Uniform boundedness of solutions to linear difference equations with periodic forcing functions

1.   Introduction

Let ${\mathbb C}$ be the set of all complex numbers and ${\mathbb R}$ the set of all real numbers. We set $\mathbb{N} = \{1, 2, \cdots\}$, $\mathbb{N}_0 = \mathbb{N}\cup\{0\}$ and ${\mathbb Z} = \{0, \pm 1, \pm 2, \cdots\}$. We denote by $\mathbb{C}^p$ the set of all $p$ dimensional complex column vectors, and by $M_{p}(\mathbb{C})$ the set of all $p\times p$ complex matrices.

In this paper we consider periodic linear difference equations of the forms

where $n\in {\mathbb Z}, H\in M_p({\mathbb C})$, $x(n)\in \mathbb{C}^p$ and $b(n)\in \mathbb{C}^p$ is a vector valued function with period $\rho\in {\mathbb N}$.

The purpose of this paper is to give criteria of the uniform boundedness of the solutions to the above equations. Criteria of the boundedness of solutions were given in ^[1,2,3,4,5]. Some related one-dimensional results can be found in ^[6] (see also the references therein).

First, we give a necessary and sufficient condition for the sequence $\{L^n\}$ of a square matrix $L$ to be bounded, from which the criterion on the uniform boundedness of the solutions to the Eq (1.1) is obtained immediately.

Second, a criterion on the uniform boundedness of the solutions for the Eq (1.2) is given by applying a certain representation of solutions developed in ^[3]. It seems that its proof is not easy to obtain from the usual representation of the solution by the variation of constants formula.

Finally, in connection with the Eq (1.2) with delay, we give the characteristic equation of a matrix under the commuting condition. In more details, making use of the simultaneous diagonalization theorem under the commuting condition $AB = BA$, we can apply the preceding results to the periodic linear difference equation with delay of the form

where $A, \ B\in M_p({\mathbb C})$, $x(n)\in \mathbb{C}^p$ and $f (n)\in \mathbb{C}^p$ is a vector-valued function with period $\rho\in {\mathbb N}$. But we only consider the characteristic equation of the matrix $M$ in a reduced equation $y(n+1) = My(n)+g(n)$ derived from the Eq (1.3).

2.   Boundedness of the sequence $\{L^n\}$

2.1.   Spectral decomposition theorem

We define $(n)_k$ as follows.

Denoting by ${{n \choose k}}$ a binomial coefficient, we have

$E$ or $E_p$ is the identity matrix in $M_{p}(\mathbb{C})$. We denote by ${\it O}$ and $0$ the zero matrix and the zero vector, respectively.

Moreover, we denote by $\sigma(L)$ the set of all eigenvalues of a matrix $L\in M_{p}(\mathbb{C})$ and by $h_{\eta}(L)$ the index of $\eta\in \sigma(L)$. Then $G_\eta(L) = {\cal N}((L-\eta E)^{h_\eta(L)})$ is the generalized eigenspace of $\eta\in \sigma(L)$, where ${\cal N}(L) = \{x \in \mathbb{C}^p \ : \ Lx = 0\}$. Clearly, ${\mathbb C}^p$ is decomposed as ${\mathbb C}^p = \bigoplus_{\eta \in \sigma (L)} G_{\eta}(L)$. We denote by $Q_\eta(L)$ the projection from ${\mathbb C}^p$ to $G_\eta(L)$. Then $Q_\eta^2(L) = Q_\eta(L)$ and $LQ_\eta(L) = Q_\eta(L)L$.

Now, we state the spectral decomposition theorem for the matrix $L\in M_{p}(\mathbb{C})$, which plays an important role in this paper.

Lemma 1. ^[1] Let $\eta \in \sigma(L)$. If $\eta \not = 0$, then

In particular, operating $Q_{\eta}(L)$ to (2.1), we have

If $\eta = 0\in\sigma(L)$, then

2.2.   Asymptotic behavior of $L^n$

We discuss the asymptotic behavior of $L^n$ as $n \to \infty$ using Lemma 1. For $L\in M_p({\mathbb C})$ we take the operator norm $\|L\| = \sup_{\|x\|\leq 1}\|Lx\|$. Then we have

Clearly, if $\lim_{n \to \infty}\|L^nu\| = \infty$ for some $u\in {\mathbb C}^p\setminus\{0\}$, then $\lim_{n \to \infty}\|L^n\| = \infty$.

