﻿ 具有脉冲的非线性耦合积分–微分系统的周期性 Periodic and Asymptotically Periodic Solutions on Nonlinear Coupled Integro-Differential Systems with Impulses

Vol. 08  No. 01 ( 2019 ), Article ID: 28619 , 10 pages
10.12677/AAM.2019.81015

Periodic and Asymptotically Periodic Solutions on Nonlinear Coupled Integro-Differential Systems with Impulses

Qiufeng Chen, Jianli Li

School of Mathmatics and Statistics, Hunan Normal University, Changsha Hunan

Received: Jan. 2nd, 2019; accepted: Jan. 17th, 2019; published: Jan. 24th, 2019

ABSTRACT

In this paper, we study the existence of periodic and asymptotically periodic solutions for a coupled nonlinear Volterra integro-differential equation with impulses. By using Schauder’s fixed point theorem, we obtain that the system has at least one periodic solution and an asymptotically periodic solution.

Keywords:Impulsive Differential Equation, Schauder’s Fixed Point Theorem, Periodic Solutions, Asymptotic Periodic Solutions

1. 引言

$\left\{\begin{array}{l}{x}^{\prime }\left(t\right)={h}_{1}\left(t\right)x\left(t\right)+{h}_{2}\left(t\right)y\left(t\right)+\underset{-\infty }{\overset{t}{\int }}a\left(t,s\right)f\left(x\left(s\right),y\left(s\right)\right),t\ne {t}_{k}\\ {y}^{\prime }\left(t\right)={p}_{1}\left(t\right)y\left(t\right)+{p}_{2}\left(t\right)x\left(t\right)+\underset{-\infty }{\overset{t}{\int }}b\left(t,s\right)g\left(x\left(s\right),y\left(s\right)\right),t\ne {t}_{k}\\ x\left({t}_{k}^{+}\right)=x\left({t}_{k}\right)+{I}_{k}\left(x\left({t}_{k}\right)\right),t={t}_{k}\\ y\left({t}_{k}^{+}\right)=y\left({t}_{k}\right)+{J}_{k}\left(y\left({t}_{k}\right)\right),t={t}_{k}\end{array}$ (1.1)

$\begin{array}{l}a\left(t+T,s+T\right)=a\left(t,s\right),\text{\hspace{0.17em}}b\left(t+T,s+T\right)=b\left(t,s\right),\\ {p}_{i}\left(t+T\right)={p}_{i}\left(t\right),\text{\hspace{0.17em}}{h}_{i}\left(t+T\right)={h}_{i}\left(t\right),\text{\hspace{0.17em}}i=1,2\end{array}$ (1.2)

${t}_{k+q}={t}_{k}+T,\text{\hspace{0.17em}}{I}_{k+q}\left(\cdot \right)={I}_{k}\left(\cdot \right),\text{\hspace{0.17em}}{J}_{k+q}\left(\cdot \right)={J}_{k}\left(\cdot \right),\text{\hspace{0.17em}}k\in {N}^{+}$ (1.3)

$\underset{0}{\overset{T}{\int }}{h}_{1}\left(s\right)\text{d}s\ne 0,\text{\hspace{0.17em}}\underset{0}{\overset{T}{\int }}{p}_{1}\left(s\right)\text{d}s\ne 0$ (1.4)

$‖\left(x,y\right)‖=\mathrm{max}\left\{\underset{t\in \left[0,T\right]}{\mathrm{max}}|x\left(t\right)|,\underset{t\in \left[0,T\right]}{\mathrm{max}}|y\left(t\right)|\right\}$

$x\left(t\right)=\left\{\begin{array}{l}{x}^{\ast }\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right]\\ {x}^{\ast }\left(t-nT\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }t\in \left[nT,\left(n+1\right)T\right]\end{array}$

$y\left(t\right)=\left\{\begin{array}{l}{y}^{\ast }\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right]\\ {y}^{\ast }\left(t-nT\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[nT,\left(n+1\right)T\right]\end{array}$

${x}^{*}\left(t\right)=\underset{0}{\overset{T}{\int }}{G}_{1}\left(t,s\right)\left({h}_{2}\left(s\right)y\left(s\right)+\underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s+\underset{0<{t}_{k}

