On the Solutions of the System of Difference Equations xn+1=max{A/xn,yn/xn}, yn+1=max{A/yn,xn/yn}

Abstract

We study the behavior of the solutions of the following system of difference equations 𝑥_𝑛+1=max{𝐴/𝑥_𝑛,𝑦_𝑛/𝑥_𝑛}, 𝑦_𝑛+1=max{𝐴/𝑦_𝑛,𝑥_𝑛/𝑦_𝑛} where the constant A and the initial conditions are positive real numbers.

1. Introduction

Recently, there has been a great interest in studying the periodic nature of nonlinear difference equations. Although difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the periodic behavior of their solutions. The periodic nature of nonlinear difference equations of the max type has been investigated by many authors. See, for example [1–24].

In this paper we study the behavior of the solutions of the following system of difference equations: 𝑥_𝑛+1𝐴=max{𝑥_𝑛,𝑦_𝑛𝑥_𝑛},𝑦_𝑛+1𝐴=max{𝑦_𝑛,𝑥_𝑛𝑦_𝑛}, (1.1) where the constant 𝐴 and the initial conditions are positive real numbers.

2. Main Result

Definition 2.1. Fibonacci sequence is 𝑓₁=1, 𝑓₂=1 and for 𝑛≥3,𝑓_𝑛=𝑓_𝑛−1+𝑓_𝑛−2.

Definition 2.2. The symbol [[]] symbolizes the greatest integer function.

Definition 2.3. The sequence of 𝑎(𝑛)≡𝑛(mod2).

Definition 2.4. The sequence of 𝑘(𝑛)=𝑛,𝑛=0,2,4,…,𝑛+1,𝑛=1,3,5,…. (2.1)

Theorem 2.5. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 𝐴<𝑥₀<𝑦₀ and 𝑦₀/𝑥₀>𝐴.
If 𝑛≥1, then 𝑥_𝑛𝐴=(^{𝑓𝑘(𝑛)−1−𝑎(𝑛)}𝑥^{𝑓𝑘(𝑛)}₀𝑦^{𝑓𝑘(𝑛)}₀)^(−1)𝑛, (2.2)𝑦₁=𝑥₀/𝑦₀, and if 𝑛≥2𝑦_𝑛𝑦=(^{𝑓𝑘(𝑛−1)0+1}𝐴^{𝑓𝑘(𝑛−1)+𝑎(𝑛)−1}𝑥^{𝑓𝑘(𝑛−1)}₀⁺¹)^(−1)𝑛. (2.3)

Proof. Let 𝑦₀>𝐴, then 𝑥₁𝐴=max{𝑥₀,𝑦₀𝑥₀𝑦}=⁰𝑥₀𝑦>𝐴,₁𝐴=max{𝑦₀,𝑥₀𝑦₀𝑥}=⁰𝑦₀𝑥<𝐴,₂𝐴=max{𝑥₁,𝑦₁𝑥₁}=𝐴𝑥₀𝑦₀𝑦<𝐴,₂𝐴=max{𝑦₁,𝑥₁𝑦₁𝑥}=¹𝑦₁=𝑦²₀𝑥²₀𝑥>𝐴,₃𝐴=max{𝑥₂,𝑦₂𝑥₂𝑦}=²𝑥₂=𝑦³₀𝐴𝑥³₀𝑦>𝐴,₃𝐴=max{𝑦₂,𝑥₂𝑦₂𝐴}=𝑦₂=𝐴𝑥²₀𝑦²₀𝑥<𝐴,₄𝐴=max{𝑥₃,𝑦₃𝑥₃𝐴}=²𝑥³₀𝑦³₀𝑦<𝐴,₄𝐴=max{𝑦₃,𝑥₃𝑦₃𝑥}=³𝑦₃=𝑦⁵₀𝐴²𝑥⁵₀𝑥>𝐴,₅𝐴=max{𝑥₄,𝑦₄𝑥₄𝑦}=⁴𝑥₄=𝑦⁸₀𝐴³𝑥⁸₀𝑦>𝐴,₅𝐴=max{𝑦₄,𝑥₄𝑦₄𝐴}=𝑦₄=𝐴³𝑥⁵₀𝑦⁵₀⋮<𝐴, (2)𝑛≥1 then 𝑥_𝑛𝐴=(^{𝑓𝑘(𝑛)−1−𝑎(𝑛)}𝑥^{𝑓𝑘(𝑛)}₀𝑦^{𝑓𝑘(𝑛)}₀)^(−1)𝑛, (2.5)𝑦₁=𝑥₀/𝑦₀ then 𝑛≥2, 𝑦_𝑛𝑦=(^{𝑓𝑘(𝑛−1)0+1}𝐴^{𝑓𝑘(𝑛−1)+𝑎(𝑛)−1}𝑥^{𝑓𝑘(𝑛−1)}₀⁺¹)^(−1)𝑛. (2.6)