For $\lambda\in \sigma(L)$, we set $P_\lambda = Q_\lambda(L)$. We also set

and

The following lemma, which slightly modifies 1) and 2) of Theorem 6.1 in ^[3], is the most probably known. We give a proof of it for completeness.

Lemma 2. Let $\lambda\in \sigma(L)$.

$\mathrm 1)$ If $\lambda \in \sigma_S(L)$, then

where $C(\lambda) = h_{\lambda}(L)\max_{0\leq j\leq h_{\lambda}(L)-1}\|(L-\lambda E)^j\|$, hence

$\mathrm 2)$ If $\lambda \in \sigma_U(L)$, then

for all $u\in {\mathbb C}^p$ satisfying $P_{\lambda}u\not = 0$.

Proof. 1) Let $\lambda \in \sigma_S(L)$.

(a) The case $\lambda\neq 0$: Since $\lim_{n\to\infty} \frac{(n)_j}{r^n} = 0, (r > 1)$, it follows from Lemma 1 that for a sufficiently large $n$,

Thus $\lim_{n \to \infty}\|L^nP_{\lambda}\| = 0$.

(b) The case $\lambda = 0$: Clearly, $\lambda^{0} = 1, \lambda^{n} = 0\ (n\not = 0)$ and hence $L^nP_0 = {\it O}$.

Combining (a) and (b), we conclude that $\lim_{n\to\infty} L^nP_\lambda = {\it O}$ holds.

2) Let $\lambda \in \sigma_U(L)$.

For every $u\in {\mathbb C}^p$ satisfying $P_{\lambda}u\not = 0$, there is a $d, \ 1\leq d \leq h_{\lambda}(L)$ such that

and hence,

Since $|\lambda| > 1$, we have $\lim_{n \to \infty}\|L^n P_\lambda u\| = \infty$. Therefore, the proof is complete. □

The following lemma is certainly known, but we also give a proof.

Lemma 3. Let $\lambda\in \sigma_N(L)$. If $h_{\lambda}(L) > 1$, then there exists a $u\in {\mathbb C}^p$ such that $P_\lambda u\not = 0$ and $\lim_{n \to \infty}\|L^nP_\lambda u\| = \infty$.

Proof. Since $h_{\lambda}(L) > 1$, there exists a $v\not = 0$ such that

It follows by induction that

Hence we obtain

Thus there exists a $u\in {\mathbb C}^p$ such that $v = P_\lambda u$. Therefore, $\lim_{n \to \infty}\|L^nP_{\lambda}u\| = \infty$ holds. □

Now we consider the case when $\lambda\in\sigma_N(L)$ with $h_{\lambda}(L) = 1$.

Theroem 1. Let $L\in M_p(\mathbb{C})$ and $\lambda\in\sigma_N(L)$. Then the following statements are equivalent$\mathrm :$

$\mathrm 1) \; h_{\lambda}(L) = 1$.

$\mathrm 2)$ $\|L^nP_{\lambda}u \| = \|P_\lambda u\|$ for all $u\in\mathbb{C}^p$ and $n\in {\mathbb N}$.

$\mathrm 3)$ $\|L^nP_{\lambda}\| = \|P_{\lambda} \|$ for all $n\in {\mathbb N}$.

Proof. $1) \Longrightarrow 2)$. For any vector $u\in \mathbb{C}^p$ we have $L^n P_\lambda u = \lambda^n P_\lambda u$ by using (2.2). Thus

$2) \Longrightarrow 1)$. Assume that 1) does not hold. Then we have $h_\lambda(L) > 1$. It follows from Lemma 3 that there exists a $u\in {\mathbb C}^p$ such that $\lim_{n \to \infty}\|L^nP_\lambda u\| = \infty$. Thus we obtain that $\sup_{n\in\mathbb{N}} \| L^nP_{\lambda}u \| = \infty$ holds, which contradicts the assertion 2).

