$\mathcal{H}$-representation method for solving reduced biquaternion matrix equation

1.   Introduction

The purpose of this paper is to study the global behavior of the following max-type system of difference equations of the second order with four variables and period-two parameters

where $A_n, B_n, C_n, D_n\in {\bf R}^+\equiv (0, +\infty)$ are periodic sequences with period 2 and the initial values $x_{-i}, y_{-i}, z_{-i}, w_{-i}\in {\bf R}^+ \ (1\leq i\leq 2)$. To do this we will use some methods and ideas which stems from ^[1,2]. For a more complex variant of the method, see ^[3]. A solution $\{(x_n, y_n, z_n, w_n)\}^{+\infty}_{n = -2}$ of (1.1) is called an eventually periodic solution with period $T$ if there exists $m\in {\bf N}$ such that $(x_n, y_n, z_n, w_n) = (x_{n+T}, y_{n+T}, z_{n+T}, w_{n+T})$ holds for all $n\geq m$.

When $x_n = y_n$ and $z_n = w_n$ and $A_0 = A_1 = B_0 = B_1 = \alpha$ and $C_0 = C_1 = D_0 = D_1 = \beta$, (1.1) reduces to following max-type system of difference equations

Fotiades and Papaschinopoulos in ^[4] investigated the global behavior of (1.2) and showed that every positive solution of (1.2) is eventually periodic.

When $x_n = z_n$ and $y_n = w_n$ and $A_n = C_n$ and $B_n = D_n$, (1.1) reduces to following max-type system of difference equations

Su et al. in ^[5] investigated the periodicity of (1.3) and showed that every solution of (1.3) is eventually periodic.

In 2020, Su et al. ^[6] studied the global behavior of positive solutions of the following max-type system of difference equations

where $A, B\in {\bf R}^+$.

In 2015, Yazlik et al. ^[7] studied the periodicity of positive solutions of the max-type system of difference equations

where $p\in {\bf \bf {R}}^+$ and obtained in an elegant way the general solution of (1.4).

In 2016, Sun and Xi ^[8], inspired by the research in ^[5], studied the following more general system

where $p, q\in {\bf \bf {R}}^+$, $m, r, t\in {\bf {N}}\equiv\{1, 2, \cdots\}$ and the initial conditions $x_{-i}, y_{-i}\in {\bf \bf {R}}^+$ ($1\leq i\leq s)$ with $s = \max\{m, r, t\}$ and showed that every positive solution of (1.5) is eventually periodic with period $2m$.

In ^[9], Stević studied the boundedness character and global attractivity of the following symmetric max-type system of difference equations

where $B, p\in \bf {R}^+$ and the initial conditions $x_{-i}, y_{-i}\in {\bf R}^+\ (1\leq i\leq 2)$.

In 2014, motivated by results in ^[9], Stević ^[10] further study the behavior of the following max-type system of difference equations

where $B, p\in \bf {R}^+$ and the initial conditions $x_{-i}, y_{-i}, z_{-i}\in {\bf R}^+\ (1\leq i\leq 2)$, and showed that system (1.6) is permanent when $p\in (0, 4)$.

For more many results for global behavior, eventual periodicity and the boundedness character of positive solutions of max-type difference equations and systems, please readers refer to ^{[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]} and the related references therein.

2.   Main results and proofs

In this section, we study the global behavior of system (1.1). For any $n\geq -1$, write

Then, (1.1) reduces to the following system

From (2.1) we see that it suffices to consider the global behavior of positive solutions of the following system

where $a, b, A, B\in {\bf R}^+$, the initial conditions $u_{-1}, U_{-1}, v_{-1}, V_{-1}\in {\bf R}^+$. If $(u_{n}, U_{n}, v_{n}, V_{n}, a, A, b, B) = (X_n, Y_n, Z'_n, W'_n, A_{2n}, B_{2n}, C_{2n-1}, D_{2n-1})$, then (2.2) is the first four equations of (2.1). If $(u_{n}, U_{n}, v_{n}, V_{n}, a, A, b, B) = (Z_n, W_n, X'_n, Y'_n, C_{2n}, D_{2n}, A_{2n-1}, B_{2n-1})$, then (2.2) is the next four equations of (2.1). In the following without loss of generality we assume $\bf a\leq A$ and $\bf b\leq B$. Let $\{(u_{n}, U_{n}, v_{n}, V_{n})\}^{\infty}_{n = -1}$ be a positive solution of (2.2).