${G}_{1}\left(t,s\right)=\frac{1}{1-{\text{e}}^{\underset{0}{\overset{T}{\int }}{h}_{1}\left(s\right)\text{d}s}}\left\{\begin{array}{l}{\text{e}}^{\underset{s}{\overset{t}{\int }}{h}_{1}\left(u\right)\text{d}u},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le s

${y}^{*}\left(t\right)=\underset{0}{\overset{T}{\int }}{G}_{2}\left(t,s\right)\left({p}_{2}\left(s\right)x\left(s\right)+\underset{-\infty }{\overset{s}{\int }}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s+\underset{0<{t}_{k}

${G}_{2}\left(t,s\right)=\frac{1}{1-{\text{e}}^{\underset{0}{\overset{T}{\int }}{p}_{1}\left(s\right)\text{d}s}}\left\{\begin{array}{l}{\text{e}}^{\underset{s}{\overset{t}{\int }}{p}_{1}\left(u\right)\text{d}u},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le s

$\begin{array}{c}x\left(t\right){\text{e}}^{-\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}-x\left(0\right)=\underset{0}{\overset{t}{\int }}{\text{e}}^{-\underset{0}{\overset{s}{\int }}{h}_{1}\left(u\right)du}\left({h}_{2}\left(s\right)y\left(s\right)+\underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{0<{t}_{k}

$x\left(0\right)=x\left(T\right)$

$\begin{array}{c}{x}^{*}\left(t\right)=x\left(0\right){\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(u\right)\text{d}u}+\underset{0}{\overset{t}{\int }}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(u\right)\text{d}u}\left({h}_{2}\left(s\right)y\left(s\right)+\underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s+\underset{0<{t}_{k}

${y}^{*}\left(t\right)=\underset{0}{\overset{T}{\int }}{G}_{2}\left(t,s\right)\left({p}_{2}\left(s\right)x\left(s\right)+\underset{-\infty }{\overset{s}{\int }}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s+\underset{0<{t}_{k}

2. 周期解

$|f\left(x,y\right)|\le {M}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}|g\left(x,y\right)|\le {M}_{2}$ (2.1)

$\underset{0}{\overset{T}{\int }}|{G}_{1}\left(t,s\right){h}_{2}\left(s\right)|\text{d}s\le {L}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{0}{\overset{T}{\int }}|{G}_{2}\left(t,s\right){p}_{2}\left(s\right)|\text{d}s\le {L}_{2}$ (2.2)

$\underset{0}{\overset{T}{\int }}|{G}_{1}\left(t,s\right)|\underset{-\infty }{\overset{s}{\int }}|a\left(s,u\right)|\text{d}u\text{d}s\le {N}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{0}{\overset{T}{\int }}|{G}_{2}\left(t,s\right)|\underset{-\infty }{\overset{s}{\int }}|b\left(s,u\right)|\text{d}u\text{d}s\le {N}_{2}$ (2.3)

$\left\{\begin{array}{l}|{I}_{k}\left(x\right)|\le {b}_{k}|x|,\text{\hspace{0.17em}}|\underset{k=1}{\overset{p}{\sum }}{G}_{1}\left(t,{t}_{k}\right){b}_{k}|\le {L}_{1}^{\ast },\text{\hspace{0.17em}}t\in \left[0,T\right]\\ |{J}_{k}\left(x\right)|\le {c}_{k}|y|,\text{\hspace{0.17em}}|\underset{k=1}{\overset{p}{\sum }}{G}_{2}\left(t,{t}_{k}\right){c}_{k}|\le {L}_{2}^{\ast },\text{\hspace{0.17em}}t\in \left[0,T\right]\end{array}$ (2.4)

${\Omega }_{M}=\left\{\left(x,y\right):\left(x,y\right)\in {P}_{T},‖\left(x,y\right)‖\le M\right\}$${P}_{T}$ 的子集，则 ${\Omega }_{M}$${P}_{T}$ 中的有界闭子集。

$F\left(x,y\right)=\left({F}_{1}\left(x,y\right)\left(t\right),{F}_{2}\left(x,y\right)\left(t\right)\right)$

${F}_{1}\left(x,y\right)\left(t\right)=\underset{0}{\overset{T}{\int }}{G}_{1}\left(t,s\right)\left({h}_{2}\left(s\right)y\left(s\right)+\underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s+\underset{0<{t}_{k}

