Multi-Delayed Discrete Matrix Functions and Their Applications in Solving Higher-Order Difference Equations

Abstract

A new class of linear non-homogeneous discrete matrix equations with multiple delays and second-order differences is considered, where the coefficient matrices satisfy pairwise permutability conditions. First, new multi-delayed discrete matrix sine- and cosine-type functions are introduced, which generalize existing delayed discrete matrix functions. Based on the introduced matrix functions and appropriate commutativity conditions, an explicit matrix representation of the solution is derived. The importance of our results is shown by comparing them with related previous works, along with suggestions about some new open problems. Finally, an example is provided to illustrate the importance of the results.

1. Introduction

Delayed discrete equations are of significant importance in both theoretical and applied mathematics, particularly in the fields of stability theory [1,2], control theory [3,4,5,6,7], dynamical systems and modeling [8], signal processing [9], and networked and distributed systems [10]. Such equations arise naturally in systems where the current state depends not only on the immediate past but also on multiple delayed states, and where second-order dynamics (involving accelerations or second differences) are present. The inclusion of matrix formulations allows for the modeling of multi-variable (vector-valued) systems, making the analysis highly relevant to real-world engineering and scientific problems.

Obtaining explicit or closed-form solutions provides deep insight into the behavior of the system and enables the precise characterization of key dynamical properties. Moreover, a powerful approach to solving initial value problems for linear discrete delay systems involves the use of delayed discrete matrix functions, which encode the system’s dynamics in a compact matrix form. For discrete systems with first-order differences, Diblík [11,12,13] investigated the solutions of equations involving a single delay, while systems with two delays and coefficient matrices that commute were studied in [14,15]. Medved’ and Pospisil [16,17] extended the analysis to discrete systems with multiple delays under the assumption of pairwise commutative matrices, while Jin and Mahmudov [18,19,20] investigated similar systems without requiring commutativity conditions, thus accommodating non-commutative matrix coefficients. For discrete systems with second-order differences, Diblík and Mencakova [21,22] investigated the representation of solutions for equations with a single delay, while Elshenhab and Wang [23] extended the analysis to systems with multiple delays, considering both permutable and nonpermutable matrix coefficients. For a comprehensive overview of the results achieved in this field, we refer to the recent article [24]. This foundational understanding has led to significant developments in stability problems [1,25,26], iterative learning control and learning analysis [3,27,28,29], and relative controllability [7,21].

Recently, Diblík [30] introduced a new class of discrete matrix equations with single delay and second-order forward difference, and derived explicit representations of their solutions. Its general form is

$${\Delta }^2\mathbb{X}\left(k\right)+\mathbb{S}\mathbb{X}\left(k-\omega \right)+\mathbb{X}\left(k-\omega \right)\mathbb{T}=\Pi \left(k\right),$$

(1)

where the independent variable k belongs to the discrete interval ${\mathcal{Z}}_0^{\infty }$, that is, $k\in {\mathcal{Z}}_0^{\infty }$. The set ${\mathcal{Z}}_p^q$, with $p\le q$, denotes the set of all integers satisfying $p\le k\le q$. Cases where $p=-\infty$ or $q=\infty$ are also allowed. In Equation (1), $\omega$ is a fixed positive integer known as delay, $\mathbb{X}:{\mathcal{Z}}_{-\omega }^{\infty }\to {\mathbb{R}}^{n\times n}$ is the unknown matrix function, and $\Pi :{\mathcal{Z}}_0^{\infty }\to {\mathbb{R}}^{n\times n}$ is a given matrix-valued function. The operators $\Delta$ and ${\Delta }^2$ denote the forward differences of first and second order, respectively, defined by

$${\Delta }^2\mathbb{X}\left(k\right)=\Delta \left(\Delta \mathbb{X}\left(k\right)\right)=\Delta (\mathbb{X}(k+1)-\mathbb{X}\left(k\right))=\mathbb{X}(k+2)-2\mathbb{X}(k+1)+\mathbb{X}\left(k\right),$$

(2)

and $\mathbb{S},\mathbb{T}$ are given constant $n\times n$ real matrices.

However, a significant challenge in the current literature is the lack of a unified framework for second-order difference equations with multiple delays. Therefore, Diblík formulated open problems aimed at extending and generalizing the results in [30] to the case of linear discrete systems with multiple delays and with coefficient matrices that are either permutable or non-permutable. This paper provides affirmative solutions to some of these open problems. Motivated by [30], we, therefore, investigate the exact solutions for a new class of matrix equations featuring multiple delays and second-order forward differences of the form

$$\left\{\begin{array}{c}{\Delta }^2\mathbb{X}\left(k\right)+{\displaystyle \sum _{i=1}^d}\left[{\mathbb{S}}_i\mathbb{X}\left(k-{\omega }_i\right)+\mathbb{X}\left(k-{\omega }_i\right){\mathbb{T}}_i\right]=\Pi \left(k\right),\\ \mathbb{X}\left(k\right)=\Psi \left(k\right),\mathrm{for}k\in {\mathbb{Z}}_{-\omega }^1,\end{array}\right.$$

(3)

where ${\omega }_i\in \mathbb{N}={\mathbb{Z}}_1^{\infty }$ are the delay parameters for $i=1,\dots ,d$, and $\omega :=max\{{\omega }_1,\dots ,{\omega }_d\}$. The function $\mathbb{X}:{\mathcal{Z}}_{-\omega }^{\infty }\to {\mathbb{R}}^{n\times n}$ denotes the unknown matrix-valued function satisfying (3) for all $k\in {\mathbb{Z}}_0^{\infty }$. The sequences of coefficients are given by $\mathbb{S}=({\mathbb{S}}_1,{\mathbb{S}}_2,\dots ,{\mathbb{S}}_d)$ and $\mathbb{T}=({\mathbb{T}}_1,{\mathbb{T}}_2,\dots ,{\mathbb{T}}_d)$, where each ${\mathbb{S}}_i,{\mathbb{T}}_i\in {\mathbb{R}}^{n\times n}$ is a given constant non-zero matrix. Furthermore, $\Pi :{\mathcal{Z}}_0^{\infty }\to {\mathbb{R}}^{n\times n}$ is a given inhomogeneous term, and $\Psi :{\mathcal{Z}}_{-\omega }^{-1}\to {\mathbb{R}}^{n\times n}$ is a prescribed initial function.

