Boundedness, Attractivity, and Stability of a Rational Difference Equation with Two Periodic Coefficients

Abstract

We study the boundedness, the attractivity, and the stability of the positive solutions of the rational difference equation 𝑥_𝑛+1=(𝑝_𝑛𝑥_𝑛−2+𝑥_𝑛−3)/(𝑞_𝑛+𝑥_𝑛−3), 𝑛=0,1,…, where 𝑝_𝑛,𝑞_𝑛, 𝑛=0,1,… are positive sequences of period 2.

1. Introduction

In [1], Camouzis et al. studied the global character of the positive solutions of the difference equation: 𝑥_𝑛+1=𝛿𝑥_𝑛−2+𝑥_𝑛−3𝐴+𝑥_𝑛−3,𝑛=0,1,…, (1.1) where 𝛿,𝐴 are positive parameters and the initial values 𝑥₋₃,𝑥₋₂,𝑥₋₁,𝑥₀ are positive real numbers.

The mathematical modeling of a physical, physiological, or economical problem very often leads to difference equations (for partial review of the theory of difference equations and their applications see [2–12]). Moreover, a lot of difference equations with periodic coefficients have been applied in mathematical models in biology (see [13–15]). In addition, between others in [16–19], we can see some more difference equations with periodic coefficients that have been studied.

In this paper, we investigate the difference equation𝑥_𝑛+1=𝑝_𝑛𝑥_𝑛−2+𝑥_𝑛−3𝑞_𝑛+𝑥_𝑛−3,𝑛=0,1,…, (1.2) where 𝑝_𝑛,𝑞_𝑛, 𝑛=0,1,… are positive sequences of period 2 and the initial values 𝑥_𝑖,𝑖=−3,−2,−1,0 are positive numbers.

Our goal in this paper is to extend some results obtained in [1]. More precisely, we study the existence of a unique positive periodic solution of (1.2) of prime period 2. In the sequel, we investigate the boundedness, the persistence, and the convergence of the positive solutions to the unique periodic solution of (1.2). Finally, we study the stability of the positive periodic solution and the zero solution of (1.2).

If we set 𝑦_𝑛=𝑥_2𝑛−1, 𝑧_𝑛=𝑥_2𝑛, it is easy to prove that (1.2) is equivalent to the following system of difference equations:𝑦_𝑛+1=𝑝₀𝑧_𝑛−1+𝑦_𝑛−1𝑞₀+𝑦_𝑛−1,𝑧_𝑛+1=𝑝₁𝑦_𝑛+𝑧_𝑛−1𝑞₁+𝑧_𝑛−1,𝑛=0,1,…, (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. So in order to study (1.2) we investigate system (1.3).

2. Existence of the Unique Positive Equilibrium of System (1.3)

In the following proposition, we study the existence of the unique positive equilibrium of system (1.3).

Proposition 2.1. Consider system (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. Suppose that 𝑞₀−1<𝑝₀,𝑞₁−1<𝑝₁ (2.1) are satisfied. Then system (1.3) possesses a unique positive equilibrium.

Proof. Let (𝑦,𝑧) be a positive equilibrium of system (1.3) then𝑝𝑦=⁰𝑧+𝑦𝑞₀𝑝+𝑦,𝑧=¹𝑦+𝑧𝑞₁+𝑧. (2.2)Equations (2.2) imply that 𝑧 is a solution of the equation𝑓(𝑥)=𝑥³+2(𝑞₁−1)𝑥²+[(𝑞₁−1)²+𝑝₁(𝑞₀−1)]𝑥+(𝑞₁−1)(𝑞₀−1)𝑝₁−𝑝₀𝑝²₁=0. (2.3)
Suppose that𝑞₁≥1. (2.4) Let 𝜆₁,𝜆₂, and 𝜆₃ be the solutions of (2.3). Then from (2.1), (2.3), and (2.4) we take𝜆₁+𝜆₂+𝜆₃=2(1−𝑞₁𝜆)≤0,₁𝜆₂𝜆₃=−(𝑞₁−1)(𝑞₀−1)𝑝₁+𝑝₀𝑝²₁>0, (2.5) and so (2.3) has unique positive solution 𝑧. Then from (2.2) and (2.4) we have𝑧>1−𝑞₁𝑧,𝑦=²+(𝑞₁−1)𝑧𝑝₁>0, (2.6)and so system (1.3) has a unique positive equilibrium.
Now suppose that𝑞₁<1,(𝑞₁−1)(𝑞₀−1)>𝑝₁𝑝₀. (2.7) If 𝜆₁,𝜆₂, and 𝜆₃ are the solutions of (2.3), then from (2.3) and (2.7) we take𝜆₁+𝜆₂+𝜆₃=2(1−𝑞₁𝜆)>0,₁𝜆₂𝜆₃=−(𝑞₁−1)(𝑞₀−1)𝑝₁+𝑝₀𝑝²₁<0, (2.8) and so (2.3) has a negative solution, but also (2.3) has a solution in the interval (0,1−𝑞₁), since𝑓(0)=(𝑞₁−1)(𝑞₀−1)𝑝₁−𝑝₀𝑝²₁>0,𝑓(1−𝑞₁)=−𝑝₀𝑝²₁<0. (2.9)Moreover, (2.3) has a solution 𝑧 in the interval (1−𝑞₁,∞), sincelim_𝑥→∞𝑓(𝑥)=∞, (2.10) therefore, we get (2.6) and so system (1.3) has a unique positive equilibrium.
Finally, suppose that𝑞₁<1,(𝑞₁−1)(𝑞₀−1)<𝑝₁𝑝₀. (2.11) If 𝜆₁,𝜆₂, and 𝜆₃ are the solutions of (2.3), then from (2.3) and (2.11), we take𝜆₁+𝜆₂+𝜆₃=2(1−𝑞₁𝜆)>0,₁𝜆₂𝜆₃=−(𝑞₁−1)(𝑞₀−1)𝑝₁+𝑝₀𝑝²₁>0. (2.12) We have lim_𝑥→∞𝑓(𝑥)=∞, and since 𝑓(1−𝑞₁)<0, it is obvious that (2.3) has a solution 𝑧 in the interval (1−𝑞₁,∞). From (2.3), we get𝑓′(𝑥)=3𝑥²+4𝑥(𝑞₁−1)+(𝑞₁−1)²+𝑝₁(𝑞₀−1). (2.13)
If equation 𝑓^′(𝑥)=0 has complex roots, then it is obvious that 𝑧 is the unique solution of (2.3). Therefore, we get (2.6), and so system (1.3) has a unique positive equilibrium.
Now, suppose that the roots of 𝑓^′(𝑥)=0𝜇₁=2(1−𝑞₁)−√𝐷3,𝜇₂=2(1−𝑞₁)+√𝐷3,𝐷=(1−𝑞₁)²+3𝑝₁(1−𝑞₀), (2.14) are real numbers.
Suppose that 𝑞₀<1, then it is obvious that𝜇₁<1−𝑞₁<𝜇₂, (2.15)and so we have that (2.3) has a unique solution 𝑧∈(1−𝑞₁,∞).
If 𝑞₀≥1, then it holds that0<𝜇₁≤𝜇₂≤1−𝑞₁, (2.16) which implies that (2.3) has a unique solution 𝑧∈(1−𝑞₁,∞).
Therefore, we can take (2.6) and so system (1.3) has a unique positive equilibrium. This completes the proof of the proposition.