${F}_{2}\left(x,y\right)\left(t\right)=\underset{0}{\overset{T}{\int }}{G}_{2}\left(t,s\right)\left({p}_{2}\left(s\right)x\left(s\right)+\underset{-\infty }{\overset{s}{\int }}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s+\underset{0<{t}_{k}

${L}_{1}+{L}_{1}^{\ast }<1,\text{\hspace{0.17em}}{L}_{2}+{L}_{2}^{\ast }<1$ (2.5)

$M=\mathrm{max}\left\{\frac{{N}_{1}{M}_{1}}{1-{L}_{1}-{L}_{1}^{\ast }},\frac{{N}_{2}{M}_{2}}{1-{L}_{2}-{L}_{2}^{\ast }}\right\}$

$\begin{array}{c}|{F}_{1}\left(x,y\right)\left(t\right)|\le \underset{0}{\overset{T}{\int }}|{G}_{1}\left(t,s\right){h}_{2}\left(s\right)||y\left(s\right)|\text{d}s+\underset{0<{t}_{k}

$\begin{array}{c}|{F}_{2}\left(x,y\right)\left(t\right)|\le \underset{0}{\overset{T}{\int }}|{G}_{2}\left(t,s\right){p}_{2}\left(s\right)||x\left(s\right)|\text{d}s+\underset{0<{t}_{k}

$\underset{n\to \infty }{\mathrm{lim}}‖\left({x}^{n},{y}^{n}\right)-\left(x,y\right)‖=0$

$\begin{array}{l}‖F\left({x}^{n},{y}^{n}\right)-F\left(x,y\right)‖\\ =\mathrm{max}\left\{\underset{t\in \left[0,T\right]}{\mathrm{max}}|{F}_{1}\left({x}^{n},{y}^{n}\right)\left(t\right)-{F}_{1}\left(x,y\right)\left(t\right)|,\underset{t\in \left[0,T\right]}{\mathrm{max}}|{F}_{2}\left({x}^{n},{y}^{n}\right)\left(t\right)-{F}_{2}\left(x,y\right)\left(t\right)|\right\}\end{array}$

$\begin{array}{l}|{F}_{1}\left({x}^{n},{y}^{n}\right)\left(t\right)-{F}_{1}\left(x,y\right)\left(t\right)|\\ =|\underset{0}{\overset{T}{\int }}{G}_{1}\left(t,s\right)\left({h}_{2}\left(s\right){y}^{n}\left(s\right)+\underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left({x}^{n}\left(u\right),{y}^{n}\left(u\right)\right)\text{d}u\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\underset{0}{\overset{T}{\int }}{G}_{1}\left(t,s\right)\left({h}_{2}\left(s\right)y\left(s\right)+\underset{-\infty }{\overset{s}{\int }}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\underset{0<{t}_{k}

$f,{I}_{k}$ 的连续性和Lebesgue控制收敛定理，有

$\underset{n\to \infty }{\mathrm{lim}}\underset{t\in \left[0,T\right]}{\mathrm{max}}|{F}_{1}\left({x}^{n},{y}^{n}\right)\left(t\right)-{F}_{1}\left(x,y\right)\left(t\right)|=0$

$\underset{n\to \infty }{\mathrm{lim}}\underset{t\in \left[0,T\right]}{\mathrm{max}}|{F}_{2}\left({x}^{n},{y}^{n}\right)\left(t\right)-{F}_{2}\left(x,y\right)\left(t\right)|=0$

$|f\left(x,y\right)|\le {Q}_{1}W\left(|y|\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{W\left(u\right)}{u}\le \frac{1-{L}_{1}-{L}_{1}^{\ast }}{{N}_{1}{Q}_{1}},\text{\hspace{0.17em}}u>0$

$|g\left(x,y\right)|\le {Q}_{2}G\left(|x|\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{G\left(u\right)}{u}\le \frac{1-{L}_{2}-{L}_{2}^{\ast }}{{N}_{2}{Q}_{2}},\text{\hspace{0.17em}}u>0$

$M=\mathrm{max}\left\{\frac{W\left(M\right){N}_{1}{Q}_{1}}{1-{L}_{1}-{L}_{1}^{\ast }},\frac{G\left(M\right){N}_{2}{Q}_{2}}{1-{L}_{2}-{L}_{2}^{\ast }}\right\}$

