.1).

Consider the difference equation𝑥_𝑛+1=𝛼+𝛾𝑥_𝑛−1𝐴+𝐵𝑥_𝑛+𝐶𝑥_𝑛−1,𝑛=0,1,… (1.2) with 𝛼,𝛾,𝐴,𝐵,𝐶∈(0,∞), and the initial conditions 𝑥₋₁, 𝑥₀ are nonnegative real numbers. Note that, the authors [1, 2] investigated this equation and studied (1.2).

In this paper, we will consider the above open problem. Actually, we will investigate the global asymptotic stability and the invariant interval for all positive solutions of (1.1).

For the global behavior of solutions of some related equations, see [3–9]. Other related results can be found in [10–19]. For the sake of convenience, we recall some definitions and theorems which will be useful in the sequel.

Definition 1.1. Let 𝐼 be some interval of real numbers and let 𝑓∶𝐼^𝑚+1⟶𝐼 (1.3) be a continuously differential function. Then for every set of initial conditions 𝑦_−𝑘,…,𝑦₋₁, 𝑦₀∈𝐼, the difference equation 𝑦_𝑛+1=𝑓𝑦_𝑛,𝑦_𝑛−1,…,𝑦_𝑛−𝑘,𝑛=0,1,…, (1.4) has a unique solution {𝑦_𝑛}^∞_𝑛=−𝑘.
A point 𝑦 is called an equilibrium point of (1.4) if𝑦=𝑓𝑦,𝑦,…,𝑦, (1.5) that is, 𝑦_𝑛=𝑦for𝑛≥0 (1.6) is a solution of (1.4), or equivalently 𝑦 is a fixed point of 𝑓.

Definition 1.2. Let 𝑦 be an equilibrium point of (1.4). (i)The equilibrium 𝑦 is called locally stable (or stable) if for every 𝜀>0, there exists 𝛿>0 such that for all 𝑦_−𝑘,…,𝑦₋₁,𝑦₀∈𝐼 with ∑^𝑖=0_𝑖=−𝑘|𝑦_𝑖−𝑦|<𝛿, we have |𝑦_𝑛−𝑦|<𝜀 for all 𝑛≥𝑘.(ii)The equilibrium 𝑦 of (1.4) is called locally asymptotically stable (asymptotic stable) if it is locally stable, and if there exists 𝛾>0 such that for all 𝑦_−𝑘,…,𝑦₋₁,𝑦₀∈𝐼 with ∑^𝑖=0_𝑖=−𝑘|𝑦_𝑖−𝑦|<𝛾, we have lim_𝑛→∞𝑦_𝑛=𝑦.(iii)The equilibrium 𝑦 of (1.4) is called a global attractor if for every 𝑦_−𝑘,…,𝑦₋₁,𝑦₀∈𝐼, we have lim_𝑛→∞𝑦_𝑛=𝑦.(iv)The equilibrium 𝑦 of (1.4) is globally asymptotically stable if it is locally stable and is a global attractor. (v)The equilibrium 𝑦 of (1.4) is called unstable if it is not stable. (vi)The equilibrium 𝑦 of (1.4) is called a source, or a repeller, if there exists 𝑟>0 such that for all 𝑦_−𝑘,…,𝑦₋₁,𝑦₀∈𝐼 with ∑^𝑖=0_𝑖=−𝑘|𝑦_𝑖−𝑦|<𝛾, there exists 𝑁≥1 such that |𝑦_𝑁−𝑦|≥𝑟.

An interval 𝐽⊆𝐼 is called an invariant interval for (1.4) if 𝑦_−𝑘,…,𝑦₀∈𝐽⟹𝑦_𝑛∈𝐽∀𝑛>0. (1.7) That is, every solution of (1.4) with initial conditions in 𝐽 remains in 𝐽.

The linearized equation associated with (1.4) about the equilibrium 𝑦 is 𝑦_𝑛+1=^𝑘_𝑖=𝑜𝜕𝑓𝜕𝑢_𝑖𝑦,…,𝑦𝑦_𝑛−𝑖,𝑛=0,1,…. (1.8)

Its characteristic equation is𝜆^𝑘+1=^𝑘_𝑖=𝑜𝜕𝑓𝜕𝑢_𝑖𝑦,…,𝑦𝜆^𝑘−𝑖. (1.9)

Theorem 1.3 (see [20]). Assume that 𝑓 is a 𝐶¹ function and let 𝑦 be an equilibrium of (1.4). Then the following statements are true: (i)If all the roots of (1.9) lie in the open unit disk |𝜆|<1, then the equilibrium 𝑦 of (1.4) is asymptotically stable.(ii)If at least one root of (1.9) has absolute value greater than one, then the equilibrium 𝑦 of (1.4) is unstable.

Theorem 1.4 (see [20]). Assume that 𝑃,𝑄∈𝑅 and 𝑘∈{1,2,…}. Then |𝑃|+|𝑄|<1 (1.10) is a sufficient condition for the asymptotic stability of the difference equation 𝑦_𝑛+1=𝑃𝑦_𝑛+𝑄𝑦_𝑛−𝑘,𝑛=0,1,…. (1.11)

Lemma 1.5 (see [4]). Let 𝑝≥2 be a positive integer and assume that every positive solution of equation 𝑥_𝑛+1=𝛼+𝛽𝑥_𝑛+𝛾𝑥_𝑛−𝑘𝐴+𝐵𝑥_𝑛+𝐶𝑛_𝑛−𝑘,𝑛=0,1,… (1.12) is periodic with period 𝑝. If 𝐶>0, then 𝐴=𝐵=0.