3. Boundedness and Persistence of the Solutions of System (1.3)

In the following propositions we study the boundedness and the persistence of the positive solutions of system (1.3). In the sequel we will use the following result which has proved in [20].

Theorem 3.1. Assume that all roots of the polynomial𝑃(𝑡)=𝑡^𝑁−𝑠₁𝑡^𝑁−1−⋯−𝑠_𝑁, (3.1) where 𝑠₁,𝑠₂,…,𝑠_𝑁≥0 have absolute value less than 1, and let 𝑦_𝑛 be a nonnegative solution of the inequality𝑦_𝑛+𝑁≤𝑠₁𝑦_{𝑛+𝑁−1}+⋯+𝑠_𝑁𝑦_𝑛+𝑧_𝑛. (3.2) Then, the following statements are true.
(i) If 𝑧_𝑛 is a nonnegative bounded sequence, then 𝑦_𝑛 is also bounded. (ii) If lim_𝑛→∞𝑧_𝑛=0, then lim_𝑛→∞𝑦_𝑛=0.

Proposition 3.2. One considers the system of difference equations (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. Then the following statements are true.
(i) If𝑞₀𝑞₁𝑝₀𝑝₁≥1, (3.3) then every solution of (1.3) is bounded. (ii) If𝑞₀−1<𝑝₀≤𝑞₀,𝑞₁−1<𝑝₁≤𝑞₁, (3.4)then every solution of (1.3) is bounded and persists.