$\left(x,y\right)\in {\Omega }_{M},t\in \left[0,T\right]$

$\begin{array}{c}|{F}_{1}\left(x,y\right)\left(t\right)|\le \underset{0}{\overset{T}{\int }}|{G}_{1}\left(t,s\right){h}_{2}\left(s\right)||y\left(s\right)|\text{d}s+\underset{0<{t}_{k}

$\begin{array}{c}|{F}_{2}\left(x,y\right)\left(t\right)|\le \underset{0}{\overset{T}{\int }}|{G}_{2}\left(t,s\right){p}_{2}\left(s\right)||x\left(s\right)|\text{d}s+\underset{0<{t}_{k}

3. 渐近周期解

$\underset{0}{\overset{T}{\int }}{h}_{1}\left(s\right)\text{d}s=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{0}{\overset{T}{\int }}{p}_{1}\left(s\right)\text{d}s=0$ (3.1)

${m}_{1}\le {\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}\le {M}_{1}^{\ast },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{2}\le {\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}\le {M}_{2}^{\ast }$ (3.2)

$\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}|a\left(s,u\right)|\text{d}u\text{d}s\le A,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}|b\left(s,u\right)|\text{d}u\text{d}s\le B$ (3.3)

$\underset{t}{\overset{\infty }{\int }}|{h}_{2}\left(s\right)|\text{d}s\le {M}_{3}^{\ast },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{t}{\overset{\infty }{\int }}|{p}_{2}\left(s\right)|\text{d}s\le {M}_{4}^{\ast }$ (3.4)

${b}_{k}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{k}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{k=1}{\overset{\infty }{\sum }}{b}_{k}<\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{k=1}{\overset{\infty }{\sum }}{c}_{k}<\infty$ (3.5)

$|f\left(x,y\right)|\le {\stackrel{¯}{Q}}_{1}|y|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}|g\left(x,y\right)|\le {\stackrel{¯}{Q}}_{2}|x|$

$\begin{array}{l}1-{M}_{3}^{\ast }{M}_{1}^{\ast }{m}_{1}^{-1}-{M}_{1}^{\ast }{m}_{1}^{-1}A{\stackrel{¯}{Q}}_{1}-{M}_{1}^{\ast }{M}_{5}^{\ast }>0,\\ 1-{M}_{4}^{\ast }{M}_{2}^{\ast }{m}_{2}^{-1}-{M}_{2}^{\ast }{m}_{2}^{-1}B{\stackrel{¯}{Q}}_{2}-{M}_{2}^{\ast }{M}_{6}^{\ast }>0\end{array}$

$x\left(t\right)={x}_{1}\left(t\right)+{x}_{2}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\left(t\right)={y}_{1}\left(t\right)+{y}_{2}\left(t\right)$

${x}_{1}\left(t\right)={c}_{1}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{1}\left(t\right)={c}_{2}{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s},\text{\hspace{0.17em}}t\in R$

${c}_{1},{c}_{2}$ 为固定的非零常数，

$\underset{t\to \infty }{\mathrm{lim}}{x}_{2}\left(t\right)=\underset{t\to \infty }{\mathrm{lim}}{y}_{2}\left(t\right)=0$ .

$‖\left(x,y\right)‖=\mathrm{max}\left\{\underset{t\in \left[0,T\right]}{\mathrm{max}}|x\left(t\right)|,\underset{t\in \left[0,T\right]}{\mathrm{max}}|y\left(t\right)|\right\}$

${H}_{T}$ 是一个Banach空间。记 ${\Omega }_{{M}^{*}}=\left\{\left(x,y\right):\left(x,y\right)\in {H}_{T},‖\left(x,y\right)‖\le {M}^{*}\right\}$ ，则 ${\Omega }_{{M}^{*}}$${H}_{T}$ 中的有界闭凸子集。

$\left(x,y\right)\in {\Omega }_{{M}^{*}}$ ，定义算子 $E:{\Omega }_{{M}^{*}}\to {H}_{T}$

$E\left(x,y\right)\left(t\right)=\left({E}_{1}\left(y\right)\left(t\right),{E}_{2}\left(x\right)\left(t\right)\right)$

$\begin{array}{l}{E}_{1}\left(y\right)\left(t\right)\\ ={c}_{1}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}-\underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{h}_{1}\left(l\right)\text{d}l}}{h}_{2}\left(s\right)y\left(s\right)\text{d}s-\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{h}_{1}\left(l\right)\text{d}l}}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\underset{t<{t}_{k}<\infty }{\sum }{\text{e}}^{\underset{{t}_{k}}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}{I}_{k}\left(x\left({t}_{k}\right)\right)\end{array}$