The paper is structured as follows: In Section 2, we present some basic notions and lemmas, two novel multi-delayed discrete matrix functions, and their main properties used in our subsequent discussions. In Section 3, the derivation of an explicit solution to problem (3) is achieved utilizing the newly introduced multi-delayed discrete matrix functions, which form the core contribution of this paper. Finally, we present an example with simulations to exemplify our theoretical results.

2. Preliminaries

In this section, we introduce fundamental definitions and auxiliary results related to multi-delayed discrete matrix functions and discrete calculus that will be used throughout the subsequent analysis. For integers $n,m\in \mathbb{Z}$ with $n\le m$, we define the discrete intervals ${\mathbb{Z}}_n^m:=\{n,n+1,\dots ,m\}$ and ${\mathbb{Z}}_n^{\infty }:=\{n,n+1,\dots \}$. If $n>m$, we set ${\mathbb{Z}}_n^m=⌀$. The $n\times n$ identity matrix and zero matrix are denoted by $\mathbb{E}$ and $\Theta$, respectively. We adopt the standard convention that an empty sum ${\sum }_{i=s_1}^{s_2}\dots$, where $s_1>s_2$, evaluates to zero, and an empty product ${\prod }_{i=s_1}^{s_2}\dots$ evaluates to one. When dealing with matrix operations, the empty sum yields the zero matrix $\Theta$, and the empty product yields the identity matrix $\mathbb{E}$. We define Binomial numbers by the formula

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a}{b}}\right):=\left\{\begin{array}{cc}{\displaystyle \frac{a!}{b!·(a-b)!}}\hfill & \mathrm{if}a\ge b\ge 0,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and satisfy the following identities:

$$\begin{array}{cc}\hfill \left(\begin{array}{c}a+1\\ b\end{array}\right)& =\left(\begin{array}{c}a\\ b\end{array}\right)+\left(\begin{array}{c}a\\ b-1\end{array}\right),\left(\begin{array}{c}a\\ a\end{array}\right)=\left(\begin{array}{c}a-1\\ a-1\end{array}\right),\left(\begin{array}{c}a\\ 0\end{array}\right)=\left(\begin{array}{c}a-1\\ 0\end{array}\right),\hfill \end{array}$$

where a, b are integers and $0!=1$.

Lemma 1.

([31]). Let a, $b\in \mathbb{Z}$, $a<b$. For given functions u and v, the following identities hold:

$${\Delta }_{k}u\left(k,r\right):=u\left(k+1,r\right)-u\left(k,r\right),$$

$$\sum _{r=a}^{b}u\left(r\right)\Delta v\left(r\right)=u\left(b\right)v\left(b+1\right)-u\left(a\right)v\left(a\right)-\sum _{r=a+1}^b\Delta u\left(r-1\right)v\left(r\right),$$

$${\Delta }_k\left[\sum _{r=1}^{k}u\left(k,r\right)\right]=u\left(k+1,k+1\right)+\sum _{r=1}^k{\Delta }_{k}u\left(k,r\right).$$

Definition 1.

Let $\mathbb{S}$$=\left({\mathbb{S}}_1,\dots ,{\mathbb{S}}_d\right)$ and $\mathbb{T}$$=\left({\mathbb{T}}_1,\dots ,{\mathbb{T}}_d\right)$ be $n\times n$ constant real nonzero matrices. We introduce the determining matrix equation ${\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k;\mathbf{i})\in {\mathbb{R}}^{n\times n}$ of the form

$${\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k+1;\mathbf{i})={\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k;\mathbf{i}-{\mathbf{e}}_j)+{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k;\mathbf{i}-{\mathbf{e}}_j){\mathbb{T}}_j],fork\in {\mathbb{Z}}_0^{\infty },\mathbf{i}\in {\left({\mathbb{Z}}_0^{\infty }\right)}^d,$$

(4)

such that

$$\begin{array}{cc}\hfill & {\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(0;\mathbf{i})=\Theta ,{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(1;0)=\mathbb{E},\hfill \\ \hfill & {\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k;-1,i_2,\dots ,i_d)={\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k;i_1,-1,\dots ,i_d)=\dots ={\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(k;i_1,i_2,\dots ,-1)=\Theta ,\hfill \end{array}$$

where ${\mathbf{e}}_j$, $j=1,\dots ,d$ is the canonical basis of ${\mathbb{R}}^d$.

Definition 2.

The multi-delayed discrete matrix sine-type ${\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)$, and cosine-type ${\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)$ functions are defined as follows:

$$\begin{array}{c}{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho }),\\ {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k-1\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{1+2l}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho }),\end{array}$$

(5)

for any $k\in {\mathbb{Z}}_{-\omega }^{\infty }$, $\mathit{\rho }=\left({\rho }_1,{\rho }_2,\dots ,{\rho }_d\right)$, $l={\sum }_{i=1}^d{\rho }_i$, and ${\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho })$ is given by (4).

Lemma 2.

Let $\mathbb{S}$$=\left({\mathbb{S}}_1,\dots ,{\mathbb{S}}_d\right)$ and  $\mathbb{T}$$=\left({\mathbb{T}}_1,\dots ,{\mathbb{T}}_d\right)$ be $n\times n$ constant real nonzero matrices. Then, ${\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)$ and ${\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)$ satisfy the following equations:

(i) 

$\Delta {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)=-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j];$

(ii) 

$\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right);$

(iii) 

${\Delta }^2{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)=-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j];$

(iv) 

${\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)=-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j].$

Proof. 

