Existence and Uniqueness of Second-Order Impulsive Delay Differential Systems

Abstract

In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results.

1. Introduction

Delay differential equations are used to explain many real-world models [1,2,3], such as networked control systems and high-speed communication networks. At the same time, impulses have a wide range of applications in various fields, such as mechanics [4] (e.g., clock mechanisms and impulsive circuit) and control theory [5] (e.g., impulsive control and robotics). Impulsive delay differential systems (IDDSs) can be used to describe situations in which the system’s state experiences abrupt changes relative to the previous time intervals and are applied in many fields, such as biological systems [6] (e.g., sudden population changes due to external effects occur frequently) and population dynamics [7]. In population dynamics, IDDSs are powerful tools for simulating the growth and variation processes of populations. The IDDS model can provide some prediction and analysis when considering factors such as population reproduction, migration, and natural environment changes. For recent developments in theory and application, one can refer to [8,9,10,11].

In recent decades, many important results on the exact solution of delay systems have been obtained by using delay matrix functions. In [12], Khusainov and Shuklin obtained the solutions to linear delay systems with permutation matrices by constructing a delay exponential matrix function. In [13], the authors studied the exact solutions of the Cauchy problem for oscillating systems with pure delay by constructing cosine and sine delay matrix functions. Inspired by Khusainov and Shuklin, You and Wang [14] considered the stability of impulsive delay differential equations. Dibliík et al. [15] gave the exact solutions for oscillating systems with two delays. Shah and Zada [16] studied the relative controllability of oscillating systems with two delays. Liu et al. [17] researched the exact solutions to fractional oscillating systems with pure delay. However, there are few results on the exact solutions for second-order differential systems.

Motivated by [13,15,17], we firstly seek the exact solutions of the following delay differential systems:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)=-A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa ),\kappa >0,\varsigma \in J,\hfill \\ & \mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\varsigma \in [-\kappa ,0],\hfill \end{array}\right.\end{array}$$

(1)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)=-A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa )+q\left(\varsigma \right),\kappa >0,\varsigma \in J,\hfill \\ & \mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\varsigma \in [-\kappa ,0].\hfill \end{array}\right.\end{array}$$

(2)

Considering the wide application of impulsive systems, it is necessary to study second-order impulsive delay systems. It is worth noting that linear impulsive usually means that the shape and intensity of the impulse remain unchanged throughout the transmission process, which makes the linear impulse more suitable for systems that require precise control. Based on this, we extend system (2) to the following linear second-order impulsive delay differential system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)=-A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa )+q\left(\varsigma \right),\varsigma \ne {\varsigma }_m,\varsigma \in J,\hfill \\ & \mathrm{\Delta }\mathrm{\Psi }\left({\varsigma }_m\right)=D_m\mathrm{\Psi }\left({\varsigma }_m\right),\varsigma ={\varsigma }_m,m=1,2,\dots ,r(T,0),\hfill \\ & \mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\varsigma \in [-\kappa ,0],\hfill \end{array}\right.\end{array}$$

(3)

where $A,B,D_m\in {\mathbb{R}}^{n\times n}$, $AB=BA$, $q\in C(J,{\mathbb{R}}^n)$, $J=[0,T]$, $T={\ell }^{*}\kappa$, ${\ell }^{*}$ is a finite positive integer, and $\vartheta \in C^2([-\kappa ,0]$, ${\mathbb{R}}^n)$. $r(T,0)$ represents the finite number of impulsive points in $(0,T)$ and $\mathrm{\Delta }\mathrm{\Psi }\left({\varsigma }_m\right)=\mathrm{\Psi }\left({\varsigma }_m^{+}\right)-\mathrm{\Psi }\left({\varsigma }_m^{-}\right)$, $\mathrm{\Psi }\left({\varsigma }_m^{-}\right)=\underset{\epsilon \to 0^{-}}{lim}\mathrm{\Psi }({\varsigma }_m+\epsilon )$, and $\mathrm{\Psi }\left({\varsigma }_m^{+}\right)=\underset{\epsilon \to 0^{+}}{lim}\mathrm{\Psi }({\varsigma }_m+\epsilon )$ denote the left and right limits of $\mathrm{\Psi }\left(\varsigma \right)$ at $\varsigma ={\varsigma }_m$.

Next, we are concerned with the existence and uniqueness of solutions for nonlinear IDDSs as follows:

$$\left\{\begin{array}{cc}\hfill & {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)=-A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa )+q(\varsigma ,\mathrm{\Psi }\left(\varsigma \right)),\varsigma \ne {\varsigma }_m,\varsigma \in J,\hfill \\ \hfill & \mathrm{\Delta }\mathrm{\Psi }\left({\varsigma }_m\right)=D_m\mathrm{\Psi }\left({\varsigma }_m\right),\varsigma ={\varsigma }_m,m=1,2,\dots ,r(T,0),\hfill \\ \hfill & \mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\varsigma \in [-\kappa ,0],\hfill \end{array}\right.$$

(4)

where $q\in C(J\times {\mathbb{R}}^n,{\mathbb{R}}^n)$.

The main novelty and contribution of this paper can be divided into two parts. Firstly, we construct delay matrix functions $C_{\kappa }$ and $S_{\kappa }$ for linear delay differential homogeneous system (1), which is different from the delay matrix function in [13], which only depends on one delay and one index (see [13], Definitions 1 and 2); the delay matrix function constructed in this paper depends on one delay and two indices. Further, we construct the exact solution corresponding to systems (1) and (2). Secondly, combined with the works by Dibliík [15] and Shah [16], we establish the representation of the solution corresponding to impulsive delay system (3). Based on this, we extend the case to the nonlinear system and obtain the existence and uniqueness of the solution.

The structure of the paper is as follows: In Section 2, we introduce the preliminaries and lemmas required for the subsequent sections. In Section 3, we derive the representation of the solution to system (3) by using cosine-type and sine-type delay matrix functions. In Section 4, the existence and uniqueness of the solution to system (4) are considered by virtue of the Schauder and Banach fixed-point theorems. Finally, we give several examples to verify the correctness of our conclusion in Section 5.

2. Preliminaries

Let ${\parallel \mathrm{\Psi }\parallel }_C=\underset{\varsigma \in J}{max}\parallel \mathrm{\Psi }\left(\varsigma \right)\parallel$ be the norm of the continuous space $C(J,{\mathbb{R}}^n)$, the space of piecewise left-continuous $PC(J,{\mathbb{R}}^n)=\{\mathrm{\Psi }:J\to {\mathbb{R}}^n:\varsigma \in C(({\varsigma }_m,{\varsigma }_{m+1}],{\mathbb{R}}^n),m=1,2,\cdots ,r(T,0)\}$, and let there exist $\mathrm{\Psi }\left({\varsigma }_m^{+}\right)$ and $\mathrm{\Psi }\left({\varsigma }_m^{-}\right)=\mathrm{\Psi }\left({\varsigma }_m\right)$ with ${\parallel \mathrm{\Psi }\parallel }_{PC}=\underset{\varsigma \in J}{sup}\parallel \mathrm{\Psi }\left(\varsigma \right)\parallel$, $C^1(J,{\mathbb{R}}^n)=\{\mathrm{\Psi }\in C(J,{\mathbb{R}}^n):{\mathrm{\Psi }}^{\prime }\in C(J,{\mathbb{R}}^n)\}$, and $C^2(J,{\mathbb{R}}^n)=\{{\mathrm{\Psi }}^{\prime }\in C(J,{\mathbb{R}}^n):{\mathrm{\Psi }}^{\prime \prime }\in C(J,{\mathbb{R}}^n)\}$. We set $x\in {\mathbb{R}}^n$ and $A\in {\mathbb{R}}^{n\times n}$ and introduce vector norm $\parallel x\parallel ={\displaystyle \sum _{♭=1}^n}\left|x_{♭}\right|$ and matrix norm $\parallel A\parallel =\underset{1\le \ell \le n}{max}{\displaystyle \sum _{♭=1}^n}\left|a_{♭\ell }\right|$. We denote ${\parallel \vartheta \parallel }_C=\underset{\varsigma \in [-\kappa ,0]}{max}\parallel \vartheta \left(\varsigma \right)\parallel$, $\parallel {\vartheta }^{\prime }{\parallel }_C=\underset{\varsigma \in [-\kappa ,0]}{max}\parallel {\vartheta }^{\prime }\left(\varsigma \right)\parallel$, and $\parallel {\vartheta }^{\prime \prime }{\parallel }_C=\underset{\varsigma \in [-\kappa ,0]}{max}\parallel {\vartheta }^{\prime \prime }\left(\varsigma \right)\parallel$.

Definition 1. 