$1) \Longrightarrow 3)$. It follows from (2.2) that $L^n P_\lambda = \lambda^n P_\lambda.$ Hence $\| L^nP_{\lambda} \| = \|P_{\lambda}\|$ holds.

$3) \Longrightarrow 2)$. This is obvious from the property of the operator norm. □

The following result is easily derived from Theorems 1–3.

Proposition 1. Let $\sigma(L) = \sigma_S(L)\cup\sigma_N(L)$. Then the following statements are equivalent${\rm:}$

$\mathrm 1)$ $h_\lambda(L) = 1$ for all $\lambda\in \sigma_N(L)$.

$\mathrm 2) \; \sup_{n\in {\mathbb N}}\|L^n\| < \infty$.

$\mathrm 3) \; \sup_{n\in {\mathbb N}}\|L^nu\| < \infty$ for all $u\in {\mathbb C}^p$.

Proof. $1) \Longleftrightarrow$ 2). Assume that 1) holds. Since $\sigma(L) = \sigma_S(L)\cup\sigma_N(L)$, we have $E = \sum_{\lambda\in \sigma(L)}P_\lambda$. Thus $L^n = \sum_{\lambda\in \sigma(L)}L^nP_\lambda$. It follows from Lemma 2 and Theorem 1 that $\|L^nP_\lambda\| < \infty$ for all $n \in {\mathbb N}$ and all $\lambda\in \sigma(L)$. Therefore, we have

Conversely, we assume that 2) holds. Since $\sup_{n\in {\mathbb N}}\|L^n\| < \infty$, we have $\|L^nP_\lambda\|\leq \|L^n\|\|P_\lambda\| < \infty$ for $\lambda\in \sigma_N(L)$. On the other hand, if $\lambda\in \sigma_N(L)$ with $h_\lambda(L) > 1$, then $\lim_{n \to \infty}\|L^nP_\lambda\| = \infty$ by Lemma 3, which leads to a contradiction. Hence 1) holds.

$2) \Longrightarrow$ 3) is obvious. $3) \Longrightarrow$ 2) follows from the principle of uniform boundedness in Functional Analysis ([7, p.249]). □

A spectral radius $r_{\sigma}(L)$ of a matrix $L\in M_p(\mathbb{C})$ is defined by $r_{\sigma}(L) = \max\{|\lambda|\ : \ \lambda \in \sigma(L)\}$ and the spectral radius $r_{\sigma}(L)$ of $L$ is given as follows:

Clearly, $r_\sigma (L)\leq \|L\|$.

The following results follow from Theorems 1–3.

Lemma 4. The following statements are equivalent${\rm:}$

$\mathrm 1) \; r_{\sigma}(L) < 1$.

$\mathrm 2) \; \lim_{n\to\infty}\|L^nP_{\lambda}\| = 0$ for all $\lambda\in \sigma(L)$ and $n\in {\mathbb N}$.

$\mathrm 3) \; \lim_{n\to\infty}\|L^n\| = 0$ for all $n\in {\mathbb N}$.

Lemma 5. Let $r_{\sigma}(L) > 1$. Then there exists a $\lambda\in \sigma_U(L)$ such that $\lim_{n \to \infty}\|L^nP_\lambda u\| = \infty$ for all $u\in {\mathbb C}^p$ satisfying $P_{\lambda}u\not = 0$.

The following proposition gives a relationship between the spectral radius of a square matrix $L$ and $\lim_{n \to \infty}L^n$.

Proposition 2. Let $r_{\sigma}(L) = 1$. Then the following statements hold.

$\mathrm (1) \; \lim_{n \to \infty}\|L^nP_\lambda\| = 0$ for all $\lambda\in \sigma_S(L)$.

$\mathrm (2) \; \sup_{n\in \mathbb N}\|L^nP_\lambda\| = \|P_\lambda\| < \infty$ for all $\lambda\in \sigma_N(L)$ with $h_{\lambda}(L) = 1$.