It is a consequence of Definitions 1 and 2; first, we prove the identity (i) as follows:

$$\begin{array}{cc}\hfill \Delta {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)& ={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k+1\right)-{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k+1\end{array}}}{\left(-1\right)}^l\left[\left({\displaystyle \genfrac{}{}{0pt}{}{k+1-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l}}\right)\right.\hfill \\ \hfill & -\left.\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l}}\right)\right]{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho })\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 1\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l-1}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho })\hfill \\ \hfill & ={\displaystyle \sum _{j=1}^d}{\mathbb{S}}_j{\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 1\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l-1}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l;\mathit{\rho }-{\mathbf{e}}_j)\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^d}{\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 1\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l-1}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l;\mathit{\rho }-{\mathbf{e}}_j){\mathbb{T}}_j\hfill \\ \hfill & =-{\displaystyle \sum _{j=1}^d}{\mathbb{S}}_j{\displaystyle \sum _{\begin{array}{c}{\rho }_1^{\prime },\dots ,{\rho }_d^{\prime }\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }^{\prime }\le k-1\end{array}}}{\left(-1\right)}^{l^{\prime }}\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\omega }_j-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }^{\prime }}{2l^{\prime }+1}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l^{\prime }+1;{\mathit{\rho }}^{\prime })\hfill \\ \hfill & -{\displaystyle \sum _{j=1}^d}{\displaystyle \sum _{\begin{array}{c}{\rho }_1^{\prime },\dots ,{\rho }_d^{\prime }\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }^{\prime }\le k-1\end{array}}}{\left(-1\right)}^{l^{\prime }}\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\omega }_j-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }^{\prime }}{2l^{\prime }+1}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l^{\prime }+1;{\mathit{\rho }}^{\prime }){\mathbb{T}}_j\hfill \\ \hfill & =-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j].\hfill \end{array}$$

Second, the proof of the identity (ii) becomes

$$\begin{array}{cc}\hfill \Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)& ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k+1\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left[\left({\displaystyle \genfrac{}{}{0pt}{}{k+1-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l+1}}\right)\right.\hfill \\ \hfill & -\left.\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l+1}}\right)\right]{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho })\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l+1;\mathit{\rho })\hfill \\ \hfill & ={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right).\hfill \end{array}$$

Finally, using (i) and (ii), we obtain (iii) and (iv) as follows:

$$\begin{array}{cc}\hfill {\Delta }^2{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)& =\Delta \left(\Delta {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\right)=-\Delta \left({\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j]\right)\hfill \\ \hfill & =-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)& =\Delta \left(\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\right)=\Delta \left({\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\right)\hfill \\ \hfill & =-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j].\hfill \end{array}$$

This completes the proof. □

3. Exact Solutions of a Multiple-Delayed Matrix Equation

In this section, we derive the representation of a solution to the problem (3) using the multi-delayed discrete matrix sine- and cosine-type functions given in Definition 2.

Theorem 1.

Let ${\omega }_{\varrho }\ge 1$ for $\varrho \in \left\{1,\dots ,d\right\}$, $\omega :=max\left\{{\omega }_1,\dots ,{\omega }_d\right\}$. Suppose that $\mathbb{S}$$=\left({\mathbb{S}}_1,\dots ,{\mathbb{S}}_d\right)$ and $\mathbb{T}$$=\left({\mathbb{T}}_1,\dots ,{\mathbb{T}}_d\right)$ are $n\times n$ permutable constant real nonzero matrices, $\Psi \left(k\right){\mathbb{T}}_{\varrho }={\mathbb{T}}_{\varrho }\Psi \left(k\right)$, for $k\in {\mathbb{Z}}_{-\omega }^1$, and ${\mathbb{T}}_{\varrho }\Pi \left(k\right)=\Pi \left(k\right){\mathbb{T}}_{\varrho }$. Then the solution $\mathbb{X}\left(k\right)$ of (3) has the form

$$\mathbb{X}\left(k\right)=\left\{\begin{array}{c}\Psi \left(k\right),k\in {\mathbb{Z}}_{-\omega }^1,\\ {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi \left(0\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi \left(0\right)\\ -{\displaystyle \sum _{\varrho =1}^d}{\displaystyle \sum _{i=-{\omega }_{\varrho }+1}^0}\left[{\mathbb{S}}_{\varrho }{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{\varrho }-i\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{\varrho }-i\right){\mathbb{T}}_{\varrho }\right]\Psi \left(i-1\right)\\ +{\displaystyle \sum _{j=0}^{k-2}}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-j-1\right)\Pi \left(j\right),k\in {\mathbb{Z}}_0^{\infty }.\end{array}\right.$$

(6)

Proof. 

Fix $k\in {\mathbb{Z}}_0^{\infty }$. We prove that $\mathbb{X}\left(k\right)$, defined by (6), satisfies (3). First, if ${\omega }_{r_1}\le k<{\omega }_{r_2}$ for each $r_1\in {\Omega }_1$, $r_2\in {\Omega }_2$ such that ${\Omega }_1$, ${\Omega }_2\subset \left\{1,2,\dots ,d\right\}$, then

$${\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right)=\left\{\begin{array}{c}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right),i\in {\mathbb{Z}}_{-{\omega }_{r_2}+1}^{k-{\omega }_{r_2}},\\ \Theta ,i\in {\mathbb{Z}}_{k-{\omega }_{r_2}+1}^0,\end{array}\right.$$

and

$$\begin{array}{cc}\hfill & \mathbb{X}\left(k\right)\hfill \\ \hfill & ={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi \left(0\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi \left(0\right)\hfill \\ \hfill & -{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}\left[{\mathbb{S}}_{r_1}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-i\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-i\right){\mathbb{T}}_{r_1}\right]\Psi \left(i-1\right)\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^{k-{\omega }_{r_2}}}\left[{\mathbb{S}}_{r_2}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right){\mathbb{T}}_{r_2}\right]\Psi \left(i-1\right)\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^{k-2}}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-j-1\right)\Pi \left(j\right).\hfill \end{array}$$