The cosine-type delay matrix function $C_{\kappa }(·):\mathbb{R}\to {\mathbb{R}}^{n\times n}$ is defined by

$$\begin{array}{c}\hfill C_{\kappa }\left(\varsigma \right)=\left\{\begin{array}{c}\Theta ,-\infty <\varsigma <0,\hfill \\ I,\varsigma =0,\hfill \\ I+{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}},0<\varsigma \le \kappa ,\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -\kappa )}^{2(♭+1)}}{\left(2\right(♭+1\left)\right)!}},\kappa <\varsigma \le 2\kappa ,\hfill \\ \dots \hfill \\ {\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+\lambda }{C}_{♭+\lambda }^{♭}{A}^{2♭}{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(♭+\lambda )}}{\left(2\right(♭+\lambda \left)\right)!}},(\ell -1)\kappa <\varsigma \le \ell \kappa ,\ell \in {\mathbb{N}}^{+},\hfill \end{array}\right.\end{array}$$

and the sine-type delay matrix function $S_{\kappa }(·):\mathbb{R}\to {\mathbb{R}}^{n\times n}$ is defined by

$$\begin{array}{c}\hfill S_{\kappa }\left(\varsigma \right)=\left\{\begin{array}{c}\Theta ,-\infty <\varsigma <0,\hfill \\ \varsigma I,\varsigma =0,\hfill \\ \varsigma I+{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}},0<\varsigma \le \kappa ,\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -\kappa )}^{2(♭+1)+1}}{\left(2\right(♭+1)+1)!}},\kappa <\varsigma \le 2\kappa ,\hfill \\ \dots \hfill \\ {\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+\lambda }{C}_{♭+\lambda }^{♭}{A}^{2♭}{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(♭+\lambda )+1}}{\left(2\right(♭+\lambda )+1)!}},(\ell -1)\kappa <\varsigma \le \ell \kappa ,\ell \in {\mathbb{N}}^{+},\hfill \end{array}\right.\end{array}$$

where Θ and I are the zero and identity matrices, respectively.

Lemma 1. 

For any $\varsigma \in \left(\right(\ell -1)\kappa ,\ell \kappa ]$ and $\ell \in {\mathbb{N}}^{+}$, one has

$$\begin{array}{c}\hfill \parallel C_{\kappa }\left(\varsigma \right)\parallel \le {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}},\parallel S_{\kappa }\left(\varsigma \right)\parallel \le {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}.\end{array}$$

Proof. 

By Definition 1, for $(\ell -1)\kappa <\varsigma \le \ell \kappa$, we have

$$\begin{array}{ccc}\hfill \parallel C_{\kappa }(\varsigma )\parallel & \le & {\displaystyle \sum _{♭=0}^{\infty }}\parallel A^2{\parallel }^{♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+1}^{♭}\parallel A^2{\parallel }^{♭}\parallel B^2\parallel {\displaystyle \frac{{(\varsigma -\kappa )}^{2(♭+1)}}{(2(♭+1\left)\right)!}}+\cdots \hfill \\ & & +{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+(\ell -1)}^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\ell -1}{\displaystyle \frac{{(\varsigma -(\ell -1)\kappa )}^{2(♭+(\ell -1\left)\right)}}{(2(♭+(\ell -1)\left)\right)!}}\hfill \\ & \le & {\displaystyle \sum _{♭=0}^{\infty }}\parallel A^2{\parallel }^{♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+1}^{♭}\parallel A^2{\parallel }^{♭}\parallel B^2\parallel {\displaystyle \frac{{\varsigma }^{2(♭+1)}}{(2(♭+1\left)\right)!}}\hfill \\ & & +\cdots +{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+(\ell -1)}^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\ell -1}{\displaystyle \frac{{\varsigma }^{2(♭+(\ell -1\left)\right)}}{(2(♭+(\ell -1)\left)\right)!}}\hfill \end{array}$$

$$\begin{array}{ccc}& =& 1+(\parallel A^2\parallel +\parallel B^2\parallel ){\displaystyle \frac{{\varsigma }^2}{2!}}+{\displaystyle \sum _{♭=0}^2}(C_2^{♭}\parallel A^2{\parallel }^{♭}\parallel B^2{\parallel }^{2-♭}){\displaystyle \frac{{\varsigma }^{2\times 2}}{(2\times 2)!}}\hfill \\ & & +{\displaystyle \sum _{♭=0}^3}(C_3^{♭}\parallel A^2{\parallel }^{♭}\parallel B^2{\parallel }^{3-♭}){\displaystyle \frac{{\varsigma }^{2\times 3}}{(2\times 3)!}}+\cdots \hfill \\ & & +{\displaystyle \sum _{♭=0}^{\ell -1}}(C_{\ell -1}^{♭}\parallel A^2{\parallel }^{♭}\parallel B^2{\parallel }^{\ell -1-♭}){\displaystyle \frac{{\varsigma }^{2(\ell -1)}}{(2(\ell -1\left)\right)!}}\hfill \\ & & +{\displaystyle \sum _{♭=\ell }^{\infty }}{\displaystyle \sum _{\lambda =0}^{\ell -1}}(C_{♭}^{\lambda }\parallel A^2{\parallel }^{♭-\lambda }\parallel B^2{\parallel }^{\lambda }){\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}\hfill \\ & \le & (\parallel A^2\parallel +\parallel B^2{\parallel )}^0+(\parallel A^2\parallel +\parallel B^2\parallel ){\displaystyle \frac{{\varsigma }^2}{2!}}+(\parallel A^2\parallel +\parallel B^2{\parallel )}^2{\displaystyle \frac{{\varsigma }^{2\times 2}}{(2\times 2)!}}+\cdots \hfill \\ & & +(\parallel A^2\parallel +\parallel B^2{\parallel )}^{\ell -1}{\displaystyle \frac{{\varsigma }^{2(\ell -1)}}{(2(\ell -1\left)\right)!}}+{\displaystyle \sum _{♭=\ell }^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}\hfill \\ & =& {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}.\hfill \end{array}$$

Similarly, one has

$$\begin{array}{cc}\hfill & \parallel S_{\kappa }\left(\varsigma \right)\parallel \le {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}.\hfill \end{array}$$

□

Lemma 2. 

Let $(\ell -1)\kappa <\varsigma \le \ell \kappa$ and $\ell \in {\mathbb{N}}^{+}$; $C_{\kappa }\left(\varsigma \right)$ and $S_{\kappa }\left(\varsigma \right)$ satisfy system (1) such that

$$\begin{array}{c}\hfill C_{\kappa }^{\prime \prime }\left(\varsigma \right)=-A^2{C}_{\kappa }\left(\varsigma \right)-B^2{C}_{\kappa }(\varsigma -\kappa ),S_{\kappa }^{\prime \prime }\left(\varsigma \right)=-A^2{S}_{\kappa }\left(\varsigma \right)-B^2{S}_{\kappa }(\varsigma -\kappa ).\end{array}$$

(5)

Proof. 

The proof method is similar to ([15], Lemma 2.2). For $0<\varsigma \le \kappa$, the left expression of Equation (5) is as follows:

$$\begin{array}{ccc}\hfill C_{\kappa }^{\prime \prime }\left(\varsigma \right)& =& {\left(I+{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}\right)}^{\prime \prime }={\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2(♭-1)}}{\left(2\right(♭-1\left)\right)!}}\hfill \\ & =& -A^2{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭-1}{A}^{2(♭-1)}{\displaystyle \frac{{\varsigma }^{2(♭-1)}}{\left(2\right(♭-1\left)\right)!}}=-A^2{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}.\hfill \end{array}$$

The right expression of Equation (5) is as follows:

$$\begin{array}{c}\hfill -A^2{C}_{\kappa }\left(\varsigma \right)-B^2{C}_{\kappa }(\varsigma -\kappa )=-A^2\left(I+{\displaystyle \sum _{♭=1}^{\infty }}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}\right)=-A^2{\displaystyle \sum _{♭=0}^{\infty }}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}.\end{array}$$

For $(\ell -1)\kappa <\varsigma \le \ell \kappa$ and $\ell =2,3,\cdots ,{\ell }^{*}$, we have

$$C_{\kappa }\left(\varsigma \right)={\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+\lambda }{C}_{♭+\lambda }^{♭}{A}^{2♭}{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(♭+\lambda )}}{\left(2\right(♭+\lambda \left)\right)!}}=I_{{\chi }_{(0,\infty )}}\left(\varsigma \right)+{\mathcal{Z}}_1\left(\varsigma \right)+{\mathcal{Z}}_2\left(\varsigma \right)+{\mathcal{Z}}_3\left(\varsigma \right),$$

