﻿ 一类脉冲微分方程解的增长

一类脉冲微分方程解的增长Growth of Solutions for a Class of Impulsive Differential Equations

Abstract: In this paper, by using the equivalent relation between the impulsive differential equation and measure differential equation, and according to the asymptotic behavior of the solution for the measure differential equation, we get the asymptotic behavior of the solution for the impulsive differential equation.

1. 引言

${x}^{\prime }\left(t\right)=f\left(x\left(t\right),t\right)$

${\Delta }^{+}x\left({\tau }_{j}\right)={I}_{j}\left(x\left({\tau }_{j}\right)\right),\begin{array}{c}\end{array}j\in {Z}_{+}$

${x}^{\Delta }\left(t\right)=f\left(x\left(t\right),t\right),\begin{array}{c}\end{array}t\in T$

${x}^{\prime }\left(t\right)=f\left(x\left(t\right),t\right)$ (1.1)

${\Delta }^{+}x\left({\tau }_{j}\right)={I}_{j}\left(x\left({\tau }_{j}\right)\right),\begin{array}{c}\end{array}j\in {Z}_{+}$ (1.2)

$x\left(t\right)=x\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}f\left(x\left(s\right),s\right)\text{d}s+\underset{\begin{array}{c}j\in {Z}_{+}\\ {t}_{0}\le {\tau }_{j} (1.3)

(1.3)式右端积分为Lebsuge积分。本文中，我们考虑更为一般的情形，(1.3)式右端积分可被理解为Kurzweil-Henstock意义下的积分。由于 $\underset{j\to +\infty }{\mathrm{lim}}{\tau }_{j}=+\infty$，所以脉冲点至多是有限个，(1.3)式右端和式是有意义的。

$X$ 按范数 $‖\cdot ‖$ 成为一个Banach空间， ${B}_{c}=\left\{x\in X|‖x‖\le c\right\}$$\Omega ={B}_{c}×\left[{t}_{0},+\infty \right)$。记 $x\left(t,{s}_{0},{x}_{0}\right)$ 为满足 $x\left({s}_{0},{s}_{0},{x}_{0}\right)={x}_{0}$ 的饱和解。 $G\left(\left[a,b\right],{B}_{c}\right)$ 表示所有的正则函数 $f:\left[a,b\right]\to {B}_{c}$$G\left(\left[a,b\right],{B}_{c}\right)$ 表示由 $f:\left[{t}_{0},+\infty \right)\to X$ 所构成的向量空间，并且满足对所有的 $\left[a,b\right]\subset \left[{t}_{0},+\infty \right)$，有 ${f|}_{\left[a,b\right]}\in G\left(\left[a,b\right],{B}_{c}\right)$

${G}_{0}\left(\left[{t}_{0},+\infty \right),X\right)=\left\{f\in G\left(\left[{t}_{0},+\infty \right),X\right):\underset{s\in \left[{t}_{0},+\infty \right)}{\mathrm{sup}}{\text{e}}^{-\gamma \left(s-{t}_{0}\right)}‖f\left(s\right)‖<\infty \right\}$${f|}_{\left[a,b\right]}$ 表示函数 $f$ 在区间 $\left[a,b\right]$ 上的限制， ${Z}_{+}$ 表示正整数集， ${R}^{+}$ 表示正实数集。

2. 预备知识

$D=\left\{\left({\tau }_{j},\left[{\alpha }_{j-1},{\alpha }_{j}\right]\right),j=1,2,\cdots ,k\right\}$，其中 ${\tau }_{j}\in \left[{\alpha }_{j-1},{\alpha }_{j}\right]\subset \left[{\tau }_{j}-\delta \left({\tau }_{j}\right),{\tau }_{j}+\delta \left({\tau }_{j}\right)\right]$，有

$‖\underset{j=1}{\overset{k}{\sum }}\left[U\left({\tau }_{j},{\alpha }_{j}\right)-U\left({\tau }_{j},{\alpha }_{j-1}\right)\right]-I‖<\epsilon$