Proof. Let (𝑦_𝑛,𝑧_𝑛) be an arbitrary solution of (1.3).
(i) From (3.3), we get that one of the three following conditions holds:𝑞₀𝑝₀𝑞>1,₁𝑝₁𝑝>1,₀=𝑞₀=𝑝,𝑝₁=𝑞₁=𝑞. (3.5)(3.6)(3.7)
We assume that (3.5) is satisfied. We prove that there exists a positive integer 𝑁 such that𝑦_𝑛<1,𝑧_𝑛<𝑞₀𝑝₀,𝑛≥𝑁. (3.8) First, we show that if there exists a positive integer 𝑛₀ such that𝑧_𝑛0<𝑞₀𝑝₀, (3.9) then𝑧_𝑛0+3𝑝<𝑞₀𝑝₀,𝑝=0,1,…. (3.10) In contradiction, we assume that𝑧_𝑛0+3=𝑝₁𝑦_𝑛0+2+𝑧_𝑛0+1𝑞₁+𝑧_𝑛0+1≥𝑞₀𝑝₀. (3.11)Using relations (1.3), (3.5), and (3.11), we get that𝑦_𝑛0+2=𝑝₀𝑧_𝑛0+𝑦_𝑛0𝑞₀+𝑦_𝑛0>𝑞₀𝑞₁𝑝₀𝑝₁, (3.12) and so relations (1.3) and (3.3) imply that𝑧_𝑛0>𝑞₀²𝑞₁𝑝₀²𝑝₁>𝑞₀𝑝₀, (3.13) which contradicts (3.9). So 𝑧_𝑛0+3<𝑞₀/𝑝₀ and working inductively, we get (3.10).
If 𝑧₋₁<𝑞₀/𝑝₀, then from the analogous relations (3.9) and (3.10), we get𝑧_−1+3𝑝<𝑞₀𝑝₀,𝑝=0,1,…. (3.14)Now, suppose that𝑧₋₁≥𝑞₀𝑝₀, (3.15) we prove that there exists a positive integer 𝑞 such that𝑧_−1+3𝑞<𝑞₀𝑝₀. (3.16) From (3.3), there exists a positive integer ℎ such that𝑧₋₁𝑞<(⁰𝑞₁𝑝₀𝑝₁)^ℎ. (3.17) If 𝑧₂<𝑞₀/𝑝₀, then (3.16) is true for 𝑞=1.
Now, suppose that𝑧₂≥𝑞₀𝑝₀. (3.18) Then from (1.3), (3.5), and (3.18), we get 𝑦₁>𝑞₀𝑞₁/𝑝₀𝑝₁ and so from (1.3), (3.3), and (3.5), we have that𝑧₋₁>𝑞₁𝑞₀²𝑝₁𝑝₀²>𝑞₁𝑞₀𝑝₁𝑝₀. (3.19) If 𝑧₅<𝑞₀/𝑝₀, then (3.16) is true for 𝑞=2.
Now, suppose that𝑧₅≥𝑞₀𝑝₀. (3.20) Using (1.3), (3.3), (3.5), (3.20) and arguing as to prove (3.19) we get𝑧₋₁𝑞>(¹𝑞₀𝑝₁𝑝₀)². (3.21)Working inductively, we get thatif𝑧_−1+3𝑤≥𝑞₀𝑝₀,𝑤=1,2,…,then𝑧₋₁𝑞>(¹𝑞₀𝑝₁𝑝₀)^𝑤. (3.22) From (3.22) for 𝑤=ℎ, we get 𝑧₋₁>(𝑞₁𝑞₀/𝑝₁𝑝₀)^ℎ which contradicts (3.17). So 𝑧_−1+3ℎ<𝑞₀/𝑝₀ which means that (3.16) holds for 𝑞=ℎ.
Arguing as for 𝑧₋₁, we can prove that there exist positive integers 𝑘,𝑙 such that𝑧_0+3𝑘<𝑞₀𝑝₀,𝑧_1+3𝑙<𝑞₀𝑝₀. (3.23) From (3.16) and (3.23), we get that there exists a positive integer 𝑟 such that𝑧_𝑟<𝑞₀𝑝₀,𝑛≥𝑟. (3.24) Finally, from (1.3) and (3.24), we get 𝑦_𝑟+2<1 and so (3.8) is true for 𝑁=𝑟+2.
Similarly, we can prove that if (3.6) holds, then there exists a positive integer 𝑁 such that𝑧_𝑛<1,𝑦_𝑛<𝑞₁𝑝₁,𝑛≥𝑁. (3.25)
Finally, suppose that (3.7) hold. From (1.3) and (3.7), we have𝑦_𝑛+1−1=𝑝(𝑧_𝑛−1−1)𝑝+𝑦_𝑛−1,𝑧_𝑛+1−1=𝑞(𝑦_𝑛−1)𝑞+𝑧_𝑛−1, (3.26) and so,𝑦_𝑛+1𝑝−1=𝑝+𝑦_𝑛−1𝑞𝑞+𝑧_𝑛−3(𝑦_𝑛−2−1). (3.27) From (3.27), we get0≤𝑦_𝑛+1−1≤𝑦_𝑛−2−1,or0≥𝑦_𝑛+1−1≥𝑦_𝑛−2−1, (3.28) and so the subsequences 𝑦_3𝑛,𝑦_3𝑛+1,𝑦_3𝑛+2 either are bounded from below by 1 and decreasing or bounded from above by 1 and increasing. Hence, 𝑦_𝑛 is bounded and persists. Similarly, we can prove that 𝑧_𝑛 is bounded and persists. This completes the proof of part (i) of the proposition.
(ii) In statement (i), we have already proved that if (3.7) hold, then every solution of (1.3) is bounded and persists. So, from (3.4), it remains to show that if either𝑞₀−1<𝑝₀<𝑞₀,𝑞₁−1<𝑝₁≤𝑞₁, (3.29) or𝑞₀−1<𝑝₀≤𝑞₀,𝑞₁−1<𝑝₁<𝑞₁, (3.30)holds, then the solution (𝑦_𝑛,𝑧_𝑛) persists. From (3.3), (3.8), (3.25), (3.29), and (3.30), we get that𝑦_𝑛<𝑞₁𝑝₁,𝑧_𝑛<𝑞₀𝑝₀,𝑛≥𝑁. (3.31) We consider the positive number 𝑚 such that𝑚<min{𝑦_𝑁,𝑧_𝑁,𝑦_𝑁+1,𝑧_𝑁+1,𝑝₀+1−𝑞₀,𝑝₁+1−𝑞₁}. (3.32) Moreover, if𝑝𝑓(𝑦,𝑧)=⁰𝑧+𝑦𝑞₀𝑝+𝑦,𝑔(𝑦,𝑧)=¹𝑦+𝑧𝑞₁+𝑧, (3.33) then it is easy to see that for the functions (3.33), 𝑓 is increasing with respect to 𝑦 for any 𝑧, 𝑧<𝑞₀/𝑝₀ and 𝑔 is increasing with respect to 𝑧 for any 𝑦,𝑦<𝑞₁/𝑝₁.
Therefore, from (1.3), (3.31), and (3.32) we have𝑦_𝑁+2>(𝑝₀+1)𝑚𝑞₀+𝑚>𝑚,𝑧_𝑁+2>(𝑝₁+1)𝑚𝑞₁+𝑚>𝑚, (3.34) and working inductively, we take𝑦_𝑁+𝑠≥𝑚,𝑧_𝑁+𝑠≥𝑚,𝑠=0,1,…. (3.35) Therefore, (𝑦_𝑛,𝑧_𝑛) persists and using statement (i), then (𝑦_𝑛,𝑧_𝑛) is bounded and persists. This completes the proof of the proposition.

Proposition 3.3. One considers the system of difference equations (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants, and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. Then, the following statements are true.
(i) If𝑞₀𝑞₁𝑝₀𝑝₁<1, (3.36) then every solution of (1.3) persists. (ii) If𝑞₀≤𝑝₀≤𝑞₀+1,𝑞₁≤𝑝₁≤𝑞₁+1, (3.37) then every solution of (1.3) is bounded and persists.