where ${\chi }_M$ is defined as the characteristic function of the set M. We set

$$\begin{array}{ccc}\hfill {\mathcal{Z}}_1\left(\varsigma \right)& =& {\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}},\hfill \\ \hfill {\mathcal{Z}}_2\left(\varsigma \right)& =& {\displaystyle \sum _{\lambda =1}^{\ell -1}}{(-1)}^{\lambda }{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2\lambda }}{\left(2\lambda \right)!}},\hfill \\ \hfill {\mathcal{Z}}_3\left(\varsigma \right)& =& {\displaystyle \sum _{\lambda =1}^{\ell -1}}{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭+\lambda }{C}_{♭+\lambda }^{♭}{A}^{2♭}{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(♭+\lambda )}}{\left(2\right(♭+\lambda \left)\right)!}}.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill {\mathcal{Z}}_1^{\prime \prime }\left(\varsigma \right)& =& {\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2(♭-1)}}{\left(2\right(♭-1\left)\right)!}}\hfill \\ & =& -A^2{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}=-A^2-A^2{\mathcal{Z}}_1\left(\varsigma \right),\hfill \\ \hfill {\mathcal{Z}}_2^{\prime \prime }\left(\varsigma \right)& =& {\displaystyle \sum _{\lambda =1}^{\ell -1}}{(-1)}^{\lambda }{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(\lambda -1)}}{\left(2\right(\lambda -1\left)\right)!}}\hfill \\ & =& -B^2{\displaystyle \sum _{\lambda -1=0}^{\ell -1}}{(-1)}^{\lambda -1}{B}^{2(\lambda -1)}{\displaystyle \frac{{(\varsigma -\kappa -(\lambda -1)\kappa )}^{2(\lambda -1)}}{\left(2\right(\lambda -1\left)\right)!}}\hfill \\ & =& -B^2-B^2{\mathcal{Z}}_2(\varsigma -\kappa ),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{Z}}_3^{\prime \prime }\left(\varsigma \right)& =& {\displaystyle \sum _{\lambda =1}^{\ell -1}}{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭+\lambda }{C}_{♭+\lambda }^{♭}{A}^{2♭}{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(♭+\lambda -1)}}{\left(2\right(♭+\lambda -1\left)\right)!}}\hfill \\ & =& -A^2{\displaystyle \sum _{\lambda =1}^{\ell -1}}{\displaystyle \sum _{♭-1=0}^{\infty }}{(-1)}^{♭-1+\lambda }{C}_{♭-1+\lambda }^{♭-1}{A}^{2(♭-1)}{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2(♭-1+\lambda )}}{\left(2\right(♭-1+\lambda \left)\right)!}}\hfill \\ & & -B^2{\displaystyle \sum _{\lambda -1=0}^{\ell -1}}{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭+\lambda -1}{C}_{♭+\lambda -1}^{\lambda -1}{A}^{2♭}{B}^{2(\lambda -1)}{\displaystyle \frac{{(\varsigma -\kappa -(\lambda -1)\kappa )}^{2(♭+\lambda -1)}}{\left(2\right(♭+\lambda -1\left)\right)!}}\hfill \\ & =& -A^2{\displaystyle \sum _{\lambda =1}^{\ell -1}}{(-1)}^{\lambda }{B}^{2\lambda }{\displaystyle \frac{{(\varsigma -\lambda \kappa )}^{2\lambda }}{\left(2\lambda \right)!}}-A^2{\mathcal{Z}}_3\left(\varsigma \right)\hfill \\ & & -B^2{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{(\varsigma -\kappa )}^{2♭}}{(2♭)!}}-B^2{\mathcal{Z}}_3(\varsigma -\kappa )\hfill \\ & =& -A^2{\mathcal{Z}}_2\left(\varsigma \right)-A^2{\mathcal{Z}}_3\left(\varsigma \right)-B^2{\mathcal{Z}}_1(\varsigma -\kappa )-B^2{\mathcal{Z}}_3(\varsigma -\kappa ),\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill C_{\kappa }^{\prime \prime }\left(\varsigma \right)=-A^2{C}_{\kappa }\left(\varsigma \right)-B^2{C}_{\kappa }(\varsigma -\kappa ).\end{array}$$

Similarly, one has

$$\begin{array}{c}\hfill S_{\kappa }^{\prime \prime }\left(\varsigma \right)=-A^2{S}_{\kappa }\left(\varsigma \right)-B^2{S}_{\kappa }(\varsigma -\kappa ).\end{array}$$

□

Lemma 3. 

Let $(\ell -1)\kappa <\varsigma \le \ell \kappa$ and $\ell \in {\mathbb{N}}^{+}$; we have

$$\begin{array}{c}\hfill {\int }_{-\kappa }^0\parallel S_{\kappa }(\varsigma -s)\parallel ds\le {\displaystyle \sum _{♭=0}^{\infty }}{\displaystyle \frac{(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}}{(2♭+2)!}}\left({\varsigma }^{2♭+2}-{(\varsigma -(\ell -1)\kappa )}^{2♭+2}\right).\end{array}$$

Proof. 

By Definition 1, we have

$$\begin{array}{ccc}& & {\int }_{-\kappa }^0\parallel S_{\kappa }(\varsigma -s)\parallel ds\hfill \\ & \le & {\int }_{-\kappa }^{\varsigma -\ell \kappa }\parallel S_{\kappa }(\varsigma -s)\parallel ds+{\int }_{\varsigma -\ell \kappa }^0\parallel S_{\kappa }(\varsigma -s)\parallel ds\hfill \\ & \le & {\displaystyle \sum _{\lambda =0}^{\ell }}{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\lambda }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\lambda }{\displaystyle \frac{1}{\left(2\right(♭+\lambda )+1)!}}{\int }_{-\kappa }^{\varsigma -\ell \kappa }{(\varsigma -s-\lambda \kappa )}^{2(♭+\lambda )+1}ds\hfill \\ & & +{\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\lambda }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\lambda }{\displaystyle \frac{1}{\left(2\right(♭+\lambda )+1)!}}{\int }_{\varsigma -\ell \kappa }^0{(\varsigma -s-\lambda \kappa )}^{2(♭+\lambda )+1}ds\hfill \\ & \le & {\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\ell }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\ell }{\displaystyle \frac{1}{\left(2\right(♭+\ell )+1)!}}{\int }_{-\kappa }^{\varsigma -\ell \kappa }{(\varsigma -s-\ell \kappa )}^{2(♭+\ell )+1}ds\hfill \\ & & +{\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\lambda }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\lambda }{\displaystyle \frac{1}{\left(2\right(♭+\lambda )+1)!}}{\int }_{-\kappa }^0{(\varsigma -s-\lambda \kappa )}^{2(♭+\lambda )+1}ds\hfill \\ & \le & {\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\ell }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\ell }{\displaystyle \frac{1}{\left(2\right(♭+\ell )+2)!}}{(\varsigma -(\ell -1)\kappa )}^{2(♭+\ell )+2}\hfill \\ & & +{\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\lambda }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\lambda }{\displaystyle \frac{1}{\left(2\right(♭+\lambda )+2)!}}\hfill \\ & & \times \left({(\varsigma -(\lambda -1)\kappa )}^{2(♭+\lambda )+2}-{(\varsigma -\lambda \kappa )}^{2(♭+\lambda )+2}\right)\hfill \\ & \le & {\displaystyle \sum _{\lambda =0}^{\ell }}{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\lambda }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\lambda }{\displaystyle \frac{1}{\left(2\right(♭+\lambda )+2)!}}{(\varsigma -(\lambda -1)\kappa )}^{2(♭+\lambda )+2}\hfill \\ & & -{\displaystyle \sum _{\lambda =0}^{\ell -1}}{\displaystyle \sum _{♭=0}^{\infty }}{C}_{♭+\lambda }^{♭}\parallel A^2{\parallel }^{♭}{\parallel B^2\parallel }^{\lambda }{\displaystyle \frac{1}{\left(2\right(♭+\lambda )+2)!}}{(\varsigma -(\ell -1)\kappa )}^{2(♭+\lambda )+2}\hfill \\ & \le & {\displaystyle \sum _{♭=0}^{\infty }}{\displaystyle \frac{(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}}{(2♭+2)!}}\left({\varsigma }^{2♭+2}-{(\varsigma -(\ell -1)\kappa )}^{2♭+2}\right).\hfill \end{array}$$

□

3. Representation of Solutions

Theorem 1. 

The solution to system (1) with the initial conditions $\mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right)$, ${\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right)$, and $\varsigma \in [-\kappa ,0]$ has the following form:

$$\begin{array}{c}\hfill \mathrm{\Psi }\left(\varsigma \right)=C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds.\end{array}$$

Proof. 

The proof of this theorem is analogous to ([17], Theorem 2). The general solution to system (1) can be obtained in the following form:

$$\mathrm{\Psi }\left(\varsigma \right)=C_{\kappa }(\varsigma +\kappa )G_1+S_{\kappa }(\varsigma +\kappa )G_2+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)f\left(s\right)ds,$$

where $f(·)$ is an unknown function, and $G_1$ and $G_2$ are unknown constant vectors.

