Euler Method for a Class of Linear Impulsive Neutral Differential Equations

Abstract

This paper presents a new numerical scheme for a class of linear impulsive neutral differential equations with constant coefficients based on the Euler method. We rigorously establish the first-order convergence of the proposed numerical approach. Additionally, the asymptotical stability of the exact solutions and numerical solutions of impulsive neutral differential equations are studied. To substantiate our findings, two illustrative examples are provided, demonstrating the theoretical conclusions of this paper.

1. Introduction

Impulsive delay differential equations (IDDEs) have garnered significant attention due to their applicability across various domains, such as in neural networks [1,2,3], dynamics [4], control theory [5,6], engineering [7], etc. In particular, the theoretical exploration of impulsive neutral delay differential equations (INDDEs) has been enriched by numerous researchers, focusing on aspects like existence [8,9,10,11,12], oscillation [13,14,15], and stability: in [16], Xiaodi Li et al. apply the Razumikhin method for impulsive functional differential equations of the neutral type to analyze the stability; Bainov Drumi Dimitrov et al., in [17], studied the uniform asymptotic stability of impulsive differential-difference equations of the neutral type via Lyapunov’s direct method; and Refs. [18,19] discussed the exponential stability of INDDEs.

In general, it is difficult or even impossible to obtain explicit solutions for INDDEs. Therefore, it is necessary to study numerical solutions of INDDEs. But there is very little research on the numerical solutions of INDDEs. The stability and asymptotical stability of numerical methods for linear and nonlinear INDDEs with special fixed impulsive moments are studied by applying the method of transformation in [20,21]. The convergence of a numerical format of the Euler method for INDDEs is studied in [22]. However, in Ref. [22], the authors ignore the fact that the exact solution of an INDDE is continuous everywhere except the points at the moments of impulsive effect. Hence, our present paper introduces a new numerical method based on the Euler method for INDDEs, addressing this oversight in the aforementioned study.

The structure of this paper is as follows. Section 2 details the construction of the numerical method and its convergence proof. In Section 3, according to the distribution of the roots of a characteristic INDDE, the conditions of stability, asymptotical stability, and instability for the exact solution of INDDEs are given. Moreover, according to the distribution of the roots of the characteristic equation of the discrete equation obtained from the Euler method for INDDEs, the conditions of stability, asymptotical stability, and instability for the numerical solution of INDDEs are provided. Section 4 presents two examples to validate the main results. Finally, Section 5 concludes the paper with a summary of the findings and suggestions for future research directions.

2. Convergence of Euler’s Method for INDDEs

In this paper, we investigate the following impulsive neutral differential equation:

$${\left(x\left(t\right)+cx(t-\sigma )\right)}^{\prime }=ax\left(t\right)+bx(t-\tau ),t\ge 0,t\ne {\tau }_k,$$

(1)

$$\Delta x\left({\tau }_k\right)=l_k,k\in {\mathbb{Z}}^{+}=\{1,2,\cdots \},$$

(2)

$$x\left(t\right)=\varphi \left(t\right),-r\le t\le 0,$$

(3)

where $\sigma$ and $\tau$ are positive constants and $r=max\{\sigma ,\tau \}$; a, b, c, and $l_k$ are real constants; $x^{\prime }\left(t\right)$ denotes the right-hand derivative of $x\left(t\right)$; and $\Delta x\left(t\right)=x\left(t^{+}\right)-x\left(t^{-}\right)$. The impulse times ${\tau }_k$ satisfy $0<{\tau }_1<\cdots <{\tau }_k<{\tau }_{k+1}<\cdots$ and $\underset{k\to \infty }{lim}{\tau }_k=\infty$. The initial function $\varphi :[-r,0]\to \mathbb{R}$ is a given continuous function.

Definition 1

([23]). A real valued right continuous function x(t) is said to be the solution of the initial value problem (1)–(3) if the following conditions are satisfied:

(a) 

$x\left(t\right)$ is continuous everywhere except the points ${\tau }_k$, $k\in {\mathbb{Z}}^{+}$;

(b) 

the function $x\left(t\right)+cx(t-\sigma )$ is continuously differentiable for $t\ge 0$ and ${\tau }_k$, $k\in {\mathbb{Z}}^{+}$;

(c) 

$x\left({\tau }_k^{+}\right)$ and $x\left({\tau }_k^{-}\right)$ exist and $x\left({\tau }_k^{+}\right)=x\left({\tau }_k\right)$, $k\in {\mathbb{Z}}^{+}$;

(d) 

$x\left(t\right)$ satisfies the differential Equation (1) for $t\ge 0$, satisfies the impulsive conditions at $t={\tau }_k$, and satisfies (3).

Based on Euler’s method, the new numerical scheme for Equations (1)–(3) is constructed as follows:

$$\left\{\begin{array}{c}{X}_{n+1}=X_n+ah{X}_n+bh{X}_{n-m_2}-c{X}_{n-m_1+1}+c{X}_{n-m_1},n\ne {\eta }_k-1,n-m_1\ne {\eta }_j-1,\hfill \\ Y_k=X_n+ah{X}_n+bh{X}_{n-m_2}-c{X}_{n-m_1+1}+c{X}_{n-m_1},n={\eta }_k-1,\hfill \\ X_{n+1}=X_n+ah{X}_n+bh{X}_{n-m_2}-c{Y}_j+c{X}_{n-m_1},n-m_1={\eta }_j-1,\hfill \\ X_{{\eta }_k}=Y_k+l_k,\hfill \\ X_i={\varphi }_i=\varphi \left(ih\right),ih\in \left[-r,0\right],i=-m,\cdots ,-1,0,\hfill \end{array}\right.$$

(4)

where $h=\frac{\sigma }{m_1},m_2=\lfloor \frac{\tau }{h}\rfloor$, $m_1,m_2,j,k\in {\mathbb{Z}}^{+}$, h is a stepsize, and $0<h<min\{\sigma ,\tau \}$; the floor function $\lfloor \frac{\tau }{h}\rfloor$ denotes the largest integer less than or equal to $\frac{\tau }{h}$. Let ${\eta }_0=0$ and

$$\begin{array}{c}\hfill {\eta }_k=\left\{\begin{array}{cc}\frac{{\tau }_k}{h},\hfill & \frac{{\tau }_k}{h}\in {\mathbb{Z}}^{+},\hfill \\ \lfloor \frac{{\tau }_k}{h}\rfloor +1,\hfill & otherwise.\hfill \end{array}\right.\end{array}$$

(5)

$X_n$ is an approximation of exact solutions $x\left(t_n\right)$ for arbitrary $t_n=nh$, $n\in {\mathbb{Z}}^{+}$. $Y_k$ is an approximation of exact solutions $x\left({\tau }_k^{-}\right)$ if ${\eta }_k=\frac{{\tau }_k}{h}$; otherwise, $Y_k$ is a virtual value obtained from the Euler method. Here, $Y_k$ is a virtual value, meaning that $Y_k$ is not an approximation of the exact solution at any given time. The reason why $Y_k$, $k\in \mathbb{N}$ are calculated is so that the numerical solution does not make additional jumps, but instead jumps only once in the vicinity of each moment of impulsive effect.

In order to study the convergence of INDDEs, Equations (1)–(3) are considered on the finite interval $[-r,T]$, where T is a given positive constant. For convenience, we assume that there exist $p,N\in {\mathbb{Z}}^{+}$ such that $T=pmh$ and $0<{\tau }_1<\cdots <{\tau }_k<\cdots <{\tau }_N\le T<{\tau }_{N+1}$, where $m=max\{m_1,m_2\}$. From [9], we know that $x\left(t\right)$ and $x^{\prime }\left(t\right)$ are bounded. Therefore, we assume that there exists an $M>0$, such that the solution $x\left(t\right)$ of Equations (1) and (2) satisfies $|x(t)|\le M$ and $|x^{\prime }\left(t\right)|\le M$ for $t\in [-r,T]$. For the sake of simplicity, we also assume that

$$|\varphi (t)-\varphi (s)|\le M|t-s|$$

(6)

and

$$|{\varphi }^{\prime }\left(t\right)-{\varphi }^{\prime }\left(s\right)|\le M|t-s|.$$

(7)

Let $e_n=|x\left(nh\right)-X_n|$, which denotes the global error. The following theorem will demonstrate that the convergence order of the Euler method for Equations (1)–(2) is 1 by analyzing the global error $e_n$.

Theorem 1.