Theorem 2.6. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 𝐴<𝑦₀<𝑥₀.
𝑥₁=𝑥₀/𝑦₀ and if 𝑛≥2, 𝑥_𝑛𝐴=(^{𝑓𝑘(𝑛−1)+𝑎(𝑛)−1}𝑥^{𝑓𝑘(𝑛−1)}₀⁺¹𝑦^{𝑓𝑘(𝑛−1)}₀⁺¹)^(−1)𝑛. (2.7)
If 𝑛≥1, then 𝑦_𝑛𝑦=(^{𝑓𝑘(𝑛)0}𝐴^{𝑓𝑘(𝑛)−1−𝑎(𝑛)}𝑥^{𝑓𝑘(𝑛)}₀)^(−1)𝑛. (2.8)

Proof. Similarly we can obtain the proof as the proof of Theorem 2.5.

Theorem 2.7. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 𝐴<𝑥₀<𝑦₀ and (𝑦₀/𝑥₀)>𝐴. (a)lim_𝑛→∞𝑥_2𝑛=0,lim_𝑛→∞𝑦_2𝑛(=∞.b)lim_𝑛→∞𝑥_2𝑛+1=∞,lim_𝑛→∞𝑦_2𝑛+1=0. (2.9)

Proof. (a) We obtain that lim_𝑛→∞𝑥_2𝑛=lim_𝑛→∞(𝐴^{𝑓𝑘(2𝑛)−1−𝑎(2𝑛)}𝑥^{𝑓𝑘(2𝑛)}₀𝑦^{𝑓𝑘(2𝑛)}₀)^(−1)2𝑛=lim_𝑛→∞(𝐴^{𝑓𝑘(2𝑛)−1−𝑎(2𝑛)}𝑥^{𝑓𝑘(2𝑛)}₀𝑦^{𝑓𝑘(2𝑛)}₀)^(−1)2𝑘=lim_𝑛→∞(𝐴^{𝑓2𝑛−1−𝑎(2𝑛)}𝑥^𝑓2𝑛₀𝑦^𝑓2𝑛₀)=0,lim_𝑛→∞𝑦_2𝑛=lim_𝑛→∞(𝑦^{𝑓𝑘(2𝑛−1)}₀⁺¹𝐴^{𝑓𝑘(2𝑛−1)+𝑎(2𝑛)−1}𝑥^{𝑓𝑘(2𝑛−1)}₀⁺¹)^(−1)2𝑛=lim_𝑛→∞(𝑦^{𝑓𝑘(2𝑛−1)}₀⁺¹𝐴^{𝑓𝑘(2𝑛−1)+𝑎(2𝑛)−1}𝑥^{𝑓𝑘(2𝑛−1)}₀⁺¹)^(−1)2𝑘=lim_𝑘→∞(𝑦^𝑓2𝑛₀⁺¹𝐴^{𝑓2𝑛−1}𝑥^𝑓2𝑛₀⁺¹)=∞. (2.10)
(b) Similarly we can obtain the proof of (b) as the proof of (a).