By setting $\varsigma =-\kappa$, one has

$$\begin{array}{ccc}& & C_{\kappa }\left(0\right)G_1+S_{\kappa }\left(0\right)G_2+{\int }_{-\kappa }^0{S}_{\kappa }(-\kappa -s)f\left(s\right)ds=\vartheta (-\kappa ),\hfill \\ & & {\displaystyle \frac{d}{d\varsigma }}{C}_{\kappa }(\varsigma +\kappa ){|}_{\varsigma =-\kappa }{G}_1+{\displaystyle \frac{d}{d\varsigma }}{S}_{\kappa }(\varsigma +\kappa ){|}_{\varsigma =-\kappa }{G}_2+{\int }_{-\kappa }^0{\displaystyle \frac{d}{d\varsigma }}\left(S_{\kappa }(\varsigma -s)\right){|}_{\varsigma =-\kappa }f\left(s\right)ds\hfill \\ & & ={\displaystyle \frac{d}{d\varsigma }}\vartheta \left(\varsigma \right){|}_{\varsigma =-{\kappa }^{\prime }}\hfill \end{array}$$

thus, $G_1=\vartheta (-\kappa )$, $G_2={\vartheta }^{\prime }(-\kappa )$. By picking any $\varsigma \in [-\kappa ,0]$ and $s\in [\varsigma ,0]$, we have

$$\begin{array}{c}\hfill C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -s)f\left(s\right)ds=\vartheta \left(\varsigma \right),-\kappa \le \varsigma \le 0.\end{array}$$

(6)

For arbitrary $\varsigma \in [-\kappa ,0]$, one has

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{d^2}{d{\varsigma }^2}}\vartheta \left(\varsigma \right)\hfill \\ & =& {\displaystyle \frac{d^2}{d{\varsigma }^2}}\left(C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -s)f\left(s\right)ds\right)\hfill \\ & =& -A^2{C}_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )-A^2{S}_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\displaystyle \frac{d^2}{d{\varsigma }^2}}\left({\int }_{-\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -s)f\left(s\right)ds\right),\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d^2}{d{\varsigma }^2}}\left({\int }_{-\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -s)f\left(s\right)ds\right)& =& {\displaystyle \frac{d}{d\varsigma }}\left({\int }_{-\kappa }^{\varsigma }{\displaystyle \frac{d}{d\varsigma }}\left(S_{\kappa }(\varsigma -s)\right)f\left(s\right)ds+S_{\kappa }\left(0\right)f\left(\varsigma \right)\right)\hfill \\ & =& {\int }_{-\kappa }^{\varsigma }{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{(\varsigma -s)}^{2(♭-1)+1}}{\left(2\right(♭-1)+1)!}}f\left(s\right)ds+f\left(\varsigma \right)\hfill \\ & =& -A^2{\int }_{-\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -s)f\left(s\right)ds+f\left(\varsigma \right).\hfill \end{array}$$

(8)

By Equations (6)–(8), we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d^2}{d{\varsigma }^2}}\vartheta \left(\varsigma \right)& =& -A^2{C}_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )-A^2{S}_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )-A^2{\int }_{-\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -s)f\left(s\right)ds+f\left(\varsigma \right)\hfill \\ & =& -A^2\vartheta \left(\varsigma \right)+f\left(\varsigma \right),\hfill \end{array}$$

then, $f\left(\varsigma \right)={\vartheta }^{\prime \prime }\left(\varsigma \right)+A^2\vartheta \left(\varsigma \right)$. □

Theorem 2. 

The particular solution $\widehat{\mathrm{\Psi }}(·)\in C([-\kappa ,T],{\mathbb{R}}^n)$ to system (2) with $\widehat{\mathrm{\Psi }}\left(\varsigma \right)\equiv \mathbf{0}={(0,0,\cdots ,0)}^{\top }$, ${\widehat{\mathrm{\Psi }}}^{\prime }\left(\varsigma \right)\equiv \mathbf{0}$, $-\kappa \le \varsigma \le 0$, satisfies

$$\widehat{\mathrm{\Psi }}\left(\varsigma \right)={\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)q\left(s\right)ds.$$

Proof. 

The solution $\widehat{\mathrm{\Psi }}(·)$ to system (2) should satisfy the following form by using the formula for the variation of constants:

$$\widehat{\mathrm{\Psi }}\left(\varsigma \right)={\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds,$$

where $c(·)$ is an unknown function.

(i) For any $0<\varsigma \le \kappa$, by system (2), we have

$${\widehat{\mathrm{\Psi }}}^{\prime \prime }\left(\varsigma \right)=-A^2\widehat{\mathrm{\Psi }}\left(\varsigma \right)-B^2\widehat{\mathrm{\Psi }}(\varsigma -\kappa )+q\left(\varsigma \right)=-A^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds+q\left(\varsigma \right).$$

By Definition 1 and Lemma 2, one has

$$\begin{array}{ccc}\hfill {\widehat{\mathrm{\Psi }}}^{\prime \prime }\left(\varsigma \right)& =& {\displaystyle \frac{d^2}{d{\varsigma }^2}}\left({\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds\right)\hfill \\ & =& {\int }_0^{\varsigma }{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{(\varsigma -s)}^{2(♭-1)+1}}{\left(2\right(♭-1)+1)!}}c\left(s\right)ds+c\left(\varsigma \right)\hfill \\ & =& -A^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds+c\left(\varsigma \right).\hfill \end{array}$$

Thus, we obtain $c\left(\varsigma \right)=q\left(\varsigma \right)$.

(ii) For any $(\ell -1)\kappa <\varsigma \le \ell \kappa$ and $\ell =2,3,\cdots ,{\ell }^{*}$, we have

$$\begin{array}{ccc}\hfill {\widehat{\mathrm{\Psi }}}^{\prime \prime }\left(\varsigma \right)& =& -A^2\widehat{\mathrm{\Psi }}\left(\varsigma \right)-B^2\widehat{\mathrm{\Psi }}(\varsigma -\kappa )+q\left(\varsigma \right)\hfill \\ & =& -A^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds-B^2{\int }_0^{\varsigma -\kappa }{S}_{\kappa }(\varsigma -\kappa -s)c\left(s\right)ds+q\left(\varsigma \right).\hfill \end{array}$$

For any $0\le s\le \varsigma$, we have $-\kappa \le \varsigma -\kappa -s\le 0$ and $0\le \varsigma -\kappa -s\le \varsigma -\kappa$. By Definition 1 and Lemma 2, we have

$$\begin{array}{ccc}\hfill {\widehat{\mathrm{\Psi }}}^{\prime \prime }\left(\varsigma \right)& =& {\int }_0^{\varsigma }{\displaystyle \frac{d^2}{d{\varsigma }^2}}{S}_{\kappa }(\varsigma -s)c\left(s\right)ds+c\left(\varsigma \right)\hfill \\ & =& -A^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds-B^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -\kappa -s)c\left(s\right)ds+c\left(\varsigma \right)\hfill \\ & =& -A^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds-B^2({\int }_0^{\varsigma -\kappa }{S}_{\kappa }(\varsigma -\kappa -s)c\left(s\right)ds\hfill \\ & & +{\int }_{\varsigma -\kappa }^{\varsigma }{S}_{\kappa }(\varsigma -\kappa -s)c\left(s\right)ds)+c\left(\varsigma \right)\hfill \\ & =& -A^2{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)c\left(s\right)ds-B^2{\int }_0^{\varsigma -\kappa }{S}_{\kappa }(\varsigma -\kappa -s)c\left(s\right)ds+c\left(\varsigma \right),\hfill \end{array}$$

where $S_{\kappa }(\varsigma -\kappa -s)=0$ for $-\kappa \le \varsigma -\kappa -s\le 0$. Thus, we obtain $c\left(\varsigma \right)=q\left(\varsigma \right)$. □

By using the superposition principle to connect Theorems 1 and 2, we can obtain the following results.

Theorem 3. 

The solution to system (2) has the following form:

$$\begin{array}{ccc}\hfill \tilde{\mathrm{\Psi }}\left(\varsigma \right)& =& C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)q\left(s\right)ds.\hfill \end{array}$$

By combining this with Theorems 1–3, we obtain the representation of the solution to system (3).

Theorem 4. 

The solution $\mathrm{\Psi }\in PC([-\kappa ,T],{\mathbb{R}}^n)$ to system (3) with initial conditions $\mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right)$, $-\kappa \le \varsigma \le 0$, can be given by

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\varsigma \right)& =& C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)q\left(s\right)ds+\sum _{0<{\varsigma }_m<\varsigma }{C}_{\kappa }(\varsigma -{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right).\hfill \end{array}$$

Proof. 