Proof. Let (𝑦_𝑛,𝑧_𝑛) be an arbitrary solution of (1.3).
(i)From (3.36), we have𝑞₀𝑝₀<1, (3.38) or𝑞₁𝑝₁<1. (3.39)
Arguing as in the proof of statement (i) of Proposition 3.2, we can easily prove that if (3.38) holds, then there exists a positive integer 𝑀 such that𝑦_𝑛>1,𝑧_𝑛>𝑞₀𝑝₀,𝑛≥𝑀, (3.40) and if (3.39) holds, then there exists a positive integer 𝑀 such that𝑧_𝑛>1,𝑦_𝑛>𝑞₁𝑝₁,𝑛≥𝑀. (3.41)
(ii)From Proposition 3.2, we have that if (3.7) holds, then every solution of (1.3) is bounded and persists. So, from (3.37), it remains to show that if either𝑞₀<𝑝₀≤𝑞₀+1,𝑞₁≤𝑝₁≤𝑞₁+1, (3.42)or𝑞₀≤𝑝₀≤𝑞₀+1,𝑞₁<𝑝₁≤𝑞₁+1, (3.43) holds, then the solution (𝑦_𝑛,𝑧_𝑛) is bounded and persists.
From (3.36), (3.40), (3.41), (3.42), and (3.43), we get that𝑦_𝑛>𝑞₁𝑝₁,𝑧_𝑛>𝑞₀𝑝₀,𝑛≥𝑀. (3.44) Suppose that𝑝₀≠𝑞₀+1or𝑝₁≠𝑞₁+1. (3.45)
From (1.3) and (3.44), we have𝑧_𝑀+1>1,𝑦_𝑀+3>1. (3.46)We have for the functions (3.33) that 𝑓 is decreasing with respect to 𝑦 for any 𝑧, 𝑧>𝑞₀/𝑝₀ and 𝑔 is decreasing with respect to 𝑧 for any 𝑦, 𝑦>𝑞₁/𝑝₁. Therefore, relations (1.3), (3.44), and (3.46) imply that𝑧_𝑀+3≤𝑝₁𝑦_𝑀+2+1𝑞₁+1, (3.47) and so from (1.3) and (3.46),𝑦_𝑀+5≤𝑝₀𝑝₁(𝑞₀+1)(𝑞₁𝑦+1)^𝑀+2+𝑝₀(𝑞₀+1)(𝑞₁+1)+1. (3.48) Working inductively, we can prove that𝑦_𝑛+5≤𝑝₀𝑝₁(𝑞₀+1)(𝑞₁𝑦+1)^𝑛+2+𝑝₀(𝑞₀+1)(𝑞₁+1)+1,𝑛≥𝑀. (3.49)Then from (3.42), (3.43), (3.45), and Theorem 3.1, 𝑦_𝑛 is bounded. Similarly, we take that 𝑧_𝑛 is bounded. Therefore, from (3.44), the solution (𝑦_𝑛,𝑧_𝑛) is bounded and persists.
Now, suppose that𝑝₀=𝑞₀+1,𝑝₁=𝑞₁+1. (3.50) We claim that 𝑦_𝑛 is bounded. For the sake of contradiction, we assume that 𝑦_𝑛 is not bounded. Then, there exists a subsequence 𝑛_𝑖 such thatlim_𝑖→∞𝑦_𝑛𝑖+1𝑦=∞,_𝑛𝑖+1>max{𝑦_𝑗,𝑗<𝑛_𝑖}. (3.51)(3.52) Moreover, from (1.3) and (3.50), we get𝑦_𝑛𝑖+1<𝑞₀+1𝑞₀𝑧_𝑛𝑖−1+1, (3.53) and so from (3.51),lim_𝑖→∞𝑧_𝑛𝑖−1=∞. (3.54) Moreover, from (1.3) and (3.50),𝑧_𝑛𝑖−1<𝑞₁+1𝑞₁𝑦_𝑛𝑖−2+1, (3.55) and so from (3.54),lim_𝑖→∞𝑦_𝑛𝑖−2=∞. (3.56) Working inductively, we can prove thatlim_𝑖→∞𝑦_{𝑛𝑖+1−3𝑠}=∞,lim_𝑖→∞𝑧_{𝑛𝑖−1−3𝑠}=∞,𝑠=0,1,…. (3.57) We claim that 𝑦_𝑛𝑖−6 is a bounded sequence. Suppose on the contrary that there exists an unbounded subsequence of 𝑦_𝑛𝑖−6 and without loss of generality, we may suppose thatlim_𝑖→∞𝑦_𝑛𝑖−6=∞. (3.58) Arguing as above, we can easily prove