The convergence order of Euler scheme (4) is 1—that is, there exists a $C>0$, such that $e_n\le Ch$, $1\le n\le pm$.

Proof. 

We shall show that there exists a $C_k>0$, such that

$$e_n\le C_{k}h,n\in I_k=[{\eta }_{k-1}+1,{\eta }_k]\bigcap \mathbb{Z},k=1,2,3,\cdots ,N.$$

(8)

First, we show there exists a $C_1>0$ such that $e_n\le C_{1}h$, $n\in I_1$. For the sake of simplicity, we only consider the following situation, but others can be proved similarly. Assume that $\tau =m_{2}h+\delta$, $0\le \delta <1$, $r=max\{\sigma ,\tau \}=\sigma$, $m=m_1$ and $0<\tau <\sigma <{\tau }_1$.

When $0\le t_n\le \tau <\sigma <{\tau }_1$, we have

$$\begin{array}{cc}\hfill & e_n=|x\left(nh\right)-X_n|\hfill \\ \hfill & =|x\left((n-1)h\right)+{\int }_{(n-1)h}^{nh}ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt-X_{n-1}-ah{X}_{n-1}\hfill \\ \hfill & -bh{X}_{n-m_2-1}+c{X}_{n-m_1}-c{X}_{n-m_1-1}|\hfill \\ \hfill & \le e_{n-1}+|{\int }_0^{h}ax((n-1)h+t)dt-ah{X}_{n-1}+{\int }_0^{h}bx((n-m_2-\delta -1)h+t)dt\hfill \\ \hfill & -bh{X}_{n-m_2-1}-{\int }_0^{h}c{x}^{\prime }((n-m_1-1)h+t)dt+c{X}_{n-m_1}-c{X}_{n-m_1-1}|\hfill \\ \hfill & \le e_{n-1}+{\int }_0^h|a||x((n-1)h+t)-X_{n-1}|+|b||x((n-m_2-\delta -1)h+t)\hfill \\ \hfill & -X_{n-m_2-1}|dt+|c||{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|,\hfill \end{array}$$

(9)

where

$$\begin{array}{cc}\hfill & |x((n-1)h+t)-X_{n-1}|\hfill \\ \hfill & =|x\left((n-1)h\right)+{\int }_{(n-1)h}^{t+(n-1)h}ax\left(u\right)+bx(u-\tau )-c{x}^{\prime }(u-\sigma )du-X_{n-1}|\hfill \\ \hfill & \le e_{n-1}+{\int }_0^t|a||x((n-1)h+u)|+|b||x((n-m_2-\delta -1)h+u)|\hfill \\ \hfill & +|c||x^{\prime }((n-m_1-1)h+u)|du\hfill \\ \hfill & \le e_{n-1}+(|a|+|b|+|c|)Mh,\hfill \end{array}$$

(10)

based on Equation (6)

$$\begin{array}{cc}\hfill & |x((n-m_2-\delta -1)h+t)-X_{n-m_2-1}|\hfill \\ \hfill & =|\varphi (t+(n-m_2-\delta -1)h)-\varphi \left((n-m_2-1)h\right)|\hfill \\ \hfill & \le M|t-\delta h|\hfill \\ \hfill & \le Mh\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill & |{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|\hfill \\ \hfill & =|x\left((n-m_1-1)h\right)-x\left((n-m_1)h\right)+X_{n-m_1}-X_{n-m_1-1}|\hfill \\ \hfill & =|\varphi \left((n-m_1-1)h\right)-\varphi \left((n-m_1)h\right)+\varphi \left((n-m_1)h\right)-\varphi \left((n-m_1-1)h\right)|\hfill \\ \hfill & =0.\hfill \end{array}$$

(12)

Substituting Equations (10)–(12) into Equation (9), we find that

$$\begin{array}{cc}\hfill & e_n\le e_{n-1}+|a|h[e_{n-1}+(|a|+|b|+|c|)Mh]+|b|M{h}^2+|c|·0\hfill \\ \hfill & =e_{n-1}(1+|a|h)+M{h}^2[|a|(|a|+|b|+|c|)+|b|]\hfill \\ \hfill & \le C_{1,1,1}h.\hfill \end{array}$$

(13)

Let $e_0=0$, and, according to Gronwall inequality, we can calculate

$$C_{1,1,1}=\frac{[|a|(|a|+|b|+|c|)+|b|]M}{|a|}{e}^{|a|T}.$$

When $\tau <t_n\le 2\tau <\sigma <{\tau }_1$, we have

$$\begin{array}{cc}\hfill & e_n=|x\left(nh\right)-X_n|\hfill \\ \hfill & \le e_{n-1}+{\int }_0^h|a||x((n-1)h+t)-X_{n-1}|+|b||x((n-m_2-\delta -1)h+t)\hfill \\ \hfill & -X_{n-m_2-1}|dt+|c||{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|.\hfill \end{array}$$

(14)

As discussed in Equation (10), for $t\in [0,h]$, we find that

$$\begin{array}{c}\hfill |x((n-1)h+t)-X_{n-1}|\le e_{n-1}+(|a|+|b|+|c|)Mh,\end{array}$$

(15)

$$\begin{array}{cc}\hfill & |x((n-m_2-\delta -1)h+t)-X_{n-m_2-1}|\hfill \\ \hfill & =|x\left((n-m_2-1)h\right)+{\int }_{(n-m_2-1)h}^{(n-m_2-\delta -1)h+t}ax\left(u\right)+bx(u-\tau )-c{x}^{\prime }(u-\sigma )du\hfill \\ \hfill & -X_{n-m_2-1}|\hfill \\ \hfill & \le e_{n-m_2-1}+|{\int }_0^{t-\delta h}ax(u+(n-m_2-1)h)+bx(u+(n-2m_2-\delta -1)h)\hfill \\ \hfill & -c{x}^{\prime }(u+(n-m_1-m_2-1)h)du|\hfill \\ \hfill & \le e_{n-m_2-1}+(|a|+|b|+|c|)Mh.\hfill \end{array}$$

(16)

Similar to Equation (12), we obtain

$$|{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|=0.$$

(17)

Substituting Equations (15)–(17) into Equation (14) yields

$$\begin{array}{cc}\hfill & e_n\le e_{n-1}+|a|h(e_{n-1}+(|a|+|b|+|c|)Mh)+|b|h(e_{n-m_2-1}+(|a|+|b|+|c|)Mh)\hfill \\ \hfill & \le e_{n-1}(1+|a|h)+M{h}^2(|a|+|b|)(|a|+|b|+|c|)+|b|C_{1,1,1}{h}^2\hfill \\ \hfill & \le C_{1,1,2}h,\hfill \end{array}$$

where

$$C_{1,1,2}=\frac{M(|a|+|b|)(|a|+|b|+|c|)+|b|C_{1,1,1}}{|a|}{e}^{|a|T}.$$

When $(k-1)\tau <t_n\le k\tau$ for some $k\in [3,\lfloor \frac{\sigma }{\tau }\rfloor ]$, similarly to the discussion above, we can obtain

$$C_{1,1,k}=\frac{M(|a|+|b|)(|a|+|b|+|c|)+|b|C_{1,1,k-1}}{|a|}{e}^{|a|T},$$

such that $e_n\le C_{1,1,k}h$.

When $\lfloor \frac{\sigma }{\tau }\rfloor \tau <t_n<\sigma$, an analogous calculation can be performed to calculate $C_{1,1,\lfloor \frac{\sigma }{\tau }\rfloor +1}$, such that the inequality $e_n\le C_{1,1,\lfloor \frac{\sigma }{\tau }\rfloor +1}h$ holds. Taking $C_{1,1}=max\{C_{1,1,1},C_{1,1,2},\cdots ,C_{1,1,\lfloor \frac{\sigma }{\tau }\rfloor +1}\}$, we have $e_n\le C_{1,1}h$ for $1\le n<m_1<{\eta }_1$.