By picking any $\varsigma \in \left(\right(\ell -1)\kappa ,\ell \kappa ]$, $\ell \in {\mathbb{N}}^{+}$ and ${\varsigma }_m\in (0,\varsigma )$, by Lemma 2 and Theorem 3, one has

$$\begin{array}{ccc}\hfill {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)& =& {\displaystyle \frac{d^2}{d{\varsigma }^2}}(C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)q\left(s\right)ds+\sum _{0<{\varsigma }_m<\varsigma }{C}_{\kappa }(\varsigma -{\varsigma }_m)\left(D_m\mathrm{\Psi }\left({\varsigma }_m\right)\right))\hfill \\ & =& -A^2\tilde{\mathrm{\Psi }}\left(\varsigma \right)-B^2\tilde{\mathrm{\Psi }}(\varsigma -\kappa )+q\left(\varsigma \right)+(-A^2\sum _{0<{\varsigma }_m<\varsigma }{C}_{\kappa }(\varsigma -{\varsigma }_m)\left(D_m\mathrm{\Psi }\left({\varsigma }_m\right)\right)\hfill \\ & & -B^2\sum _{0<{\varsigma }_m<\varsigma -\kappa }{C}_{\kappa }(\varsigma -\kappa -{\varsigma }_m)\left(D_m\mathrm{\Psi }\left({\varsigma }_m\right)\right))\hfill \\ & =& -A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa )+q\left(\varsigma \right).\hfill \end{array}$$

Let ${\varsigma }_m\in (0,\varsigma )$ and $m=1,2,\cdots ,r(T,0)$; we have

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left({\varsigma }_m^{+}\right)& =& C_{\kappa }({\varsigma }_m^{+}+\kappa )\vartheta (-\kappa )+S_{\kappa }({\varsigma }_m^{+}+\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }({\varsigma }_m^{+}-s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{{\varsigma }_m^{+}}{S}_{\kappa }({\varsigma }_m^{+}-s)q\left(s\right)ds+\sum _{0<{\varsigma }_h<{\varsigma }_m^{+}}{C}_{\kappa }({\varsigma }_m^{+}-{\varsigma }_h)\left(D_h\mathrm{\Psi }\left({\varsigma }_h\right)\right)\hfill \\ & =& C_{\kappa }({\varsigma }_m^{+}+\kappa )\vartheta (-\kappa )+S_{\kappa }({\varsigma }_m^{+}+\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }({\varsigma }_m^{+}-s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{{\varsigma }_m^{+}}{S}_{\kappa }({\varsigma }_m^{+}-s)q\left(s\right)ds+\sum _{0<{\varsigma }_h<{\varsigma }_m^{-}}{C}_{\kappa }({\varsigma }_m^{-}-{\varsigma }_h)\left(D_h\mathrm{\Psi }\left({\varsigma }_h\right)\right)+D_m\mathrm{\Psi }\left({\varsigma }_m\right),\hfill \\ \hfill \mathrm{\Psi }\left({\varsigma }_m^{-}\right)& =& C_{\kappa }({\varsigma }_m^{-}+\kappa )\vartheta (-\kappa )+S_{\kappa }({\varsigma }_m^{-}+\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }({\varsigma }_m^{-}-s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{{\varsigma }_m^{-}}{S}_{\kappa }({\varsigma }_m^{-}-s)q\left(s\right)ds+\sum _{0<{\varsigma }_h<{\varsigma }_m^{-}}{C}_{\kappa }({\varsigma }_m^{-}-{\varsigma }_h)\left(D_h\mathrm{\Psi }\left({\varsigma }_h\right)\right),\hfill \end{array}$$

where $C_{\kappa }\left({\varsigma }_m^{+}\right)=C_{\kappa }\left({\varsigma }_m^{-}\right)=C_{\kappa }\left({\varsigma }_m\right)$ and $S_{\kappa }\left({\varsigma }_m^{+}\right)=S_{\kappa }\left({\varsigma }_m^{-}\right)=S_{\kappa }\left({\varsigma }_m\right)$, which implies that $\mathrm{\Delta }\mathrm{\Psi }\left({\varsigma }_m\right)=D_m\mathrm{\Psi }\left({\varsigma }_m\right)$. □

4. Existence and Uniqueness of Solution to System (4)

In this section, the aim of our work is to establish conditions guaranteeing the existence and uniqueness of the solution to system (4).

According to Theorems 2 and 4, the solution $\mathrm{\Psi }(·)$ to system (4) is as follows:

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\varsigma \right)& =& C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)q(s,\mathrm{\Psi }\left(s\right))ds+\sum _{0<{\varsigma }_m<\varsigma }{C}_{\kappa }(\varsigma -{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right).\hfill \end{array}$$

(9)

Let $\varsigma \in \left(\right(\ell -1)\kappa ,\ell \kappa ]$ and $\ell \in {\mathbb{N}}^{+}$; we define

$$\begin{array}{ccc}\hfill {{\rm Y}}_1\left(\varsigma \right)& =& {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}},\hfill \\ \hfill {{\rm Y}}_2\left(\varsigma \right)& =& {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{(\varsigma +\kappa )}^{2♭}}{(2♭)!}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {{\rm Y}}_3\left(\varsigma \right)& =& {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{{(\varsigma +\kappa )}^{2♭+1}}{(2♭+1)!}},\hfill \\ \hfill {{\rm Y}}_4\left(\varsigma \right)& =& {\displaystyle \sum _{♭=0}^{\infty }}{\displaystyle \frac{(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}}{(2♭+2)!}}\left({\varsigma }^{2♭+2}-{(\varsigma -(\ell -1)\kappa )}^{2♭+2}\right).\hfill \end{array}$$

We give the following assumptions:

$\left[K_1\right]$ There exist $L,N>0$ such that $\parallel q(\varsigma ,\mathrm{\Psi })\parallel \le L\parallel \mathrm{\Psi }\parallel +N$, $\forall \varsigma \in J$ and $\mathrm{\Psi }\in {\mathbb{R}}^n$.

$\left[K_2\right]$ Let $\varrho =LT{{\rm Y}}_1\left(T\right)<1$.

$\left[K_3\right]$ There exists $\tilde{L}>0$ such that $\parallel q(\varsigma ,\mathrm{\Psi })-q(\varsigma ,\tilde{\mathrm{\Psi }})\parallel \le \tilde{L}\parallel \mathrm{\Psi }-\tilde{\mathrm{\Psi }}\parallel$, $\forall \varsigma \in J$ and $\mathrm{\Psi },\tilde{\mathrm{\Psi }}\in {\mathbb{R}}^n$.

Theorem 5. 

System (4) has at least one solution $\mathrm{\Psi }\in PC([-\kappa ,T],{\mathbb{R}}^n)$ if $\left[K_1\right],\left[K_2\right]$ and $\left[K_3\right]$ hold.

Proof. 

Define an operator $\mathcal{F}:{\mathfrak{B}}_{\gamma }\to {\mathfrak{B}}_{\gamma }$ as

$$\begin{array}{ccc}\hfill \left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)& =& C_{\kappa }(\varsigma +\kappa )\vartheta (-\kappa )+S_{\kappa }(\varsigma +\kappa ){\vartheta }^{\prime }(-\kappa )+{\int }_{-\kappa }^0{S}_{\kappa }(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{\varsigma }{S}_{\kappa }(\varsigma -s)q(s,\mathrm{\Psi }\left(s\right))ds+\sum _{0<{\varsigma }_m<\varsigma }{C}_{\kappa }(\varsigma -{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right),\hfill \end{array}$$

(10)

where ${\mathfrak{B}}_{\gamma }=\{\mathrm{\Psi }\in PC([-\kappa ,T],{\mathbb{R}}^n),\parallel \mathrm{\Psi }{\parallel }_{PC}\le \gamma and\gamma >{\displaystyle \frac{\zeta }{1-\varrho }}\}$, and $\zeta ={{\rm Y}}_2\left(T\right)\parallel \vartheta (-\kappa )\parallel +{{\rm Y}}_3\left(T\right)\parallel {\vartheta }^{\prime }(-\kappa )\parallel +{{\rm Y}}_4\left(T\right)(\parallel {\vartheta }^{\prime \prime }{\parallel }_C+\parallel A^2{\parallel \parallel \vartheta \parallel }_C)+NT{{\rm Y}}_1\left(T\right)+{\displaystyle \sum _{0<{\varsigma }_m<T}}\parallel C_{\kappa }(T-{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel$. According to the definition of operator $\mathcal{F}$, the solution to system (4) is equivalent to the fixed point of $\mathcal{F}$.

We divide the whole proof into three steps.

Firstly, we show that $\mathcal{F}\left({\mathfrak{B}}_{\gamma }\right)\subset {\mathfrak{B}}_{\gamma }$. For any $\mathrm{\Psi }\in {\mathfrak{B}}_{\gamma }$, one has

$$\begin{array}{ccc}\hfill \parallel \left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)\parallel & \le & \parallel C_{\kappa }(\varsigma +\kappa )\parallel \parallel \vartheta (-\kappa )\parallel +{\int }_{-\kappa }^0\parallel S_{\kappa }(\varsigma -s)\parallel (\parallel {\vartheta }^{\prime \prime }\left(s\right)\parallel +\parallel A^2\parallel \parallel \vartheta \left(s\right)\parallel )ds\hfill \\ & & +\parallel S_{\kappa }(\varsigma +\kappa )\parallel \parallel {\vartheta }^{\prime }(-\kappa )\parallel +{\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma -s)\parallel \parallel q(s,\mathrm{\Psi }\left(s\right))\parallel ds\hfill \\ & & +\parallel \sum _{0<{\varsigma }_m<\varsigma }{C}_{\kappa }(\varsigma -{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel \hfill \\ & \le & {{\rm Y}}_2\left(\varsigma \right)\parallel \vartheta (-\kappa )\parallel +{{\rm Y}}_3\left(\varsigma \right)\parallel {\vartheta }^{\prime }(-\kappa )\parallel +{{\rm Y}}_4\left(\varsigma \right)(\parallel {\vartheta }^{\prime \prime }{\parallel }_C+\parallel A^2{\parallel \parallel \vartheta \parallel }_C)\hfill \\ & & +{\int }_0^{\varsigma }\parallel S_{\kappa }{(\varsigma -s)\parallel (L\parallel \mathrm{\Psi }\parallel }_{PC}+N)ds+\sum _{0<{\varsigma }_m<\varsigma }\parallel C_{\kappa }(\varsigma -{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel \hfill \\ & \le & {{\rm Y}}_2\left(T\right)\parallel \vartheta (-\kappa )\parallel +{{\rm Y}}_3\left(T\right)\parallel {\vartheta }^{\prime }(-\kappa )\parallel +{{\rm Y}}_4\left(T\right)(\parallel {\vartheta }^{\prime \prime }{\parallel }_C+\parallel A^2{\parallel \parallel \vartheta \parallel }_C)\hfill \\ & & +{{\rm Y}}_1{\left(T\right)T(L\parallel \mathrm{\Psi }\parallel }_{PC}+N)+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel \hfill \\ & \le & \zeta +\varrho \gamma <\gamma ,\hfill \end{array}$$

then, $\mathcal{F}\left({\mathfrak{B}}_{\gamma }\right)\subset {\mathfrak{B}}_{\gamma }$.