thatlim_𝑖→∞𝑦_𝑛𝑖−9=lim_𝑖→∞𝑦_{𝑛𝑖−12}=∞. (3.59) Also, since from (1.3),𝑦_𝑛𝑖−6=(𝑞₀+1)((𝑧_𝑛𝑖−8)/(𝑦_𝑛𝑖−8))+1𝑞₀/𝑦_𝑛𝑖−8<+1(𝑞₀+1)𝑧_𝑛𝑖−8𝑦_𝑛𝑖−8+1, (3.60) from (3.58), we have that lim_𝑖→∞(𝑧_𝑛𝑖−8/𝑦_𝑛𝑖−8)=∞ and so eventually,𝑧_𝑛𝑖−8>𝑦_𝑛𝑖−8. (3.61) From (1.3), (3.50), and (3.61), we have𝑦_𝑛𝑖+1=(𝑞₀+1)𝑧_𝑛𝑖−1+𝑦_𝑛𝑖−1𝑞₀+𝑦_𝑛𝑖−1<𝑞₀+1𝑞₀𝑧_𝑛𝑖−1=𝑞+1₀+1𝑞₀((𝑞₁+1)𝑦_𝑛𝑖−2+𝑧_𝑛𝑖−3𝑞₁+𝑧_𝑛𝑖−3𝑞)+1<1+⁰+1𝑞₀+𝑞₀+1𝑞₀𝑞₁+1𝑞₁𝑦_𝑛𝑖−2<⋯<𝐴+𝐵𝑦_𝑛𝑖−8<𝐴+𝐵𝑧_𝑛𝑖−8, (3.62) where𝑞𝐴=1+⁰+1𝑞₀+𝑞₀+1𝑞₀𝑞₁+1𝑞₁𝑞+(⁰+1𝑞₀)²𝑞₁+1𝑞₁𝑞+(⁰+1𝑞₀)²(𝑞₁+1𝑞₁)²𝑞+(⁰+1𝑞₀)³(𝑞₁+1𝑞₁)²,𝑞𝐵=(⁰+1𝑞₀)³(𝑞₁+1𝑞₁)³. (3.63) Therefore, using (1.3) and (3.50), we get𝑦_𝑛𝑖+1<𝐴+𝐵((𝑞₁+1)𝑦_𝑛𝑖−9+𝑧_{𝑛𝑖−10}𝑞₁+𝑧_{𝑛𝑖−10}), (3.64) and since from (3.57) and (3.59), we have that 𝑦_𝑛𝑖−9→∞,𝑧_{𝑛𝑖−10}→∞ as 𝑖→∞, we can easily prove that eventually, 𝑦_𝑛𝑖+1<𝑦_𝑛𝑖−9, (3.65) which contradicts to (3.52).
Therefore, 𝑦_𝑛𝑖−6 is a bounded sequence. From (1.3), (3.50), and (3.57), we get𝑧_𝑛𝑖−5=(𝑞₁+1)𝑦_𝑛𝑖−6+𝑧_𝑛𝑖−7𝑞₁+𝑧_𝑛𝑖−7=(𝑞₁+1)(𝑦_𝑛𝑖−6/𝑧_𝑛𝑖−7)+1𝑞₁/𝑧_𝑛𝑖−7+1→1,𝑖→∞. (3.66) Similarly, from (1.3), (3.50) and (3.57) and (3.66) follows,𝑦_𝑛𝑖−3=(𝑞₀+1)𝑧_𝑛𝑖−5+𝑦_𝑛𝑖−5𝑞₀+𝑦_𝑛𝑖−5=(𝑞₀+1)(𝑧_𝑛𝑖−5/𝑦_𝑛𝑖−5)+1𝑞₀/𝑦_𝑛𝑖−5+1→1,𝑖→∞. (3.67) Now, we prove thatliminf_𝑖→∞𝑦_𝑛𝑖−1>1. (3.68) Otherwise, and without loss of generality, we may suppose that lim_𝑖→∞𝑦_𝑛𝑖−1≤1. So, relations (1.3), (3.50), and (3.67) imply thatlim_𝑖→∞𝑦_𝑛𝑖−1=(𝑞₀+1)lim_𝑖→∞𝑧_𝑛𝑖−3+lim_𝑖→∞𝑦_𝑛𝑖−3𝑞₀+lim_𝑖→∞𝑦_𝑛𝑖−3≤1, (3.69) and so lim_𝑖→∞𝑧_𝑛𝑖−3≤𝑞₀𝑞₀+1. (3.70)Moreover, from (1.3), (3.44), and (3.50), we get eventually𝑧_𝑛𝑖−3=(𝑞₁+1)𝑦_𝑛𝑖−4+𝑧_𝑛𝑖−5𝑞₁+𝑧_𝑛𝑖−5>(𝑞₁+1)(𝑞₁/(𝑞₁+1))+𝑧_𝑛𝑖−5𝑞₁+𝑧_𝑛𝑖−5=1, (3.71) and so from (3.66), lim_𝑖→∞𝑧_𝑛𝑖−3≥1 which contradicts to (3.70).
Hence, (3.68) is true.
Similarly, we can prove thatliminf_𝑖→∞𝑧_𝑛𝑖−3>1. (3.72) Therefore, from (3.68) and (3.72), we have eventually𝑦_𝑛𝑖−1>1+𝑘,𝑧_𝑛𝑖−3>1+𝑚, (3.73) where 𝑘,𝑚 are positive real numbers.
Hence, from (1.3), (3.50), and (3.73) we have𝑦_𝑛𝑖+1=(𝑞₀+1)[((𝑞₁+1)𝑦_𝑛𝑖−2+𝑧_𝑛𝑖−3)/(𝑞₁+𝑧_𝑛𝑖−3)]+𝑦_𝑛𝑖−1𝑞₀+𝑦_𝑛𝑖−1<(𝑞₀+1)(𝑞₁+1)(𝑞₁+1+𝑚)(𝑞₀𝑦+1+𝑘)^𝑛𝑖−2+𝑞₀+1𝑞₀+1. (3.74) Then from (3.57), we can prove that eventually𝑦_𝑛𝑖+1<𝑦_𝑛𝑖−2, (3.75) which contradicts to (3.52).
Therefore, 𝑦_𝑛 is a bounded sequence. Moreover, from (1.3), (3.50), we take that 𝑧_𝑛 is bounded. Therefore, the solution (𝑦_𝑛,𝑧_𝑛) is bounded and persists. This completes the proof of the proposition.