Theorem 2.8. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 𝐴<𝑦₀<𝑥₀ and (𝑥₀/𝑦₀)>𝐴. (a)lim_𝑛→∞𝑥_2𝑛=∞,lim_𝑛→∞𝑦_2𝑛(=0.b)lim_𝑛→∞𝑥_2𝑛+1=0,lim_𝑛→∞𝑦_2𝑛+1=∞. (2.11)

Proof. Similarly we can obtain the proof as the proof of Theorem 2.7.

Theorem 2.9. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 1<𝑥₀<𝑦₀<𝐴.
If 𝑛≥1, then 𝑥_𝑛𝑥=(⁰𝐴^𝑎(𝑛))^(−1)𝑛. (2.12)
If 𝑛≥1, then 𝑦_𝑛𝑦=(⁰𝐴^𝑎(𝑛))^(−1)𝑛. (2.13)

Proof. Let 𝑥₁𝐴=max{𝑥₀,𝑦₀𝑥₀𝐴}=𝑥₀𝑦<𝐴,₁𝐴=max{𝑦₀,𝑥₀𝑦₀𝐴}=𝑦₀𝑥<𝐴,₂𝐴=max{𝑥₁,𝑦₁𝑥₁}=𝐴𝑥₀𝐴=𝑥₀𝑦<𝐴,₂𝐴=max{𝑦₁,𝑥₁𝑦₁𝐴}=𝑦₁=𝑦₀𝑥<𝐴,₃𝐴=max{𝑥₂,𝑦₂𝑥₂𝐴}=𝑥₂=𝐴𝑥₀𝑦<𝐴,₃𝐴=max{𝑦₂,𝑥₂𝑦₂𝐴}=𝑦₂=𝐴𝑦₀𝑥<𝐴,₄𝐴=max{𝑥₃,𝑦₃𝑥₃𝐴}=𝑥₃=𝑥₀𝑦<𝐴,₄𝐴=max{𝑦₃,𝑥₃𝑦₃𝐴}=𝑦₃=𝑦₀𝑥<𝐴,₅𝐴=max{𝑥₄,𝑦₄𝑥₄𝐴}=𝑥₄=𝐴𝑥₀𝑦<𝐴,₅𝐴=max{𝑦₄,𝑥₄𝑦₄𝐴}=𝑦₄=𝐴𝑦₀⋮<𝐴, (2)𝑛≥1, then 𝑥_𝑛𝑥=(⁰𝐴^𝑎(𝑛))^(−1)𝑛,𝑦_𝑛𝑦=(⁰𝐴^𝑎(𝑛))^(−1)𝑛. (2.15)

Theorem 2.10. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 1<𝑥₀<𝑦₀<𝐴. (a)lim_𝑛→∞𝑥_2𝑛=𝑥₀,lim_𝑛→∞𝑦_2𝑛=𝑦₀.(b)lim_𝑛→∞𝑥_2𝑛+1=𝐴𝑥₀,lim_𝑛→∞𝑦_2𝑛+1=𝐴𝑦₀. (2.16)

Proof. (a) We obtain that lim_𝑛→∞𝑥_2𝑛=lim_𝑛→∞(𝑥₀𝐴^𝑎(2𝑛))^(−1)2𝑛=lim_𝑛→∞(𝑥₀𝐴⁰)^(−1)2𝑛=lim_𝑛→∞(𝑥₀𝐴⁰)=𝑥₀,lim_𝑛→∞𝑦_2𝑛=lim_𝑛→∞(𝑦₀𝐴^𝑎(2𝑛))^(−1)2𝑛=lim_𝑛→∞(𝑦₀𝐴⁰)^(−1)2𝑛=lim_𝑛→∞(𝑦₀𝐴⁰)=𝑦₀. (2.17)
(b) Similarly we can obtain the proof of (b) as the proof of (a).

Lemma 2.11. Let (𝑥₀,𝑦₀) be the initial condition of (1.1) for 0<𝑥₀<1<𝑦₀<𝐴; there is at least an 𝑖₀∈𝑁 such that every 𝑛∈𝑁 for 𝑛>𝑖₀, 𝑦₀/𝑥^𝑛₀>𝐴.