Secondly, We prove that $\mathcal{F}$ is continuous. Let ${\left\{{\mathrm{\Psi }}_n(·)\right\}}_{n=1}^{\infty }$ in ${\mathfrak{B}}_{\gamma }$ such that ${\mathrm{\Psi }}_n(·)\to \mathrm{\Psi }(·)(n\to \infty )$, $q_n(·)=q(·,{\mathrm{\Psi }}_n(·))$, and $q(·)=q(·,\mathrm{\Psi }(·\left)\right)$. For any $\varsigma \in J$, one has

$$\begin{array}{ccc}& & \parallel \left(\mathcal{F}{\mathrm{\Psi }}_n\right)\left(\varsigma \right)-\left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)\parallel \hfill \\ & \le & {\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma -s)\parallel \parallel q(s,{\mathrm{\Psi }}_n\left(s\right))-q(s,\mathrm{\Psi }\left(s\right))\parallel ds\hfill \\ & & +\sum _{0<{\varsigma }_m<\varsigma }\parallel C_{\kappa }(\varsigma -{\varsigma }_m)D_m\left({\mathrm{\Psi }}_n\left({\varsigma }_m\right)-\mathrm{\Psi }\left({\varsigma }_m\right)\right)\parallel \hfill \\ & \le & \tilde{L}T{{\rm Y}}_1\left(T\right)\parallel {\mathrm{\Psi }}_n{-\mathrm{\Psi }\parallel }_{PC}+\sum _{0<{\varsigma }_m<\varsigma }\parallel C_{\kappa }(\varsigma -{\varsigma }_m)D_m\left({\mathrm{\Psi }}_n\left({\varsigma }_m\right)-\mathrm{\Psi }\left({\varsigma }_m\right)\right)\parallel \hfill \\ & \le & \tilde{L}T{{\rm Y}}_1\left(T\right)\parallel {\mathrm{\Psi }}_n{-\mathrm{\Psi }\parallel }_{PC}+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)\parallel \parallel D_m\parallel \parallel {\mathrm{\Psi }}_n{-\mathrm{\Psi }\parallel }_{PC},\hfill \end{array}$$

which yields

$$\begin{array}{c}\hfill \parallel \left(\mathcal{F}{\mathrm{\Psi }}_n\right)-\left(\mathcal{F}\mathrm{\Psi }\right){\parallel }_{PC}\le \left(\tilde{L}T{{\rm Y}}_1\left(T\right)+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)\parallel \parallel D_m\parallel \right){\parallel {\mathrm{\Psi }}_n-\mathrm{\Psi }\parallel }_{PC}.\end{array}$$

Thus, the operator $\mathcal{F}$ is continuous.

Finally, we show that $\mathcal{F}$ is relatively compact on ${\mathfrak{B}}_{\gamma }$. For any $\mathrm{\Psi }\in {\mathfrak{B}}_{\gamma }$, $0<(\ell -1)\kappa <\varsigma <\varsigma +\mathrm{\Delta }\varsigma \le \ell \kappa \le T$, and $\mathrm{\Delta }\varsigma \to 0$, we have

$$\begin{array}{ccc}& & \parallel \left(\mathcal{F}\mathrm{\Psi }\right)(\varsigma +\mathrm{\Delta }\varsigma )-\left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)\parallel \hfill \\ & \le & \parallel C_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma +\kappa )-C_{\kappa }(\varsigma +\kappa )\parallel \parallel \vartheta (-\kappa )\parallel +\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma +\kappa )-S_{\kappa }(\varsigma +\kappa )\parallel \parallel {\vartheta }^{\prime }(-\kappa )\parallel \hfill \\ & & +{\int }_{-\kappa }^0\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)-S_{\kappa }(\varsigma -s)\parallel (\parallel {\vartheta }^{\prime \prime }\left(s\right)\parallel +\parallel A^2\parallel \parallel \vartheta \left(s\right)\parallel )ds\hfill \\ & & +{\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)-S_{\kappa }(\varsigma -s)\parallel \parallel q(s,\mathrm{\Psi }\left(s\right))\parallel ds\hfill \\ & & +\sum _{0<{\varsigma }_m<\varsigma }\parallel C_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -{\varsigma }_m)-C_{\kappa }(\varsigma -{\varsigma }_m)\parallel \parallel D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel \hfill \\ & & +{\int }_{\varsigma }^{\varsigma +\mathrm{\Delta }\varsigma }\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)\parallel \parallel q(s,\mathrm{\Psi }\left(s\right))\parallel ds\hfill \\ & \le & {\mathcal{M}}_1+{\mathcal{M}}_2+{\mathcal{M}}_3+{\mathcal{M}}_4+{\mathcal{M}}_5,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\mathcal{M}}_1& =& \parallel C_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma +\kappa )-C_{\kappa }(\varsigma +\kappa )\parallel \parallel \vartheta (-\kappa )\parallel +\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma +\kappa )-S_{\kappa }(\varsigma +\kappa )\parallel \parallel {\vartheta }^{\prime }(-\kappa )\parallel ,\hfill \\ \hfill {\mathcal{M}}_2& =& {\int }_{-\kappa }^0\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)-S_{\kappa }(\varsigma -s)\parallel (\parallel {\vartheta }^{\prime \prime }\left(s\right)\parallel +\parallel A^2\parallel \parallel \vartheta \left(s\right)\parallel )ds\hfill \\ & & \le (\parallel {\vartheta }^{\prime \prime }{\parallel }_C+\parallel A^2{\parallel \parallel \vartheta \parallel }_C){\int }_{-\kappa }^0\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)-S_{\kappa }(\varsigma -s)\parallel ds,\hfill \\ \hfill {\mathcal{M}}_3& =& {\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)-S_{\kappa }(\varsigma -s)\parallel \parallel q(s,\mathrm{\Psi }\left(s\right))\parallel ds\hfill \\ & & \le {(L\parallel \mathrm{\Psi }\parallel }_{PC}+N){\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)-S_{\kappa }(\varsigma -s)\parallel ds,\hfill \\ \hfill {\mathcal{M}}_4& =& \sum _{0<{\varsigma }_m<\varsigma }\parallel C_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -{\varsigma }_m)-C_{\kappa }(\varsigma -{\varsigma }_m)\parallel \parallel D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel ,\hfill \\ \hfill {\mathcal{M}}_5& =& {\int }_{\varsigma }^{\varsigma +\mathrm{\Delta }\varsigma }\parallel S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)\parallel \parallel q(s,\mathrm{\Psi }\left(s\right))\parallel ds.\hfill \end{array}$$

Let $(\ell -1)\kappa <\varsigma <\varsigma +\mathrm{\Delta }\varsigma \le \ell \kappa$ as $\mathrm{\Delta }\varsigma \to 0$; we obtain

$$\begin{array}{c}\hfill C_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma +\kappa )\to C_{\kappa }(\varsigma +\kappa ),S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma +\kappa )\to S_{\kappa }(\varsigma +\kappa ),S_{\kappa }(\varsigma +\mathrm{\Delta }\varsigma -s)\to S_{\kappa }(\varsigma -s),\end{array}$$

which yields ${\mathcal{M}}_1\to 0$, ${\mathcal{M}}_2\to 0$, ${\mathcal{M}}_3\to 0$, and ${\mathcal{M}}_4\to 0$ as $\mathrm{\Delta }\varsigma \to 0$. For ${\mathcal{M}}_5$, one has

$$\begin{array}{ccc}\hfill {\mathcal{M}}_5& \le & \parallel S_{\kappa }{(\varsigma +\mathrm{\Delta }\varsigma -s)\parallel (L\parallel \mathrm{\Psi }\parallel }_{PC}+N){\int }_{\varsigma }^{\varsigma +\mathrm{\Delta }\varsigma }ds\hfill \\ & \le & \parallel S_{\kappa }{(\varsigma +\mathrm{\Delta }\varsigma -s)\parallel (L\parallel \mathrm{\Psi }\parallel }_{PC}+N)\mathrm{\Delta }\varsigma \to 0as\mathrm{\Delta }\varsigma \to 0,\hfill \end{array}$$

which implies that $\parallel \left(\mathcal{F}\mathrm{\Psi }\right)(\varsigma +\mathrm{\Delta }\varsigma )-\left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)\parallel \to 0$ as $\mathrm{\Delta }\varsigma \to 0$. Hence, $\mathcal{F}$ is relatively compact on ${\mathfrak{B}}_{\gamma }$ by the Arzela–Ascoli Lemma.

