## 脉冲无穷时滞中立型测度微分方程mild解的存在性

1 安徽城市管理职业学院公共教学部, 合肥 230011

2 安徽建筑大学数理学院, 合肥 230601

## Existence Results of Mild Solutions for Impulsive Neutral Measure Differential Equations with Infinite Delay

Liu Wenjie,1, Xie Shengli,2

1 Public Curriculum Department, Anhui Vocational College of City Management, Hefei 230011

2 School of Mathematics & Physics, Anhui University of Architecture, Hefei 230601

 基金资助: 安徽省自然科学基金.  1508085MA08安徽省教育厅自然科学基金.  KJ2014A043安徽城市管理职业学院重点科研项目.  2021zrkx03

 Fund supported: NSF of Anhui Province.  1508085MA08the NSF of Anhui Provincial Education Department.  KJ2014A043the Key Project of Anhui Vocational College of City Management.  2021zrkx03

Abstract

In this paper, we mainly examine the existence of mild solutions for impulsive neutral measure differential equations with infinite delay. Under the condition that semigroups are non-compact, we obtain sufficient conditions for the existence of mild solutions by using operator semigroup theory, Kuratowski measure of noncompactness, Mönch fixed point theorem and piecewise estimation. Without utilizing a priori estimation and non-compact constraints, we generalize many existing results. Finally, an example is delivered to illustrate the feasibility of the result.

Keywords： Impulsive neutral measure differential equations with infinite delay ; Mild solution ; Kuratowski measure of noncompactness ; Fixed point theorem

Liu Wenjie, Xie Shengli. Existence Results of Mild Solutions for Impulsive Neutral Measure Differential Equations with Infinite Delay. Acta Mathematica Scientia[J], 2022, 42(6): 1671-1681 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} d(x(t)+g(t, x_t))=Ax(t){\rm d}t+f(t, x_t){\rm d}u(t), \ t\in J=[0, a], t\neq t_i, \\ \Delta x(t_i)=I_i(x_{t_i}), i=1, 2, \cdots , n, \\ x_0=\phi\in {\cal B}, \end{array} \right. \end{eqnarray}$

$$$M_1c+\varphi\sup\limits_{0\leq t\leq b}\bigg(\int_{0}^{t}[(t-s)^{\alpha-1}]^q{\rm d}g(s)\bigg)^\frac{1}{q}\frac{M_1}{\Gamma(\alpha)}\|h\|_{HLS}<1,$$$

$$$\|(-A)^{-\beta}\|Q^*+2\frac{M_{1-\beta}}{\beta}\|l_g\|_{L^1(J, {{\Bbb R}}^+)}+M \bigg(T\|l_f\|_{L^1(J, {{\Bbb R}}^+)}+\|l_h\|_{L^2(J, {{\Bbb R}}^+)}+\sum\limits_{n=1}^ml_k\bigg)<1,$$$

## 2 预备知识

(H$_2) $$x\in{\cal B}, t\leq 0 时, 存在局部有界函数 H:(-\infty, 0]\rightarrow {{\Bbb R}}^+ , 使得 \|x(t)\|\leq H\|x_t\|_{{\cal B}} . (H _3)$$ x\in {\cal B}, \mu\leq t\leq0$时, 存在函数$K:[0, \infty)\rightarrow [1, \infty), N:[0, \infty)\rightarrow {{\Bbb R}}^+$, 使得

(H$_4) $$x\in {\cal B} , 那么函数 t\rightarrow \|x_t\|_{{\cal B}}$$ (-\infty, 0]$上是正则的.