(7)

Taking the first and second forward differences of (7), and applying Lemma 1, we obtain

$$\begin{array}{cc}\hfill & \Delta X\left(k\right)\hfill \\ \hfill & =\Delta {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi \left(0\right)+\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi \left(0\right)\hfill \\ \hfill & -{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}\left[{\mathbb{S}}_{r_1}\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-i\right)\Psi \left(i-1\right)\right.\hfill \\ \hfill & \left.+\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-i\right){\mathbb{T}}_{r_1}\Psi \left(i-1\right)\right]\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^{k-{\omega }_{r_2}}}\left[{\mathbb{S}}_{r_2}\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right)\Psi \left(i-1\right)\right.\hfill \\ \hfill & \left.+\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right){\mathbb{T}}_{r_2}\Psi \left(i-1\right)\right]\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^{k-2}}\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-j-1\right)\Pi \left(j\right)+\Pi \left(k-1\right),\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill & {\Delta }^2\mathbb{X}\left(k\right)\hfill \\ \hfill & ={\Delta }^2{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi \left(0\right)+{\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi \left(0\right)\hfill \\ \hfill & -{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}\left[{\mathbb{S}}_{r_1}{\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-i\right)\Psi \left(i-1\right)\right.\hfill \\ \hfill & +\left.{\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-i\right){\mathbb{T}}_{r_1}\Psi \left(i-1\right)\right]\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^{k-{\omega }_{r_2}}}\left[{\mathbb{S}}_{r_2}{\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right)\Psi \left(i-1\right)\right.\hfill \\ \hfill & +\left.{\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-i\right){\mathbb{T}}_{r_2}\Psi \left(i-1\right)\right]\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}\left[{\mathbb{S}}_{r_2}\Psi \left(k-{\omega }_{r_2}\right)+\Psi \left(k-{\omega }_{r_2}\right){\mathbb{T}}_{r_2}\right]\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^{k-2}}{\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-j-1\right)\Pi \left(j\right)+\Pi \left(k\right),\hfill \end{array}$$

(9)

since ${\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(0\right)=\Theta$, ${\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(1\right)=\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(0\right)=\Delta {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(1\right)=\mathbb{E}$. In setting $\Pi \left(k\right)\equiv \Theta$, and using Lemma 2, it follows that ${\mathcal{H}}_c^{\mathbf{S},\mathbf{T}}\left(k\right)$ and ${\mathcal{M}}_s^{\mathbf{S},\mathbf{T}}\left(k\right)$ are solutions of (3) that satisfy the prescribed initial conditions; specifically,

$$\begin{array}{cc}\hfill {\Delta }^2{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)& =-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j],\hfill \\ \hfill {\Delta }^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)& =-{\displaystyle \sum _{j=1}^d}[{\mathbb{S}}_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_j\right){\mathbb{T}}_j],\hfill \end{array}$$

(10)

whereas ${\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}\right)={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}\right)=\Theta$, for $r_2\in {\Omega }_2$. It follows from (9) and (10) that

$$\begin{array}{cc}\hfill & {\Delta }^2\mathbb{X}\left(k\right)\hfill \\ \hfill & =-\sum _{p\in {\Omega }_1}[{\mathbb{S}}_p{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right)+{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right){\mathbb{T}}_p]\Psi \left(0\right)\hfill \\ \hfill & -\sum _{p\in {\Omega }_1}[{\mathbb{S}}_p{\mathcal{M}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right)+{\mathcal{M}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right){\mathbb{T}}_p]\Delta \Psi \left(0\right)\hfill \\ \hfill & +{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}{\mathbb{S}}_{r_1}\sum _{p\in {\Omega }_1}\left[[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right)\right.\hfill \\ \hfill & \left.+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right){\mathbb{T}}_p\right]\Psi \left(i-1\right)\hfill \\ \hfill & +{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}\sum _{p\in {\Omega }_1}\left[[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right)\right.\hfill \\ \hfill & \left.+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right){\mathbb{T}}_p\right]{\mathbb{T}}_{r_1}\Psi \left(i-1\right)\hfill \\ \hfill & +{\displaystyle \sum _{r_2\in {\Omega }_2}}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^0}{\mathbb{S}}_{r_2}\sum _{p\in {\Omega }_1}\left[[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i\right)\right.\hfill \\ \hfill & \left.+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i-1\right){\mathbb{T}}_p\right]\Psi \left(i\right)\hfill \\ \hfill & +{\displaystyle \sum _{r_2\in {\Omega }_2}}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^0}\sum _{p\in {\Omega }_1}\left[[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i\right)\right.\hfill \\ \hfill & \left.+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i\right){\mathbb{T}}_p\right]{\mathbb{T}}_{r_2}\Psi \left(i-1\right)\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}\left[{\mathbb{S}}_{r_2}\Psi \left(k-{\omega }_{r_2}\right)+\Psi \left(k-{\omega }_{r_2}\right){\mathbb{T}}_{r_2}\right]\hfill \\ \hfill & -{\displaystyle \sum _{j=0}^{k-2}}\sum _{p\in {\Omega }_1}[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p-j-1\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p-j-1\right){\mathbb{T}}_p]\Pi \left(j\right)\hfill \\ \hfill & +\Pi \left(k\right),\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill & {\Delta }^2\mathbb{X}\left(k\right)\hfill \\ \hfill & =-\sum _{p\in {\Omega }_1}[{\mathbb{S}}_p{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right)+{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right){\mathbb{T}}_p]\Psi \left(0\right)\hfill \\ \hfill & -\sum _{p\in {\Omega }_1}[{\mathbb{S}}_p{\mathcal{M}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right)+{\mathcal{M}}_c^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p\right){\mathbb{T}}_p]\Delta \Psi \left(0\right)\hfill \\ \hfill & +\sum _{p\in {\Omega }_1}{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}\left[{\mathbb{S}}_p{\mathbb{S}}_{r_1}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right)\Psi \left(i-1\right)\right.\hfill \\ \hfill & \left.+{\mathbb{S}}_{r_1}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right)\Psi \left(i-1\right){\mathbb{T}}_p\right]\hfill \\ \hfill & +\sum _{p\in {\Omega }_1}{\displaystyle \sum _{r_1\in {\Omega }_1}}{\displaystyle \sum _{i=-{\omega }_{r_1}+1}^0}\left[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right){\mathbb{T}}_{r_1}\Psi \left(i-1\right)\right.\hfill \\ \hfill & \left.+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_1}-{\omega }_p-i\right){\mathbb{T}}_{r_1}\Psi \left(i-1\right){\mathbb{T}}_p\right]\hfill \\ \hfill & +\sum _{p\in {\Omega }_1}{\displaystyle \sum _{r_2\in {\Omega }_2}}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^0}\left[{\mathbb{S}}_p{\mathbb{S}}_{r_2}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i-1\right)\Psi \left(i\right)\right.\hfill \\ \hfill & \left.+{\mathbb{S}}_{r_2}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i-1\right)\Psi \left(i\right){\mathbb{T}}_p\right]\hfill \\ \hfill & +\sum _{p\in {\Omega }_1}{\displaystyle \sum _{r_2\in {\Omega }_2}}{\displaystyle \sum _{i=-{\omega }_{r_2}+1}^0}\left[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i\right){\mathbb{T}}_{r_2}\Psi \left(i-1\right)\right.\hfill \\ \hfill & \left.+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_{r_2}-{\omega }_p-i\right){\mathbb{T}}_{r_2}\Psi \left(i-1\right){\mathbb{T}}_p\right]\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}\left[{\mathbb{S}}_{r_2}\Psi \left(k-{\omega }_{r_2}\right)+\Psi \left(k-{\omega }_{r_2}\right){\mathbb{T}}_{r_2}\right]\hfill \\ \hfill & -\sum _{p\in {\Omega }_1}{\displaystyle \sum _{j=0}^{k-2}}[{\mathbb{S}}_p{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p-j-1\right)\Pi \left(j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-{\omega }_p-j-1\right)\Pi \left(j\right){\mathbb{T}}_p]\hfill \\ \hfill & +\Pi \left(k\right).\hfill \end{array}$$