Lemma 1.6 (see [21]). Consider the difference equation 𝑦_𝑛+1=𝑓𝑦_𝑛,𝑦_𝑛−𝑘,𝑛=0,1,…, (1.13) where 𝑘∈{1,2,…}. Let 𝐼=[𝑎,𝑏] be some interval of real numbers and assume that 𝑓∶[𝑎,𝑏]×[𝑎,𝑏]⟶[𝑎,𝑏] (1.14) is a continuous function satisfying the following properties: (a)𝑓(𝑢,𝑣) is nonincreasing in 𝑢 and nondecreasing in 𝑣,(b)if (𝑚,𝑀)∈[𝑎,𝑏]×[𝑎,𝑏] is a solution of the system 𝑚=𝑓(𝑀,𝑚),𝑀=𝑓(𝑚,𝑀), (1.15) then 𝑚=𝑀. Then (1.13) has a unique equilibrium 𝑦∈[𝑎,𝑏], and every solution of (1.13) converges to 𝑦.

Lemma 1.7 (see [21]). Consider the difference equation 𝑦_𝑛+1=𝑓𝑦_𝑛,𝑦_𝑛−𝑘,𝑛=0,1,…, (1.16) where 𝑘∈{1,2,…}. Let 𝐼=[𝑎,𝑏] be some interval of real numbers, and assume that 𝑓∶[𝑎,𝑏]×[𝑎,𝑏]⟶[𝑎,𝑏] (1.17) is a continuous function satisfying the following properties: (a)𝑓(𝑢,𝑣) is nonincreasing in each of its arguments, (b)if (𝑚,𝑀)∈[𝑎,𝑏]×[𝑎,𝑏] is a solution of the system 𝑚=𝑓(𝑀,𝑀),𝑀=𝑓(𝑚,𝑚), (1.18) then 𝑚=𝑀. Then (1.16) has a unique equilibrium 𝑦∈[𝑎,𝑏], and every solution of (1.16) converges to 𝑦.

2. The Special Case 𝑝𝑞𝑟=0

If the parameters 𝑝𝑞𝑟=0, then (1.1) contains the following several equations. We now assume that all their parameters are positive𝑥_𝑛+1=𝑞𝑥_𝑛−𝑘1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,𝑛=0,1,…,_𝑛+1=𝑝1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,𝑛=0,1,…,_𝑛+1𝑥=0,𝑛=0,1,…,_𝑛+1=𝑝1+𝑥_𝑛𝑥,𝑛=0,1,…,_𝑛+1=𝑞𝑥_𝑛−𝑘1+𝑥_𝑛𝑥,𝑛=0,1,…,_𝑛+1=𝑝+𝑞𝑥_𝑛−𝑘1+𝑥_𝑛,𝑛=0,1,…. (2.1)(2.2)(2.3)(2.4)(2.5)(2.6)

Equation (2.2) was studied in [19], where it is shown that the unique positive equilibrium is a global attractor. Equation (2.3) is trivial. Equation (2.4) is the Riccati equation [1]. Equation (2.5) can be reduces to (2.6), which was discussed in [22], and they showed that the unique positive equilibrium of (2.6) is globally asymptotically stable when 𝑞<1. So, here we only consider (2.1).

Clearly, 𝑥=0 is always an equilibrium of (2.1) and when 𝑞>1, (2.1) also possesses the unique positive equilibrium 𝑥=(𝑞−1)/(1+𝑟).

The linearized equation associated with (2.1) about the zero equilibrium is 𝑧_𝑛+1−𝑞𝑧_𝑛−𝑘=0. (2.7) The linearized equation associated with (2.1) about the positive equilibrium is 𝑧_𝑛+1+𝑥1+𝑥+𝑟𝑥𝑧_𝑛−𝑞−𝑟𝑥1+𝑥+𝑟𝑥𝑧_𝑛−𝑘=0. (2.8)

From this and by Theorem 1.4, we have the following result.

Theorem 2.1. (a) Assume that 𝑞<1. Then 𝑥=0 of (2.1) is locally asymptotically stable.
(b) Assume that 𝑞>1 and 𝑟>1. Then the unique positive equilibrium 𝑥=(𝑞−1)/(1+𝑟) of (2.1) is locally asymptotically stable.

Theorem 2.2. Assume that 𝑞>1 and 𝑟>1. Then the unique positive equilibrium 𝑥 of (2.1) is globally asymptotically stable.

Proof. By Theorem 2.1, the positive equilibrium of (2.1) is locally asymptotically stable. It suffices to show that 𝑥 is a global attractor.
Let𝑓𝑥,𝑦=𝑞𝑦(),1+𝑥+𝑟𝑦for𝑥,𝑦∈0,∞ (2.9) then 𝑓(𝑥,𝑦) is nonincreasing in 𝑥 and nondecreasing in 𝑦. So ≤𝑞0≤𝑓𝑥,𝑦𝑟for𝑥,𝑦∈(0,∞). (2.10) From 𝑞𝑚1+𝑀+𝑟𝑚=𝑚,𝑞𝑀1+𝑚+𝑟𝑀=𝑀, (2.11) we have 𝑚=𝑀.
Hence by Lemma 1.6 the proof is complete.

In the following sections we assume that all parameters in (1.1) are positive.