$‖F\left(x,{s}_{2}\right)-F\left(x,{s}_{1}\right)‖\le |h\left({s}_{2}\right)-h\left({s}_{1}\right)|,$ (2.1)

$‖F\left(x,{s}_{2}\right)-F\left(x,{s}_{1}\right)-F\left(y,{s}_{2}\right)+F\left(y,{s}_{1}\right)‖\le |h\left({s}_{2}\right)-h\left({s}_{1}\right)|‖x-y‖.$

(A1) 函数 $g:\left[{t}_{0},+\infty \right)\to R$$\left({t}_{0},+\infty \right)$ 上是不减左连续的。

(A2) 对每个 $x\in G\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\in \left[{t}_{0},+\infty \right)$ 积分 ${\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(s\right),s\right)\text{d}g\left(s\right)$ 存在。

(A3) 存在局部Kurzweil可积函数 $M:\left[{t}_{0},+\infty \right)\to {R}_{+}$，使得对任意的 $x,z\in {G}_{0}\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\subseteq \left[{t}_{0},+\infty \right)$${u}_{1}\ge {u}_{2}$ 时，有

$‖{\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(t\right),t\right)\text{d}g\left(t\right)‖\le {\int }_{{u}_{1}}^{{u}_{2}}M\left(t\right)\text{d}g\left(t\right).$

(A4) 存在局部Kurzweil可积函数 $L:\left[{t}_{0},+\infty \right)\to {R}_{+}$，使得对任意的 $x,z\in {G}_{0}\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\subseteq \left[{t}_{0},+\infty \right)$${u}_{1}\ge {u}_{2}$ 时，有

$‖{\int }_{{u}_{1}}^{{u}_{2}}\left[f\left(x\left(t\right),t\right)-f\left(z\left(t\right),t\right)\right]dg\left(t\right)‖\le {‖x-z‖}_{\left[{t}_{0},+\infty \right)}{\int }_{{u}_{1}}^{{u}_{2}}L\left(t\right)dg\left(t\right).$

${\tau }_{0}\in \left[{t}_{0},+\infty \right)$ 并且定义 $F:{B}_{c}×\left[{\tau }_{0},+\infty \right)\to {R}^{n}$

$F\left(x,t\right)={\int }_{{\tau }_{0}}^{t}f\left(x,s\right)\text{d}g\left(s\right),\begin{array}{c}\end{array}\left(x,t\right)\in {B}_{c}×\left[{\tau }_{0},+\infty \right),$

$F\in \mathcal{F}\left(\Omega ,h\right)$。其中 $\Omega ={B}_{c}×\left[{\tau }_{0},+\infty \right)$$h:\left[{\tau }_{0},+\infty \right)\to R$ 是不减左连续的，

$h\left(t\right)={\int }_{{\tau }_{0}}^{t}\left[M\left(s\right)+L\left(s\right)\right]\text{d}g\left(s\right),\begin{array}{c}\end{array}t\in \left[{\tau }_{0},+\infty \right).$

$x\left(t\right)=x\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}f\left(x\left(s\right),s\right)\text{d}g\left(s\right)$ (2.2)

(U1) 对测度微分方程(2.2)的每个解 $x:I\to {B}_{c}$

${W}_{1}\left(‖x\left(t\right)‖\right)\le U\left(t,x\left(t\right)\right)\le {W}_{2}\left(‖x\left(t\right)‖\right),$

(U2) 对测度微分方程(2.2)的每个饱和解 $x\left(t\right)=x\left(t,{s}_{0},{x}_{0}\right)$$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，有

$U\left(t,x\left(t\right)\right)-U\left(s,x\left(s\right)\right)\ge {\int }_{s}^{t}{W}_{3}\left(‖x\left(\xi \right)‖\right)\text{d}l\left(\xi \right),$

$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，则 $\omega \left({s}_{0},{x}_{0}\right)<\infty$