When $\sigma \le t_n<2\sigma <{\tau }_1$, $m_1\le n<2m_1<{\eta }_1$ is satisfied, we prioritize $\sigma \le t_n<\sigma +\tau <2\sigma <{\tau }_1$, which gives us

$$\begin{array}{cc}\hfill & e_n=|x\left(nh\right)-X_n|\hfill \\ \hfill & \le e_{n-1}+{\int }_0^h|a||x((n-1)h+t)-X_{n-1}|+|b||x((n-m_2-\delta -1)h+t)\hfill \\ \hfill & -X_{n-m_2-1}|dt+|c||{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|.\hfill \end{array}$$

(18)

For $t\in [0,h]$, as discussed in Equation (10) and Equation (3), we have

$$|x((n-1)h+t)-X_{n-1}|\le e_{n-1}+(|a|+|b|+|c|)Mh,$$

(19)

$$\begin{array}{cc}\hfill & |x((n-m_2-\delta -1)h+t)-X_{n-m_2-1}|\hfill \\ \hfill & \le e_{n-m_2-1}+(|a|+|b|+|c|)Mh,\hfill \\ \hfill & \le C_{1,1}h+(|a|+|b|+|c|)Mh\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill & |{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|\hfill \\ \hfill & =|x\left((n-m_1-1)h\right)-x\left((n-m_1)h\right)+X_{n-m_1}-X_{n-m_1-1}|\hfill \\ \hfill & =|e_{n-m_1-1}-e_{n-m_1}|\hfill \\ \hfill & \le e_{n-m_1-1}|a|h+M{h}^2(|a|+|b|)(|a|+|b|+|c|)+|b|C_{1,1}{h}^2\hfill \\ \hfill & =(|a|+|b|)C_{1,1}{h}^2+M{h}^2(|a|+|b|)(|a|+|b|+|c|).\hfill \end{array}$$

(21)

Substituting Equations (19)–(21) into Equation (18) yields

$$\begin{array}{cc}\hfill e_n& \le e_{n-1}+|a|h(e_{n-1}+(|a|+|b|+|c|)Mh)+|b|h(C_{1,1}h+(|a|+|b|+|c|)Mh)+|c|\hfill \\ \hfill & ((|a|+|b|)C_{1,1}{h}^2+M{h}^2(|a|+|b|)(|a|+|b|+|c|))\hfill \\ \hfill & \le (1+|a|h)e_{n-1}+M{h}^2(|a|+|b|+|c|)(|a|+|b|)(1+|c|)+(|b|+|c|(|a|+|b|))C_{1,1}{h}^2\hfill \\ \hfill & \le C_{1,2,1}h,\hfill \end{array}$$

where $C_{1,2,1}=\frac{M(|a|+|b|+|c|)(|a|+|b|)(1+|c|)+(|b|+|c|(|a|+|b|))C_{1,1}}{|a|}{e}^{|a|T}.$

When $\sigma +\tau \le t_n<\sigma +2\tau$, similarly to the discussion above, we can conclude that

$$\begin{array}{cc}\hfill & e_n\le e_{n-1}+|a|h(e_{n-1}+(|a|+|b|+|c|)Mh)+|b|h(C_{1,2,1}h+(|a|+|b|+|c|)Mh)\hfill \\ & +|c|((|a|+|b|)C_{1,1}{h}^2+M{h}^2(|a|+|b|)(|a|+|b|+|c|))\hfill \\ \hfill & \le e_{n-1}(1+|a|h)+M{h}^2(|a|+|b|+|c|)(|a|+|b|)(1+|c|)+(|b|+|c|(|a|+|b|))C_{1,2,1}{h}^2\hfill \\ \hfill & \le C_{1,2,2}h,\hfill \end{array}$$

where $C_{1,2,2}=\frac{M(|a|+|b|+|c|)(|a|+|b|)(1+|c|)+C_{1,2,1}(|b|+|c|(|a|+|b|))}{|a|}{e}^{|a|T}.$

When $\sigma +(k-1)\tau \le t_n<\sigma +k\tau$ for some $k\in [3,\lfloor \frac{\sigma }{\tau }\rfloor ]$, similarly to the discussion above, we can calculate $C_{1,2,k}=\frac{M(|a|+|b|+|c|)(|a|+|b|)(1+|c|)+C_{1,2,k-1}(|b|+|c|(|a|+|b|))}{|a|}{e}^{|a|T}$.

When $\sigma +\lfloor \frac{\sigma }{\tau }\rfloor \tau \le t_n<2\sigma$, an analogous calculation can be performed to obtain $C_{1,2,\lfloor \frac{\sigma }{\tau }\rfloor +1}$, such that the inequality $e_n\le C_{1,2,\lfloor \frac{\sigma }{\tau }\rfloor +1}h$ holds.

Taking $C_{1,2}=max\{C_{1,2,1},C_{1,2,2},\cdots ,C_{1,2,\lfloor \frac{\sigma }{\tau }\rfloor +1}\}$, we have $e_n\le C_{1,2}h$ for $m_1\le n<2m_1<{\eta }_1$.

Let $B_{i+1}=\lfloor \frac{{\eta }_{i+1}-{\eta }_i}{m_1}\rfloor$. When $(j-1)\sigma <t_n\le j\sigma$, $(j-1)m_1<n\le j{m}_1$, $3\le j\le B_1$, and $e_n\le C_{1,j}h$, where $C_{1,j}=max\{C_{1,j,1},C_{1,j,2},\cdots ,C_{1,j,\lfloor \frac{\sigma }{\tau }\rfloor +1}\}$, and $C_{1,j,k}$ can thus be obtained as follows:

$$\begin{array}{cc}\hfill e_n& \le e_{n-1}(1+|a|h)+M{h}^2(|a|+|b|+|c|)(|a|+|b|)(1+|c|)\hfill \\ \hfill & +(|b|+|c|(|a|+|b|))C_{1,j,k-1}{h}^2\hfill \\ \hfill & \le C_{1,j,k}h.\hfill \end{array}$$

When $B_1\sigma \le t_n<{\tau }_1$, an analogous calculation can be performed to obtain $C_{1,B_1+1}$, such that the inequality $e_n\le C_{1,B_1+1}h$ holds.

If we take $\widetilde{C_1}=max\{C_{1,1},C_{1,2},\cdots ,C_{1,B_1+1}\}$, we have $e_n\le \widetilde{C_1}h$ for $1\le n<{\eta }_1$.

If $t_n={\tau }_1$, $n={\eta }_1$, it is possible to derive

$$\begin{array}{cc}\hfill & e_n=|x\left({\eta }_{1}h\right)-X_{{\eta }_1}|=|x\left({\eta }_{1}h\right)-Y_1-l_1|\hfill \\ \hfill & =|x\left({\tau }_1\right)+{\int }_{{\tau }_1}^{{\eta }_{1}h}ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt-X_{{\eta }_1}|\hfill \\ \hfill & =|l_1+x\left({\tau }_1^{-}\right)+{\int }_{{\tau }_1}^{{\eta }_{1}h}ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt-(l_1+X_{{\eta }_1-1}\hfill \\ \hfill & +ah{X}_{{\eta }_1-1}+bh{X}_{{\eta }_1-m_2-1}-c{X}_{{\eta }_1-m_1}+c{X}_{{\eta }_1-m_1-1})|\hfill \\ \hfill & \le |x\left(({\eta }_1-1)h\right)+{\int }_{({\eta }_1-1)h}^{{\tau }_1}ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt+{\int }_{{\tau }_1}^{{\eta }_{1}h}ax\left(t\right)\hfill \\ \hfill & +bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt-(X_{{\eta }_1-1}+ah{X}_{{\eta }_1-1}+bh{X}_{{\eta }_1-m_2-1}\hfill \\ \hfill & -c{X}_{{\eta }_1-m_1}+c{X}_{{\eta }_1-m_1-1})|\hfill \\ \hfill & \le e_{{\eta }_1-1}+|{\int }_{({\eta }_1-1)h}^{{\tau }_1}ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt+{\int }_{{\tau }_1}^{({\eta }_1-1)h}ax\left(t\right)\hfill \\ \hfill & +bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt+{\int }_{({\eta }_1-1)h}^{{\eta }_{1}h}ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )dt\hfill \\ \hfill & -ah{X}_{{\eta }_1-1}-bh{X}_{{\eta }_1-m_2-1}+c{X}_{{\eta }_1-m_1}-c{X}_{{\eta }_1-m_1-1}|\hfill \\ \hfill & \le e_{{\eta }_1-1}+{\int }_0^h|a||x(t+({\eta }_1-1)h)-X_{{\eta }_1-1}|+|b||x(t+({\eta }_1-m_2-\delta -1)h\hfill \\ \hfill & -X_{{\eta }_1-m_2-1})dt+|{\int }_0^h-c{x}^{\prime }(t+({\eta }_1-m_1-1)h)dt+c{X}_{{\eta }_1-m_1}-c{X}_{{\eta }_1-m_1-1}|\hfill \\ \hfill & \le e_{{\eta }_1-1}+|a|h(e_{{\eta }_1-1}+(|a|+|b|+|c|)Mh)+|b|h(e_{{\eta }_1-m_2-1}+(|a|+|b|\hfill \\ \hfill & +|c|\left)Mh\right)+|c||e_{{\eta }_1-m_1-1}-e_{{\eta }_1-m_1}|\hfill \\ \hfill & \le (1+|a|h)e_{{\eta }_1-1}+M{h}^2(|a|+|b|)(1+|c|)(|a|+|b|+|c|)+{\tilde{C}}_1{h}^2(|b|+|c|\hfill \\ \hfill & (|a|+|b|))\hfill \\ \hfill & \le C_{1}h.\hfill \end{array}$$

(22)

If we let $C_1=\frac{M(|a|+|b|)(1+|c|)(|a|+|b|+|c|)+{\tilde{C}}_1(|b|+|c|(|a|+|b|))}{|a|}{e}^{|a|T}$, we can obtain the inequality $e_n\le C_{1}h$, which holds for $n\in I_1$.