3. Local Stability and Period-Two Solutions

The equilibria of (1.1) are the solutions of the equation 𝑥=𝑝+𝑞𝑥1+𝑥+𝑟𝑥. (3.1) So (1.1) possesses the unique positive equilibrium 𝑥=𝑞−1+𝑞−1²+4𝑝(𝑟+1)2().𝑟+1 (3.2) The linearized equation associated with (1.1) about the positive equilibrium is𝑧_𝑛+1+𝑥1+𝑥+𝑟𝑥𝑧_𝑛−𝑞−𝑟𝑥1+𝑥+𝑟𝑥𝑧_𝑛−𝑘=0. (3.3)

By Theorem 1.4, it is sufficient to show that in 𝑥1+𝑥+𝑟𝑥+𝑞−𝑟𝑥1+𝑥+𝑟𝑥<1. (3.4) Thus𝑞−𝑟𝑥<1+𝑟𝑥. (3.5)If 𝑞−𝑟𝑥<0, then we have 𝑟𝑥−𝑞<1+𝑟𝑥, and it clearly holds.If 𝑞−𝑟𝑥≥0, then we have 𝑞−𝑟𝑥<1+𝑟𝑥, and 𝑞−1<2𝑟𝑥. (3.6)

If 𝑞≤1, the inequality (3.6) obviously holds. Suppose 𝑞>1, then we can get 𝑞−1<𝑟𝑞−1²+4𝑝(𝑟+1),𝑞−1²<𝑟²𝑞−1²+4𝑝𝑟²(𝑟+1), (3.7) from which it follows that(𝑟−1)𝑞−1²+4𝑝𝑟²>0. (3.8)

So we have the following result.

Theorem 3.1. Assume that either𝑞≤1or𝑞>1,(𝑟−1)𝑞−1²+4𝑝𝑟²>0. (3.9) Then the positive equilibrium 𝑥 of (1.1) is locally asymptotically stable.

Theorem 3.2. (a) Assume that 𝑘 is odd. Then (1.1) has a nonnegative prime period-two solution if and only if 𝑞>1,(𝑟−1)𝑞−1²+4𝑝𝑟²<0. (3.10) Further when (3.10) holds, the period-two solution is “unique” and the value of 𝜙₁ and 𝜙₂ are the positive roots of the quadratic equation 𝑡²−𝑞−1𝑟𝑝𝑡+1−𝑟=0. (3.11)
 (b) Assume that 𝑘 is even. Equation (1.1) has no nonnegative prime period-two solution.

Proof. (a) Assume that 𝑘 is odd, then 𝑥_𝑛+1=𝑥_𝑛−𝑘. Let …,𝜙₁,𝜙₂,𝜙₁,𝜙₂,… (3.12) be a nonnegative prime period-two solution of (1.1). Then 𝜙₁, 𝜙₂ satisfy the following system: 𝜙₁=𝑝+𝑞𝜙₁1+𝜙₂+𝑟𝜙₁,𝜙₂=𝑝+𝑞𝜙₂1+𝜙₁+𝑟𝜙₂. (3.13)
Substituting the above two equations, we obtain 𝜙₁−𝜙₂𝜙₁+𝜙₂−𝑞−1𝑟=0. (3.14) Thus 𝜙₁+𝜙₂=𝑞−1𝑟. (3.15)
Adding them and using the above equations, we can get𝜙₁𝜙₂=𝑝.1−𝑟 (3.16)
Clearly, in this case the discriminant of (3.11) is positive, that is()Δ=𝑟−1𝑞−1²+4𝑝𝑟²𝑟²(𝑟−1)>0, (3.17) and so 𝜙₁, 𝜙₂ are the positive roots of (3.11).
(b) Assume that 𝑘 is even, then 𝑥_𝑛=𝑥_𝑛−𝑘. If there exist distinctive nonnegative real number 𝜙₁ and 𝜙₂, such that …,𝜙₁,𝜙₂,𝜙₁,𝜙₂,… (3.18) is a prime period-two solution of (1.1) and 𝜙₁, 𝜙₂ satisfy the following system: 𝜙₁=𝑝+𝑞𝜙₂1+𝜙₂+𝑟𝜙₂,𝜙₂=𝑝+𝑞𝜙₁1+𝜙₁+𝑟𝜙₁, (3.19) which is equivalent to 𝜙₁+𝜙₁𝜙₂+𝑟𝜙₁𝜙₂=𝑝+𝑞𝜙₂,𝜙₂+𝜙₁𝜙₂+𝑟𝜙₁𝜙₂=𝑝+𝑞𝜙₁. (3.20) Subtracting these two equation, we can get 𝜙₁−𝜙₂𝑞+1=0. (3.21) Then 𝜙₁=𝜙₂. This contradicts the hypothesis that 𝜙₁≠𝜙₂.
The proof is complete.

Further, applying Lemma 1.5, we have the following result about the period solutions.

Theorem 3.3. There is no integer 𝑝≥2 such that every positive solution of (1.1) is periodic with period 𝑝.

4. Boundedness and Invariant Interval

In this section, we will investigate the boundedness and invariant interval of (1.1).

Theorem 4.1. Every solution of (1.1) is bounded from above and from below by positive constants.