4. Attractivity of the Positive Equilibrium of System (1.3)

In the following propositions, we study the convergency of the solutions of system (1.3) to its positive equilibrium.

Proposition 4.1. One considers the system of difference equations (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants, and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. If either (3.29) or (3.30) hold, then every solution of (1.3) tents to the positive equilibrium of (1.3).

Proof. Let (𝑦_𝑛,𝑧_𝑛) be an arbitrary solution of (1.3). From Proposition 3.2, there exist𝐿₁=limsup_𝑛→∞𝑦_𝑛,𝐿₂=limsup_𝑛→∞𝑧_𝑛,𝑙₁=liminf_𝑛→∞𝑦_𝑛,𝑙₂=liminf_𝑛→∞𝑧_𝑛,0<𝐿₁,𝐿₂,𝑙₁,𝑙₂<∞. (4.1) From (1.3), (3.31), and the monotony of functions (3.33), we have 𝐿₁≤𝑝₀𝐿₂+𝐿₁𝑞₀+𝐿₁,𝐿₂≤𝑝₁𝐿₁+𝐿₂𝑞₁+𝐿₂,𝑙₁≥𝑝₀𝑙₂+𝑙₁𝑞₀+𝑙₁,𝑙₂≥𝑝₁𝑙₁+𝑙₂𝑞₁+𝑙₂, (4.2) and hence 𝐿₁²+𝐿₁(𝑞₀−1)−𝑝₀𝐿₂≤0,𝐿₂²+𝐿₂(𝑞₁−1)−𝑝₁𝐿₁𝑙≤0,₁₂+𝑙₁(𝑞₀−1)−𝑝₀𝑙₂≥0,𝑙₂²+𝑙₂(𝑞₁−1)−𝑝₁𝑙₁≥0. (4.3)The third inequality of (4.3), implies that𝑙₁≥1−𝑞₀+(1−𝑞₀)²+4𝑝₀𝑙₂2, (4.4)and so from the last inequality of (4.3), we have2𝑙₂²+2𝑙₂(𝑞₁−1)+(𝑞₀−1)𝑝₁≥𝑝₁(1−𝑞₀)²+4𝑝₀𝑙₂. (4.5) Hence, we get(2𝑙₂²+2𝑙₂(𝑞₁−1)+(𝑞₀−1)𝑝₁)²≥(𝑝₁(1−𝑞₀)²+4𝑝₀𝑙₂)², (4.6) or 𝑙₂³+2𝑙₂²(𝑞₁−1)+𝑙₂[(𝑞₁−1)²+𝑝₁(𝑞₀−1)]+𝑝₁(𝑞₁−1)(𝑞₀−1)−𝑝₀𝑝₁²≥0. (4.7) The first inequality of (4.3), implies that0<𝐿₁≤1−𝑞₀+(1−𝑞₀)²+4𝑝₀𝐿₂2, (4.8) and so from second inequality of (4.3), we get2𝐿₂²+2𝐿₂(𝑞₁−1)+(𝑞₀−1)𝑝₁≤𝑝₁(1−𝑞₀)²+4𝑝₀𝐿₂. (4.9) Using (4.3), we have𝐿₁≥𝑙₁>1−𝑞₀,𝐿₂≥𝑙₂>1−𝑞₁. (4.10) Therefore, from (4.5) and (4.10), we get2𝐿₂²+2𝐿₂(𝑞₁−1)+(𝑞₀−1)𝑝₁=2𝐿₂(𝐿₂+𝑞₁−1)+(𝑞₀−1)𝑝₁≥2𝑙₂²+2𝑙₂(𝑞₁−1)+(𝑞₀−1)𝑝₁>0. (4.11)
Using (4.9) and (4.11), we have𝐿₂³+2𝐿₂²(𝑞₁−1)+𝐿₂[(𝑞₁−1)²+𝑝₁(𝑞₀−1)]+𝑝₁(𝑞₁−1)(𝑞₀−1)−𝑝₀𝑝₁²≤0. (4.12) In Proposition 2.1, we proved that (2.3) has a unique positive solution 𝑧,𝑧∈(1−𝑞₁,∞). We can write𝑓(𝑥)=(𝑥−𝑧)(𝑥²+𝑎𝑥+𝑏),𝑎,𝑏∈ℝ, (4.13) where 𝑓(𝑥) is defined in (2.3) and 𝑥²+𝑎𝑥+𝑏>0 for any 𝑥>1−𝑞₁. Then from (4.7), (4.12), and (4.13), we have(𝐿₂−𝑧)(𝐿²₂+𝑎𝐿₂+𝑏)≤0,(𝑙₂−𝑧)(𝑙²₂+𝑎𝑙₂+𝑏)≥0. (4.14) Therefore, from (4.10) and (4.14),𝐿₂≤𝑧≤𝑙₂, (1) which implies that𝐿₂=𝑙₂=𝑧. (4.15)
In addition, using (4.15), the first and the third inequalities of (4.3), we have𝐿²₁+(𝑞₀−1)𝐿₁≤𝑙²₁+(𝑞₀−1)𝑙₁, (4.16) and so (4.10) implies that𝐿₁=𝑙₁. (4.17) This completes the proof of the proposition.

Proposition 4.2. One considers the system of difference equations (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants, and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. If either (3.42) or (3.43) hold, then every solution of (1.3) tents to the positive equilibrium of (1.3).

Proof. Let (𝑦_𝑛,𝑧_𝑛) be an arbitrary solution of (1.3). From Proposition 3.3, there exist 𝐿_𝑖,𝑙_𝑖, 𝑖=1,2 such that (4.1) are satisfied.
From (1.3), the monotony of functions (3.33) and (3.44), we have𝐿₁≤𝑝₀𝐿₂+𝑙₁𝑞₀+𝑙₁,𝐿₂≤𝑝₁𝐿₁+𝑙₂𝑞₁+𝑙₂,𝑙₁≥𝑝₀𝑙₂+𝐿₁𝑞₀+𝐿₁,𝑙₂≥𝑝₁𝑙₁+𝐿₂𝑞₁+𝐿₂, (4.18) and hence𝐿₁𝑙₁+𝐿₁𝑞₀≤𝑝₀𝐿₂+𝑙₁,𝐿₁𝑙₁+𝑙₁𝑞₀≥𝑝₀𝑙₂+𝐿₁,𝐿₂𝑙₂+𝐿₂𝑞₁≤𝑝₁𝐿₁+𝑙₂,𝐿₂𝑙₂+𝑙₂𝑞₁≥𝑝₁𝑙₁+𝐿₂, (4.19) which implies that(1+𝑞₀)(𝐿₁−𝑙₁)≤𝑝₀(𝐿₂−𝑙₂),(1+𝑞₁)(𝐿₂−𝑙₂)≤𝑝₁(𝐿₁−𝑙₁). (4.20) Therefore,[(1+𝑞₀)(1+𝑞₁)−𝑝₀𝑝₁](𝐿₁−𝑙₁)≤0. (4.21) First suppose that (3.45) holds. Then from (3.42) or (3.43), and (3.45), we get 𝐿₁−𝑙₁≤0, which means that𝐿₁=𝑙₁. (4.22)Using (4.20), it is obvious that𝐿₂=𝑙₂. (4.23) So if (3.45) holds, the proof is completed.
Now, suppose that (3.50) hold. Then from (4.20), we have𝐿₂−𝑙₂=𝐿₁−𝑙₁. (4.24) Moreover, from (4.24), it follows that(𝑞₀+1)𝑙₂+𝐿₁−𝑙₁𝑞₀=(𝑞₀+1)𝐿₂+𝑙₁−𝐿₁𝑞₀. (4.25) In addition, from (3.50), the first and the second inequalities of (4.19), we get(𝑞₀+1)𝑙₂+𝐿₁−𝑙₁𝑞₀≤𝐿₁𝑙₁≤(𝑞₀+1)𝐿₂+𝑙₁−𝐿₁𝑞₀. (4.26)Therefore, from (4.25) and (4.26), we have𝐿₁=(𝑞₀+1)𝐿₂+𝑙₁𝑞₀+𝑙₁. (4.27) We may assume that there exists a positive integer 𝑛_𝑖 such thatlim_𝑖→∞𝑦_{𝑛𝑖−𝑗}=𝐴_𝑗,lim_𝑖→∞𝑧_{𝑛𝑖−𝑗}=𝐵_𝑗,lim_𝑖→∞𝑦_𝑛𝑖+1=𝐿₁. (4.28) Moreover, from (1.3), (3.50), and (4.28), we get𝐿₁=(𝑞₀+1)𝐵₁+𝐴₁𝑞₀+𝐴₁. (4.29) Since 𝑓(𝑥,𝑦)=((𝑞₀+1)𝑥+𝑦)/(𝑞₀+𝑦) is decreasing with respect to 𝑦, for any 𝑥>(𝑞₀/(𝑞₀+1)), if 𝐵₁<𝐿₂ or 𝑙₁<𝐴₁, then from (3.44), and (3.50), we get𝐿₁<(𝑞₀+1)𝐿₂+𝑙₁𝑞₀+𝑙₁, (4.30) which contradicts to (4.27). So,𝐵₁=𝐿₂,𝑙₁=𝐴₁. (4.31) Using the same argument, we can prove that𝐴₂=𝐿₁,𝐵₃=𝑙₂,𝐵₃=𝑙₂,𝐴₃=𝐿₁,𝐵₄=𝐿₂,𝐴₄=𝑙₁,𝐴₄=𝑙₁,𝐵₅=𝐿₂,𝐵₅=𝐿₂,𝐴₅=𝑙₁,𝐴₅=𝐿₁,𝐵₆=𝑙₂, (4.32) and so 𝐿₁=𝑙₁=𝐴. Also, from (4.24), we have 𝐿₂=𝑙₂=𝐵. Therefore,lim_𝑛→∞𝑦_𝑛=𝐴,lim_𝑛→∞𝑧_𝑛=𝐵, (4.33) where obviously 𝐴=𝐵=2. This completes the proof of the proposition.