Thus

$$\begin{array}{cc}\hfill {\Delta }^2\mathbb{X}\left(k\right)& =-\sum _{p\in {\Omega }_1}\left[{\mathbb{S}}_{p}X\left(k-{\omega }_p\right)+X\left(k-{\omega }_p\right){\mathbb{T}}_p\right]\hfill \\ \hfill & -\sum _{r_2\in {\Omega }_2}\left[{\mathbb{S}}_{r_2}\Psi \left(k-{\omega }_{r_2}\right)+\Psi \left(k-{\omega }_{r_2}\right){\mathbb{T}}_{r_2}\right]+\Pi \left(k\right).\hfill \end{array}$$

(11)

Since $k-{\omega }_{r_2}\in {\mathbb{Z}}_{-\omega }^1$, then $\Psi \left(k-{\omega }_{r_2}\right)=X\left(k-{\omega }_{r_2}\right)$ for each $r_2\in {\Omega }_2$. Hence

$$\begin{array}{cc}\hfill {\Delta }^2\mathbb{X}\left(k\right)& =-\sum _{\varrho =1}^d({\mathbb{S}}_{\varrho }X\left(k-{\omega }_{\varrho }\right)+X\left(k-{\omega }_{\varrho }\right){\mathbb{T}}_{\varrho })+\Pi \left(k\right).\hfill \end{array}$$

Second, if ${\Omega }_2=\left\{1,\dots ,d\right\}$, then (11) says

$$\begin{array}{cc}\hfill {\Delta }^2\mathbb{X}\left(k\right)& =-\sum _{r_2\in {\Omega }_2}({\mathbb{S}}_{r_2}\Psi \left(k-{\omega }_{r_2}\right)+\Psi \left(k-{\omega }_{r_2}\right){\mathbb{T}}_{r_2})+\Pi \left(k\right)\hfill \\ \hfill & =-\sum _{\varrho =1}^d({\mathbb{S}}_{\varrho }X\left(k-{\omega }_{\varrho }\right)+X\left(k-{\omega }_{\varrho }\right){\mathbb{T}}_{\varrho })+\Pi \left(k\right),\hfill \end{array}$$

for $0\le k<{min}_{\varrho =1,\dots ,d}{\omega }_{\varrho }$. Since $\Psi (k-{\omega }_{\varrho })=X(k-{\omega }_{\varrho })$ for each $\varrho =1,\dots ,d$, and finally, let ${max}_{\varrho =1,\dots ,d}{\omega }_{\varrho }\le k$, with ${\Omega }_1=\{1,\dots ,d\}$, then (11) implies

$$\begin{array}{cc}\hfill {\Delta }^2\mathbb{X}\left(k\right)& =-\sum _{p\in {\Omega }_1}({\mathbb{S}}_{p}X\left(k-{\omega }_p\right)+X\left(k-{\omega }_p\right){\mathbb{T}}_p)+\Pi \left(k\right)\hfill \\ \hfill & =-\sum _{\varrho =1}^d({\mathbb{S}}_{\varrho }X\left(k-{\omega }_{\varrho }\right)+X\left(k-{\omega }_{\varrho }\right){\mathbb{T}}_{\varrho })+\Pi \left(k\right).\hfill \end{array}$$

This completes the proof. □

Remark 1.