When $n\in I_2$, an analogous calculation can be performed to obtain $e_n\le C_{2,1}h$ for ${\eta }_1<n<{\eta }_1+m_1-1$, but in this paper we omit the proof of this calculation. Next, for $n={\eta }_1+m_1$,

$$\begin{array}{cc}\hfill & e_n=|x\left(nh\right)-X_n|\hfill \\ \hfill & =|x\left((n-1)h\right)+{\int }_{(n-1)h}^{nh}\left[ax\left(t\right)+bx(t-\tau )-c{x}^{\prime }(t-\sigma )\right]dt-X_{n-1}-ah{X}_{n-1}\hfill \\ \hfill & -bh{X}_{n-m_2-1}+c{Y}_1-c{X}_{n-m_1-1}|\hfill \\ \hfill & \le e_{n-1}+|a|{\int }_0^h|x((n-1)h+t)-X_{n-1}|dt+|b|{\int }_0^h|x((n-m_2-\delta -1)h+t)\hfill \\ \hfill & -X_{n-m_2-1}|dt+|c||{\int }_{(n-1)h}^{nh}-x^{\prime }(t-\sigma )dt+Y_1-X_{n-m_1-1}|.\hfill \end{array}$$

(23)

As discussed in Equations (11) and (17) for $t\in [0,h]$, we have

$$|x((n-1)h+t)-X_{n-1}|\le e_{n-1}+\left(|a|+|b|+|c|\right)Mh$$

(24)

and

$$\begin{array}{cc}\hfill & |x((n-m_2-\delta -1)h+t)-X_{n-m_2-1}|\hfill \\ \hfill & \le e_{n-m_2-1}+\left(|a|+|b|+|c|\right)Mh\hfill \\ \hfill & \le C_{2,1}h+\left(|a|+|b|+|c|\right)Mh\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill & |{\int }_{(n-1)h}^{nh}-x^{\prime }(t-\sigma )dt+Y_1-X_{n-m_1-1}|\hfill \\ \hfill & =|{\int }_{({\eta }_1-1)h}^{{\eta }_{1}h}-x^{\prime }\left(s\right)ds+Y_1-X_{n-m_1-1}|\hfill \\ \hfill & =|-{\int }_{({\eta }_1-1)h}^{{\tau }_1}{x}^{\prime }\left(s\right)ds-{\int }_{{\tau }_1}^{{\eta }_{1}h}{x}^{\prime }\left(s\right)ds+Y_1-X_{n-m_1-1}|\hfill \\ \hfill & =|{\int }_{({\eta }_1-1)h}^{{\tau }_1}\left[-ax\left(s\right)-bx(s-\tau )+c{x}^{\prime }(s-\sigma )\right]ds+{\int }_{{\tau }_1}^{{\eta }_{1}h}[-ax\left(s\right)-bx(s-\tau )\hfill \\ \hfill & +c{x}^{\prime }(s-\sigma )]ds+ah{X}_{{\eta }_1-1}+bh{X}_{{\eta }_1-m_1-1}-c{X}_{{\eta }_1-m_1}+c{X}_{{\eta }_1-m_1-1}|\hfill \\ \hfill & \le |a|{\int }_0^{{\tau }_1-({\eta }_1-1)h}|x(s+({\eta }_1-1)h)-X_{{\eta }_1-1}|ds+|b|{\int }_0^{{\tau }_1-({\eta }_1-1)h}|x(s+({\eta }_1-m_2-\delta -1)h)\hfill \\ \hfill & -X_{{\eta }_1-m_2-1}|ds+|a|{\int }_{{\tau }_1-({\eta }_1-1)h}^h|x(s+({\eta }_1-1)h)-X_{{\eta }_1-1}|ds\hfill \\ \hfill & +|b|{\int }_{{\tau }_1-({\eta }_1-1)h}^h|x(s+({\eta }_1-m_2-\delta -1)h)-X_{{\eta }_1-m_2-1}|ds+|{\int }_{({\eta }_1-1)h}^{{\tau }_1}c{x}^{\prime }(s-\sigma )ds\hfill \\ \hfill & +{\int }_{{\tau }_1}^{{\eta }_{1}h}c{x}^{\prime }(s-\sigma )ds-c{X}_{{\eta }_1-m_1}+c{X}_{{\eta }_1-m_1-1}|\hfill \\ \hfill & \le |a|h\left(e_{{\eta }_1-1}+(|a|+|b|+|c|)Mh\right)+|b|h\left(e_{{\eta }_1-m_2-1}+(|a|+|b|+|c|)Mh\right)\hfill \\ \hfill & +|c||e_{{\eta }_1-m_2}-e_{{\eta }_1-m_2-1}|\hfill \\ \hfill & \le M{h}^2(|a|+|b|)(1+|c|+{|c|}^2)(|a|+|b|+|c|)+C_1{h}^2(|a|+|b|)(1+|c|+{|c|}^2).\hfill \end{array}$$

(26)

Substituting Equations (24)–(26) into Equation (23), we find that

$$\begin{array}{cc}\hfill & e_n\le e_{n-1}+|a|h[e_{n-1}+(|a|+|b|+|c|)Mh]+|b|h[C_{2,1}h+(|a|+|b|+|c|)Mh]\hfill \\ \hfill & +|c|M{h}^2(|a|+|b|)(1+|c|+{|c|}^2)(|a|+|b|+|c|)\hfill \\ \hfill & +|c|C_1{h}^2(|a|+|b|)(1+|c|+{|c|}^2)\hfill \\ \hfill & =e_{n-1}(1+|a|h)+M{h}^2(|a|+|b|+|c|)(|a|+|b|)(1+|c|+{|c|}^2+{|c|}^3)\hfill \\ \hfill & +|b|C_{2,1}{h}^2+|c|C_1{h}^2(|a|+|b|)(1+|c|+{|c|}^2)\hfill \\ \hfill & \le D_{1}h,\hfill \end{array}$$

(27)

where $D_1=\frac{M(|a|+|b|+|c|)(|a|+|b|)(1+|c|+{|c|}^2+{|c|}^3)+C_{2,1}|b|+|c|C_1(|a|+|b|)(1+|c|+{|c|}^2)}{|a|}·{\mathrm{e}}^{|a|T}$.

The same as before, we take ${\tilde{C}}_2=max\{C_{2,1},C_{2,2},\cdots ,C_{2,B_2+1},D_1\}$. So, $e_n\le {\tilde{C}}_{2}h$ holds for $n\in I_2$.

Assume that Equation (10) holds for $n\in I_{s-1}$—that is, $e_n\le C_{s-1}h$ holds for $n\in I_{s-1}$. Now, we will show that Equation (8) holds for $n\in I_s$.