Proposition 4.3. One considers the system of difference equations (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants, and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. If relations (3.7) hold, then every solution of (1.3) tents to the positive equilibrium (1,1) of (1.3).

Proof. Let (𝑦_𝑛,𝑧_𝑛) be an arbitrary solution of (1.3). From the proof of Proposition 3.2, the subsequences 𝑦_3𝑛,𝑦_3𝑛+1,𝑦_3𝑛+2,𝑧_3𝑛,𝑧_3𝑛+1, and 𝑧_3𝑛+2 are monotone and 𝑦_𝑛, 𝑧_𝑛 are bounded and persist. So, there exist positive numbers 𝐿₁,𝐿₂,𝐿₃,𝑀₁,𝑀₂, and 𝑀₃ such that𝐿₁=lim_𝑛→∞𝑦_3𝑛,𝐿₂=lim_𝑛→∞𝑦_3𝑛+1,𝐿₃=lim_𝑛→∞𝑦_3𝑛+2,𝑀₁=lim_𝑛→∞𝑧_3𝑛,𝑀₂=lim_𝑛→∞𝑧_3𝑛+1,𝑀₃=lim_𝑛→∞𝑧_3𝑛+2, (4.34) and from (1.3) and (3.7), we get𝐿₁=𝑝𝑀₂+𝐿₂𝑝+𝐿₂,𝑀₁=𝑞𝐿₃+𝑀₂𝑞+𝑀₂,𝐿₂=𝑝𝑀₃+𝐿₃𝑝+𝐿₃,𝑀₂=𝑞𝐿₁+𝑀₃𝑞+𝑀₃,𝐿₃=𝑝𝑀₁+𝐿₁𝑝+𝐿₁,𝑀₃=𝑞𝐿₂+𝑀₁𝑞+𝑀₁. (4.35) Then, we have𝐿₁𝑝+𝐿₁𝐿₂=𝑝𝑀₂+𝐿₂,𝑀₁𝑞+𝑀₁𝑀₂=𝑞𝐿₃+𝑀₂,𝐿₂𝑝+𝐿₂𝐿₃=𝑝𝑀₃+𝐿₃,𝑀₂𝑞+𝑀₂𝑀₃=𝑞𝐿₁+𝑀₃,𝐿₃𝑝+𝐿₁𝐿₃=𝑝𝑀₁+𝐿₁,𝑀₃𝑞+𝑀₃𝑀₁=𝑞𝐿₂+𝑀₁, (4.36) and hence,(𝐿₁−𝑀₂)𝑝=𝐿₂(1−𝐿₁),(𝑀₁−𝐿₃)𝑞=𝑀₂(1−𝑀₁),(𝐿₂−𝑀₃)𝑝=𝐿₃(1−𝐿₂),(𝑀₂−𝐿₁)𝑞=𝑀₃(1−𝑀₂),(𝐿₃−𝑀₁)𝑝=𝐿₁(1−𝐿₃),(𝑀₃−𝐿₂)𝑞=𝑀₁(1−𝑀₃). (4.37) Therefore, we take1𝑝𝐿₂(1−𝐿₁1)=𝑞𝑀₃(𝑀₂1−1),𝑝𝐿₃(1−𝐿₂1)=𝑞𝑀₁(𝑀₃1−1),𝑝𝐿₁(1−𝐿₃1)=𝑞𝑀₂(𝑀₁−1). (2) So,if𝐿₁≥1(resp.,𝐿₁≤1),then𝑀₂≤1(resp.,𝑀₂≥1),if𝐿₂≥1(resp.,𝐿₂≤1),then𝑀₃≤1(resp.,𝑀₃≥1),if𝐿₃≥1(resp.,𝐿₃≤1),then𝑀₁≤1(resp.,𝑀₁≥1). (4.38) Therefore, if 𝐿₁≥1,𝑀₂≤1 (resp., 𝐿₁≤1,𝑀₂≥1), we have 𝐿₁−𝑀₂≥0 (resp., 𝐿₁−𝑀₂≤0) and so from (4.37), 𝐿₁≤1 (resp., 𝐿₁≥1). Hence, 𝐿₁=1 and from (4.37), 𝑀₂=1. Similarly, we can prove that 𝐿₂=1,𝐿₃=1,𝑀₁=1,𝑀₃=1. This completes the proof of the proposition.

5. Stability of System (1.3)

In this section we find conditions so that the positive equilibrium (𝑦,𝑧) and the zero equilibrium of (1.3) are stable.