By using the Schauder fixed-point theorem, the operator $\mathcal{F}$ has at least a fixed point which is a solution to system (4). Hence, system (4) has at least one solution. □

$\left[K_4\right]$ Let $\tilde{\varrho }=\tilde{L}T{{\rm Y}}_1\left(T\right)+{\displaystyle \sum _{0<{\varsigma }_m<T}}\parallel C_{\kappa }(T-{\varsigma }_m)\parallel \parallel D_m\parallel <1$.

Theorem 6. 

System (4) has a unique solution $\mathrm{\Psi }\in PC([-\kappa ,T],{\mathbb{R}}^n)$ if $\left[K_3\right]$ and $\left[K_4\right]$ hold.

Proof. 

By using the same ideas and method as in Theorem 5, it is easy to prove that $\mathcal{F}:{\mathfrak{B}}_{\tilde{\gamma }}\to {\mathfrak{B}}_{\tilde{\gamma }}$ defined in Equation (10) is uniformly bounded. Next, we check that $\mathcal{F}$ is a contraction operator. According to the definition of operator $\mathcal{F}$, the solution to system (4) is equivalent to the fixed point of $\mathcal{F}$. For any $\mathrm{\Psi },\tilde{\mathrm{\Psi }}\in {\mathfrak{B}}_{\tilde{\gamma }}$, where ${\mathfrak{B}}_{\tilde{\gamma }}=\{\mathrm{\Psi }\in PC([-\kappa ,T],{\mathrm{R}}^n){,\parallel \mathrm{\Psi }\parallel }_{PC}\le \tilde{\gamma }\mathrm{and}\tilde{\gamma }>{\displaystyle \frac{\tilde{\zeta }}{1-\tilde{\varrho }}}\}$, $\tilde{\zeta }={{\rm Y}}_2\left(T\right)\parallel \vartheta (-\kappa )\parallel +{{\rm Y}}_3\left(T\right)\parallel {\vartheta }^{\prime }(-\kappa )\parallel +{{\rm Y}}_4\left(T\right)(\parallel {\vartheta }^{\prime \prime }{\parallel }_C+\parallel A^2{\parallel \parallel \vartheta \parallel }_C)+T{{\rm Y}}_1\left(T\right)\tilde{q}+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right)\parallel$ and $\tilde{q}=\underset{\varsigma \in J}{sup}\parallel q(\varsigma ,\mathbf{0})\parallel$.

For any $\varsigma \in [-\kappa ,T]$, one has

$$\begin{array}{cc}\hfill & \parallel \left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)-\mathcal{F}\tilde{\mathrm{\Psi }}\left(\varsigma \right)\parallel \hfill \\ \hfill & \le {\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma -s)\parallel \parallel q(s,\mathrm{\Psi }\left(s\right))-q(s,\tilde{\mathrm{\Psi }}\left(s\right))\parallel ds+\sum _{0<{\varsigma }_m<\varsigma }\parallel C_{\kappa }(\varsigma -{\varsigma }_m)D_m\left(\mathrm{\Psi }\left({\varsigma }_m\right)-\tilde{\mathrm{\Psi }}\left({\varsigma }_m\right)\right)\parallel \hfill \\ \hfill & \le \tilde{L}\parallel \mathrm{\Psi }-\tilde{\mathrm{\Psi }}{\parallel }_{PC}{\int }_0^{\varsigma }\parallel S_{\kappa }(\varsigma -s)\parallel ds+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)D_m\parallel \parallel \mathrm{\Psi }-\tilde{\mathrm{\Psi }}{\parallel }_{PC}\hfill \\ \hfill & \le \left(\tilde{L}T{{\rm Y}}_1\left(T\right)+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)\parallel \parallel D_m\parallel \right){\parallel \mathrm{\Psi }-\tilde{\mathrm{\Psi }}\parallel }_{PC},\hfill \end{array}$$

which implies that

$$\parallel \left(\mathcal{F}\mathrm{\Psi }\right)\left(\varsigma \right)-\mathcal{F}\tilde{\mathrm{\Psi }}\left(\varsigma \right)\parallel \le \left(\tilde{L}T{{\rm Y}}_1\left(T\right)+\sum _{0<{\varsigma }_m<T}\parallel C_{\kappa }(T-{\varsigma }_m)\parallel \parallel D_m\parallel \right){\parallel \mathrm{\Psi }-\tilde{\mathrm{\Psi }}\parallel }_{PC}.$$

Assumption $\left[K_4\right]$ of this theorem guarantees that $\mathcal{F}$ is a contraction operator. By the Banach fixed-point theorem, $\mathcal{F}$ has a unique fixed point which is a solution to system (4). □

5. Examples

Example 1. 

Let $T=1.5$, $\kappa =0.3$, ${\ell }^{*}=5$, $r(T,0)=4$, ${\varsigma }_m=0.3m$, and $m=1,2,3,4$. Consider

$$\left\{\begin{array}{cc}\hfill & {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)=-A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa )+q\left(\varsigma \right),\varsigma \ne {\varsigma }_m,\varsigma \in [0,1.5],\hfill \\ \hfill & \mathrm{\Delta }\mathrm{\Psi }\left({\varsigma }_m\right)=D_m\mathrm{\Psi }\left({\varsigma }_m\right),\varsigma ={\varsigma }_m,m=1,2,3,4,\hfill \\ \hfill & \mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\varsigma \in [-0.3,0],\hfill \end{array}\right.$$

(11)

where

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\varsigma \right)& =& \left(\begin{array}{c}{\mathrm{\Psi }}_1\left(\varsigma \right)\\ {\mathrm{\Psi }}_2\left(\varsigma \right)\end{array}\right),\vartheta \left(\varsigma \right)=\left(\begin{array}{c}{\varsigma }^2+\varsigma \\ {\varsigma }^3+\varsigma \end{array}\right),A=\left(\begin{array}{cc}0.2& 0.4\\ 0& 0.2\end{array}\right),\hfill \\ \hfill B& =& \left(\begin{array}{cc}0.5& 0.5\\ 0& 0.5\end{array}\right),D_m=\left(\begin{array}{cc}{\displaystyle \frac{m}{50}}& 0\\ 0& {\displaystyle \frac{m}{60}}\end{array}\right),q\left(\varsigma \right)=\left(\begin{array}{c}\varsigma \\ {\varsigma }^2\end{array}\right).\hfill \end{array}$$

According to Theorem 4, the solution $\mathrm{\Psi }\in PC([-0.3,1.5],{\mathbb{R}}^2)$ to system (11) can be given by