Proof. Let {𝑥_𝑛}^∞_𝑛=−𝑘 be a solution of (1.1). Clearly, if the solution is bounded from above by a constant 𝑀, then 𝑥_𝑛+1≥𝑝1+(𝑟+1)𝑀∀𝑛≥−𝑘, (4.1) and so it is also bounded from below. Now for the sake of contradiction assume that the solution is not bounded from above. Then there exists a subsequences {𝑥_𝑛𝑚+1}^∞_𝑚=0 such that lim_𝑚→∞𝑛_𝑚=∞,lim_𝑚→∞𝑥_𝑛𝑚+1=∞, (4.2) and also 𝑥_𝑛𝑚+1=max𝑥_𝑛∶𝑛≤𝑛_𝑚for𝑚≥0. (4.3) From (1.1) we see that 𝑥_𝑛+1<𝑝+𝑞𝑥_𝑛−𝑘for𝑛≥0, (4.4) and so lim_𝑚→∞𝑥_𝑛𝑚+1=lim_𝑚→∞𝑥_{𝑛𝑚−𝑘}=∞. (4.5) Hence, for sufficiently large 𝑚, 0≤𝑥_𝑛𝑚+1−𝑥_{𝑛𝑚−𝑘}=𝑝+𝑞𝑥_{𝑛𝑚−𝑘}1+𝑥_𝑛𝑚+𝑟𝑥_{𝑛𝑚−𝑘}−𝑥_{𝑛𝑚−𝑘}=𝑝+𝑞−1−𝑥_𝑛𝑚−𝑟𝑥_{𝑛𝑚−𝑘}𝑥_{𝑛𝑚−𝑘}1+𝑥_𝑛𝑚+𝑟𝑥_{𝑛𝑚−𝑘}<0, (4.6) which is a contradiction. The proof is complete.

Let 𝑓(𝑢,𝑣)=𝑝+𝑞𝑣.1+𝑢+𝑟𝑣 (4.7) Then the following statements are true.

Lemma 4.2. (a) Assume that 𝑝𝑟≤𝑞. Then 𝑓(𝑢,𝑣) is decreasing in 𝑢 for each 𝑣 and increasing in 𝑣 for each 𝑢.
(b) Assume that 𝑝𝑟>𝑞. Then 𝑓(𝑢,𝑣) is decreasing in 𝑢 for each 𝑣 and decreasing in 𝑣 for 𝑢∈[0,(𝑝𝑟−𝑞)/𝑞], and increasing in 𝑣 for 𝑢∈[(𝑝𝑟−𝑞)/𝑞,∞).

Proof. The proofs of (a) and (b) are simple and will be omitted.

Lemma 4.3. Equation (1.1) possesses the following invariant intervals: (a)[0,𝑞/𝑟], when 𝑝𝑟≤𝑞;(b)[(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟], when 𝑞<𝑝𝑟<𝑞+𝑞²/𝑟;(c)[0,𝑝], when 𝑝𝑟=𝑞+𝑞²/𝑟;(d)[𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞], when 𝑞+𝑞²/𝑟<𝑝𝑟<𝑝𝑞+𝑞;(e)[𝑞/𝑟,𝑝], when 𝑝𝑟≥𝑝𝑞+𝑞.

Proof. (a) Set 𝑔(𝑥)=(𝑝+𝑞𝑥)/(1+𝑟𝑥), so 𝑔(𝑥) is increasing for 𝑥 and 𝑔(𝑞/𝑟)≤𝑞/𝑟. Using (1.1) we see that when 𝑥_−𝑘,…,𝑥₋₁,𝑥₀∈[0,𝑞/𝑟], then 𝑥₁=𝑝+𝑞𝑥_−𝑘1+𝑥₀+𝑟𝑥_−𝑘≤𝑝+𝑞𝑥_−𝑘1+𝑟𝑥_−𝑘𝑞≤𝑔𝑟≤𝑞𝑟. (4.8) The proof follows by induction.
(b) Using the monotonic character of the function 𝑓(𝑢,𝑣) which is described by Lemma 4.2(b) and the condition that 𝑞<𝑝𝑟<𝑞+𝑞²/𝑟, when 𝑥_−𝑘,…,𝑥₋₁,𝑥₀∈[(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟], we can get 𝑝𝑟−𝑞𝑞𝑞≤𝑓𝑟,𝑝𝑟−𝑞𝑞≤𝑥₁=𝑝+𝑞𝑥_−𝑘1+𝑥₀+𝑟𝑥_−𝑘=𝑓𝑥_𝑜,𝑥_−𝑘≤𝑓𝑝𝑟−𝑞𝑞,𝑞𝑟=𝑞𝑟. (4.9) The proof follows by induction.
(c) Set ℎ(𝑥)=(𝑝+𝑞𝑥)/(1+𝑝+𝑟𝑥), 𝑔(𝑥)=(𝑝+𝑞𝑥)/(1+𝑟𝑥), then ℎ(𝑥) is increasing for 𝑥 and 𝑔(𝑥) is decreasing for 𝑥 when 𝑝𝑟=𝑞+𝑞²/𝑟, we see that when 𝑥_−𝑘,…,𝑥₋₁,𝑥₀∈[0,𝑝], then𝑥₁=𝑝+𝑞𝑥_−𝑘1+𝑥₀+𝑟𝑥_−𝑘≥𝑝+𝑞𝑥_−𝑘1+𝑝+𝑟𝑥_−𝑘(0)=𝑝≥ℎ𝑥1+𝑝>0,₁=𝑝+𝑞𝑥_−𝑘1+𝑥₀+𝑟𝑥_−𝑘≤𝑝+𝑞𝑥_−𝑘1+𝑟𝑥_−𝑘≤𝑔(0)=𝑝. (4.10) The proof follows by induction.
(d) Using the monotonic character of the function 𝑓(𝑢,𝑣) which is described by Lemma 4.2(b) and the condition 𝑞+𝑞²/𝑟<𝑝𝑟<𝑝𝑞+𝑞, when 𝑥_−𝑘,…,𝑥₋₁,𝑥₀∈[𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞], we obtain𝑞𝑟=𝑓𝑝𝑟−𝑞𝑞,𝑝𝑟−𝑞𝑞≤𝑥₁=𝑝+𝑞𝑥_−𝑘1+𝑥₀+𝑟𝑥_−𝑘=𝑓𝑥_𝑜,𝑥_−𝑘𝑞≤𝑓𝑟,𝑞𝑟=𝑝𝑟+𝑞²()𝑞<𝑟+𝑟+1𝑝𝑟−𝑞𝑞. (4.11) The proof follows by induction.
(e) In view of the condition 𝑝𝑟≥𝑝𝑞+𝑞, we can get (𝑝𝑟−𝑞)/𝑞>𝑝>𝑞/𝑟. By using the monotonic character of the function 𝑓(𝑢,𝑣) which is described by Lemma 4.2(b) and the condition 𝑝𝑟≥𝑝𝑞+𝑞, when 𝑥_−𝑘,…,𝑥₋₁,𝑥₀∈[𝑞/𝑟,𝑝], we have 𝑞𝑟≤𝑝+𝑞𝑝1+𝑝+𝑝𝑟=𝑓𝑝,𝑝≤𝑥₁=𝑝+𝑞𝑥_−𝑘1+𝑥₀+𝑟𝑥_−𝑘=𝑓𝑥_𝑜,𝑥_−𝑘𝑞≤𝑓𝑟,𝑞𝑟=𝑝𝑟+𝑞²𝑟+𝑞+𝑞𝑟<𝑝. (4.12) The proof follows by induction.
The proof is complete.