When ${\tau }_{s-1}\le t_n\le {\tau }_{s-1}+\tau <{\tau }_{s-1}+\sigma$,

$$\begin{array}{cc}\hfill & e_n=|x\left(nh\right)-X_n|\hfill \\ \hfill & \le e_{n-1}+{\int }_0^h|a||x((n-1)h+t)-X_{n-1}|+|b||x((n-m_2-\delta -1)h+t)\hfill \\ \hfill & -X_{n-m_2-1}|dt+|c||{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|.\hfill \end{array}$$

According to Equation (10), Equation (3), and Equation (21) and the related discussions above, for $t\in [0,h]$, we have

$$|x((n-1)h+t)-X_{n-1}|\le e_{n-1}+(|a|+|b|+|c|)Mh,$$

$$\begin{array}{cc}\hfill & |x((n-m_2-\delta -1)h+t)-X_{n-m_2-1}|\le e_{n-m_2-1}+(|a|+|b|+|c|)Mh,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & |{\int }_0^h-x^{\prime }((n-m_1-1)h+t)dt+X_{n-m_1}-X_{n-m_1-1}|\hfill \\ \hfill & \le e_{n-m_1-1}|a|h+M{h}^2(|a|+|b|)(|a|+|b|+|c|)+|b|C_{s-1}{h}^2.\hfill \end{array}$$

Then, we find

$$\begin{array}{cc}\hfill e_n& \le (1+|a|h)e_{n-1}+M{h}^2(|a|+|b|+|c|)(|a|+|b|)(1+|c|)\hfill \\ \hfill & +(|b|+|c|(|a|+|b|))C_{s-1}{h}^2\hfill \\ \hfill & \le C_{s,1,1}h.\hfill \end{array}$$

Just as in the discussion of $C_{1,1,k}$, we can obtain $C_{s,1,k}$, such that $e_n\le C_{s,1,k}h,$ for $k=1,2,\cdots ,\lfloor \frac{\sigma }{\tau }\rfloor$. When ${\tau }_{s-1}+\lfloor \frac{\sigma }{\tau }\rfloor \tau \le t_n<{\tau }_{s-1}+\sigma$, we also find that $C_{s,1,\lfloor \frac{\sigma }{\tau }\rfloor +1}$ satisfies $e_n\le C_{s,1,\lfloor \frac{\sigma }{\tau }\rfloor +1}h$.

Taking $C_{s,1}=max\{C_{s,1,1},C_{s,1,2},\cdots ,C_{s,1,\lfloor \frac{\sigma }{\tau }\rfloor +1}\}$, we have $e_n\le C_{s,1}h$, for ${\eta }_{s-1}+1\le n<{\eta }_{s-1}+m_2$. Then, there is a finite number of $C_{s,j}$, and the number of $C_{s,j}$ does not depend on the stepsize h.

Also, when $n-m_1={\eta }_s$, similarly to Equation (27), we have

$$\begin{array}{cc}\hfill & e_n\le e_{n-1}+|a|h[e_{n-1}+(|a|+|b|+|c|)Mh]+|b|h[C_{s,1}h+(|a|+|b|+|c|)Mh]\hfill \\ \hfill & +|c|M{h}^2(|a|+|b|)(1+|c|+{|c|}^2)(|a|+|b|+|c|)\hfill \\ \hfill & +|c|C_{s-1}{h}^2(|a|+|b|)(1+|c|+{|c|}^2)\hfill \\ \hfill & =e_{n-1}(1+|a|h)+M{h}^2(|a|+|b|+|c|)(|a|+|b|)(1+|c|+{|c|}^2+{|c|}^3)\hfill \\ \hfill & +|b|C_{s,1}{h}^2+|c|C_{s-1}{h}^2(|a|+|b|)(1+|c|+{|c|}^2)\hfill \\ \hfill & \le D_{s-1}h,\hfill \end{array}$$

(28)

where $D_{s-1}=\frac{M(|a|+|b|+|c|)(|a|+|b|)(1+|c|+{|c|}^2+{|c|}^3)+C_{s,1}|b|+|c|C_{s-1}(|a|+|b|)(1+|c|+{|c|}^2)}{|a|}·{\mathrm{e}}^{|a|T}$.

Taking

$${\tilde{C}}_s=max\{C_{s,1},C_{s,2},\cdots ,C_{s,B_s+1},D_{s-1}\},$$

we have

$e_n\le {\tilde{C}}_{s}h,{\eta }_{s-1}+1\le n<{\eta }_s$.

When $t_n={\tau }_s$ and $n={\eta }_s$, performing a calculation analogous to that in Equation (22) yields the following:

$$\begin{array}{cc}\hfill & e_n=|x\left({\eta }_{s}h\right)-X_{{\eta }_s}|\hfill \\ \hfill & \le (1+|a|h)e_{{\eta }_s-1}+M{h}^2(|a|+|b|)(1+|c|)(|a|+|b|+|c|)\hfill \\ \hfill & +{\tilde{C}}_s{h}^2(|b|+|c|(|a|+|b|))\hfill \\ \hfill & \le C_{s}h.\hfill \end{array}$$

If we let $C_s=\frac{M(|a|+|b|)(1+|c|)(|a|+|b|+|c|)+{\tilde{C}}_s(|b|+|c|(|a|+|b|))}{|a|}{e}^{|a|T}$, we obtain $e_n\le C_{s}h$ for $n\in I_s$.

When ${\eta }_N<n\le pm=\frac{T}{h}$, the same as $n\in \left[{\eta }_{s-1}+1,{\eta }_s\right)$, we can calculate that $C_{N+1,1}$ satisfies $e_n\le C_{N+1,1}h$.

Taking

$$C=max\{C_1,C_2,\cdots ,C_N,C_{N+1,1}\},$$

we find that $e_n\le Ch$ holds for $1\le n\le pm$. This completes the proof. □

3. Asymptotical Stability of INDDEs

In this section, we study the asymptotical stability not only of the exact solutions of INDDEs but also of the numerical solutions of INDDEs.

3.1. Asymptotical Stability of the Exact Solutions of INDDEs

In order to study the asymptotical stability of INDDE (1) and (2), we consider the same equation with another initial function:

$${\left(\tilde{x}\left(t\right)+c\tilde{x}(t-\sigma )\right)}^{\prime }=a\tilde{x}\left(t\right)+b\tilde{x}(t-\tau ),t\ge 0,t\ne {\tau }_k,$$

(29)

$$\Delta \tilde{x}\left({\tau }_k\right)=l_k,k\in {\mathbb{Z}}^{+},$$

(30)

$$\tilde{x}\left(t\right)=\tilde{\varphi }\left(t\right),-r\le t\le 0,$$

(31)

where $\sigma$ and $\tau$ are positive constants and $r=max\{\sigma ,\tau \}$, a, b, c and $l_k$ are real constants, ${\tilde{x}}^{\prime }\left(t\right)$ denotes the right-hand derivative of $\tilde{x}\left(t\right)$, and $\Delta \tilde{x}\left(t\right)=\tilde{x}\left(t^{+}\right)-\tilde{x}\left(t^{-}\right)$. The impulse times ${\tau }_k$ satisfy $0<{\tau }_1<\cdots <{\tau }_k<{\tau }_{k+1}<\cdots$ and $\underset{k\to \infty }{lim}{\tau }_k=\infty$. The initial function $\tilde{\varphi }:[-r,0]\to \mathbb{R}$ is a given continuous function.

Definition 2.

The solutions $x\left(t\right)$ of INDDE (1)–(3) and $\tilde{x}\left(t\right)$ of (29)–(31) are said to be stable if for every $ϵ>0$, there exists a number $\delta =\delta \left(ϵ\right)>0$, such that

$$\parallel \varphi -\tilde{\varphi }\parallel =\underset{-r\le t\le 0}{max}|\varphi \left(t\right)-\tilde{\varphi }\left(t\right)|<\delta ,$$

which implies that

$$\parallel x\left(t\right)-\tilde{x}\left(t\right)\parallel <ϵ,\mathrm{for}\mathrm{all}t\ge 0.$$

The solutions $x\left(t\right)$ of INDDE (1)–(3) and $\tilde{x}\left(t\right)$ of (29)–(31) are said to be asymptotically stable if they are stable and there exists a number ${\delta }_0>0$ such that $\parallel \varphi -\tilde{\varphi }\parallel <{\delta }_0$ implies

$$\underset{t\to \infty }{lim}\parallel x\left(t\right)-\tilde{x}\left(t\right)\parallel =0.$$

Assume that $x\left(t\right)$ is the solution of INDDE (1)–(3) and $\tilde{x}\left(t\right)$ is the solution of (29)–(31). Then, $y\left(t\right)=x\left(t\right)-\tilde{x}\left(t\right)$, $t\ge -r$, satisfying the following NDDE without impulsive perturbations:

$$\left\{\begin{array}{cc}{\left(y\left(t\right)+cy(t-\sigma )\right)}^{\prime }=ay\left(t\right)+by(t-\tau ),\hfill & t\ge 0,\hfill \\ y\left(t\right)=\phi \left(t\right),\hfill & -r\le t\le 0,\hfill \end{array}\right.$$

(32)

where $a,b,c,\sigma$, and $\tau$ are real constants; $\tau >0$, $\sigma >0$, $\phi \left(t\right)=\varphi \left(t\right)-\tilde{\varphi }\left(t\right)$ for all $t\in [-r,0]$; and $\phi \in C\left(\right[-r,0],\mathbb{R})$ is the initial function.