$\mathrm (3)$ If $\lambda\in \sigma_N(L)$ with $h_{\lambda}(L) > 1$, then there exists a $v\in {\mathbb C}^p$ such that $P_\lambda v\not = 0$ and $\lim_{n \to \infty}\|L^nP_\lambda v\| = \infty$.

2.3.   Uniform boundedness of the solution to the Eq (1.1)

We denote by $x(n; \tau, w, b(\cdot))$ the solution of the Eq (1.2) through the point $(\tau, w)\in {\mathbb Z}\times {\mathbb C}^p$. Then $x(n; \tau, w): = x(n; \tau, w, 0)$ is the solution of the Eq (1.1) through the point $(\tau, w)$.

Definition 1. ^[8] The solutions to the Eq $\mathrm (1.2)$ are said to be uniformly bounded if for any $\alpha > 0$ there exists a $\beta(\alpha) > 0$ such that $\|x(n; \tau, w, b(\cdot))\| < \beta(\alpha)$ for all $(\tau, w)\in \mathbb{Z}\times B_{\alpha}$ and $n \ge \tau$, where $B_{\alpha} = \{w\in {\mathbb C}^p \ : \ \|w\| < \alpha\}.$

The solution of the Eq (1.1) through the point $(\tau, w)\in {\mathbb Z}\times {\mathbb C}^p$ is expressed as $x(n; \tau, w) = H^{n-\tau}w$. Therefore, the following result, which is concerned with [9, Theorem 4.9 and Theorem 4.13], follows immediately from Proposition 1.

Proposition 3. The solutions to the Eq $\mathrm (1.1)$ are uniformly bounded if and only if every eigenvalue $\eta$ of $H$ satisfies either $|\eta| < 1$ or $|\eta| = 1$ with the index $h_{\eta}(H) = 1$.

The following result is easily derived from Proposition 1 and Proposition 3.

Corollary 1. Let $\sigma(H) = \sigma_S(H)\cup \sigma_N(H)$. Then the following statements are equivalent$\mathrm :$

$\mathrm 1)$ $h_{\eta}(H) = 1$ for all $\eta\in\sigma_N(H)$.

$\mathrm 2)$ All the solutions of the Eq $\mathrm (1.1)$ are bounded.

$\mathrm 3)$ The solutions of the Eq $\mathrm (1.1)$ are uniformly bounded.

3.   Uniform boundedness of the solution to the Eq (1.2)

In this section, we give a criterion on the uniform boundedness of the solutions for the Eq (1.2), namely, we state and prove the main result in the paper.

Theroem 2. The solutions to the Eq $\mathrm (1.2)$ are uniformly bounded if and only if every eigenvalue $\eta$ of $H$ satisfies either $|\eta| < 1$ or $|\eta| = 1, \eta^\rho\not = 1$ with the index $h_{\eta}(H) = 1$.

To prove this theorem, we prepare some results and lemmas.

3.1.   A representation of solutions to the Eq (1.2)

First, we give a representation of solutions to the Eq (1.2), which was given in ^[3]. Hereafter, we abbreviate $Q_\eta = Q_\eta(H)$. We denote by $x(n; \tau, w, b(\cdot))$ the solution of the Eq $\mathrm (1.2)$ satisfying the initial condition $x(\tau) = w \in {\mathbb C}^p$, while by $x(n; \tau, w)$ if $b(n) = 0$. Any $n\in \mathbb N$ can be written as $n = k(n)\rho +r(n), \ k(n) = \left[\frac{n}{\rho}\right], \ 0 \leq r(n) \leq \rho -1$, where the symbol $[a]$ stands for the largest integer which is not greater than $a\in {\mathbb R}$. Set $S_n(H) = \sum _{k = 0}^{n-1}H^k, \ S_0(H) = {\it O},$ and

Then the unique solution $x(n; \tau, w, b(\cdot)), \ n \ge \tau$ of the equation $\mathrm (1.2)$ with $x(\tau) = w$ is expressed as follows$\mathrm :$

To obtain the representation of solutions to the Eq (1.2), we define the characteristic quantities $\gamma_{\eta}(\tau, w, b(\cdot))$ and $\delta_{\eta}(\tau, w, b(\cdot))$ as in ^[3].