Let $d=1$ and $\omega \ge 1$. Suppose that $\mathbb{S}$ and $\mathbb{T}$ are $n\times n$ constant real nonzero matrices. We can obtain alternative conclusions of Theorem 1 by applying summation by parts twice to the following term:

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{j=-\omega +1}^0}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j\right){\Delta }_j^2\Psi (j-1)\hfill \\ \hfill & ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right)\Delta \Psi (-\omega )\hfill \\ \hfill & +{\displaystyle \sum _{j=-\omega +1}^{-1}}{\Delta }_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j-1\right){\Delta }_j\Psi \left(j\right)\hfill \\ \hfill & ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right)\Delta \Psi (-\omega )\hfill \\ \hfill & +{\displaystyle \sum _{j=-\omega }^{-1}}{\Delta }_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j-1\right){\Delta }_j\Psi \left(j\right)-{\Delta }_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right){\Delta }_j\Psi (-\omega )\hfill \\ \hfill & ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi (-\omega )\hfill \\ \hfill & +{\displaystyle \sum _{j=-\omega }^{-1}}{\Delta }_j{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j-1\right){\Delta }_j\Psi \left(j\right)\hfill \\ \hfill & ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi (-\omega )\hfill \\ \hfill & +{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Psi \left(0\right)-{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-1\right)\Psi (-\omega )\hfill \\ \hfill & +{\displaystyle \sum _{j=-\omega +1}^{-1}}{\Delta }_j^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j\right)\Psi \left(j\right)\hfill \\ \hfill & ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi (-\omega )\hfill \\ \hfill & +{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Psi \left(0\right)-{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-1\right)\Psi (-\omega )\hfill \\ \hfill & +{\displaystyle \sum _{j=-\omega }^{-1}}{\Delta }_j^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j-1\right)\Psi \left(j\right)-{\Delta }_j^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right)\Psi (-\omega )\hfill \\ \hfill & =\left[{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi (-\omega )\right]\hfill \\ \hfill & +{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Psi \left(0\right)-{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi (-\omega )\hfill \\ \hfill & +{\displaystyle \sum _{j=-\omega }^{-1}}{\Delta }_j^2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j-1\right)\Psi \left(j\right)\hfill \\ \hfill & ={\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)-{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi (-\omega )\hfill \\ \hfill & +{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Psi \left(0\right)-{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi (-\omega )\hfill \\ \hfill & -{\displaystyle \sum _{j=-\omega +1}^0}[\mathbb{S}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-2\omega -j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-2\omega -j\right)\mathbb{T})\Psi (j-1).\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi (-\omega )+{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi (-\omega )+{\displaystyle \sum _{j=-\omega +1}^0}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega -j\right){\Delta }_j^2\Psi (j-1)\hfill \\ \hfill & ={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Psi \left(0\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-\omega \right)\Delta \Psi \left(0\right)\hfill \\ \hfill & -{\displaystyle \sum _{j=-\omega +1}^0}\left[\mathbb{S}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-2\omega -j\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-2\omega -j\right)\mathbb{T}\right]\Psi (j-1).\hfill \end{array}$$

Corollary 1.

Let $d=1$ and $\omega \ge 1$. Consider $n\times n$ constant real nonzero matrices $\mathbb{S}$ and $\mathbb{T}$, $\Psi \left(k\right)\mathbb{T}=\mathbb{T}\Psi \left(k\right)$, for $k\in {\mathbb{Z}}_{-\omega }^1$, and $\mathbb{T}\Pi \left(k\right)=\Pi \left(k\right)\mathbb{T}$. Then the solution $\mathbb{X}\left(k\right)$ of (3) can be expressed as

$$\begin{array}{cc}\hfill X\left(k\right)& ={\mathcal{N}}_c\left(k\right)\Psi (-\omega )+{\mathcal{N}}_s\left(k\right)\Delta \Psi (-\omega )+{\displaystyle \sum _{j=-\omega +1}^0}{\mathcal{N}}_s\left(k-\omega -j\right){\Delta }_j^2\Psi (j-1)\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^{k-2}}{\mathcal{N}}_s\left(k-\omega -j-1\right)\Pi \left(j\right),k\in {\mathbb{Z}}_0^{\infty },\hfill \end{array}$$

(12)

or

$$\begin{array}{cc}\hfill X\left(k\right)& ={\mathcal{N}}_c\left(k-\omega \right)\Psi \left(0\right)+{\mathcal{N}}_s\left(k-\omega \right)\Delta \Psi \left(0\right)\hfill \\ \hfill & -{\displaystyle \sum _{j=-\omega +1}^0}\left[\mathbb{S}{\mathcal{N}}_s\left(k-2\omega -j\right)+{\mathcal{N}}_s\left(k-2\omega -j\right)\mathbb{T}\right]\Psi (j-1)\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^{k-2}}{\mathcal{N}}_s\left(k-\omega -j-1\right)\Pi \left(j\right),k\in {\mathbb{Z}}_0^{\infty }.\hfill \end{array}$$

(13)

Proof. 

If $d=1$, then using the Binomial theorem, we have

$$\begin{array}{cc}\hfill {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)& ={\displaystyle \sum _{l=0}^{⌊\frac{k}{\omega +2}⌋}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\left(l-1\right)\omega }{2l}}\right)[{\mathbb{SQ}}^{\mathbb{S},\mathbb{T}}(l;l-1)+{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l;l-1)\mathbb{T}]\hfill \\ \hfill & ={\displaystyle \sum _{l=0}^{⌊\frac{k}{\omega +2}⌋}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\left(l-1\right)\omega }{2l}}\right)\sum _{s=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{s}}\right){\mathbb{S}}^{l-s}{\mathbb{T}}^s={\mathcal{N}}_c\left(k\right),\hfill \end{array}$$

similarly,

$$\begin{array}{cc}\hfill {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)& ={\displaystyle \sum _{l=0}^{⌊\frac{k-1}{\omega +2}⌋}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\left(l-1\right)\omega }{1+2l}}\right)[{\mathbb{SQ}}^{\mathbb{S},\mathbb{T}}(l;l-1)+{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}(l;l-1)\mathbb{T}]\hfill \\ \hfill & ={\displaystyle \sum _{l=0}^{⌊\frac{k-1}{\omega +2}⌋}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\left(l-1\right)\omega }{1+2l}}\right)\sum _{s=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{s}}\right){\mathbb{S}}^{l-s}{\mathbb{T}}^s={\mathcal{N}}_s\left(k\right),\hfill \end{array}$$

where ${\mathcal{N}}_c\left(k\right)$ and ${\mathcal{N}}_s\left(k\right)$ are the delayed discrete matrix functions defined in [30]. From the conclusion of Theorem 1 and Remark 1, we obtain (12) and (13). This ends the proof. □

Remark 2.