5. Semicycles Analysis

Let {𝑥_𝑛}^∞_𝑛=−𝑘 be a positive solution of (1.1). Then we have the following equations:𝑥_𝑛+1−𝑞𝑟=𝑞𝑟𝑝𝑟−𝑞/𝑞−𝑥_𝑛1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,for𝑛≥0,_𝑛+1−𝑝=−𝑝𝑥_𝑛+𝑝𝑟−𝑞𝑥_𝑛−𝑘1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,for𝑛≥0,_𝑛+1−𝑝𝑟−𝑞𝑞=1/𝑞𝑞+𝑞²/𝑟−𝑝𝑟+𝑝𝑟−𝑞/𝑞𝑞/𝑟−𝑥_𝑛+𝑟/𝑞𝑞+𝑞²/𝑟−𝑝𝑟𝑥_𝑛−𝑘1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,for𝑛≥0,_𝑛+1−𝑟𝑥=𝑥−𝑞/𝑟𝑥−𝑥_𝑛−𝑘+𝑥𝑥−𝑥_𝑛1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,for𝑛≥0,_𝑛+2−𝑥_𝑛=𝑟𝑥_{𝑛+1−𝑘}𝑞/𝑟−𝑥_𝑛𝑟𝑥_𝑛−𝑘+𝑥_𝑛+1+𝑞+𝑟𝑥_𝑛−𝑘𝑝𝑟/𝑞+𝑟−𝑥_𝑛+𝑝−𝑥_𝑛−𝑥²_𝑛1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘1+𝑟𝑥_{𝑛+1−𝑘}+𝑝+𝑞𝑥_𝑛−𝑘,for𝑛≥0. (5.1)(5.2)(5.3)(5.4)(5.5)

If 𝑝𝑟=𝑞+𝑞²/𝑟, then the unique positive equilibrium is 𝑥=𝑞/𝑟, and (5.1) and (5.5) change into𝑥_𝑛+1−𝑞𝑟=𝑞𝑟𝑞/𝑟−𝑥_𝑛1+𝑥_𝑛+𝑟𝑥_𝑛−𝑘𝑥,for𝑛≥0,_𝑛+2−𝑥_𝑛=𝑞/𝑟−𝑥_𝑛𝑟²𝑥_{𝑛+1−𝑘}𝑥_𝑛−𝑘+𝑟𝑥_𝑛𝑥_{𝑛+1−𝑘}+𝑟𝑥_{𝑛+1−𝑘}+𝑝𝑟²/𝑞𝑥_𝑛−𝑘+𝑥_𝑛+𝑞/𝑟+11+𝑥_𝑛+𝑟𝑥_𝑛−𝑘1+𝑟𝑥_{𝑛+1−𝑘}+𝑝+𝑞𝑥_𝑛−𝑘,for𝑛≥0. (5.6)(5.7)

The following lemma is straightforward consequences of identities (5.1)–(5.7).

Lemma 5.1. Assume that 𝑝𝑟≤𝑞 and let {𝑥_𝑛}^∞_𝑛=−𝑘 be a solution of (1.1). Then the following statements are true: (i)𝑥_𝑛≤𝑞/𝑟 for all 𝑛≥1; (ii)if for some 𝑁≥0, 𝑥_𝑁−𝑘≤𝑥 and 𝑥_𝑁≥𝑥, then 𝑥_𝑁+1≤𝑥; (iii)if for some 𝑁≥0, 𝑥_𝑁−𝑘>𝑥 and 𝑥_𝑁<𝑥, then 𝑥_𝑁+1>𝑥; (iv)0≤𝑥≤𝑞/𝑟.

