On a Periodic System of Difference Equations

Abstract

Here we give a very short and elegant proof of an extension of two recent results on global periodicity of two systems of difference equations in a work of Elsayed et al. 2012.

1. Introduction

For a solution (𝑥_𝑛⁽¹⁾,…,𝑥_𝑛^(𝑘)), 𝑛≥−𝑙, 𝑙∈ℕ, of the system of difference equations 𝑥_𝑛^(𝑗)=𝑓_𝑗𝑥⁽¹⁾_𝑛−1,…,𝑥^(𝑘)_𝑛−1,…,𝑥⁽¹⁾_𝑛−𝑙,…,𝑥^(𝑘)_𝑛−𝑙,𝑗=1,𝑘,𝑛∈ℕ₀, (1.1) is said that it is eventually periodic with period 𝑝, if there is an 𝑛₁≥−𝑙 such that 𝑥⁽¹⁾_𝑛+𝑝,…,𝑥^(𝑘)_𝑛+𝑝=𝑥_𝑛⁽¹⁾,…,𝑥_𝑛^(𝑘),for𝑛≥𝑛₁. (1.2) If 𝑛=−𝑙, then for the solution is said that it is periodic with period 𝑝. If all well-defined solutions of system (1.1) are eventually periodic with period 𝑝, it is said that the system is 𝑝-periodic.

For some periodic difference equations and systems see, for example, [1–11] and the references therein. Periodic systems are essentially subclasses of the systems which can be solved explicitly, an area that has again attracted attention recently (see, e.g., [12–21]). For some classical results in the topic see, for example, [22].

In the recent paper [23] are formulated two results on global periodicity of particular systems of difference equations. Namely, they gave some tedious straightforward calculations which suggest that all the solutions with nonzero real initial values of the following systems of difference equations 𝑥_𝑛+1=1𝑥_𝑛−𝑝𝑦_𝑛−𝑝,𝑦_𝑛+1=𝑥_𝑛−𝑝𝑦_𝑛−𝑝𝑥_𝑛−𝑞𝑦_𝑛−𝑞,𝑥_𝑛+1=1𝑥_𝑛−𝑝𝑦_𝑛−𝑝𝑧_𝑛−𝑝,𝑦_𝑛+1=𝑥_𝑛−𝑝𝑦_𝑛−𝑝𝑧_𝑛−𝑝𝑥_𝑛−𝑞𝑦_𝑛−𝑞𝑧_𝑛−𝑞,𝑧_𝑛+1=𝑥_𝑛−𝑞𝑦_𝑛−𝑞𝑧_𝑛−𝑞𝑥_𝑛−𝑟𝑦_𝑛−𝑟𝑧_𝑛−𝑟 (1.3)(1.4) are periodic with periods 𝑞+1 and 𝑟+1, respectively, without giving complete proofs of these statements.

2. Main Result

Here we prove a result which naturally extends the results on systems (1.3) and (1.4), and give a very short and elegant proof of it.

Theorem 2.1. Assume that 𝑘,𝑙,𝑠∈ℕ, 𝑘≥2, functions 𝑔_𝑗∶(ℝ⧵{0})^𝑙→ℝ⧵{0}, 𝑗=1,𝑘, satisfy the following condition ^𝑘_𝑗=1𝑔_𝑗𝑡₁,𝑡₂,…,𝑡_𝑙=1, (2.1) for every (𝑡₁,𝑡₂,…,𝑡_𝑙)∈(ℝ⧵{0})^𝑙.
Consider the following system of difference equations 𝑥_𝑛^(𝑗)=𝑔_𝑗^𝑘_𝑖=1𝑥^(𝑖)_𝑛−1,…,^𝑘_𝑖=1𝑥^(𝑖)_𝑛−𝑙,𝑗=𝑥1,𝑘−1,_𝑛(𝑘)=𝑔_𝑘^𝑘_𝑖=1𝑥^(𝑖)_𝑛−1,…,^𝑘_𝑖=1𝑥^(𝑖)_𝑛−𝑙^𝑘_𝑖=1𝑥^(𝑖)_𝑛−𝑠⁻¹,𝑛∈ℕ₀. (2.2)
Then every solution (𝑥_𝑛⁽¹⁾,…,𝑥_𝑛^(𝑘)) of system (2.2) whose initial values are arbitrary nonzero real numbers is eventually periodic with period 2𝑠.