(H$_5) $${\cal B} 是完备的空间. 注2.2 (H _1) –(H _5) 与文献[12]中的相空间定义几乎相同, 除(H _3) 与(H3)不同外. 其实, 若 \sigma>0, x\in H_0 , 令 t=0, \mu=-\sigma 得出 \|x\|_{H_0}\leq k_2(\sigma)\sup\limits_{t\in [\sigma, 0]}\|x(t)\| , 即是文献[12]中的(H3). 记空间 {\cal B}_a=\{x:(-\infty, a]\rightarrow X:x_k\in C(J_i, X), x$$ t\neq t_i$时连续, $x(t^-_i) =x(t_i) $$x(t^+_i) 存在, i = 1, 2, \cdots, n, x_0=\phi\} , 赋予半范数 \|x\|_{{\cal B}_a}=\|x_0\|_{{\cal B}}+\sup\limits_{t\in J}\|x(t)\| . 函数 U:[a, b]\times[a, b]\rightarrow X 被称为Kurzweil-Henstock-Stieltjes积分, 若存在 I\in X , 对任意的 \varepsilon>0 , 都有区间 [a, b] 上的一个划分 \delta , 使得 成立. 定义 \int_a^bDU(\tau, s)=I , 令 DU(\tau, s)=f(\tau)u(s) , 则积分为 \int_a^bf(s){\rm d}u(s) . K_u([0, a], X) 表示Kurzweil-Henstock-Stieltjes可积函数空间. 命题2.3 函数 f:[a, b]\rightarrow X$$ u:[a, b]\rightarrow {{\Bbb R}}$满足$u$是正则函数且$\int_{a}^{b}f(s){\rm d}u(s)$存在. 对$t_0\in [a, b]$, 函数$h(t)=\int_{t_0}^{t}f(s){\rm d}u(s), t\in[a, b]$为正则函数并满足

## 3 主要定理及其证明

$\begin{eqnarray} x(t)&=&T(t)(\phi(0)-g(0, \phi(0)))+g(t, x_t)+\int_{0}^{t}AT(t-s)g(s, x_s){\rm d}s{}\\ &&+\int_{0}^{t}T(t-s)f(s, x_s){\rm d}u(s)+\sum\limits_{0<t_i<t}T(t-t_i)I_i(x_{t_i}), \ t\in [0, a]. \end{eqnarray}$

(F$_1) $$M=\sup\limits_{t\in J}\|T(t)\| , (T(t))_{t\geq0} 是强连续的, 且有 0<\beta<1 , 存在 C_{1-\beta} , 满足 (F _2) 函数 f:J\times {\cal B}\rightarrow X 满足下列条件 (i) 对 x:(-\infty, a]\rightarrow X , 满足 x_0=\phi, x(\cdot)|_J\in PC , 映射 t\rightarrow f(t, x_t)$$ J$上是强可测的, 对几乎处处$t\in J, f(t, \cdot): {\cal B}\rightarrow X$是连续的;

(ii) 存在一个非负函数$p(t)\in K_u(J, X)$, 使得

(iii) 对任意的有界集$V\subset G(J, X)$, 都存在一个非负函数$l_f(t)\in K_u(J, X)$, 使得

(F$'_2)$函数$f(\cdot, \cdot)$是连续的, 且存在一个正常数$0\leq L_f$, 使得

(F$_3)$存在常数$0<\beta<1$, 对$(t, \phi)\in J\times X$, 有$g(t, \phi)\in X_\beta=D((-A)^\beta) $$(-A)^\beta g(t, \phi) 连续, 且存在常数 0\leq c_1<\frac{1}{M_0}, \ 0<c_2$$ 0\leq l_g^*<\frac{1}{M_0}$满足

(F$'_3)$函数$g(\cdot, \cdot)$是连续的, $g(t, 0)=0$且存在一个正常数$0\leq L_g<\frac{1}{M_0}$使得

(F$_4)$函数$I_i:{\cal B}_a\rightarrow X\ (i=1, \cdots, n)$连续并且存在常数$d\geq0, e>0$, 使得对每一个$i=1, \cdots, n$, $\|I_i(\phi)\|\leq d\|\phi\|_{{\cal B}_a}+e$.