Definition 3

([24,25]). The zero solution of (32) is stable if for every $ϵ>0$, there exists a number $\delta =\delta \left(ϵ\right)>0$ such that

$$\parallel \phi \parallel =\underset{-r\le t\le 0}{max}|\phi \left(t\right)|=\underset{-r\le t\le 0}{max}|\varphi \left(t\right)-\tilde{\varphi }\left(t\right)|<\delta$$

implies $|y(t)|<ϵ$.

The zero solution of (32) is asymptotically stable if the zero solution of (32) is stable and there exists a number ${\delta }_0>0$ such that $\parallel \phi \parallel <{\delta }_0$ implies

$$\underset{t\to \infty }{lim}|y\left(t\right)|=0.$$

Due to Definitions 2 and 3, we can easily reveal the following theorem.

Theorem 2.

The solutions $x\left(t\right)$ of INDDE (1)–(3) and $\tilde{x}\left(t\right)$ of (29)–(31) are stable if and only if the zero solution of (32) is stable.

Moreover, the solutions $x\left(t\right)$ of INDDE (1)–(3) and $\tilde{x}\left(t\right)$ of (29)–(31) are asymptotically stable if and only if the zero solution of (32) is asymptotically stable.

The characteristic equation for an NDDE (i.e., the first of (32)) is as follows:

$$\lambda (1+c{\mathrm{e}}^{-\lambda \sigma })=a+b{\mathrm{e}}^{-\lambda \tau }.$$

(33)

According to Refs. [24,25], the following lemma is a special case of their main results; the associated proof is omitted from this paper for brevity.

Lemma 1.

Assume that ${\lambda }_0$ is a real root of characteristic Equation (33) and satisfies

$$\mu \left({\lambda }_0\right)=|b|\tau {\mathrm{e}}^{-{\lambda }_0\tau }+|c|{\mathrm{e}}^{-{\lambda }_0\sigma }(1+|{\lambda }_0|\sigma )<1.$$

(34)

Then, the solution $y\left(t\right)$ of (32) satisfies

$$\underset{t\to \infty }{lim}\left[{\mathrm{e}}^{-{\lambda }_{0}t}y\left(t\right)\right]=\frac{L({\lambda }_0,\phi )}{1+\beta \left({\lambda }_0\right)},$$

(35)

where

$$L({\lambda }_0,\phi )=\phi \left(0\right)+c\phi (-\sigma )+b{\mathrm{e}}^{-{\lambda }_0\tau }{\int }_{-\tau }^0{\mathrm{e}}^{-{\lambda }_{0}s}\phi \left(s\right)ds-c{\lambda }_0{\mathrm{e}}^{-{\lambda }_0\sigma }{\int }_{-\sigma }^0{\mathrm{e}}^{-{\lambda }_{0}s}\phi \left(s\right)ds$$

and

$$\beta \left({\lambda }_0\right)=b\tau {\mathrm{e}}^{-{\lambda }_0\tau }+c{\mathrm{e}}^{-{\lambda }_0\sigma }(1-{\lambda }_0\sigma ).$$

(36)

Based on this, we can utilize the following statement to illustrate the stability and asymptotic stability of (32).

Theorem 3.

Assume that ${\lambda }_0$ is a real root of characteristic Equation (32) and satisfies $\mu \left({\lambda }_0\right)<1$. Let $\beta \left({\lambda }_0\right)$ be defined by (36) and set

$$R({\lambda }_0;\phi )=max\left\{1,\underset{-r\le t\le 0}{max}|\phi \left(t\right)|,\underset{-r\le t\le 0}{max}{\mathrm{e}}^{{\lambda }_{0}t}|\phi \left(t\right)|\right\}.$$

Then, the solution $y\left(t\right)$ of (32) satisfies

$$|y\left(t\right)|\le N\left({\lambda }_0\right)R({\lambda }_0;\phi ){\mathrm{e}}^{{\lambda }_{0}t},\forall t\ge 0,$$

where

$$N\left({\lambda }_0\right)=\mu \left({\lambda }_0\right)+k\left({\lambda }_0\right)\left(\frac{1+\mu \left({\lambda }_0\right)}{1+\beta \left({\lambda }_0\right)}\right)$$

and

$$k\left({\lambda }_0\right)=1+|b|\tau {\mathrm{e}}^{-{\lambda }_0\tau }+|c|(1+|{\lambda }_0|\sigma {\mathrm{e}}^{-{\lambda }_0\sigma }).$$

Moreover, the zero solution of (32) is described as follows:

(i) 

The solution is stable if ${\lambda }_0=0$, or, equivalently, if the following conditions are satisfied:

$$a+b=0,|b|\tau +|c|<1;$$

(ii) 

The solution is asymptotically stable if ${\lambda }_0<0$;

(iii) 

The solution is unstable if ${\lambda }_0>0$.

Proof. 

Assume that

$$z\left(t\right)={\mathrm{e}}^{-{\lambda }_{0}t}y\left(t\right),\hat{z}\left(t\right)=z\left(t\right)-\frac{L({\lambda }_0;\phi )}{1+\beta \left({\lambda }_0\right)}.$$

We can show that for $t\ge 0$,

$$z\left(t\right)\le \mu \left({\lambda }_0\right)H({\lambda }_0;\phi )+\frac{|L({\lambda }_0;\phi )|}{1+\beta \left({\lambda }_0\right)}.$$

Obviously, we can affirm that

$$\begin{array}{ccc}\hfill |L({\lambda }_0;\phi )|& \le & |\phi \left(0\right)|+|c||\phi (-\sigma )|+|b|{\mathrm{e}}^{-{\lambda }_0\tau }{\int }_{-\tau }^0{\mathrm{e}}^{-{\lambda }_{0}s}|\phi \left(s\right)|ds\hfill \\ & & +|c||{\lambda }_0|{\mathrm{e}}^{-{\lambda }_0\sigma }{\int }_{-\sigma }^0{\mathrm{e}}^{-{\lambda }_{0}s}|\phi \left(s\right)|ds\hfill \\ & \le & \left(1+|c|+|b|\tau {\mathrm{e}}^{-{\lambda }_0\tau }+|c||{\lambda }_0|\sigma {\mathrm{e}}^{-{\lambda }_0\sigma }\right)R({\lambda }_0;\phi )\hfill \\ & =& k\left({\lambda }_0\right)R({\lambda }_0;\phi ).\hfill \end{array}$$

We also can find that

$$\begin{array}{ccc}\hfill |H({\lambda }_0;\phi )|& \le & max\left\{1,R({\lambda }_0;\phi )+\frac{L({\lambda }_0;\phi )}{1+\beta \left({\lambda }_0\right)}\right\}\hfill \\ & \le & R({\lambda }_0;\phi )+\frac{L({\lambda }_0;\phi )}{1+\beta \left({\lambda }_0\right)}\hfill \\ & \le & R({\lambda }_0;\phi )+\frac{k\left({\lambda }_0\right)R({\lambda }_0;\phi )}{1+\beta \left({\lambda }_0\right)}\hfill \\ & =& \left(1+\frac{k\left({\lambda }_0\right)}{1+\beta \left({\lambda }_0\right)}\right)R({\lambda }_0;\phi ).\hfill \end{array}$$

So, for $t\ge 0$, we have

$$\begin{array}{ccc}\hfill |z(t)|& \le & \mu \left({\lambda }_0\right)\left(1+\frac{k\left({\lambda }_0\right)}{1+\beta \left({\lambda }_0\right)}\right)R({\lambda }_0;\phi )+\frac{k\left({\lambda }_0\right)R({\lambda }_0;\phi )}{1+\beta \left({\lambda }_0\right)}\hfill \\ & =& \left[\mu \left({\lambda }_0\right)\left(1+\frac{k\left({\lambda }_0\right)}{1+\beta \left({\lambda }_0\right)}\right)+\frac{k\left({\lambda }_0\right)}{1+\beta \left({\lambda }_0\right)}\right]R({\lambda }_0;\phi )\hfill \\ & =& N\left({\lambda }_0\right)R({\lambda }_0;\phi ).\hfill \end{array}$$

Finally, from the definition of z, we calculate

$$|y\left(t\right)|=N\left({\lambda }_0\right)R({\lambda }_0;\phi ){\mathrm{e}}^{-{\lambda }_{0}t},t\ge 0.$$

(37)

When ${\lambda }_0=0$,

$$|y(t)|=N\left(0\right)R(0;\phi ),t\ge 0.$$

Obviously, $\parallel \phi \parallel ={max}_{-r\le t\le 0}|\phi \left(t\right)|\le R(0;\phi )$, and thus it follows that

$$|y(t)|=N\left(0\right)R(0;\phi ),t\ge -r.$$

For arbitrary $ϵ>0$, there exists a constant $\delta =\frac{ϵ}{N\left(0\right)}$ such that $\parallel \phi \parallel \le R(0;\phi )\le \delta$. Then,

$$|y(t)|=N\left(0\right)R(0;\phi )<N\left(0\right)\delta =ϵ.$$

In addition, for ${\lambda }_0<0$, the inequality (37) guarantees that

$$\underset{t\to \infty }{lim}y\left(t\right)=0.$$

Hence, the zero solution of (32) is asymptotically stable when ${\lambda }_0<0$. □

According to Theorem 2 and Theorem 1, we can obtain the following theorem.