$\begin{array}{l}{E}_{2}\left(x\right)\left(t\right)\\ ={c}_{2}{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}-\underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{p}_{1}\left(l\right)\text{d}l}}{p}_{2}\left(s\right)x\left(s\right)\text{d}s-\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{p}_{1}\left(l\right)\text{d}l}}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\underset{t<{t}_{k}<\infty }{\sum }{\text{e}}^{\underset{{t}_{k}}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}{J}_{k}\left(y\left({t}_{k}\right)\right)\end{array}$

$|\underset{k=1}{\overset{\infty }{\sum }}{b}_{k}|\le {M}_{5}^{\ast },\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\underset{k=1}{\overset{\infty }{\sum }}{c}_{k}|\le {M}_{6}^{\ast }$

${M}^{*}=\left\{\frac{{c}_{1}{M}_{1}^{\ast }}{1-{M}_{3}^{\ast }{M}_{1}^{\ast }{m}_{1}^{-1}-{M}_{1}^{\ast }{m}_{1}^{-1}A{\stackrel{¯}{Q}}_{1}-{M}_{1}^{\ast }{M}_{5}^{\ast }},\frac{{c}_{2}{M}_{2}^{\ast }}{1-{M}_{4}^{\ast }{M}_{2}^{\ast }{m}_{2}^{-1}-{M}_{2}^{\ast }{m}_{2}^{-1}B{\stackrel{¯}{Q}}_{2}-{M}_{2}^{\ast }{M}_{6}^{\ast }}\right\}$

$\begin{array}{l}|{E}_{1}\left(y\right)\left(t\right)-{c}_{1}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)ds}|\\ \le {M}^{*}{M}_{3}^{\ast }{M}_{1}^{\ast }{m}_{1}^{-1}+{M}_{1}^{\ast }{m}_{1}^{-1}{\stackrel{¯}{Q}}_{1}\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}|a\left(s,u\right)||x|\text{d}u\text{d}s+\underset{t<{t}_{k}<\infty }{\sum }{\text{e}}^{\underset{{t}_{k}}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}|{I}_{k}\left(x\left({t}_{k}\right)\right)|\\ \le {M}^{*}{M}_{3}^{\ast }{M}_{1}^{\ast }{m}_{1}^{-1}+{M}_{1}^{\ast }{m}_{1}^{-1}{\stackrel{¯}{Q}}_{1}‖x‖\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}|a\left(s,u\right)|\text{d}u\text{d}s+|\underset{t<{t}_{k}<\infty }{\sum }{b}_{k}|{M}_{1}^{\ast }{M}^{*}\\ \le {M}^{*}{M}_{3}^{\ast }{M}_{1}^{\ast }{m}_{1}^{-1}+{M}_{1}^{\ast }{m}_{1}^{-1}A{\stackrel{¯}{Q}}_{1}{M}^{*}+{M}_{1}^{\ast }{M}^{*}{M}_{5}^{\ast }\end{array}$

$|{E}_{2}\left(x\right)\left(t\right)-{c}_{2}{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}|\le {M}^{*}{M}_{4}^{\ast }{M}_{2}^{\ast }{m}_{2}^{-1}+{M}_{2}^{\ast }{m}_{2}^{-1}B{\stackrel{¯}{Q}}_{2}{M}^{*}+{M}_{2}^{\ast }{M}^{*}{M}_{6}^{\ast }$

$|{E}_{1}\left(y\right)\left(t\right)|\le {M}^{*}{M}_{3}^{\ast }{M}_{1}^{\ast }{m}_{1}^{-1}+{M}_{1}^{\ast }{m}_{1}^{-1}A{\stackrel{¯}{Q}}_{1}{M}^{*}+{M}_{1}^{\ast }{M}^{*}{M}_{5}^{\ast }+{c}_{1}{M}_{1}^{\ast }\le {M}^{*}$

$|{E}_{2}\left(x\right)\left(t\right)|\le {M}^{*}{M}_{4}^{\ast }{M}_{2}^{\ast }{m}_{2}^{-1}+{M}_{2}^{\ast }{m}_{2}^{-1}B{\stackrel{¯}{Q}}_{2}{M}^{*}+{M}_{2}^{\ast }{M}^{*}{M}_{6}^{\ast }+{c}_{2}{M}_{2}^{\ast }\le {M}^{*}$