定义算子$\Gamma:{\cal B}_a\rightarrow {\cal B}_a$

$\begin{eqnarray} \Gamma x(t)=\left\{\begin{array}{ll} \phi(t), \ t\in(-\infty, 0], \\ { } T(t)(\phi(0)-g(0, \phi(0)))+g(t, x_t)+\int_{0}^{t}AT(t-s)g(s, x_s){\rm d}s\\ { } +\int_{0}^{t}T(t-s)f(s, x_s){\rm d}u(s)+\sum\limits_{0<t_i<t}T(t-t_i)I_i(x_{t_i}), \ t\in [0, a]. \end{array} \right. \end{eqnarray}$

$\begin{eqnarray} \widehat{\phi}(t)=\left\{\begin{array}{ll} \phi(t), \ &t\in(-\infty, 0], \\ T(t)\phi(0), \ &t\in J. \end{array} \right. \end{eqnarray}$

$t\in J_1=(t_1, t_2]$时, $\widetilde{y}_1\in C([t_1, t_2], X)$. 类比(3.6)式, 可以得出

$\begin{eqnarray} \|\widetilde{y}_1(t)\|&\leq& c_1KM_1\|y\|_t+(c_1(\|\phi\|_{\cal B}+r)+c_2)(M+1)(M_1+\frac{C_{1-\beta}}{\beta}t_2^{\beta}){}\\ &&+M\|I_1(y_{t_1}+\widehat{\phi}_{t_1})\|+Mr\int_{0}^{t_2}p(s){\rm d}u(s){}\\ &&+K(c_1\frac{C_{1-\beta}}{\beta}t_2^{\beta}+M)\int_{0}^{t}(1+p(s))\|y\|_sd(u(s)+s) \end{eqnarray}$

$\begin{eqnarray} v(t)&\leq& M\delta+c_1KM_1(N_0+v(t))+(c_1(\|\phi\|_{\cal B}+r)+c_2)(M+1)(M_1+\frac{C_{1-\beta}}{\beta}t_2^{\beta}){}\\ &&+Mr\int_{0}^{t_2}p(s){\rm d}u(s)+KN_0(c_1\frac{C_{1-\beta}}{\beta}t_2^{\beta}+M)\int_{0}^{t_1}(1+p(s)){\rm d}(u(s)+s){}\\ &&+K(c_1\frac{C_{1-\beta}}{\beta}t_2^{\beta}+M)\int_{t_1}^{t_2}(1+p(s))v(s){\rm d}(u(s)+s), \ t\in [t_1, t_2]. \end{eqnarray}$

$V\subset \overline{\Omega}_R$是一个可数集且$V\subset\overline{\rm co}(\{0\}\cup F(V))$,

$t\in J_0=[0, t_1]$时, 由根据非紧性测度性质, 条件(F$_1)$, (F$_2)$(iii), (F$_3)$和引理2.7得

$m(t)=\sup\limits_{0\leq s\leq t}\alpha(V(s)), t\in J$. 则推出

对任意有界集$V\subset PC$, 由条件(F$'_3)$得到

$$$\left\{\begin{array}{ll} { } {\rm d}(u(t, x)-\int_{-\infty}^t\int_{0}^\pi b(s-t, \xi, x)u(s, \xi){\rm d}\xi {\rm d}s) =\frac{\partial^2}{\partial x^2}u(t, x){\rm d}t\\ [3mm] { } +\int_{-\infty}^t\mu(t, s-t)u(s, x){\rm d}w(s), \ 0\leq t\leq 1, t\neq t_i, 0\leq x\leq\pi, \\ u(t, 0)=u(t, \pi)=0, \ 0\leq t\leq 1, \\ u(\theta, x)=\phi(\theta, x), \ -\infty<\theta\leq0, 0\leq x\leq\pi, \\ { } \Delta u(t_i, x)=\int_{-\infty}^{t_i}q_i(t_i-s)u(s, x){\rm d}s, \ i=1, 2, \cdots, m, \end{array} \right.$$$

(a) $\{\delta_n : n\in\mathbb{N}\}\subset X$是一个正交基;

(b) 若$\delta\in D(A)$, 则$A\delta=-\sum\limits_{n=1}^\infty n^2\langle\delta, \delta_n\rangle\delta_n$;

(c) 若$\delta\in X$, 则$(-A)^\frac{1}{2}\delta=\sum\limits_{n=1}^\infty \frac{1}{n}\langle\delta, \delta_n\rangle\delta_n$;

(d) 算子$(-A)^\frac{1}{2}$在空间$D[(-A)^\frac{1}{2}]=\{\delta\in X:\sum\limits_{n=1}^\infty n\langle\delta, \delta_n\rangle\delta_n\in X\}$上定义为$(-A)^\frac{1}{2}\delta=\sum\limits_{n=1}^\infty n\langle\delta, \delta_n\rangle\delta_n$.