(H1) 对测度微分方程(2.2)的每个解 $x:I\to {B}_{c}$

$|U\left(t,x\left(t\right)\right)|\le {W}_{1}\left(‖x\left(t\right)‖\right),$

(H2) 对测度微分方程(2.2)的每个饱和解 $x\left(t\right)=x\left(t,{s}_{0},{x}_{0}\right)$$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，函数 $t\to U\left(t,x\left(t\right)\right)$$\left[{s}_{0},\omega \left({s}_{0},{x}_{0}\right)\right)$ 上正则。有

$U\left(t,x\left(t\right)\right)-U\left(s,x\left(s\right)\right)\le -{\int }_{s}^{t}{W}_{2}\left(|U\left(\xi ,x\left(\xi \right)|\right)\text{d}l\left(\xi \right),$

$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，使得 $U\left({s}_{0},{x}_{0}\right)<0$，则 $\omega \left({s}_{0},{x}_{0}\right)<\infty$

${\int }_{a}^{b}\stackrel{˜}{f}\left(s\right)\text{d}g\left(s\right)={\int }_{a}^{b}f\left(s\right)\text{d}s+\underset{j=1}{\overset{k}{\sum }}\stackrel{˜}{f}\left({\eta }_{j}\right).$

$x\left(t\right)=x\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}\stackrel{˜}{f}\left(x\left(s\right),s\right)\text{d}g\left(s\right),\begin{array}{c}\end{array}t\in \left[a,b\right]$ (2.3)

$\stackrel{˜}{f}\left(z,t\right)=\left\{\begin{array}{cc}f\left(z,t\right),& \left[{t}_{0},+\infty \right)/\left\{{\tau }_{j};j\in {Z}_{+}\right\},\\ {I}_{j}\left(z\right),& t={\tau }_{j},j\in {Z}_{+}.\end{array}$ (2.4)

$g:\left[{t}_{0},+\infty \right)\to R$

$g\left(t\right)=t+j,\begin{array}{c}\end{array}t\in \left({\tau }_{j},{\tau }_{j+1}\right],\begin{array}{c}\end{array}j\in {Z}_{+}$ (2.5)

3. 主要结果

(B1) 对每个 $x\in G\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\in \left[{t}_{0},+\infty \right)$ 积分 ${\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(t\right),t\right)\text{d}t$ 存在。

(B2) 存在局部Kurzweil可积函数 $m:\left[{t}_{0},+\infty \right)\to {R}_{+}$，对任意的 $x,y\in G\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$$\left[{u}_{1},{u}_{2}\right]\subseteq \left[{t}_{0},+\infty \right)$${u}_{1}\ge {u}_{2}$，有

$‖{\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(t\right),t\right)\text{d}t‖\le {\int }_{{u}_{1}}^{{u}_{2}}m\left(t\right)\text{d}t,$

$‖{\int }_{{u}_{1}}^{{u}_{2}}\left[f\left(x\left(t\right),t\right)-f\left(z\left(t\right),t\right)\right]\text{d}t‖\le {‖x-z‖}_{\left[{t}_{0},+\infty \right)}{\int }_{{u}_{1}}^{{u}_{2}}m\left(t\right)\text{d}t.$

(B3) 对每个 $j\in {Z}_{+}$，存在常数 ${m}_{j}\ge 0$ 使得

$‖{I}_{j}\left(x\right)‖\le {m}_{j},\begin{array}{c}\end{array}‖{I}_{j}\left(x\right)-{I}_{j}\left(z\right)‖\le {‖x-z‖}_{\left[{t}_{0},+\infty \right)}{m}_{j}.$

(A1) 函数 $g:\left[{t}_{0},+\infty \right)\to R$$\left({t}_{0},+\infty \right)$ 上是不减左连续的。

(A2) 对每个 $x\in G\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\in \left[{t}_{0},+\infty \right)$ 积分 ${\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(s\right),s\right)\text{d}g\left(s\right)$ 存在。