Proof. We consider that 𝑥₀<1 hence lim_𝑛→∞(𝑦₀/𝑥^𝑛₀)=∞ and that proofs the existing of 𝑖₀ defined in hypothesis.

Theorem 2.12. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 0<𝑥₀<1<𝑦₀<𝐴, and 𝑖₀ is the number, defined by Lemma 2.11.
1≤𝑛≤𝑖₀𝑥_𝑛=𝐴^𝑎(𝑛)(𝑥₀)^(−1)𝑛, (2.18)
1≤𝑛≤𝑖₀𝑦_𝑛=𝐴^𝑎(𝑛)(𝑦₀𝑥₀^[[𝑛/2]])^(−1)𝑛, (2.19) and when 𝑛>𝑖₀, the solutions will be different for every different constant 𝐴.

Proof. Let 𝑦₀<𝐴, then 𝑥₁𝐴=max{𝑥₀,𝑦₀𝑥₀𝐴}=𝑥₀𝑦>1,₁𝐴=max{𝑦₀,𝑥₀𝑦₀𝐴}=𝑦₀𝑥>1,₂𝐴=max{𝑥₁,𝑦₁𝑥₁𝐴}=𝑥₁=𝑥₀𝑦<1,₂𝐴=max{𝑦₁,𝑥₁𝑦₁𝑥}=¹𝑦₁=𝑦₀𝑥₀𝑥>1,₃𝐴=max{𝑥₂,𝑦₂𝑥₂𝐴}=𝑥₂=𝐴𝑥₀𝑦>1,₃𝐴=max{𝑦₂,𝑥₂𝑦₂𝐴}=𝑦₂=𝐴𝑥₀𝑦₀,𝑥₄𝐴=max{𝑥₃,𝑦₃𝑥₃𝐴}=𝑥₃=𝑥₀𝑦<1,₄𝐴=max{𝑦₃,𝑥₃𝑦₃𝑥}=³𝑦₃=𝑦₀𝑥²₀,𝑥₅𝐴=max{𝑥₄,𝑦₄𝑥₄𝐴}=𝑥₄=𝐴𝑥₀𝑦>1,₅𝐴=max{𝑦₄,𝑥₄𝑦₄𝐴}=𝑦₄=𝐴𝑥²₀𝑦₀, (2.20)1≤𝑛≤𝑖₀, 𝑥_𝑛=𝐴^𝑎(𝑛)(𝑥₀)^(−1)𝑛,𝑦_𝑛=𝐴^𝑎(𝑛)(𝑦₀𝑥₀^[[𝑛/2]])^(−1)𝑛. (2.21)

Lemma 2.13. Let (𝑥₀,𝑦₀) be the initial condition of (1.1) for 0<𝑦₀<1<𝑥₀<𝐴; there is at least an 𝑖₀∈𝑁 such that every 𝑛∈𝑁 for 𝑛>𝑖₀, 𝑥₀/𝑦^𝑛₀>𝐴.

Proof. Similarly we can obtain the proof as the proof of Lemma 2.11.

Theorem 2.14. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 0<𝑦₀<1<𝑥₀<𝐴, and 𝑖₀ is the number, defined by Lemma 2.13.
1≤𝑛≤𝑖₀𝑥_𝑛=𝐴^𝑎(𝑛)(𝑥₀𝑦₀^[[𝑛/2]])^(−1)𝑛, (2.22)
1≤𝑛≤𝑖₀𝑦_𝑛=𝐴^𝑎(𝑛)(𝑦₀)^(−1)𝑛, (2.23) and when 𝑛>𝑖₀, the solutions will be different for every different constant 𝐴.

Proof. Similarly we can obtain the proof of be as the proof of Theorem 2.12.

Lemma 2.15. Let (𝑥₀,𝑦₀) be the initial condition of (1.1) for 0<𝑦₀<𝑥₀<1; there is at least an 𝑖₀∈𝑁 such that every 𝑛∈𝑁 for 𝑛>𝑖₀, (𝑥₀/𝑦₀)^𝑛>𝐴.