Lemma 5.2. Assume that 𝑞<𝑝𝑟<𝑞+𝑞²/𝑟 and let {𝑥_𝑛}^∞_𝑛=−𝑘 be a solution of (1.1). Then the following statements are true: (i)if for some 𝑁≥0, 𝑥_𝑁<𝑞/𝑟, then 𝑥_𝑁+1>(𝑝𝑟−𝑞)/𝑞; (ii)if for some 𝑁≥0, 𝑥_𝑁<(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1>𝑞/𝑟; (iii)if for some 𝑁≥0, 𝑥_𝑁=(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1=𝑞/𝑟; (iv)if for some 𝑁≥0, 𝑥_𝑁>(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1<𝑞/𝑟; (v)if for some 𝑁≥0, (𝑝𝑟−𝑞)/𝑞≤𝑥_𝑁≤𝑞/𝑟, then (𝑝𝑟−𝑞)/𝑞≤𝑥_𝑛≤𝑞/𝑟 for 𝑛≥𝑁; (vi)if for some 𝑁≥0, 𝑥_𝑁−𝑘≤𝑥, and 𝑥_𝑁≥𝑥, then 𝑥_𝑁+1≤𝑥; (vii)if for some 𝑁≥0, 𝑥_𝑁−𝑘>𝑥, and 𝑥_𝑁<𝑥, then 𝑥_𝑁+1>𝑥; (viii)if for some 𝑁≥0, 𝑥_𝑁<(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+2>𝑥_𝑁; (ix)if for some 𝑁≥0, 𝑥_𝑁>𝑞/𝑟, then 𝑥_𝑁+2<𝑥_𝑁; (x)(𝑝𝑟−𝑞)/𝑞≤𝑥≤𝑞/𝑟.

Lemma 5.3. Assume that 𝑝𝑟=𝑞+𝑞²/𝑟 and let {𝑥_𝑛}^∞_𝑛=−𝑘 be a solution of (1.1). Then the following statements are true: (i)if for some 𝑁≥0, 𝑥_𝑁>𝑞/𝑟, then 𝑥_𝑁+1<𝑞/𝑟; (ii)if for some 𝑁≥0, 𝑥_𝑁=𝑞/𝑟, then 𝑥_𝑁+1=𝑞/𝑟; (iii)if for some 𝑁≥0, 𝑥_𝑁<𝑞/𝑟, then 𝑥_𝑁+1>𝑞/𝑟; (iv)if for some 𝑁≥0, 𝑥_𝑁≥𝑥, then 𝑥_𝑁+1≤𝑥; (v)if for some 𝑁≥0, 𝑥_𝑁<𝑥, then 𝑥_𝑁+1>𝑥; (vi)if for some 𝑁≥0, 𝑥_𝑁>𝑞/𝑟, then 𝑥_𝑁+2<𝑥_𝑁; (vii)if for some 𝑁≥0, 𝑥_𝑁<𝑞/𝑟, then 𝑥_𝑁+2>𝑥_𝑁; (viii)𝑥=𝑞/𝑟.

Lemma 5.4. Assume that 𝑞+𝑞²/𝑟<𝑝𝑟<𝑝𝑞+𝑞 and let {𝑥_𝑛}^∞_𝑛=−𝑘 be a solution of (1.1). Then the following statements are true: (i)if for some 𝑁≥0, 𝑥_𝑁>𝑞/𝑟, then 𝑥_𝑁+1<(𝑝𝑟−𝑞)/𝑞; (ii)if for some 𝑁≥0, 𝑥_𝑁<(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1>𝑞/𝑟; (iii)if for some 𝑁≥0, 𝑥_𝑁=(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1=𝑞/𝑟; (iv)if for some 𝑁≥0, 𝑥_𝑁>(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1<𝑞/𝑟; (v)if for some 𝑁≥0, 𝑞/𝑟≤𝑥_𝑁≤(𝑝𝑟−𝑞)/𝑞, then 𝑞/𝑟≤𝑥_𝑛≤(𝑝𝑟−𝑞)/𝑞 for 𝑛≥𝑁; (vi)if for some 𝑁≥0, 𝑥_𝑁−𝑘≤𝑥 and 𝑥_𝑁≤𝑥, then 𝑥_𝑁+1≥𝑥; (vii)if for some 𝑁≥0, 𝑥_𝑁−𝑘>𝑥 and 𝑥_𝑁>𝑥, then 𝑥_𝑁+1<𝑥; (viii)if for some 𝑁≥0, 𝑥_𝑁<(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+1>𝑞/𝑟; (ix)if for some 𝑁≥0, 𝑥_𝑁<𝑞/𝑟, then 𝑥_𝑁+2>𝑥_𝑁; (x)if for some 𝑁≥0, 𝑥_𝑁>(𝑝𝑟−𝑞)/𝑞, then 𝑥_𝑁+2<𝑥_𝑁; (xi)𝑞/𝑟≤𝑥≤(𝑝𝑟−𝑞)/𝑞.

Lemma 5.5. Assume that 𝑝𝑟≥𝑝𝑞+𝑞 and let {𝑥_𝑛}^∞_𝑛=−𝑘 be a solution of (1.1). Then the following statements are true: (i)𝑥_𝑛≤𝑝 for all 𝑛≥1; (ii)if for some 𝑁≥0, 𝑥_𝑁≤𝑝, then 𝑥_𝑁+1≥𝑞/𝑟; (iii)if for some 𝑁≥0, 𝑞/𝑟≤𝑥_𝑁≤𝑝, then 𝑞/𝑟≤𝑥_𝑛≤𝑝 for 𝑛≥𝑁; (iv)if for some 𝑁≥0, 𝑥_𝑁−𝑘≤𝑥 and 𝑥_𝑁≤𝑥, then 𝑥_𝑁+1≥𝑥; (v)if for some 𝑁≥0, 𝑥_𝑁−𝑘>𝑥 and 𝑥_𝑁>𝑥, then 𝑥_𝑁+1<𝑥; (vi)if for some 𝑁≥0, 𝑥_𝑁<𝑞/𝑟, then 𝑥_𝑁+2>𝑥_𝑁; (vii)if for some 𝑁≥0, 𝑥_𝑁>𝑝, then 𝑥_𝑁+2<𝑥_𝑁; (viii)𝑞/𝑟≤𝑥≤𝑝.