Proposition 5.1. Consider system (1.3) where 𝑝_𝑖,𝑞_𝑖, 𝑖=0,1 are positive constants and the initial values 𝑦_𝑖,𝑧_𝑖, 𝑖=−1,0 are positive numbers. Then, the following statements are true.
(i) If 𝑞₀−1<𝑝₀≤𝑞₀,𝑞₁−1<𝑝₁≤𝑞₁,𝑞₀+𝑞₁+𝑝₀𝑝₁+𝑞₀𝑞₁<1, (5.1) then the unique positive equilibrium (𝑦,𝑧) of (1.3) is globally asymptotically stable. (ii) If 𝑞₀+𝑞₁+𝑝₀𝑝₁+1<𝑞₀𝑞₁, (5.2) then the zero equilibrium of (1.3) is locally asymptotically stable.

Proof. (i) Since (𝑦,𝑧) is the unique positive positive equilibrium of (1.3), we have𝑝𝑦=⁰𝑧+𝑦𝑞₀𝑝+𝑦,𝑧=¹𝑦+𝑧𝑞₁+𝑧. (5.3) Then from (5.1) and (5.3), we get𝑞𝑦≤⁰𝑧+𝑦𝑞₀𝑞+𝑦,𝑧≤¹𝑦+𝑧𝑞₁+𝑧. (5.4) Without loss of generality we assume that 𝑧≤𝑦. Then from (5.4), it results that𝑞𝑦≤⁰𝑦+𝑦𝑞₀+𝑦, (5.5) which means that𝑦≤1. (5.6) Moreover, from (5.4) and (5.6), we get𝑞𝑧≤¹+𝑧𝑞₁+𝑧=1. (5.7) In addition, from (5.3), we have𝑦𝑦>𝑞₀𝑧+𝑦,𝑧>𝑞₁+𝑧, (5.8) and so𝑦>1−𝑞₀,𝑧>1−𝑞₁. (5.9) Then the linearized system of (1.3) about the positive equilibrium (𝑦,𝑧) is𝑍_𝑛+1=𝐴𝑍_𝑛, (5.10) where𝐴=𝑞00100001₀−𝑝₀𝑧(𝑞₀+𝑦)²𝑝₀𝑞₀0𝑞+𝑦00₁−𝑝₁𝑦(𝑞₁+𝑧)²𝑝₁𝑞₁0+𝑧,𝑍_𝑛=𝑤_𝑛−1𝑣_𝑛−1𝑤_𝑛𝑣_𝑛. (5.11) The characteristic equation of 𝐴 is𝜆⁴𝑞−(⁰−𝑝₀𝑧(𝑞₀+𝑦)²+𝑞₁−𝑝₁𝑦(𝑞₁+𝑧)²)𝜆²−𝑝₁𝑝₀(𝑞₀+𝑦)(𝑞₁+𝑧)𝜆+(𝑞₀−𝑝₀𝑧)(𝑞₁−𝑝₁𝑦)(𝑞₀+𝑦)²(𝑞₁+𝑧)²=0. (5.12) According to Remark 1.3.1 of [7], all the roots of (5.12) are of modulus less than 1 if and only if|𝑞₀−𝑝₀𝑧(𝑞₀+𝑦)²+𝑞₁−𝑝₁𝑦(𝑞₁+𝑧)²𝑝|+¹𝑝₀(𝑞₀+𝑦)(𝑞₁+𝑧)+|(𝑞₀−𝑝₀𝑧)(𝑞₁−𝑝₁𝑦)(𝑞₀+𝑦)²(𝑞₁+𝑧)²|<1. (5.13) From (5.3), we get𝑞₀−𝑝₀𝑧=(1−𝑦)(𝑦+𝑞₀),𝑞₁−𝑝₁𝑦=(1−𝑧)(𝑧+𝑞₁). (5.14) Then from (5.6), (5.7), and (5.14), inequality (5.13) is equivalent to1−𝑦𝑞₀++𝑦1−𝑧𝑞₁+𝑝+𝑧¹𝑝₀(𝑞₀+𝑦)(𝑞₁++𝑧)(1−𝑦)(1−𝑧)(𝑞₀+𝑦)(𝑞₁+𝑧)<1. (5.15) Using (5.9), inequality (5.15) holds if (5.1) are satisfied. Using Propositions 4.1 and 4.3, we have that the unique positive equilibrium (𝑦,𝑧) of (1.3) is globally asymptotically stable.
(ii) Arguing as above, we can prove that the linearized system of (1.3) about the zero equilibrium is𝑍_𝑛+1=𝐴𝑍_𝑛, (5.16) where𝐴=100100001𝑞₀𝑝₀𝑞₀0100𝑞₁𝑝₁𝑞₁0,𝑍_𝑛=𝑤_𝑛−1𝑣_𝑛−1𝑤_𝑛𝑣_𝑛. (5.17) The characteristic equation of 𝐴 is𝜆⁴1−(𝑞₀+1𝑞₁)𝜆²−𝑝₁𝑝₀𝑞₀𝑞₁1𝜆+𝑞₀𝑞₁=0. (5.18) Using [7, Remark 1.3.1], all the roots of (5.18) are of modulus less than 1 if and only if relation (5.2) holds. This completes the proof of the proposition.

6. Conclusion

In this paper, in order to investigate (1.2), we study the equivalent system (1.3). Summarizing the results of Sections 2, 3, 4, we get the following statements, concerning (1.2).

(i)If (2.1) hold, then (1.2) has a unique positive periodic solution of period 2.(ii)If either (3.4) or (3.37) holds, then every positive solution of (1.2) is bounded and persists and tends to the unique positive periodic solution.(iii)If (5.1) hold, then the unique periodic solution of (1.2) is globally asymptotically stable and if (5.2) holds, then the zero solution of (1.2) is locally asymptotically stable.

Open Problem
Consider the difference equation (1.2) where 𝑝_𝑛,𝑞_𝑛, 𝑛=0,1,… are positive sequences of period 2, and the initial values 𝑥_𝑖, 𝑖=−3,−2,−1,0 are positive numbers. Prove that
(i)if 𝑞₀−1<𝑝₀≤𝑞₀+1,𝑞₁−1<𝑝₁≤𝑞₁+1 (6.1) are satisfied, then every positive solution of (1.2) is bounded and persists;(ii)if relations (6.1) are satisfied, then every positive solution of (1.2) tends to the unique positive equilibrium (𝑦,𝑧) of (1.2) as 𝑛→∞.

Acknowledgment

The authors would like to thank the referees for their helpful suggestions.