${x}_{1}\left(t\right)={c}_{1}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{1}\left(t\right)={c}_{2}{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s},\text{\hspace{0.17em}}t\in R$

$\begin{array}{c}{x}_{2}\left(t\right)=-\underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{h}_{1}\left(l\right)\text{d}l}}{h}_{2}\left(s\right)y\left(s\right)\text{d}s-\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{h}_{1}\left(l\right)\text{d}l}}a\left(s,u\right)f\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{t<{t}_{k}<\infty }{\sum }{\text{e}}^{\underset{{t}_{k}}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}{I}_{k}\left(x\left({t}_{k}\right)\right)\end{array}$

$\begin{array}{c}{y}_{2}\left(t\right)=-\underset{t}{\overset{\infty }{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{p}_{1}\left(l\right)\text{d}l}}{p}_{2}\left(s\right)x\left(s\right)\text{d}s-\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}\frac{{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(l\right)\text{d}l}}{{\text{e}}^{\underset{0}{\overset{s}{\int }}{p}_{1}\left(l\right)\text{d}l}}b\left(s,u\right)g\left(x\left(u\right),y\left(u\right)\right)\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{t<{t}_{k}<\infty }{\sum }{\text{e}}^{\underset{{t}_{k}}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}{J}_{k}\left(y\left({t}_{k}\right)\right)\end{array}$

${x}_{1}\left(t+T\right)={c}_{1}{\text{e}}^{\underset{0}{\overset{t+T}{\int }}{h}_{1}\left(s\right)\text{d}s}={c}_{1}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}{\text{e}}^{\underset{t}{\overset{t+T}{\int }}{h}_{1}\left(s\right)\text{d}s}={c}_{1}{\text{e}}^{\underset{0}{\overset{t}{\int }}{h}_{1}\left(s\right)\text{d}s}$

${y}_{1}\left(t+T\right)={c}_{2}{\text{e}}^{\underset{0}{\overset{t+T}{\int }}{p}_{1}\left(s\right)\text{d}s}={c}_{2}{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}{\text{e}}^{\underset{t}{\overset{t+T}{\int }}{p}_{1}\left(s\right)\text{d}s}={c}_{2}{\text{e}}^{\underset{0}{\overset{t}{\int }}{p}_{1}\left(s\right)\text{d}s}$

$\begin{array}{l}\underset{t\to \infty }{\mathrm{lim}}|{x}_{2}\left(t\right)|\le {M}^{*}{M}_{1}^{\ast }{m}_{1}^{-1}\underset{t\to \infty }{\mathrm{lim}}\underset{t}{\overset{\infty }{\int }}|{h}_{2}\left(s\right)|\text{d}s+{M}_{1}^{\ast }{m}_{1}^{-1}{\stackrel{¯}{Q}}_{1}{M}^{*}\underset{t\to \infty }{\mathrm{lim}}\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}|a\left(s,u\right)|\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{M}^{*}{M}_{1}^{\ast }\underset{t\to \infty }{\mathrm{lim}}|\underset{t<{t}_{k}<\infty }{\sum }{b}_{k}|=0\end{array}$

$\begin{array}{l}\underset{t\to \infty }{\mathrm{lim}}|{y}_{2}\left(t\right)|\le {M}^{*}{M}_{2}^{\ast }{m}_{2}^{-1}\underset{t\to \infty }{\mathrm{lim}}\underset{t}{\overset{\infty }{\int }}|{p}_{2}\left(s\right)|\text{d}s+{M}_{2}^{\ast }{m}_{2}^{-1}{\stackrel{¯}{Q}}_{2}{M}^{*}\underset{t\to \infty }{\mathrm{lim}}\underset{t}{\overset{\infty }{\int }}\underset{-\infty }{\overset{s}{\int }}|b\left(s,u\right)|\text{d}u\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{M}^{*}{M}_{2}^{\ast }\underset{t\to \infty }{\mathrm{lim}}|\underset{t<{t}_{k}<\infty }{\sum }{c}_{k}|=0\end{array}$

