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A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.

In this paper, we study the stability of nonlinear impulsive differential equations:

Impulsive differential equations are widely used in actual modeling such as epidemic, optimal control and population dynamics (see [

The rest of this paper is organized as follows. In Section

Recently, a new technique has been introduced in the stability analysis of exact and numerical solutions to impulsive differential equations by constructing equivalent equations (see [

Assume that the function

It is easy to check that

Consider the equation

In the case that

Assume that Hypothesis

If

If

(1) It is obvious that

(2) On each interval

Let

To study the stability of (

The zero solution of (

The zero solution of (

Assume that

if

if

Note that

By Theorem

Assume that

if

if

To investigate the stability of (

In this section, some specific stability conditions are given based on the choice of

Taking

Assume that

The zero solution of (

Assume that

Taking

If there exists a constant

Note that it is difficult to find sequences

Assume that the following holds:

Note that while we take

By Theorem

In this section, we establish a convergent numerical process of

The application of one-stage

Let

If we take

If we take

The convergence of numerical processes (

Assume that

On each interval

In this section, we study the stability property of numerical solutions. As usual, we expect that the numerical solution can reproduce the property of the exact solutions. The stability of the numerical solutions can be defined according to Theorem

A numerical method is said to be stable for (

A numerical method is said to be asymptotically stable for (

Denote

It follows from (

Assume that (

By difference equation (

Using a similar technique to the one in [

Assume that

It follows from (

The implicit Euler method can preserve the asymptotic stability of (

Assume that

It is necessary to find other stability criteria of numerical process (

Assume that

Note that (

Therefore,

The implicit Euler method can preserve the asymptotic stability of (

If

In this section, we will do some numerical experiments to illustrate the conclusion.

Consider the following equations:

Numerical solutions of (

The difference of numerical solutions to (

The authors declare that they have no conflicts of interest.

This work is supported by the National Natural Science Foundation of China (Grant no. 11505090) and Shandong Provincial Natural Science Foundation (no. ZR2017BA026 and no. BS2015SF009).