Proposition 2.1. If $ab < 1$, then there exists a solution $\{(u_n, U_n, v_n, V_n)\}^{\infty}_{n = -1}$ of (2.2) such that $u_n = v_n = 1$ for any $n\geq -1$ and $\lim_{n\longrightarrow\infty}U_n = \lim_{n\longrightarrow\infty}V_n = \infty$.

Proof. Let $u_{-1} = v_{-1} = 1$ and $U_{-1} = V_{-1} = \max\{\frac{b}{aA}, \frac{aA}{B}, \frac{a}{bB}\}+1$. Then, from (2.2) we have

and

Suppose that for some $k\in {\bf N}$, we have

Then,

By mathematical induction, we can obtain the conclusion of Proposition 2.1. The proof is complete.

Now, we assume that $ab\geq 1$. Then, from (2.2) it follows that

Lemma 2.1. The following statements hold:

(1) For any $n\in {\bf {N}}_0$,

(2) If $ab\geq 1$, then for any $k\in \bf N$ and $n\geq k+2$,

(3) If $ab\geq 1$, then for any $k\in \bf N$ and $n\geq k+4$,

Proof. (1) It follows from (2.2).

(2) Since $AB\geq ab\geq 1$, it follows from (2.2) and (2.3) that for any $k\in \bf N$ and $n\geq k+2$,

In a similar way, also we can obtain the other three formulas.

(3) By (2.5) one has that for any $k\in \bf N$ and $n\geq k+2$,

from which and (2.4) it follows that for any $n\geq k+4$,

The proof is complete.

Proposition 2.2. If $ab = AB = 1$, then $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$.

Proof. By the assumption we see $a = A$ and $b = B$. By (2.5) we see that for any $k\in {\bf N}$ and $n\geq k+2$,

(1) If $a = b = 1$, then it follows from (2.7) and (2.4) that for any $n\geq k+4$,

We claim that $v_n = 1$ for any $n\geq 6$ or $V_n = 1$ for any $n\geq 6$. Indeed, if $v_n > 1$ for some $n\geq 6$ and $V_m > 1$ for some $m\geq 6$, then

which implies

A contradiction.

If $v_n = 1$ for any $n\geq 6$, then by (2.8) we see $u_n = 1$ for any $n\geq 10$, which implies $U_n = V_n = V_{10}$.

If $V_n = 1$ for any $n\geq 6$, then by (2.8) we see $U_n = 1$ for any $n\geq 10$, which implies $v_n = u_n = v_{10}$.

Then, $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$.

(2) If $a < 1 < b$, then it follows from (2.7) that for any $n\geq k+4$,

It is easy to verify $v_n = 1$ for any $n\geq 6$ or $V_n = 1$ for any $n\geq 6$.

If $V_n = v_n = 1$ eventually, then by (2.9) we have

Since $U_{n} \geq \frac{b^3}{ u_{n-1}}$ and $u_{n} \geq \frac{b^3}{ U_{n-1}}$, we see

which implies

If $V_{n} > 1 = v_n$ eventually, then by (2.9) we have

Thus,

which implies

If $V_{n} = 1 < v_n$ eventually, then by (2.9) we have $U_{n-2} = U_{n}\ {\mbox{eventually}}$ and $u_{n} = u_{n-1}\ {\mbox{eventually}}.$ By the above we see that $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$.

(3) If $b < 1 < a$, then for any $k\in \bf N$ and $n\geq k$+2,

It is easy to verify $u_n = 1$ for any $n\geq 3$ or $U_n = 1$ for any $n\geq 3$.

If $u_n = U_n = 1 \ {\mbox{eventually}}$, then

Thus, by (2.6) we have

If $u_n = 1 < U_n\ {\mbox{eventually}}$, then

Thus,

If $u_n > 1 = U_n\ {\mbox{eventually}}$, then we have $V_{n} = V_{n-2}\ {\mbox{eventually}}$ and $v_n = 1\ {\mbox{eventually}}$ or $v_n = v_k\ {\mbox{eventually}}$.