$(t, \phi)\in[0, 1]\times{\cal B}, $$\phi(\theta, x)=\phi(\theta)(x), (\theta, x)\in(-\infty, 0]\times[0, \pi] , u(t, x)=u(t)(x), 为了研究系统(4.1), 我们假设以下条件成立 (h_1)$$ b(s, \xi, x), \frac{\partial b(s, \xi, x)}{\partial x}$是可测的, 且$b(s, \xi, 0)=b(s, \xi, \pi)=0$

$(h_2) $$\mu\in C({{\Bbb R}}^2, {{\Bbb R}})$$ (\int_{-\infty}^0\mu^2(t, \theta)\rho^{-1}(\theta) {\rm d}\theta)^\frac{1}{2}=p(t)\in C(J, {{\Bbb R}}^+)$;

$(h_3) $$\mu\in C({{\Bbb R}}, {{\Bbb R}}^+)$$ c_i=(\int_{-\infty}^0q^2_i(\theta)\rho^{-1}(\theta) {\rm d}\theta)^\frac{1}{2}<\infty, i=1, 2, \cdots, m$.

$(t, \phi)\in[0, 1]\times {\cal B}\rightarrow X$,

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Brogliato B . Nonsmooth Mechanics: Models, Dynamics and Control. Berlin: Springer, 1999

Xu Y T . Functional Differential Equations and Measure Differential Equations. Guangzhou: Sun Yat-Sen University Press, 1988

Van De Wouw N, Leine R I. Tracking control for a class of measure differential inclusions[C]. Proceedings of the 47th IEEE Conference on Decision and Control, Mexico, 2008: 2526-2532

Piccoli B .

Measure differential equations

Archive for Rational Mechanics and Analysis, 2019, 233 (3): 1289- 1317

Józef B , Tomasz Z .

On a measure of noncompactness in the space of regulated functions and its applications

Advances in Nonlinear Analysis, 2019, 8 (1): 1099- 1110

Federson M , Grau R , Mesquita J G , et al.

Lyapunov stability for measure differential equations and dynamic equations on time scales

Journal of Differential Equations, 2019, 267 (7): 4192- 4223

Cao Y J , Sun J T .

Measures of noncompactness in spaces of regulated functions with application to semilinear measure driven equations

Boundary Value Problems, 2016, 1, 1- 17

Gu H B , Sun Y .

Nonlocal controllability of fractional measure evolution equation

Journal of Inequalities and Applications, 2020, 60 (1): 1- 18

Deng S F , Shu X B , Mao J Z .

Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mónch fixed point

Journal of Mathematical Analysis and Applications, 2018, 467 (1): 398- 420

Pazy A . Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 2012

Eduardo H M , Rabello M , Henríquez H R .

Existence of solutions for impulsive partial neutral functional differential equations

Journal of Mathematical Analysis and Applications, 2007, 331 (2): 1135- 1158

Slavík A .

Measure functional differential equations with infinite delay

Nonlinear Analysis: Theory, Methods Applications, 2013, 79, 140- 155

Heinz H P .

On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions

Nonlinear Analysis: Theory, Methods Applications, 1983, 7 (12): 1351- 1371

Rao S H .

Integral inequalities of Gronwall type for distributions

Journal of Mathematical Analysis and Applications, 1979, 72 (2): 545- 550

Guo Y C , Chen M Q , Shu X B , et al.

The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm

Stochastic Analysis and Applications, 2021, 39 (4): 643- 666

Mönch H .

Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces

Nonlinear Analysis: Theory, Methods Applications, 1980, 4 (5): 985- 999

Hino Y , Murakami S , Naito T . Functional Differential Equations with Infinite Delay. Berlin: Springer-Verlag, 1991

/

 〈 〉