The following results are consequences of Lemmas 5.1–5.5.

Theorem 5.6. Let {𝑥_𝑛}^∞_𝑛=−𝑘 be a nontrivial solution of (1.1) and 𝑥 is the unique positive equilibrium point of (1.1). Then the following statements are true. (a)Assume that 𝑝𝑟≤𝑞. Then except possibly for the first semicycle, every oscillatory solution of (1.1) has semicycles of length at most 𝑘.(b)Assume that 𝑞<𝑝𝑟<𝑞+𝑞²/𝑟. Then, except possibly for the first semicycle, every oscillatory solution of (1.1) which lies in the invariant interval [(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟] has semicycles of length at most 𝑘.(c)Assume that 𝑝𝑟=𝑞+𝑞²/𝑟. Then after the first semicycle, every oscillatory solution of (1.1) about the equilibrium point 𝑥 with semicycle of length one. (d)Assume that 𝑞+𝑞²/𝑟<𝑝𝑟<𝑝𝑞+𝑞. Then, except possibly for the first semicycle, every oscillatory solution of (1.1) which lies in the invariant interval [𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞] has semicycles that is either of length at least 𝑘−1, or of length at most 𝑘+1.(e)Assume that 𝑝𝑟≥𝑝𝑞+𝑞. Then, except possibly for the first semicycle, every oscillatory solution of (1.1) which lies in the invariant interval [𝑞/𝑟,𝑝] has semicycles that is either of length at least 𝑘−1, or of length at most 𝑘+1.

6. Global Stability

In this section, we will investigate global stability of the positive equilibrium point 𝑥 of (1.1).

Theorem 6.1. Assume that (3.9) holds and let {𝑥_𝑛}^∞_𝑛=−𝑘 be a positive solution of (1.1). Then every solution of (1.1) eventually enters the invariant interval (a)[0,𝑞/𝑟] if 𝑝𝑟≤𝑞; (b)[(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟] if 𝑞<𝑝𝑟<𝑞+𝑞²/𝑟; (c)[𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞] if 𝑞+𝑞²/𝑟<𝑝𝑟<𝑝𝑞+𝑞; (d)[𝑞/𝑟,𝑝] if 𝑝𝑟≥𝑝𝑞+𝑞.

Proof. (a) The proof is a direct consequence of Lemma 5.1.
(b) From Lemma 5.2(v) we know that if there exist an integer 𝑁 such that 𝑥_𝑁∈[(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟], then 𝑥_𝑛∈[(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟] for 𝑛≥𝑁 and the result follows. Now assume for the sake of contradiction that all terms of {𝑥_𝑛} never enter the invariant interval [(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟] for 𝑛≥0. Notice that Lemma 5.2(ii) implies that 𝑥_𝑛+1>𝑞/𝑟 for 𝑥_𝑛<(𝑝𝑟−𝑞)/𝑞. Further using Lemma 5.2(viii) and (ix), we obtain that the subsequence {𝑥_2𝑛}^∞_𝑛=0 and {𝑥_2𝑛+1}^∞_𝑛=0 are both monotonous. If one of them is decreasing, then it is bounded above by (𝑝𝑟−𝑞)/𝑞, and if one of them is decreasing, then it is bounded below by 𝑞/𝑟. Thus lim_𝑛→∞𝑥_2𝑛 and lim_𝑛→∞𝑥_2𝑛+1 exist. Setlim_𝑛→∞𝑥_2𝑛=𝐿,lim_𝑛→∞𝑥_2𝑛+1=𝑀, (6.1) then 𝐿≤(𝑝𝑟−𝑞)/𝑞 and 𝑀≥𝑞/𝑟, or vice versa. From which it follows that …,𝐿,𝑀,𝐿,𝑀,… (6.2) is a period-two solution of (1.1), which is a contradiction, since when (2.3) holds, (1.1) has no period-two solution.
(c) The proof is similar to (b), so will be omitted.
(d) In view of Lemma 5.5(i) and (iii), we know that 𝑥_𝑛≤𝑝 for all 𝑛≥1 and [𝑞/𝑟,𝑝] is an invariant interval of (1.1). If there exist an integer 𝑁 such that 𝑥_𝑁∈[𝑞/𝑟,𝑝], then 𝑥_𝑛∈[𝑞/𝑟,𝑝] for 𝑛≥𝑁, from which it follows that the result is true. Now assume for the sake of contradiction that terms of {𝑥_𝑛} never enter the invariant interval [𝑞/𝑟,𝑝], then they all lie in the interval [0,𝑞/𝑟]. Noticing that 𝑥₁≤𝑞/𝑟 and 𝑝𝑟≥𝑝𝑞+𝑞 hold, we get𝑥₂−𝑥₁=𝑝−𝑥₁+𝑟𝑥_1−𝑘𝑞/𝑟−𝑥₁−𝑥²₁1+𝑥₁+𝑟𝑥_1−𝑘≥𝑝−𝑞/𝑟−𝑞/𝑟²1+𝑥₁+𝑟𝑥_1−𝑘>0, (6.3) from which it follows by induction that the sequence {𝑥_𝑛} is increasing in the interval [0,𝑞/𝑟]. Hence, lim_𝑛→∞𝑥_𝑛 exists and lim_𝑛→∞𝑥_𝑛≤𝑞/𝑟, which is a contradiction because (1.1) has no equilibrium point in the interval [0,𝑞/𝑟].
The proof is complete.

Theorem 6.2. Assume that (3.9) holds. Then the positive equilibrium 𝑥 is a global attractor of (1.1).