Proof. Since 𝑥_𝑛^(𝑗)≠0 for every −max{𝑙,𝑠}≤𝑛≤−1 and 𝑗=1,𝑘, functions 𝑔_𝑗, 𝑗=1,𝑘, map the set (ℝ⧵{0})^𝑙 to ℝ⧵{0} and ℎ𝑡₁,…,𝑡_𝑘=^𝑘_𝑖=1𝑡_𝑖⁻¹, (2.3) map the set (ℝ⧵{0})^𝑘 to ℝ⧵{0}, from (2.2) with 𝑛=0, we see that 𝑥₀^(𝑗)≠0 for every 𝑗=1,𝑘.
Assume we have proved that 𝑥_𝑖^(𝑗)≠0 for −max{𝑙,𝑠}≤𝑖≤𝑛₀ and every 𝑗=1,𝑘. Then from this, (2.2) and since functions 𝑔_𝑗, 𝑗=1,𝑘, map the set (ℝ⧵{0})^𝑙 to ℝ⧵{0}, and ℎ map the set (ℝ⧵{0})^𝑘 to ℝ⧵{0}, we get 𝑥_𝑛^(𝑗)₀₊₁≠0 for every 𝑗=1,𝑘. So, by induction it follows that 𝑥_𝑛^(𝑗)≠0 for every 𝑛∈ℕ₀ and 𝑗=1,𝑘. Hence, every solution (𝑥_𝑛⁽¹⁾,…,𝑥_𝑛^(𝑘)) of system (2.2) whose initial values are arbitrary nonzero real numbers is well-defined.
Multiplying all the equations in (2.2), then using condition (2.1) and the above proven fact that ∏^𝑘_𝑖=1𝑥_𝑛^(𝑖)≠0, for every 𝑛≥−max{𝑙,𝑠}, we get ^𝑘_𝑗=1𝑥_𝑛^(𝑗)=∏^𝑘_𝑗=1𝑔_𝑗∏^𝑘_𝑗=1𝑥^(𝑖)_𝑛−1,…,∏^𝑘_𝑗=1𝑥^(𝑖)_𝑛−𝑙∏^𝑘_𝑖=1𝑥^(𝑖)_𝑛−𝑠=1∏^𝑘_𝑖=1𝑥^(𝑖)_𝑛−𝑠, (2.4) from which it follows that ^𝑘_𝑗=1𝑥_𝑛^(𝑗)=^𝑘_𝑗=1𝑥^(𝑗)_𝑛−2𝑠, (2.5) that is, the sequence ∏^𝑘_𝑗=1𝑥_𝑛^(𝑗) is eventually periodic with period 2𝑠. In particular, the sequences ∏^𝑘_𝑗=1𝑥^(𝑗)_𝑛−𝑞 are eventually periodic with period 2𝑠, for every 𝑞∈ℕ.
This implies that the sequences 𝑔_𝑗^𝑘_𝑖=1𝑥^(𝑖)_𝑛−1,…,^𝑘_𝑖=1𝑥^(𝑖)_𝑛−𝑙,𝑗=1,𝑘,_𝑘𝑖=1𝑥^(𝑖)_𝑛−𝑠⁻¹ (2.6) are eventually periodic with period 2𝑠, which along with (2.2) implies that the sequences (𝑥_𝑛^(𝑗))_{𝑛≥−max{𝑙,𝑠}} are eventually periodic with period 2𝑠, for 𝑗=1,𝑘, from which the theorem follows.