Corollary 1 coincides with the corresponding results in [30].

Remark 3.

If  $\mathbb{T}$$=\Theta$, and $\mathbb{S}$$=\left({\mathbb{S}}_1,\dots ,{\mathbb{S}}_d\right)$ are $n\times n$ permutable constant real nonzero matrices, then

$$\begin{array}{cc}\hfill & {\mathcal{H}}_c^{\mathbb{S},\Theta }\left(k\right)\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^d\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{{\rho }_1,\dots ,{\rho }_d}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{2l}}\right)\prod _{i=1}^d{\mathbb{S}}_i^{\rho i}\mathcal{X}\left(k\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathcal{M}}_s^{\mathbb{S},\Theta }\left(k\right)\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ {\sum }_{\varrho =1}^n\left({\omega }_{\varrho }+2\right){\rho }_{\varrho }\le k-1\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{{\rho }_1,\dots ,{\rho }_d}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\sum }_{\varrho =1}^d{\omega }_{\varrho }{\rho }_{\varrho }}{1+2l}}\right)\prod _{i=1}^d{\mathbb{S}}_i^{\rho i}=\mathcal{Y}\left(k\right).\hfill \end{array}$$

where $\mathcal{X}\left(k\right)$ and $\mathcal{Y}\left(k\right)$ are the delayed discrete matrix functions defined in [23]. In this case, our results coincide with the corresponding results in [23].

Remark 4.

If $d=1$,$\mathbb{T}$$=\Theta$, and $\mathbb{S}$ is an $n\times n$ constant real nonzero matrix, then

$$\begin{array}{cc}\hfill {\mathcal{H}}_c^{\mathbb{S},\Theta }\left(k\right)& ={\displaystyle \sum _{\begin{array}{c}\rho \ge 0\\ \left(\omega +2\right)\rho \le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\omega \rho }{2l}}\right){\mathbb{Q}}^{\mathbb{S},\Theta }(l+1;\mathit{\rho })\hfill \\ \hfill & ={\displaystyle \sum _{\begin{array}{c}l\ge 0\\ \left(\omega +2\right)l\le k\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-ml}{2l}}\right){\mathbb{SQ}}^{\mathbb{S},\Theta }(l;l-1)\hfill \\ \hfill & ={\displaystyle \sum _{l=0}^{⌊\frac{k}{\omega +2}⌋}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\left(l-1\right)\omega }{2l}}\right){\mathbb{S}}^l={\mathcal{M}}_c\left(k,\mathbb{S},\omega \right),\hfill \end{array}$$

similarly,

$$\begin{array}{cc}\hfill {\mathcal{M}}_s^{\mathbb{S},\Theta }\left(k\right)& ={\displaystyle \sum _{\begin{array}{c}\rho \ge 0\\ \left(\omega +2\right)\rho \le k-1\end{array}}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-ml}{1+2l}}\right){\mathbb{Q}}^{\mathbb{S},\Theta }(l+1;\mathit{\rho })\hfill \\ \hfill & ={\displaystyle \sum _{l=0}^{⌊\frac{k-1}{\omega +2}⌋}}{\left(-1\right)}^l\left({\displaystyle \genfrac{}{}{0pt}{}{k-\left(l-1\right)\omega }{1+2l}}\right){\mathbb{S}}^l={\mathcal{M}}_s\left(k,\mathbb{S},\omega \right),\hfill \end{array}$$

where ${\mathcal{M}}_c\left(k,\mathbb{S},\omega \right)$ and ${\mathcal{M}}_s\left(k,\mathbb{S},\omega \right)$ are the delayed discrete matrix functions defined in [22]. In this case, our results coincide with the corresponding results in [22].

4. An Example

To illustrate the validity and utility of formula (6), we provide the following example.

Example 1.

Consider the nonhomogeneous discrete matrix equation involving two delays:

$$\begin{array}{c}{\Delta }^2\mathbb{X}\left(k\right)+{\mathbb{S}}_{1}X\left(k-1\right)+X\left(k-1\right){\mathbb{T}}_1+{\mathbb{S}}_{2}X\left(k-2\right)+X\left(k-2\right){\mathbb{T}}_2=\Pi \left(k\right),\mathrm{for}k\in {\mathbb{Z}}_0^{\infty },\\ \mathbb{X}\left(k\right)=\Psi \left(k\right)=\left(\begin{array}{cc}k& 0\\ 0& k\end{array}\right),\mathrm{for}k\in {\mathbb{Z}}_{-2}^1,\end{array}$$

(14)

where

$$\begin{array}{cc}\hfill {\mathbb{S}}_1& =\left(\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right),{\mathbb{T}}_1=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right),{\mathbb{S}}_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\hfill \\ \hfill {\mathbb{T}}_2& =\left(\begin{array}{cc}3& 0\\ 0& 3\end{array}\right),\Pi \left(k\right)=\left(\begin{array}{cc}0& k+2\\ k+2& 0\end{array}\right).\hfill \end{array}$$