Theorem 4.

Assume that ${\lambda }_0$ is a real root of characteristic Equation (33) and satisfies (34). Then, the solutions $x\left(t\right)$ of (1)–(3) and $\tilde{x}\left(t\right)$ of (29)–(31) satisfy

$$\underset{t\to \infty }{lim}\left[{\mathrm{e}}^{-{\lambda }_{0}t}\left(x\left(t\right)-\tilde{x}\left(t\right)\right)\right]=\frac{L({\lambda }_0;\varphi -\tilde{\varphi })}{1+\beta \left({\lambda }_0\right)}.$$

Using Theorem 2 and Theorem 3, we can obtain the following theorem.

Theorem 5.

Assume that ${\lambda }_0$ is a real root of the characteristic Equation (32) and satisfies $\mu \left({\lambda }_0\right)<1$. Then, the solution $x\left(t\right)$ of (1)–(3) and $\tilde{x}\left(t\right)$ of (29)–(31) satisfies

$$|x\left(t\right)-\tilde{x}\left(t\right)|\le N\left({\lambda }_0\right)R({\lambda }_0;\varphi -\tilde{\varphi }){\mathrm{e}}^{{\lambda }_{0}t},\forall t\ge 0,$$

where

$$N\left({\lambda }_0\right)=\mu \left({\lambda }_0\right)+k\left({\lambda }_0\right)\left(\frac{1+\mu \left({\lambda }_0\right)}{1+\beta \left({\lambda }_0\right)}\right)$$

and

$$k\left({\lambda }_0\right)=1+|b|\tau {\mathrm{e}}^{-{\lambda }_0\tau }+|c|(1+|{\lambda }_0|\sigma {\mathrm{e}}^{-{\lambda }_0\sigma }).$$

Moreover, the solution of (1)–(3) is described as follows:

(i) 

The solution is stable if ${\lambda }_0=0$, or, equivalently, if the following conditions are satisfied:

$$a+b=0,|b|\tau +|c|<1;$$

(38)

(ii) 

The solution is asymptotically stable if ${\lambda }_0<0$;

(iii) 

The solution is unstable if ${\lambda }_0>0$.

3.2. Asymptotical Stability of Euler’s Method for IDDEs

To analyze the stability of the numerical method (4) for (1)–(3), we also consider the Euler method for IDDE (29)–(31), as follows:

$$\left\{\begin{array}{c}{\tilde{X}}_{n+1}={\tilde{X}}_n+ah{\tilde{X}}_n+bh{\tilde{X}}_{n-m_2}-c{\tilde{X}}_{n-m_1+1}+c{\tilde{X}}_{n-m_1},n\ne {\eta }_k-1,n-m_1\ne {\eta }_j-1\hfill \\ {\tilde{Y}}_k={\tilde{X}}_n+ah{\tilde{X}}_n+bh{\tilde{X}}_{n-m_2}-c{\tilde{X}}_{n-m_1+1}+c{\tilde{X}}_{n-m_1},n={\eta }_k-1,\hfill \\ {\tilde{X}}_{n+1}={\tilde{X}}_n+ah{\tilde{X}}_n+bh{\tilde{X}}_{n-m_2}-c{\tilde{Y}}_j+c{\tilde{X}}_{n-m_1},n-m_1={\eta }_j-1,\hfill \\ {\tilde{X}}_{{\eta }_k}={\tilde{Y}}_k+l_k,\hfill \\ {\tilde{X}}_i={\tilde{\varphi }}_i=\tilde{\varphi }\left(ih\right),ih\in \left[-r,0\right],i\in \left[-m,0\right].\hfill \end{array}\right.$$

(39)

Definition 4.

The Euler method for INDDE (1)–(3) is said to be stable if for every $ϵ>0$, there exists a number $\delta =\delta \left(ϵ\right)>0$ such that

$$\underset{-m\le i\le 0}{max}|{\varphi }_i-{\tilde{\varphi }}_i|<\delta ,$$

which implies that

$$|X_n-{\tilde{X}}_n|<ϵ,\mathrm{for}\mathrm{all}n\ge 0,$$

where $X_n$ and ${\tilde{X}}_n$ are obtained from (4) and (39), respectively.

The Euler method for INDDE (1)–(3) is said to be asymptotically stable if it is stable and there exists a number ${\delta }_1>0$ such that ${max}_{-m\le i\le 0}|\varphi \left(ih\right)-\tilde{\varphi }\left(ih\right)|<{\delta }_1$ implies

$$\underset{n\to \infty }{lim}|X_n-{\tilde{X}}_n|=0.$$

Denote $y_n=\delta X_n=X_n-{\tilde{X}}_n$,$n\ge -m$, $\delta {\varphi }_i=\varphi \left(ih\right)-\tilde{\varphi }\left(ih\right)$, $i=-m,\cdots ,0$. It is very interesting that we can obtain the following neat difference equation from (4) and (39):

$$y_{n+1}=y_n+ah{y}_n+bh{y}_{n-m_2}-c{y}_{n-m_1+1}+c{y}_{n-m_1},\forall n\in \mathbb{N},$$

(40)

$$y_i=\delta {\varphi }_i,i=-m,\cdots ,0.$$

(41)

The characteristic equation of (40) is

$$(\lambda -1)(1+c{\lambda }^{-m_1})=ha+hb{\lambda }^{-m_2}.$$

(42)

Applying [26] (Theorem 1) to the differential Equations (40) and (41), we can obtain the following theorem of the Euler method for INDDEs.

Theorem 6.

Assume that ${\lambda }_1$ is a positive root of characteristic Equation (42) and satisfies

$${\mu }_{{\lambda }_1}=|c|-m_1{\lambda }_1^{-m_1}|c||1-\frac{1}{{\lambda }_1}|+\frac{1}{{\lambda }_1}h{m}_2|b|{\lambda }_1^{-m_2}<1.$$

(43)

Then, the solutions $X_n$ obtained from (4) and ${\tilde{X}}_n$ obtained from (39) satisfy

$$\underset{n\to \infty }{lim}{\lambda }_1^{-n}|X_n-{\tilde{X}}_n|=\frac{L_{{\lambda }_1}\left(\varphi \right)}{1+{\gamma }_{{\lambda }_1}},$$

where

$$L_{{\lambda }_1}\left(\varphi \right)={\varphi }_0+c{\varphi }_{-m_1}-(1-\frac{1}{{\lambda }_1})c{\lambda }_1^{-m_1}\sum _{s=-m_1}^{-1}{\lambda }_1^{-s}{\varphi }_s+\frac{1}{{\lambda }_1}hb{\lambda }_1^{-m_2}\sum _{s=-m_2}^{-1}{\lambda }_1^{-s}{\varphi }_s$$

and

$${\gamma }_{{\lambda }_1}=\sum c\left(1-(1-\frac{1}{{\lambda }_1})m_1\right){\lambda }_1^{-m_1}+\frac{1}{{\lambda }_1}b{m}_2{\lambda }_1^{-m_2}.$$

Similarly, applying [26] (Theorem 2) to (40) and (41), we can obtain the following theorem.

Theorem 7.