For $k, m, n\in\mathbb{N}_0$, $p(k, m, n)$ stands for the set of all finite sequences $\alpha: = (\alpha_1, \alpha_2, \cdots, \alpha_k), \ \alpha_i \in \mathbb{N}_0 \, (i = 1, 2, \cdots, k)$$\mathrm :$

For $k, m\in\mathbb{N}_0$ and $j\in\mathbb{N}$ we define

Let $f^{(k)}(t)$ be the $k$-th derivative of a function $f(t)$ and $f^{(0)}(t) = f(t)$. If $a(w) = (w-1)^{-1}, \ (w \not = 1)$ and $w = z^\rho$, then the $k$-th derivative of the composite function $c(z) = a(z^{\rho})$ is given as follows:

By using Fa$\acute{\rm a}$ di Bruno's formula ^[10] the $k$-th derivative $c^{(k)}(z)$ at $\eta$ is expressed as

For an $\eta \in \sigma(H)$ we set

Then we define $Z_{\eta}(H, b(\tau+\cdot))$ by

Based on this, we can define the characteristic quantities $\gamma_{\eta}(\tau, w, b(\cdot))$ and $\delta_{\eta}(\tau, w, b(\cdot))$ for the Eq (1.2) as follows:

Furthermore, we set $H_{[k, \eta]} = \frac{1}{k!\eta^k}(H-\eta E)^k, \ (\eta \not = 0)$ and

Clearly, $B_\eta(r(n); \tau, w, b(\cdot))$ is a function with period $\rho$.

A representation of solutions to the Eq (1.2) is given by the following lemma.

Lemma 6. ^[3] Let $\eta\in \sigma(H)$. Then the component $Q_{\eta}x(n, \tau, w, b(\cdot))$ of the solution $x(n; \tau, w, b(\cdot))$ to the Eq $\mathrm (1.2)$ is expressed as follows$\mathrm :$

$\mathrm 1)$ If $\eta^\rho \neq 1$, then

In particular,

$\mathrm 2)$ If $\eta^\rho = 1$, then

3.2.   Lemmas

Next, we give some lemmas.

We set $b = \max_{0\leq n\leq\rho}\|b(n)\|$. Then by the definition of $Z_\eta^0(H)$ there exists a constant $K(\eta) > 0$ such that $\|Z_\eta^0(H)\|\leq K(\eta)$. We also set $S(b) = \sum_{k = 0}^{\rho}\|H\|^kb$.

Lemma 7. Let $\eta\in \sigma(H)$. Then the following inequalities hold$\mathrm :$

$\mathrm 1) \; \max_{0\leq n\leq\rho}\|S_{n}(H, Q_{\eta}b(\tau+\cdot))\|\leq S(b)$.

$\mathrm 2) \; \|Z_\eta (H, b(\tau+\cdot))\|\leq K(\eta)S(b)$.

$\mathrm 3) \; \|\gamma_\eta(\tau, w, b(\cdot))\|\leq \|Q_\eta\| \|w\|+K(\eta)S(b), \ (\eta^\rho \not = 1)$.

$\mathrm 4) \; \|\delta_\eta(\tau, w, b(\cdot))\|\leq\|H^\rho-E\|\|Q_\eta\| \|w\|+S(b), \ (\eta^\rho = 1)$.

Proof. The proof follows from the definitions of $S_{n}(H, Q_{\eta}b(\tau+\cdot)), Z_\eta (\tau, H, b(\cdot))$, $\gamma_\eta(\tau, w, b(\cdot))$ and $\delta_\eta (\tau, w, b(\cdot))$. □

Lemma 8. Let $\eta\in \sigma(H)$. Then the following statements hold.

$\mathrm 1)$ There exists a $\beta_\eta > 0$ such that

holds for all $\tau \in \mathbb{Z}$ and $n\ge\tau$.