Proof. The proof is finished by considering the following five cases.Case 1 (when 𝑝𝑟≤𝑞). By Lemma 4.3(a) and Theorem 6.1(a), we know that (1.1) possesses an invariant interval [0,𝑞/𝑟] and every solution of (1.1) eventually enters the interval [0,𝑞/𝑟]. Further, it is easy to see that 𝑓(𝑢,𝑣) decreases in 𝑢 and increases in 𝑣 in [0,𝑞/𝑟].Finally observe that when (3.9) holds, the only solution of the system 𝑝+𝑞𝑚1+𝑀+𝑟𝑚=𝑚,𝑝+𝑞𝑀1+𝑚+𝑟𝑀=𝑀, (6.4) is 𝑚=𝑀. Further, Lemma 5.1 implies that (1.1) has a unique equilibrium 𝑥∈[0,𝑞/𝑟]. Thus, in view of Lemma 1.6, every solution of (1.1) converges to 𝑥. So the unique positive equilibrium 𝑥 is a global attractor of (1.1).Case 2 (when 𝑞<𝑝𝑟<𝑞+𝑞²/𝑟). By Lemma 4.3(b) and Theorem 6.1(b), we know that (1.1) possesses an invariant interval [(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟] and every solution of (1.1) eventually enters the interval [(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟]. Further, it is easy to see that 𝑓(𝑢,𝑣) decreases in 𝑢 and increases in 𝑣 in [(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟]. Then using the same argument in Case 1, (1.1) has a unique equilibrium 𝑥∈[(𝑝𝑟−𝑞)/𝑞,𝑞/𝑟] and every solution of (1.1) converges to 𝑥. So the unique positive equilibrium 𝑥 is a global attractor of (1.1).Case 3 (when 𝑝𝑟=𝑞+𝑞²/𝑟). In view of part (c) of Theorem 5.6, we know that, after the first semicycle, the nontrivial solution oscillates about 𝑥 with semicycles of length one. Considering the subsequences {𝑥_2𝑛}^∞_𝑛=0 and {𝑥_2𝑛+1}^∞_𝑛=0, we have 𝑥_2𝑛>𝑞𝑟,𝑥_2𝑛+1<𝑞𝑟for𝑛≥0, (6.5) or 𝑥_2𝑛<𝑞𝑟,𝑥_2𝑛+1>𝑞𝑟for𝑛≥0. (6.6) Let us consider Case 1. Case 2 can be handled in a similar way. In view of Theorem 4.1 and Lemma 5.3, we know that {𝑥_𝑛}^∞_𝑛=−𝑘 is bounded and 𝑥_𝑛+1=𝑞/𝑟 if 𝑥_𝑛=𝑞/𝑟 and 𝑥₁<𝑥₃<𝑥₅𝑞<⋯<𝑟<⋯<𝑥₄<𝑥₂<𝑥₀. (6.7) So lim_𝑛→∞𝑥_2𝑛 and lim_𝑛→∞𝑥_2𝑛+1 exist.Let 𝐿=lim_𝑛→∞𝑥_2𝑛,𝑀=lim_𝑛→∞𝑥_2𝑛+1, (6.8) then 𝐿≥𝑞/𝑟,𝑀≤𝑞/𝑟. If 𝐿≠𝑀, then 𝐿,𝑀 is a period-two solution of (1.1). Furthermore, the condition (3.9) holds. This contradicts Theorem 3.2. Thus 𝐿=𝑀 and lim_𝑛→∞𝑥_𝑛=𝑞/𝑟. So 𝑥=𝑞/𝑟 is a global attractor of (1.1).Case 4 (when 𝑞+𝑞²/𝑟<𝑝𝑟<𝑝𝑞+𝑞). By Theorem 6.1(c), we know that every solution of (1.1) eventually enters the interval [𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞]. Furthermore, it is easy to see that the function 𝑓(𝑢,𝑣) decreases in each of its arguments in the interval [𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞]. Let 𝑚,𝑀∈[𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞] is a solution of the system 𝑝+𝑞𝑚1+𝑚+𝑟𝑚=𝑀,𝑝+𝑞𝑀1+𝑀+𝑟𝑀=𝑚, (6.9) that is, the solution of the system 𝑝+𝑞𝑚=𝑀+𝑚𝑀+𝑟𝑚𝑀,𝑝+𝑞𝑀=𝑚+𝑚𝑀+𝑟𝑚𝑀. (6.10) Then (𝑚−𝑀)(𝑞+1)=0, which implies that 𝑚=𝑀. Employing Lemma 1.7, we see that (1.1) has a unique equilibrium 𝑥∈[𝑞/𝑟,(𝑝𝑟−𝑞)/𝑞] and every solution of (1.1) converges to 𝑥. Thus the unique positive equilibrium 𝑥 is a global attractor of (1.1).Case 5. When 𝑝𝑟≥𝑝𝑞+𝑞. By Theorem 6.1(d), we know that every solution of (1.1) eventually enters the interval [𝑞/𝑟,𝑝]. Further, it is clear to see that the function 𝑓(𝑢,𝑣) decreases in each of its arguments in the interval [𝑞/𝑟,𝑝]. Then, using the same argument as in Case 4, (1.1) has a unique equilibrium 𝑥∈[𝑞/𝑟,𝑝] and every solution of (1.1) converges to 𝑥. Thus the unique positive equilibrium 𝑥 is a global attractor of (1.1).
The proof is complete.

In view of Theorems 3.1 and 6.2, we have the following result, which solves Open Problem 1 when conditions (3.9) holds.

Theorem 6.3. Assumed that (3.9) holds. Then the positive equilibrium of (1.1) is globally asymptotically stable.