Assume that ${\lambda }_1$ is a real root of characteristic Equation (42) and satisfies ${\mu }_{{\lambda }_1}<1$. Then, the solutions $X_n$ obtained from (4) and ${\tilde{X}}_n$ obtained from (39) satisfy

$$|X_n-{\tilde{X}}_n|\le N_{{\lambda }_1}\parallel \varphi \parallel {\lambda }_1^n,\forall n\ge 0,$$

where

$$N_{{\lambda }_1}=\frac{1+{\mu }_{{\lambda }_1}}{1+{\gamma }_{{\lambda }_1}}+\left(1+\frac{1+{\mu }_{{\lambda }_1}}{1+{\gamma }_{{\lambda }_1}}\right){\mu }_{{\lambda }_1}max\{1,{\lambda }_1^r\}.$$

Moreover, the Euler method (4) for INDDE (1)–(3) is described as follows:

(i) 

The solution is stable if ${\lambda }_1=1$, or, equivalently, if the following conditions are satisfied:

$$a+b=0,|c|+h|b|m_2<1;$$

(44)

(ii) 

The solution is asymptotically stable if ${\lambda }_1<1$;

(iii) 

The solution is unstable if ${\lambda }_1>1$.

Corollary 1.

If the condition (38) holds, the Euler method (4) preserves the stable property of INDDE (1)–(3) without additional restrictions on the stepsize.

Proof. 

Using Theorem 5, we find that the solution $x\left(t\right)$ of INDDE (1)–(3) is stable. Because $m_2=\lfloor \frac{\tau }{h}\rfloor$, it is easy to conclude that $m_{2}h\le \tau$. Hence, the condition (38) holds, implying that (44) holds. Based on Theorem 7, we can affirm that the Euler method (4) for INDDE (1)–(3) is also stable. □

4. Numerical Examples

In this section, two examples are given to illustrate the conclusions of this paper.

Example 1.

Consider the following INDDE:

$${\left(x\left(t\right)-\frac{1}{3\mathrm{e}}x(t-\frac{1}{2})\right)}^{\prime }=\frac{1}{3}x\left(t\right)-\frac{1}{\mathrm{e}}x(t-\frac{1}{5}),t\ge 0,t\ne k,$$

(45)

$$\Delta x\left({\tau }_k\right)={\left(-\frac{1}{\mathrm{e}}\right)}^k,{\tau }_k=k,k\in {\mathbb{Z}}^{+},$$

(46)

$$x\left(t\right)=\varphi \left(t\right),-\frac{1}{2}\le t\le 0.$$

(47)

The characteristic equation of (45) is

$$\lambda \left(1-\frac{1}{3\mathrm{e}}\right)=\frac{1}{3}-\frac{1}{\mathrm{e}}{\mathrm{e}}^{-\frac{\lambda }{5}}.$$

(48)

Solving (48), we find that ${\lambda }_1\approx -3.94$ and ${\lambda }_2\approx -0.043$. Let ${\lambda }_0={\lambda }_2$, which implies that $\mu \left({\lambda }_0\right)<1$. Using Theorem 5, we can conclude that the exact solution of (45)–(47) is asymptotically stable in the sense of Definition 2.

Let $h=\frac{1}{i}$, $i\in {\mathbb{Z}}^{+}$, with i being divisible by 10. The characteristic Equation (42) of (40) for (45) is then changed into

$$(\lambda -1)(1-\frac{{\lambda }^{-\frac{i}{2}}}{3\mathrm{e}})=\frac{1}{3i}-\frac{{\lambda }^{-\frac{i}{5}}}{i\mathrm{e}}.$$

(49)

When $i=10$, (49) is changed into $f\left(\lambda \right)=0$, where

$$f\left(\lambda \right)=(\lambda -1)(1-\frac{{\lambda }^{-5}}{3\mathrm{e}})-\frac{1}{30}+\frac{{\lambda }^{-2}}{10\mathrm{e}}.$$

(50)

A root of $f\left(\lambda \right)=0$ is ${\lambda }_1\approx 0.995685180437323<1$. It is easy to verify that ${\mu }_{{\lambda }_1}<1$. Applying Theorem 7, we can conclude that the Euler method (4) for (45)–(47) is asymptotically stable for $h=\frac{1}{10}$ (see Figure 1).

In

Table 1. The global errors of the Euler scheme for INDDE (45)–(47).

Stepsize	${\mathit{e}}_{6/5}$	Ratio	${\mathit{e}}_{7/5}$	Ratio
1/20	0.0066590383		6.0532924880 $\times {10}^{-4}$
1/40	0.0033576411	0.5042231270	3.3600485518 $\times {10}^{-4}$	0.5550778454
1/80	0.0016859400	0.5021203734	1.7654071490 $\times {10}^{-4}$	0.52541120220
1/160	8.4476115143 $\times {10}^{-4}$	0.5010624028	9.0430888108 $\times {10}^{-5}$	0.5122381438
1/320	4.2282978351 $\times {10}^{-4}$	0.5005317572	4.5758856992 $\times {10}^{-5}$	0.5060091519

, the global errors at $t=\frac{6}{5}$ and $t=\frac{7}{5}$ of the Euler method (4) for (45)–(47) are represented by $e_{6/5}$ and $e_{7/5}$, respectively. As can be seen from the ratios in this table, when the stepsizes are halved, the global errors become about half of the originals, which roughly shows that the Euler method (4) for (45)–(47) is convergent of order 1.

According to Figure 2, the numerical solution obtained from (4) for (45)–(47) does not make additional jumps, but rather only jumps near $t=k$, $k\in \mathbb{N}$, which is consistent with the nature of the exact solution (see (a) of Definition 1).

Example 2.

Consider the following INDDE:

$${\left(x\left(t\right)+\frac{1}{2}x(t-1)\right)}^{\prime }=x\left(t\right)+\frac{1}{2}x(t-1),t\ge 0,t\ne 2k$$

(51)

$$\Delta x\left({\tau }_k\right)=k,{\tau }_k=2k,k\in {\mathbb{Z}}^{+},$$

(52)

$$x\left(t\right)=\varphi \left(t\right),-1\le t\le 0.$$

(53)

The characteristic equation of (51) is

$$\lambda \left(2+{\mathrm{e}}^{-\lambda }\right)=2+{\mathrm{e}}^{-\lambda },$$

(54)

which implies that $\lambda =1$ is the unique real root of (54). For ${\lambda }_0=\lambda =1$,

$$\mu \left({\lambda }_0\right)=\frac{1}{2\mathrm{e}}+\frac{1}{\mathrm{e}}<1.$$

Consequently, the exact solution of (51)–(53) is unstable, based on Theorem 3.

Let $h=\frac{1}{i}$, $i\in {\mathbb{Z}}^{+}$, with i being divisible by 10. The characteristic Equation (42) of (40) for (51) is then changed into

$$(\lambda -1)(1+\frac{{\lambda }^{-i}}{2})=\frac{1}{i}+\frac{{\lambda }^{-i}}{2i}.$$

(55)

When $i=10$, (55) is changed into $g\left(\lambda \right)=0$, where

$$g\left(\lambda \right)=(\lambda -1)(1+\frac{{\lambda }^{-10}}{2})-\frac{1}{10}-\frac{{\lambda }^{-10}}{20}.$$

(56)

A root of $g\left(\lambda \right)=0$ is ${\lambda }_1\approx 1.1>1$. It is easy to verify that ${\mu }_{{\lambda }_1}<1$. Applying Theorem 7, we can affirm that the Euler method (4) for (51)–(53) is unstable for $h=\frac{1}{10}$ (see Figure 3).

In

Table 2. The global errors of the Euler scheme for INDDE (51)–(53).

Stepsize	${\mathit{e}}_3$	Ratio	${\mathit{e}}_4$	Ratio
1/100	0.0134679990		0.0730382471
1/200	0.0067647055	0.5022799238	0.0367309910	0.5029007739
1/400	0.0033900841	0.5011428898	0.0184189152	0.5014543502
1/800	0.0016969818	0.5005721798	0.0092228697	0.5007281696
1/1600	8.4897669001 $\times {10}^{-4}$	0.5002862740	0.0046147951	0.5003643339

, the global errors at $t=3$ and $t=4$ of the Euler method (4) for (51)–(53) are represented by $e_3$ and $e_4$, respectively. As can be seen from the ratios in this table, when the stepsizes are halved, the global errors decrease to about half of the originals, which roughly shows that the Euler method (4) for (51)–(53) is convergent of order 1.

5. Conclusions and Future Work

In this research, we have introduced a new numerical scheme based on the Euler method for solving linear impulsive neutral differential equations. It is rigorously proven that the constructed numerical method is convergent of order 1. Additionally, we have determined the conditions under which the numerical solutions maintain the asymptotic stability of the exact solutions.

Overall, we find that the numerical methods constructed in this article are only convergent of order 1. Future work will focus on developing higher-order numerical methods for INDDEs.