By the above we see that $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$.

Proposition 2.3. If $ab = 1 < AB$, then $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$.

Proof. Note that $U_{n}\geq \frac{B}{aA u_{n-1}}$ and $V_{n} \geq \frac{A}{bB v_{n-1}}$. By (2.5) we see that there exists $N\in {\bf N}$ such that for any $n\geq N$,

It is easy to verify that $u_n = 1$ for any $n\geq N+1$ or $v_n = 1$ for any $n\geq N+1$.

If $u_n = v_n = 1\ {\mbox{eventually}}$, then by (2.11) we see that $U_{n} = U_{n-1}\ {\mbox{eventually}}$ and $V_n = V_{n-1}\ {\mbox{eventually}}$.

If $u_{M+2n} > 1 = v_n\ {\mbox{eventually}}$ for some $M\in {\bf N}$, then by (2.11) and (2.4) we see that

By (2.11) we see that $U_n$ is bounded, which implies $B = b$.

If $U_{M+2n-1}\leq \frac{BV_{k}}{aA u_{M+2n}u_{M+2n-1}v_{M+2n-1}\cdots u_{k}v_{k}}\ {\mbox{eventually}}$, then

Thus, $U_{M+2n+1} = U_{M+2n-1}\ {\mbox{eventually}}$ and $u_{M+2n} = u_{M+2n-2}\ {\mbox{eventually}}$. Otherwise, we have $U_{M+2n+1} = U_{M+2n-1}\ {\mbox{eventually}}$ and $u_{M+2n} = u_{M+2n-2}$ eventually. Thus, $V_{n} = V_{n-1} = \max\Big\{1, \frac{A}{bB}\Big\}\ {\mbox{eventually}}$ since $\lim_{n\longrightarrow\infty}\frac{V_{k}}{ u_{n-1}v_{n-1}\cdots u_{k}v_{k}} = 0$. By (2.2) it follows $U_{M+2n} = U_{M+2n-2}$ eventually and $u_{M+2n+1} = u_{M+2n-1}$ eventually.

If $v_{M+2n} > 1 = u_n\ {\mbox{eventually}}$ for some $M\in {\bf N}$, then we may show that $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$. The proof is complete.

Proposition 2.4. If $ab > 1$, then $\{(u_n, U_n, v_n, V_n)\}^{+\infty}_{n = -1}$ is eventually periodic with period $2$.

Proof. By (2.5) we see that there exists $N\in {\bf N}$ such that for any $n\geq N$,

If $a < A$, then for $n\geq 2k+N$ with $k\in {\bf N}$,

which implies $v_n = 1$ eventually and $V_{n} = \max\Big\{1, \frac{A}{bB }\Big\}$ eventually.

If $a = A$, then

Which implies

Thus, $V_n, v_n$ are eventually periodic with period $2$. In a similar way, we also may show that $U_n, u_n$ are eventually periodic with period $2$. The proof is complete.

From (2.1), (2.2), Proposition 2.1, Proposition 2.2, Proposition 2.3 and Proposition 2.4 one has the following theorem.

Theorem 2.1. (1) If $\min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\} < 1$, then system (1.1) has unbounded solutions.

(2) If $\min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\}\geq 1$, then every solution of system (1.1) is eventually periodic with period $4$.

3.   Conclusions

In this paper, we study the eventual periodicity of max-type system of difference equations of the second order with four variables and period-two parameters (1.1) and obtain characteristic conditions of the coefficients under which every positive solution of (1.1) is eventually periodic or not. For further research, we plan to study the eventual periodicity of more general max-type system of difference equations by the proof methods used in this paper.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Project supported by NSF of Guangxi (2022GXNSFAA035552) and Guangxi First-class Discipline SCPF(2022SXZD01, 2022SXYB07) and Guangxi Key Laboratory BDFE(FED2204) and Guangxi University of Finance and Economics LSEICIC(2022YB12).

Conflict of interest

There are no conflict of interest in this article.