Using Definitions 1 and 2, we have

$$\begin{array}{cc}\hfill {\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(0\right)& ={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(1\right)=\mathbb{E},{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(4\right)=4\mathbb{E}\hfill \\ \hfill {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(0\right)& =\Theta ,{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(1\right)=\mathbb{E},{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(2\right)=2\mathbb{E},{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(3\right)=3\mathbb{E},\hfill \end{array}$$

for $k\ge 2$, we compute recursively:

$$\begin{array}{c}{\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)={\displaystyle \sum _{\begin{array}{c}{\rho }_1,{\rho }_2\ge 0\\ 3{\rho }_1+4{\rho }_2\le k\end{array}}}{\left(-1\right)}^{\left({\rho }_1+{\rho }_2\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\rho }_1-2{\rho }_2}{2\left({\rho }_1+{\rho }_2\right)}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2+1;{\rho }_1,{\rho }_2),\\ {\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)={\displaystyle \sum _{\begin{array}{c}{\rho }_1,\dots ,{\rho }_d\ge 0\\ 3{\rho }_1+4{\rho }_2\le k-1\end{array}}}{\left(-1\right)}^{\left({\rho }_1+{\rho }_2\right)}\left({\displaystyle \genfrac{}{}{0pt}{}{k-{\rho }_1-2{\rho }_2}{1+2\left({\rho }_1+{\rho }_2\right)}}\right){\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2+1;{\rho }_1,{\rho }_2),\end{array}$$

(15)

where

$$\begin{array}{cc}\hfill & {\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2+1;{\rho }_1,{\rho }_2)\hfill \\ \hfill & =[{\mathbb{S}}_1{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2;{\rho }_1-1,{\rho }_2)+{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2;{\rho }_1-1,{\rho }_2){\mathbb{T}}_1]\hfill \\ \hfill & +[{\mathbb{S}}_2{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2;{\rho }_1,{\rho }_2-1)+{\mathbb{Q}}^{\mathbb{S},\mathbb{T}}({\rho }_1+{\rho }_2;{\rho }_1,{\rho }_2-1){\mathbb{T}}_2],\hfill \end{array}$$

such that

$$\begin{array}{cc}\hfill & \mathbb{Q}(0;{\rho }_1,{\rho }_2)=\Theta ,\mathbb{Q}(1;0)=\mathbb{E},\hfill \\ \hfill & \mathbb{Q}(k;-1,{\rho }_2)=\mathbb{Q}(k;{\rho }_1,-1)=\Theta ,\mathrm{for}k\ge 2.\hfill \end{array}$$

Using Theorem 1, for $k\in {\mathbb{Z}}_2^8$, we can compute the explicit solutions $\mathbb{X}\left(k\right)$ of (14) as follows:

$$\begin{array}{cc}\hfill \mathbb{X}\left(k\right)& ={\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)\Psi \left(0\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)\Delta \Psi \left(0\right)\hfill \\ \hfill & -\left[{\mathbb{S}}_1{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right){\mathbb{T}}_1\right]\Psi \left(-1\right)\hfill \\ \hfill & -\left[{\mathbb{S}}_2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-1\right){\mathbb{T}}_2\right]\Psi \left(-2\right)\hfill \\ \hfill & -\left[{\mathbb{S}}_2{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-2\right)+{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-2\right){\mathbb{T}}_2\right]\Psi \left(-1\right)\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^{k-2}}{\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k-j-1\right)\Pi \left(j\right).\hfill \end{array}$$

(16)

Under the initial conditions, the graph of the solutions $\mathbb{X}\left(k\right)=\left(\begin{array}{cc}{X}_{11}\left(k\right)& X_{12}\left(k\right)\\ X_{21}\left(k\right)& X_{22}\left(k\right)\end{array}\right)$ in Figure 1, and its values are in

Table 1. $\mathbb{X}\left(k\right)$ for Figure 1.

k	$\mathbb{X}\left(\mathit{k}\right)$
2	$\left(\begin{array}{cc}8& 4\\ 4& 8\end{array}\right)$
3	$\left(\begin{array}{cc}18& 12\\ 12& 18\end{array}\right)$
4	$\left(\begin{array}{cc}28& 24\\ 24& 28\end{array}\right)$
5	$\left(\begin{array}{cc}35& 40\\ 40& 35\end{array}\right)$
6	$\left(\begin{array}{cc}14& 42\\ 42& 14\end{array}\right)$
7	$\left(\begin{array}{cc}-73& -3\\ -3& -73\end{array}\right)$
8	$\left(\begin{array}{cc}-268& -140\\ -140& -268\end{array}\right)$

. Figure 1 illustrates the evolution of the four elements of the solution matrix $\mathbb{X}\left(k\right)$ from time step $k=2$ to $k=8$. The plot clearly shows the dynamic, oscillatory behavior of the system introduced by the multiple delays and commutative coefficient matrices. The values in Table 1 provide a precise numerical confirmation of the solution calculated using our explicit formula (16). As shown by direct computation, the solution satisfies the equation. Additionally, from (16), we recover the initial condition $\mathbb{X}\left(k\right)=\Psi \left(k\right)$ for $k\in {\mathbb{Z}}_{-2}^1$.

5. Conclusions

In this paper, a significant open problem proposed in recent literature was addressed through the extension of second-order delayed discrete matrix equations from the single-delay case to the general multiple-delay setting. Two novel multi-delayed discrete matrix functions were introduced: the multi-delayed discrete matrix cosine-type function ${\mathcal{H}}_c^{\mathbb{S},\mathbb{T}}\left(k\right)$ and the multi-delayed discrete matrix sine-type function ${\mathcal{M}}_s^{\mathbb{S},\mathbb{T}}\left(k\right)$, which generalized existing delayed discrete matrix functions. These functions were shown to be crucial for solving the considered equation. Under specific commutativity conditions, an explicit representation of the solution to the initial value problem (3) was derived in Theorem 1. The generalization and extension of previous works were demonstrated through several important special cases. The practical application of the theoretical results was illustrated through a numerical example in which explicit solutions were computed and represented graphically.

The introduction of these multi-delayed discrete matrix functions was found to open new avenues for future research in stability analysis, controllability, and iterative learning control for higher-order discrete systems with multiple delays. The developed methodology was recognized as potentially extendable to systems with non-permutable matrices, time-varying coefficients, or fractional-order differences (see monograph [32]), which would further enrich the theory of discrete dynamical systems with memory effects.