$\mathrm 2)$ There exists a $\beta_\eta(\alpha) > 0$ such that

holds for all $\tau \in \mathbb{Z}$, $n\ge\tau$ and $w\in B_\alpha$.

Proof. Note that for any $\eta\in \sigma(H)$, we have, by Lemma 7,

$\mathrm 1)$ Let $\eta^{\rho}\not = 1$. Since $\|Z_{\eta}^0(H)\|\leq K(\eta)$ and $\|Z_{\eta}(H, b(\tau+\cdot))\|\leq K(\eta)S(b)$, we obtain that

$\mathrm 2)$ Let $\eta^{\rho} = 1$. If $\|w\| < \alpha$, then it follows that for any $n\ge\tau$

Since the remainder is obvious, the proof is complete. □

Lemma 9. Let $\eta\in \sigma(H)$.

$\mathrm 1)$ If $\eta\in \sigma_S(H)$, then

where $T(\eta) = \max_{\tau\leq n < \infty}(n-\tau)_{h_{\eta}(H)}|\eta|^{n-\tau-h_{\eta}(H)}C(\eta)$.

$\mathrm 2)$ If $\eta\in \sigma_U(H)$, then $\lim_{n \to \infty}\|Q_\eta x(n; \tau, w, b(\cdot))\| = \infty$ if $\gamma_{\eta}(\tau, w, b(\cdot))\not = 0$.

Proof. 1) Let $\eta\in \sigma_S(H)$. Then it follows from Lemma 2 and Lemma 7 that

where $C(\eta) = h_{\lambda}(H)\max_{0\leq j\leq h_{\eta}(H)-1}\|(H-\eta E)^j\|$. Since

we have

Thus for all $n\ge\tau$ we obtain

2) Let $\eta\in \sigma_U(H)$.

If $\gamma_{\eta}(\tau, w, b(\cdot))\not = 0$, then there is a $d\geq 1$ such that

It follows from Lemma 6 and Lemma 8 that if $\gamma_{\eta}(\tau, w, b(\cdot))\not = 0$, then

Thus $Q_\eta x(n; \tau, w, b(\cdot))\to \infty$ as $n \to \infty$. Therefore, the proof is complete. □

3.3.   Proof of Theorem 2

We are now in a position to prove Theorem 2.

We set

and

Then $\sigma_{N}(H) = \sigma_{N_0}(H)\cup \sigma_{N_1}(H)$.

For any $(\tau, w)\in {\mathbb Z}\times {\mathbb C}^p$ the component $Q_\eta x(n; \tau, w, b(\cdot))$ of the solution to the Eq (1.2) is given by Lemma 6.

I. First we prove "if" part of Theorem 2.

If $|\eta| < 1$, then (3.2) holds.

Let $\eta\in \sigma_{N_0}(H)$ with $h_\eta(H) = 1$. Then it follows from Lemma 6 that

Applying Lemma 7 and Lemma 8, we obtain that if $w\in B_\alpha$, then

Indeed, we have

Since the hypothesis yields

any vector $w\in \mathbb {C}^p$ can be represented as

Set

where

and

If $w\in B_\alpha$, then it follows from Lemma 9 and (3.4) that

This implies that $x(n; \tau, w, b(\cdot))$ is uniformly bounded.

II. Next, we prove "only if" part of Theorem 2.

It suffices to prove that if the solutions of the Eq (1.2) are uniformly bounded, then

Since $\sigma(H) = \sigma_S(H)\cup\sigma_U(H)\cup \sigma_{N_0}(H)\cup \sigma_{N_1}(H)$, any vector $w\in \mathbb {C}^p$ can be represented as

The uniform boundedness of the solutions is equivalent to the uniform boundedness of the components $Q_\eta x(n; \tau, w, b(\cdot))$ of solutions for every $\eta\in \sigma(H)$.