(A3) 存在局部Kurzweil可积函数 $M:\left[{t}_{0},+\infty \right)\to {R}_{+}$，使得对任意的 $x,z\in {G}_{0}\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\subseteq \left[{t}_{0},+\infty \right),$${u}_{1}\ge {u}_{2}$ 时，有

$‖{\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(t\right),t\right)\text{d}g\left(t\right)‖\le {\int }_{{u}_{1}}^{{u}_{2}}M\left(t\right)\text{d}g\left(t\right).$

(A4) 存在局部Kurzweil可积函数 $L:\left[{t}_{0},+\infty \right)\to {R}_{+}$，使得对任意的 $x,z\in {G}_{0}\left(\left[{t}_{0},+\infty \right),{B}_{c}\right)$${u}_{1},{u}_{2}\subseteq \left[{t}_{0},+\infty \right)$${u}_{1}\ge {u}_{2}$ 时，有

$‖{\int }_{{u}_{1}}^{{u}_{2}}\left[f\left(x\left(t\right),t\right)-f\left(z\left(t\right),t\right)\right]\text{d}g\left(t\right)‖\le {‖x-z‖}_{\left[{t}_{0},+\infty \right)}{\int }_{{u}_{1}}^{{u}_{2}}L\left(t\right)\text{d}g\left(t\right).$

$j\in {Z}_{+}$ 时，由(2.5)式有

$\begin{array}{c}{\Delta }^{+}g\left({\tau }_{j}\right)=\underset{t\to {\tau }_{j}{}^{+}}{\mathrm{lim}}g\left(t\right)-g\left({\tau }_{j}\right)\\ ={\tau }_{j}^{+}+j-\left({\tau }_{j}+j-1\right)\\ =1.\end{array}$

$\left[u,v\right]\cap \left\{{\tau }_{1},\cdots ,{\tau }_{k}\right\}=0/$ 时，由(2.5)式有

$g\left(t\right)-g\left(u\right)=t-u.$

$\stackrel{˜}{m}\left(t\right)=\left\{\begin{array}{cc}m\left(t\right),& t\in \left[{t}_{0},+\infty \right)/\left\{{\tau }_{j};j\in {Z}_{+}\right\},\\ {m}_{j},& t={\tau }_{j},j\in {Z}_{+}.\end{array}$

$\begin{array}{c}‖{\int }_{{u}_{1}}^{{u}_{2}}\stackrel{˜}{f}\left(x\left(t\right),t\right)\text{d}g\left(t\right)‖=‖{\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(t\right),t\right)\text{d}t+\underset{j=1}{\overset{k}{\sum }}\stackrel{˜}{f}\left(x\left({\tau }_{j}\right),{\tau }_{j}\right)‖\\ =‖{\int }_{{u}_{1}}^{{u}_{2}}f\left(x\left(t\right),t\right)\text{d}t+\underset{\begin{array}{c}j\in {Z}_{+}\\ {u}_{1}\le {\tau }_{j}<{u}_{2}\end{array}}{\sum }{I}_{j}\left(x\right)‖\\ ={\int }_{{u}_{1}}^{{u}_{2}}m\left(t\right)\text{d}t+\underset{\begin{array}{c}j\in {Z}_{+}\\ \text{u}{u}_{1}\le {\tau }_{j}<{u}_{2}\end{array}}{\sum }{m}_{j}\\ ={\int }_{{u}_{1}}^{{u}_{2}}m\left(t\right)\text{d}t+\underset{j=1}{\overset{k}{\sum }}\stackrel{˜}{m}\left({\tau }_{j}\right)\\ ={\int }_{{u}_{1}}^{{u}_{2}}\stackrel{˜}{m}\left(t\right)\text{d}g\left(t\right).\end{array}$