Proof. We consider that 𝑦₀<𝑥₀ hence lim_𝑛→∞(𝑥₀/𝑦₀)^𝑛=∞ and that proofs the existing of 𝑖₀ defined in hypothesis.

Theorem 2.16. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 0<𝑦₀<𝑥₀<1, 𝐴>1, and 𝑖₀ is the number, defined by Lemma 2.15.
𝑥₁=𝐴/𝑥₀ and if 1<𝑛≤𝑖₀, 𝑥_𝑛=𝐴^𝑎(𝑛)(𝑥₀𝑦₀)^{[[𝑛/2]](−1)𝑛}, (2.24)
𝑦₁=𝐴/𝑦₀, and if 1<𝑛≤𝑖₀, 𝑦_𝑛=𝐴^𝑎(𝑛)(𝑦₀𝑥₀)^(−1)𝑛, (2.25) and when 𝑛>𝑖₀, the solutions will be different for every different constant 𝐴.

Proof. Let 𝑦₀<𝐴, then 𝑥₁𝐴=max{𝑥₀,𝑦₀𝑥₀𝐴}=𝑥₀𝑦>1,₁𝐴=max{𝑦₀,𝑥₀𝑦₀𝐴}=𝑦₀𝑥>1,₂𝐴=max{𝑥₁,𝑦₁𝑥₁𝑦}=¹𝑥₁=𝑥₀𝑦₀𝑦>1,₂𝐴=max{𝑦₁,𝑥₁𝑦₁𝑥}=¹𝑦₁=𝑦₀𝑥₀𝑥<1,₃𝐴=max{𝑥₂,𝑦₂𝑥₂𝐴}=𝑥₂=𝐴𝑦₀𝑥₀,𝑦₃𝐴=max{𝑦₂,𝑥₂𝑦₂𝐴}=𝑦₂=𝐴𝑥₀𝑦₀𝑥>1,₄𝐴=max{𝑥₃,𝑦₃𝑥₃𝑦}=³𝑥₃=𝑥²₀𝑦²₀𝑦>1,₄𝐴=max{𝑦₃,𝑥₃𝑦₃𝐴}=𝑦₃=𝑦₀𝑥₀<1, (2.26)𝑛>1 then 𝑥_𝑛=𝐴^𝑎(𝑛)(𝑥₀𝑦₀)^{[[𝑛/2]](−1)𝑛}, (2.27)𝑛≥1 then 𝑦_𝑛=𝐴^𝑎(𝑛)(𝑦₀𝑥₀)^(−1)𝑛. (2.28)

Lemma 2.17. Let (𝑥₀,𝑦₀) be the initial condition of (1.1) for 0<𝑦₀<𝑥₀<1; there is at least an 𝑖₀∈𝑁 such that every 𝑛∈𝑁 for 𝑛>𝑖₀, (𝑦₀/𝑥₀)^𝑛>𝐴.

Proof. Similarly we can obtain the proof as the proof of Lemma 2.15.

Theorem 2.18. Let (𝑥_𝑛,𝑦_𝑛) be the solution of the system of difference equations (1.1) for 0<𝑥₀<𝑦₀<1, 𝐴>1, and 𝑖₀ is the number, defined by Lemma 2.17.
𝑥₁=𝐴/𝑥₀, and if 1<𝑛≤𝑖₀, 𝑥_𝑛=𝐴^𝑎(𝑛)(𝑥₀𝑦₀)^(−1)𝑛, (2.29)
𝑦₁=𝐴/𝑦₀, and if 1<𝑛≤𝑖₀, 𝑦_𝑛=𝐴^𝑎(𝑛)(𝑦₀𝑥₀)^{[[𝑛/2]](−1)𝑛}, (2.30) and when 𝑛>𝑖₀, the solutions will be different for every different constant 𝐴.

Proof. Similarly we can obtain the proof as the proof of Theorem 2.16, which completes the proofs of theorems.