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\varsigma \right)& =& C_{0.3}(\varsigma +0.3)\vartheta (-0.3)+S_{0.3}(\varsigma +0.3){\vartheta }^{\prime }(-0.3)+{\int }_{-0.3}^0{S}_{0.3}(\varsigma -s)({\vartheta }^{\prime \prime }\left(s\right)+A^2\vartheta \left(s\right))ds\hfill \\ & & +{\int }_0^{\varsigma }{S}_{0.3}(\varsigma -s)q\left(s\right)ds+\sum _{0<{\varsigma }_m<\varsigma }{C}_{0.3}(\varsigma -{\varsigma }_m)D_m\mathrm{\Psi }\left({\varsigma }_m\right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill S_{0.3}\left(\varsigma \right)=\left\{\begin{array}{c}\varsigma I+{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}},\varsigma \in (0,0.3],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)+1}}{\left(2\right(♭+1)+1)!}},\varsigma \in (0.3,0.6],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)+1}}{\left(2\right(♭+1)+1)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+2}{C}_{♭+2}^{♭}{A}^{2♭}{B}^4{\displaystyle \frac{{(\varsigma -0.6)}^{2(♭+2)+1}}{\left(2\right(♭+2)+1)!}},\varsigma \in (0.6,0.9],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)+1}}{\left(2\right(♭+1)+1)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+2}{C}_{♭+2}^{♭}{A}^{2♭}{B}^4{\displaystyle \frac{{(\varsigma -0.6)}^{2(♭+2)+1}}{\left(2\right(♭+2)+1)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+3}{C}_{♭+3}^{♭}{A}^{2♭}{B}^6{\displaystyle \frac{{(\varsigma -0.9)}^{2(♭+3)+1}}{\left(2\right(♭+3)+1)!}},\varsigma \in (0.9,1.2],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭+1}}{(2♭+1)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)+1}}{\left(2\right(♭+1)+1)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+2}{C}_{♭+2}^{♭}{A}^{2♭}{B}^4{\displaystyle \frac{{(\varsigma -0.6)}^{2(♭+2)+1}}{\left(2\right(♭+2)+1)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+3}{C}_{♭+3}^{♭}{A}^{2♭}{B}^6{\displaystyle \frac{{(\varsigma -0.9)}^{2(♭+3)+1}}{\left(2\right(♭+3)+1)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+4}{C}_{♭+4}^{♭}{A}^{2♭}{B}^8{\displaystyle \frac{{(\varsigma -1.2)}^{2(♭+4)+1}}{\left(2\right(♭+4)+1)!}},\varsigma \in (1.2,1.5],\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill C_{0.3}\left(\varsigma \right)=\left\{\begin{array}{c}I+{\displaystyle \sum _{♭=1}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}},\varsigma \in (0,0.3],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)}}{\left(2\right(♭+1\left)\right)!}},\varsigma \in (0.3,0.6],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)}}{\left(2\right(♭+1\left)\right)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+2}{C}_{♭+2}^{♭}{A}^{2♭}{B}^4{\displaystyle \frac{{(\varsigma -0.6)}^{2(♭+2)}}{\left(2\right(♭+2\left)\right)!}},\varsigma \in (0.6,0.9],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)}}{\left(2\right(♭+1\left)\right)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+2}{C}_{♭+2}^{♭}{A}^{2♭}{B}^4{\displaystyle \frac{{(\varsigma -0.6)}^{2(♭+2)}}{\left(2\right(♭+2\left)\right)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+3}{C}_{♭+3}^{♭}{A}^{2♭}{B}^6{\displaystyle \frac{{(\varsigma -0.9)}^{2(♭+3)}}{\left(2\right(♭+3\left)\right)!}},\varsigma \in (0.9,1.2],\hfill \\ {\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭}{A}^{2♭}{\displaystyle \frac{{\varsigma }^{2♭}}{(2♭)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+1}{C}_{♭+1}^{♭}{A}^{2♭}{B}^2{\displaystyle \frac{{(\varsigma -0.3)}^{2(♭+1)}}{\left(2\right(♭+1\left)\right)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+2}{C}_{♭+2}^{♭}{A}^{2♭}{B}^4{\displaystyle \frac{{(\varsigma -0.6)}^{2(♭+2)}}{\left(2\right(♭+2\left)\right)!}}+{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+3}{C}_{♭+3}^{♭}{A}^{2♭}{B}^6{\displaystyle \frac{{(\varsigma -0.9)}^{2(♭+3)}}{\left(2\right(♭+3\left)\right)!}}\hfill \\ +{\displaystyle \sum _{♭=0}^{\infty }}{(-1)}^{♭+4}{C}_{♭+4}^{♭}{A}^{2♭}{B}^8{\displaystyle \frac{{(\varsigma -1.2)}^{2(♭+4)}}{\left(2\right(♭+4\left)\right)!}},\varsigma \in (1.2,1.5].\hfill \end{array}\right.\end{array}$$

Example 2. 

Let $\kappa =0.3$, ${\ell }^{*}=5$, $T=1.5$, $r(T,0)=4$, ${\varsigma }_m=0.3m$, and $m=1,2,3,4$. Consider

$$\left\{\begin{array}{cc}\hfill & {\mathrm{\Psi }}^{\prime \prime }\left(\varsigma \right)=-A^2\mathrm{\Psi }\left(\varsigma \right)-B^2\mathrm{\Psi }(\varsigma -\kappa )+q(\varsigma ,\mathrm{\Psi }\left(\varsigma \right)),\varsigma \ne {\varsigma }_m,\varsigma \in [0,1.5],\hfill \\ \hfill & \mathrm{\Delta }\mathrm{\Psi }\left({\varsigma }_m\right)=D_m\mathrm{\Psi }\left({\varsigma }_m\right),\varsigma ={\varsigma }_m,m=1,2,3,4,\hfill \\ \hfill & \mathrm{\Psi }\left(\varsigma \right)=\vartheta \left(\varsigma \right),{\mathrm{\Psi }}^{\prime }\left(\varsigma \right)={\vartheta }^{\prime }\left(\varsigma \right),\varsigma \in [-0.3,0],\hfill \end{array}\right.$$

(12)

where $A,B$, $D_m$, and $\vartheta \left(\varsigma \right)$ are defined in Example 1 and $q(\varsigma ,\mathrm{\Psi }\left(\varsigma \right))={\left({\displaystyle \frac{\varsigma }{10}}{\mathrm{\Psi }}_1\left(\varsigma \right)+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\varsigma }{10}}{\mathrm{\Psi }}_2\left(\varsigma \right)+{\displaystyle \frac{1}{2}}\right)}^{\top }$.

Let $\varsigma \in [0,1.5]$ and $\mathrm{\Psi },\tilde{\mathrm{\Psi }}\in {\mathbb{R}}^2$; one has

$$\begin{array}{ccc}\hfill \parallel q(\varsigma ,\mathrm{\Psi }(\varsigma \left)\right)\parallel & \le & \left({\displaystyle \frac{\varsigma }{10}}\right|{\mathrm{\Psi }}_1\left(\varsigma \right)|+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{\varsigma }{10}}|{\mathrm{\Psi }}_2\left(\varsigma \right)|+{\displaystyle \frac{1}{2}})\hfill \\ & \le & {\displaystyle \frac{\varsigma }{10}}\left(\right|{\mathrm{\Psi }}_1\left(\varsigma \right)|+|{\mathrm{\Psi }}_2\left(\varsigma \right)\left|\right)+1\le {\displaystyle \frac{3}{20}}{\parallel \mathrm{\Psi }\parallel }_{PC}+1,\hfill \\ \hfill \parallel q(\varsigma ,\mathrm{\Psi }\left(\varsigma \right))-q(\varsigma ,\tilde{\mathrm{\Psi }}\left(\varsigma \right))\parallel & \le & {\displaystyle \frac{\varsigma }{10}}\left(\right|{\mathrm{\Psi }}_1\left(\varsigma \right)-{\tilde{\mathrm{\Psi }}}_1\left(\varsigma \right)|+|{\mathrm{\Psi }}_2\left(\varsigma \right)-{\tilde{\mathrm{\Psi }}}_2\left(\varsigma \right)\left|\right)\hfill \\ & \le & {\displaystyle \frac{3}{20}}{\parallel \mathrm{\Psi }-\tilde{\mathrm{\Psi }}\parallel }_{PC}.\hfill \end{array}$$

By numerical calculation, one has $\parallel A^2\parallel =0.2$, $\parallel B^2\parallel =0.75$, $L=\tilde{L}={\displaystyle \frac{3}{20}}$,

$$\begin{array}{ccc}\hfill {{\rm Y}}_1\left(T\right)& =& {\displaystyle \sum _{♭=0}^{\infty }}(\parallel A^2\parallel +\parallel B^2{\parallel )}^{♭}{\displaystyle \frac{T^{2♭+1}}{(2♭+1)!}}=2.0945;\hfill \\ \hfill \sum _{0<{\varsigma }_m<T}\parallel C_{0.3}(T-{\varsigma }_m)\parallel \parallel D_m\parallel & =& \parallel C_{0.3}\left(1.2\right)\parallel \parallel D_1\parallel +\parallel C_{0.3}\left(0.9\right)\parallel \parallel D_2\parallel \hfill \\ & & +\parallel C_{0.3}\left(0.6\right)\parallel \parallel D_3\parallel +\parallel C_{0.3}\left(0.3\right)\parallel \parallel D_4\parallel \hfill \\ & =& 0.2457;\hfill \end{array}$$

thus,

$$\begin{array}{ccc}\hfill \varrho & =& LT{{\rm Y}}_1\left(T\right)=0.4713<1,\hfill \\ \hfill \tilde{\varrho }& =& \tilde{L}T{{\rm Y}}_1\left(T\right)+\sum _{0<{\varsigma }_m<T}\parallel C_{0.3}(T-{\varsigma }_m)\parallel \parallel D_m\parallel =0.7170<1,\hfill \end{array}$$

Hence, $\left[K_1\right],\left[K_2\right],\left[K_3\right]$, and $\left[K_4\right]$ are satisfied. By Theorems 5 and 6, the existence and uniqueness of the solution to system (12) can be guaranteed.

6. Conclusions

In this paper, we first derive the exact solution of the second-order delay differential system by using the sine-type and cosine-type delay matrix functions. Next, considering the effect of impulses on the system, the solution of the second-order impulsive delay differential system is derived. Then, we use the fixed-point theorem to prove the existence and uniqueness of the solution to the nonlinear impulsive delay differential system. Finally, an example is given to illustrate the effectiveness of our results. In the future, we will consider constructing the fundamental solution matrix of second-order impulsive delay differential systems to derive the exact solution expression of second-order impulsive delay differential systems, as well as its related properties, such as stability and controllability.