$\begin{array}{c}‖{\int }_{{u}_{1}}^{{u}_{2}}\left[\stackrel{˜}{f}\left(x,t\right)-\stackrel{˜}{f}\left(z,t\right)\right]\text{d}g\left(t\right)‖=‖{\int }_{{u}_{1}}^{{u}_{2}}\left(f\left(x,t\right)-f\left(z,t\right)\right)\text{d}t+\underset{j=1}{\overset{k}{\sum }}\left[\stackrel{˜}{f}\left(x\left({\tau }_{j}\right),{\eta }_{j}\right)-\stackrel{˜}{f}\left(z\left({\tau }_{j}\right),{\eta }_{j}\right)\right]‖\\ =‖{\int }_{{u}_{1}}^{{u}_{2}}\left(f\left(x,t\right)-f\left(z,t\right)\right)\text{d}t+\underset{\begin{array}{c}j\in {Z}_{+}\\ \text{u}{u}_{1}\le {\tau }_{j}<{u}_{2}\end{array}}{\sum }\left[{I}_{j}\left(x\right)-{I}_{j}\left(z\right)\right]‖\\ ={‖x-z‖}_{\left[{t}_{0},+\infty \right)}{\int }_{{u}_{1}}^{{u}_{2}}m\left(t\right)\text{d}t+\underset{\begin{array}{c}j\in {Z}_{+}\\ {u}_{1}\le {\tau }_{j}<{u}_{2}\end{array}}{\sum }{‖x-z‖}_{\left[{t}_{0},+\infty \right)}{m}_{\text{j}j}\\ ={‖x-z‖}_{\left[{t}_{0},+\infty \right)}{\int }_{{u}_{1}}^{{u}_{2}}m\left(t\right)\text{d}t+\underset{j=1}{\overset{k}{\sum }}{‖x-z‖}_{\left[{t}_{0},+\infty \right)}\stackrel{˜}{m}\left({\tau }_{j}\right)\\ ={‖x-z‖}_{\left[{t}_{0},+\infty \right)}{\int }_{{u}_{1}}^{{u}_{2}}\stackrel{˜}{m}\left(t\right)\text{d}g\left(t\right).\end{array}$

(J1) 对脉冲微分方程(1.3)的每个解 $x:I\to {B}_{c}$

${W}_{1}\left(‖x\left(t\right)‖\right)\le U\left(t,x\left(t\right)\right)\le {W}_{2}\left(‖x\left(t\right)‖\right),$

(J2) 对脉冲微分方程(1.3)的每个饱和解 $x\left(t\right)=x\left(t,{s}_{0},{x}_{0}\right)$$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，有

$U\left(t,x\left(t\right)\right)-U\left(s,x\left(s\right)\right)\ge {\int }_{s}^{t}{W}_{3}\left(‖x\left(\xi \right)‖\right)\text{d}l\left(\xi \right),$

$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，则 $\omega \left({s}_{0},{x}_{0}\right)<\infty$

${s}_{0}\in \left[{t}_{0},+\infty \right)/\left\{{\tau }_{j},j\in {Z}_{+}\right\}$ 时，由(2.5)式有

$\begin{array}{c}{\Delta }^{+}g\left({s}_{0}\right)=\underset{t\to {s}_{0}{}^{+}}{\mathrm{lim}}g\left(t\right)-g\left({s}_{0}\right)\\ ={s}_{0}{}^{+}+j-\left({s}_{0}+j\right)\\ =0.\end{array}$

${x}_{0}+\stackrel{˜}{f}\left({x}_{0},{s}_{0}\right){\Delta }^{+}g\left({s}_{0}\right)={x}_{0}\in {B}_{c}$

${s}_{0}={\tau }_{j}$ 时， $j\in {Z}_{+}$ 由(2.5)式有

$\begin{array}{c}{\Delta }^{+}g\left({s}_{0}\right)=\underset{t\to {s}_{0}{}^{+}}{\mathrm{lim}}g\left(t\right)-g\left({s}_{0}\right)\\ ={s}_{0}{}^{+}+j-\left({s}_{0}+j-1\right)\\ =1.\end{array}$