Now we present the main results in paper [23] as two corollaries.

Corollary 2.2. Assume that (𝑥_𝑛,𝑦_𝑛), 𝑛≥−max{𝑝,𝑞}, is an arbitrary solution of system (1.3) whose initial values are arbitrary nonzero real numbers. Then the solution is eventually periodic with period 2𝑞+2.

Proof. Note that for system (1.3), 𝑙=𝑝+1, 𝑠=𝑞+1, 𝑔₁(𝑡₁,…,𝑡_𝑝+1)=1/𝑡_𝑝+1 and 𝑔₂(𝑡₁,…,𝑡_𝑝+1)=𝑡_𝑝+1, so that 𝑔₁𝑡₁,…,𝑡_𝑝+1𝑔₂𝑡₁,…,𝑡_𝑝+1=1for𝑡₁,…,𝑡_𝑝+1∈(ℝ⧵{0})^𝑝+1. (2.7) Hence, all the conditions of Theorem 2.1 are fulfilled from which the corollary follows.

Corollary 2.3. Assume that (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛), 𝑛≥−max{𝑝,𝑞,𝑟}, is an arbitrary solution of system (1.4) whose initial values are arbitrary nonzero real numbers. Then the solution is eventually periodic with period 2𝑟+2.

Proof . For system (1.4), we have that 𝑙=max{𝑝+1,𝑞+1}, 𝑠=𝑟+1, 𝑔₁𝑡₁,…,𝑡_𝑙=1𝑡_𝑝+1,𝑔₂𝑡₁,…,𝑡_𝑙=𝑡_𝑝+1𝑡_𝑞+1,𝑔₃𝑡₁,…,𝑡_𝑙=𝑡_𝑞+1, (2.8) so that 𝑔₁𝑡₁,…,𝑡_𝑙𝑔₂𝑡₁,…,𝑡_𝑙𝑔₃𝑡₁,…,𝑡_𝑙=1for𝑡₁,…,𝑡_𝑙∈(ℝ⧵{0})^𝑙. (2.9) Hence, all the conditions of Theorem 2.1 are fulfilled from which the corollary follows.

The next corollary, which corresponds to case 𝑘=4 in Theorem 2.1, is obtained similarly, so we omit the proof.

Corollary 2.4. Assume that (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑤_𝑛), 𝑛≥−max{𝑝,𝑞,𝑟,𝑠}, is an arbitrary solution of the following system 𝑥_𝑛+1=1𝑥_𝑛−𝑝𝑦_𝑛−𝑝𝑧_𝑛−𝑝𝑤_𝑛−𝑝,𝑦_𝑛+1=𝑥_𝑛−𝑝𝑦_𝑛−𝑝𝑧_𝑛−𝑝𝑤_𝑛−𝑝𝑥_𝑛−𝑞𝑦_𝑛−𝑞𝑧_𝑛−𝑞𝑤_𝑛−𝑞,𝑧_𝑛+1=𝑥_𝑛−𝑞𝑦_𝑛−𝑞𝑧_𝑛−𝑞𝑤_𝑛−𝑞𝑥_𝑛−𝑟𝑦_𝑛−𝑟𝑧_𝑛−𝑟𝑤_𝑛−𝑟,𝑤_𝑛+1=𝑥_𝑛−𝑟𝑦_𝑛−𝑟𝑧_𝑛−𝑟𝑤_𝑛−𝑟𝑥_𝑛−𝑠𝑦_𝑛−𝑠𝑧_𝑛−𝑠𝑤_𝑛−𝑠, (2.10)𝑛∈ℕ₀, whose initial values are arbitrary nonzero real numbers. Then the solution is eventually periodic with period 2𝑠+2.

Acknowledgments

The second author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government Grant MSM 00 216 30519. The fourth author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.