$E\left(x,y\right)\left(t\right)=\left({E}_{1}\left(y\right)\left(t\right),{E}_{2}\left(x\right)\left(t\right)\right)=\left(x\left(t\right),y\left(t\right)\right)$

$\left\{\begin{array}{l}{x}^{\prime }\left(t\right)=\mathrm{cos}\left(t\right)x\left(t\right)+\frac{2t}{{\left({t}^{2}+10\right)}^{2}}y\left(t\right)+\underset{-\infty }{\overset{t}{\int }}{\text{e}}^{-6t+4s}\mathrm{sin}\left(x\left(t\right)+y\left(t\right)\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ne {t}_{k}\\ {y}^{\prime }\left(t\right)=\mathrm{cos}\left(t\right)y\left(t\right)+\frac{2t}{{\left({t}^{2}+10\right)}^{2}}x\left(t\right)+\underset{-\infty }{\overset{t}{\int }}{\text{e}}^{-6t+4s}\mathrm{cos}\left(x\left(t\right)+y\left(t\right)\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ne {t}_{k}\\ x\left({t}_{k}^{+}\right)=x\left({t}_{k}\right)+{\left(\frac{1}{12}\right)}^{k}\left(x\left({t}_{k}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t={t}_{k}\\ y\left({t}_{k}^{+}\right)=y\left({t}_{k}\right)+{\left(\frac{1}{13}\right)}^{k}\left(y\left({t}_{k}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t={t}_{k}\end{array}$

Periodic and Asymptotically Periodic Solutions on Nonlinear Coupled Integro-Differential Systems with Impulses[J]. 应用数学进展, 2019, 08(01): 135-144. https://doi.org/10.12677/AAM.2019.81015

1. 1. Li, J., Nieto, J.J. and Shen, J. (2007) Impulsive Periodic Boundary Value Problems of First-Order Differential Equations. Journal of Mathematical Analysis and Applications, 325, 226-236. https://doi.org/10.1016/j.jmaa.2005.04.005

2. 2. Gao, S., Chen, L., Nieto, J.J. and Terres, A. (2006) Analysis of a Delayed Epidemic Model with Pulse Vaccination and Staturation Incidence. Vaccine, 24, 6037-6045. https://doi.org/10.1016/j.vaccine.2006.05.018

3. 3. Dai, B. and Su, H. (2009) Periodic Solution of a Delayed Ra-tio-Dependent Predator-Prey Model with Monotonic Functional Response and Impulse. Nonlinear Analysis: Theory, Methods & Applications, 70, 126-134. https://doi.org/10.1016/j.na.2007.11.036

4. 4. Georescu, P. and Morosanu, G. (2007) Pest Regulation by Means of Impulsive Controls. Applied Mathematics and Computation, 190, 790-803. https://doi.org/10.1016/j.amc.2007.01.079

5. 5. Lenci, S. and Rega, G. (2000) Periodic Solutions and Bifurcations in an Impact Inverted Pendulum under Impulsive Excitation. Chaos Solutions Fractals, 11, 2453-2472. https://doi.org/10.1016/S0960-0779(00)00030-8

6. 6. Diblik, J., Schmeidel, E. and Ruzickova, M. (2010) As-ymptotically Periodic Solutions of Volterra System of Difference Equations. Computers & Mathematics with Applica-tions, 59, 2854-2867. https://doi.org/10.1016/j.camwa.2010.01.055

7. 7. Diblik, J., Schmeidel, E. and Ruzickova, M. (2009) Existence of Asymptotically Periodic Solutions of System of Volterra Difference Equations. Journal of Difference Equations and Applications, 15, 1165-1177. https://doi.org/10.1080/10236190802653653

8. 8. Islam, M.N. and Raffoul, Y.N. (2014) Periodic and Asymptot-ically Periodic Solutions in Coupled Nonlinear Systems of Volterra Integro-Differential Equations. Dynamic Systems and Applications, 23, 235-244.

9. 9. Myslo, Y.M. and Tkachenko, V.I. (2017) Asymptotically Almost Periodic Solutions of Equations with Delays and Nonfixed Times of Pulse Action. Journal of Mathematical Sciences, 228, 1-16.

10. 10. Raffoul, Y. (2018) Analysis of Periodic and Asymptotically Periodic Solutions in Nonlinear Coupled Volterra Integro-Differential Systems. Turkish Journal of Mathematics, 42, 108-120. https://doi.org/10.3906/mat-1611-123