${x}_{0}+\stackrel{˜}{f}\left({x}_{0},{s}_{0}\right){\Delta }^{+}g\left({s}_{0}\right)={x}_{0}+{I}_{j}\left({x}_{0}\right)\in {B}_{c}$

$U:\left[{t}_{0},+\infty \right)×{R}^{n}\to R$ 和楔子 ${W}_{\text{i}i}:\left[0,+\infty \right)\to \left[0,+\infty \right)$$i=1,2,3$，满足(J1)，(J2)其中 ${B}_{c}$${R}^{n}$ 代替。若 $\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{R}^{n}$。则 $\omega \left({s}_{0},{x}_{0}\right)=+\infty$。并且当 $t\to +\infty$ 时， $‖x\left(t,{s}_{0},{x}_{0}\right)‖\to +\infty$

$\gamma$ 是正数满足 ${W}_{2}\left(\gamma \right)={W}_{1}\left(‖{x}_{0}‖\right)$ 这样的 $\gamma$ 的存在性由 ${W}_{2}$ 的连续性保证。由(J2)有函数 $\left[{s}_{0},\omega \left({s}_{0},{x}_{0}\right)\right)∍↦V\left(t,x\left(t\right)\right)$ 是不减的。则由(J1)对每个 $t\in \left[{s}_{0},\omega \right)$，有

${W}_{2}\left(‖x\left(t\right)‖\right)\ge V\left(t,x\left(t\right)\right)\ge V\left({s}_{0},x\left({s}_{0}\right)\right)\ge {W}_{1}\left(‖x\left({s}_{0}\right)‖\right)={W}_{2}\left(\gamma \right).$ (3.1)

$‖x\left(t\right)‖\ge \gamma ,\begin{array}{c}\end{array}t\in \left[{s}_{0},+\infty \right).$

$\begin{array}{c}V\left(t,x\left(t\right)\right)\ge V\left({s}_{0},x\left({s}_{0}\right)\right)+{\int }_{{s}_{0}}^{t}{W}_{3}\left(‖x\left(s\right)‖\right)\text{d}l\left(s\right)\\ \ge V\left({s}_{0},x\left({s}_{0}\right)\right)+{W}_{3}\left(\gamma \right)\left(l\left(t\right)-l\left({s}_{0}\right)\right).\end{array}$

${W}_{2}\left(‖x\left(t\right)‖\right)\ge V\left({s}_{0},x\left({s}_{0}\right)\right)+{W}_{3}\left(\gamma \right)\left(l\left(t\right)-l\left({s}_{0}\right)\right).$

(K1) 对脉冲微分方程(1.3)的每个解 $x:I\to {B}_{c}$

$|U\left(t,x\left(t\right)\right)|\le {W}_{1}\left(‖x\left(t\right)‖\right),$

(K2) 对脉冲微分方程(1.3)的每个饱和解 $x\left(t\right)=x\left(t,{s}_{0},{x}_{0}\right)$$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$，函数 $t↦U\left(t,x\left(t\right)\right)$$\left[{s}_{0},\omega \left({s}_{0},{x}_{0}\right)\right)$ 上是正则的，有

$U\left(t,x\left(t\right)\right)-U\left(s,x\left(s\right)\right)\le {\int }_{s}^{t}{W}_{2}\left(|U\left(\xi ,x\left(\xi \right)\right)|\right)\text{d}l\left(\xi \right),$

$\left({s}_{0},{x}_{0}\right)\in \left[{t}_{0},+\infty \right)×{B}_{c}$ 使得 $U\left({s}_{0},{x}_{0}\right)<0$，则 $\omega \left({s}_{0},{x}_{0}\right)<\infty$

$|V\left(t,x\left(t\right)\right)|\le {W}_{1}\left(‖x\left(t\right)‖\right)\le {W}_{1}\left(c\right),\begin{array}{c}\end{array}t\in \left[{s}_{0},+\infty \right).$

$V\left(t,x\left(t\right)\right)-V\left({s}_{0},x\left({s}_{0}\right)\right)\le {\int }_{{s}_{0}}^{t}{W}_{2}\left(|V\left(\xi ,x\left(\xi \right)\right)|\right)\text{d}l\left(\xi \right)\le 0.$