(1) The case $\eta\in \sigma_U(H)$: It follows from Lemma 9 that $Q_\eta x(n; \tau, w, b(\cdot))\to \infty$ as $n \to \infty$ if $\gamma_{\eta}(\tau, w, b(\cdot))\not = 0$, which is a contradiction. If $\gamma_{\eta}(\tau, w, b(\cdot)) = 0$, then $Q_\eta w = -Z_\eta(H, b(\tau+\cdot))$. Thus $Q_\eta x(n; \tau, w, b(\cdot))$ is bounded, but it is not uniformly bounded, which is a contradiction. Therefore, $\sigma_U(H) = \emptyset$, hence $\oplus_{\eta \in\sigma_U(H)}G_\eta(H) = \{0\}$.

(2) The case $\eta\in \sigma_S(H)$: It follows from Lemma 9 that

is uniformly bounded.

(3) The case $\eta\in \sigma_{N_0}(H)$: It suffices to prove $G_\eta(H) = W_\eta(H)$. Assume $h_\eta(H)\geq 2$. By Lemma 6 we have

which is a contradiction. Also, $h_\eta(H) = 0$ yields a contradiction. If $h_\eta(H) = 1$, then (3.3) holds, which is uniformly bounded.

(4) The case $\eta\in \sigma_{N_1}(H)$: Assume $G_\eta(H) = W_\eta(H)$. Then using the same argument as in (1), we have $G_\eta(H) = \{0\}$. Indeed, we obtain from Lemma 6

It follows from Lemma 8 that $Q_{\eta}x(n; \tau, w, b(\cdot))\to \infty \ {\rm as} \ n \to \infty$. This is a contradiction. Therefore, the proof is complete.□

4.   A characteristic equation

Making use of the simultaneous diagonalization theorem under the commuting condition $AB = BA$, we can apply the preceding results to the Eq (1.3). But we only consider the characteristic equation of the matrix $M$ in a reduced equation

derived from the Eq (1.3). Indeed, making a change of variables $y_i(n) = x(n-\rho+i), \ i\in \{0, 1, \cdots, \rho\}$, we have

Therefore, the Eq (1.3) is transformed to the Eq (4.1), where

We denote by $M_{mn}(\mathbb{C})$ the set of all $m\times n$ complex matrices.

First, we give the characteristic equation $\det(zE-M) = 0$ of the matrix $M$ in the Eq (4.1) in the following proposition.

Proposition 4. The characteristic equation of $M$ is given by

Proof. Let $z\not = 0$. Then we have

Repeating this procedure we have

Thus the characteristic equation of $M$ becomes

If $z = 0$, then $\det (zE-M) = \det(-M) = \det(-B)$. □

Corollary 2. The matrix $M$ is nonsingular if and only if the matrix $B$ is nonsingular.

Next, we find the eigenvalues of $M$ under the commuting condition (C): $AB = BA.$

Definition 2. Let two matrices $A$ and $B$ be semisimple matrices in $M_p({\mathbb{C}})$. Then two matrices $A$ and $B$ are said to be simultaneously diagonalizable if there exists a nonsingular matrix $P \in M_p({\mathbb{C}})$ such that $P^{-1}AP, P^{-1}BP$ are diagonal matrices.

Let $\alpha_1, \ldots, \alpha_p$ (not necessarily distinct) be all the eigenvalues of $A$ and $\mu_1, \ldots, \mu_p$ (not necessarily distinct) all the eigenvalues of $B$. By the assumption $(C)$ the simultaneous diagonalization theorem holds, that is, there exists a nonsingular matrix $P\in M_p(\mathbb{C})$ such that

Proposition 5. Let $A, B$ be semisimple matrices in $M_p({\mathbb{C}})$ and satisfy the commuting condition $\mathrm (C)$. Then

Proof. Using Proposition 4, we obtain

Therefore, the proof is complete. □

5.   Conclusions

We have given a criterion on the uniform boundedness of the solutions to linear difference equations (LEs) with periodic forcing functions. In particular, we have shown a subtle difference on the uniform boundedness of the solutions between the nonhomogenuous equation (1.2) and the corresponding homogenuous equation (1.1).

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

Dohan Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korean Government(MSIT) (No. 2021R1A2C1092945).

Conflict of interest

All authors declare no conflicts of interest that could affect the publication of this paper.