$V\left(t,x\left(t\right)\right)\le V\left({s}_{0},x\left({s}_{0}\right)\right)=V\left({s}_{0},{x}_{0}\right)<0,\begin{array}{c}\end{array}t\in \left[{s}_{0},+\infty \right),$

$|V\left(t,x\left(t\right)\right)|\ge |V\left({s}_{0},x\left({s}_{0}\right)\right)|>0,\begin{array}{c}\end{array}t\in \left[{s}_{0},+\infty \right).$ (3.4)

$\begin{array}{c}V\left(t,x\left(t\right)\right)\le V\left({s}_{0},x\left({s}_{0}\right)\right)-{\int }_{{s}_{0}}^{t}{W}_{2}\left(|V\left(s,x\left(s\right)\right)|\right)\text{d}l\left(s\right)\\ \le V\left({s}_{0},x\left({s}_{0}\right)\right)-{\int }_{{s}_{0}}^{t}{W}_{2}\left(|V\left({s}_{0},x\left({s}_{0}\right)\right)|\right)\text{d}l\left(s\right)\\ =V\left({s}_{0},x\left({s}_{0}\right)\right)-{W}_{2}\left(|V\left({s}_{0},x\left({s}_{0}\right)\right)|\right)\left(l\left(t\right)-l\left({s}_{0}\right)\right)<0\end{array}$

$\begin{array}{c}|V\left(t,x\left(t\right)\right)|\ge -V\left({s}_{0},x\left({s}_{0}\right)\right)+{W}_{2}\left(|V\left({s}_{0},x\left({s}_{0}\right)\right)|\right)\left(l\left(t\right)-l\left({s}_{0}\right)\right)\\ >{W}_{2}\left(|V\left({s}_{0},x\left({s}_{0}\right)\right)|\right)\left(l\left(t\right)-l\left({s}_{0}\right)\right)\end{array}$

[1] Bainov, D.D. Simeonvo, P.S. (1989) Theory of Impulsive Differential Equations. World Scientific, Singa-pore.

[2] Schwabik, Š. (1992) Generalized Ordinary Differential Equations. World Scientific, Singapore.
https://doi.org/10.1142/1875

[3] Borysenko, S.D. and Speranza, T. (2009) Impulsive Differential Systems: The Problem of Stability and Practical Stability. Nonlinear Analysis: Theory, Methods & Applications, 71, 1843-1849.
https://doi.org/10.1016/j.na.2009.02.084

[4] Dishlieva, K.G. (2012) Continuous Dependence of the Solution of Impulsive Differential Equations on the Initial Condition and Barrier Curves. Act Mathematic Scientia, 32, 1035-1052.
https://doi.org/10.1016/S0252-9602(12)60077-0

[5] Li, X., Bohner, M. and Wang, C. (2015) Impulsive Differ-ential Equations: Periodic Solutions and Applications. Automatica, 52, 173-178.
https://doi.org/10.1016/j.automatica.2014.11.009

[6] Gallegos, C.A., Henríquez, H.R. and Mesquita, J.G. (2019) Growth of Solution for Measure Differential Equations and Dynamic Equations and Dynamic Equations on Time Scales. Journal of Mathematical Analysis and Applications, 479, 941-962.
https://doi.org/10.1016/j.jmaa.2019.06.059

[7] Fleury, M., Mesquita, J.G. and Slavík, A. (2020) Massera’s The-orems for Various Types of Equations with Discontinuous Solutions. Journal of Differential Equations, 269, 11667-11693.
https://doi.org/10.1016/j.jde.2020.08.043

[8] Federson, M., Grau, R. and Mesquita, J.G. (2019) Prolongation of Solutions of Measure Differential Equations and Dynamic Equations on Time Scales. Mathematische Nachrichten, 292, 22-55.
https://doi.org/10.1002/mana.201700420

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