## The Averaging Method of Set Impulsive Differential Equations with Initial and Boundary Value Conditions

Wang Peiguang,, Yang Kaiyu,

College of Mathematics and Information Science, Hebei University, Hebei Baoding 071002

 基金资助: 国家自然科学基金.  11771115国家自然科学基金.  12171135

Received: 2020-04-24

 Fund supported: the NSFC.  11771115the NSFC.  12171135

Abstract

This paper uses the scheme of full, and partially additive averaging method to study the set impulsive differential equations with the initial and multipoint boundary value problems in Euclidean space $\mathbb{R}^{n}$, and proves the approximate relationship of the solutions between the original equations and the average equations.

Keywords： Set differential equations ; Multipoint boundary value ; Impulse ; Averaging method

Wang Peiguang, Yang Kaiyu. The Averaging Method of Set Impulsive Differential Equations with Initial and Boundary Value Conditions. Acta Mathematica Scientia[J], 2021, 41(5): 1504-1515 doi:

## 2 预备知识

$K_{c}({{\Bbb R}} ^{n})$表示${{\Bbb R}} ^{n}$中所有非空紧致凸子集构成的集合, $A$, $B $$K_{c}({{\Bbb R}} ^{n}) 的两个非空闭子集. 定义 A , B 的Hausdorff度量如下: 其中 \parallel\cdot\parallel 是Euclidean范数, (K_{c}({{\Bbb R}} ^{n}), D) 是完备度量空间. 定义2.1 对任意给定的集合 A, B\in K_c{({{\Bbb R}} ^{n})} , 若存在集合 C\in K_c({{\Bbb R}} ^{n}) , 使得 A = B+C 成立, 则称 C$$ A $$B$$ \rm Hukuhara$差集.

$$$D_{H}U = F(t, U), \; \; U(t_{0}) = U_{0}\in K_c({{\Bbb R}} ^{n}), \; \; t_{0}\in I,$$$

$$$U(t) = U_{0}+\int_{t_{0}}^{t} F(s, U(s)){\rm d}s, \; \; \; t\in I,$$$

## 3 全局平均法

$$$\left\{ \begin{array}{ll} D_{H}X(t) = \varepsilon F(t, X(t)), \; \; \; &t\neq\tau_{k}, \\ X(\tau_{k}^{+})-X(\tau_{k}^{-}) = I_{k}(X(\tau_{k})), \; \; \; &t = \tau_{k}, \\ \end{array} \right.$$$

$$$X(0) = \Psi(0), \; \; A_{0}X(t_{0})+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)X_{i}(t_{i}) = \Phi(X_{0}(t_{0}), \cdots , X_{n}(t_{n}), \varepsilon),$$$

($A_{3.2}$)   $\Phi:\underbrace{Q\times Q\times \cdots \cdots Q}_{n+1}\times(0, \varepsilon_{0}]\to K_{c}({{\Bbb R}} ^{m})$连续, 并且关于$X$满足Lipschitz条件, 即

($A_{3.3}$)

($A_{3.4}$)   对于任意的$X\in Q$, 极限$\rm(3.3)$一致成立, 且有下式成立

($A_{3.5}$)   对任意的$0<\varepsilon\leq\varepsilon_{0}$, $0\leq t\leq L\varepsilon^{-1}$, $X(0)$, $Y(0)\in Q\subset K_{c}({{\Bbb R}} ^{n})$, 系统(3.4)–(3.5) 存在解$Y(t)$, 且$O(Y(t), \rho)\subset Q$, 其中$O(Y(t), \rho) $$Y(t)$$ \rho$邻域, $\rho>0$为常数.

容易得到, 具有初值和多点边值条件的系统(3.1)–(3.2) 的解满足下列积分方程

$$$\left\{ \begin{array}{ll} { } X(t) = X_{0}+\varepsilon \int^{t}_{t_{0}}F(s, X(s)){\rm d}s+\varepsilon\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), \\ X(0) = \Psi(0), \end{array} \right.$$$

$$$A_{0}(X_{0}+\varepsilon \zeta_{0})+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)(X_{0} +\varepsilon \zeta_{i}) = \Phi(X_{0}+\varepsilon \zeta_{0}, \cdots , X_{0}+\varepsilon \zeta_{n}, \varepsilon),$$$

$$$\left\{ \begin{array}{ll} { } Y(t) = Y_{0}+\varepsilon \int^{t}_{t_{0}}\overline{F}(Y(s)){\rm d}s, \\ Y(0) = \Psi(0), \end{array} \right.$$$

$$$A_{0}(Y_{0}+\varepsilon \overline{\zeta}_{0})+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)(Y_{0}+\varepsilon \overline{\zeta}_{i}) = \Phi(Y_{0}+\varepsilon \overline{\zeta}_{0}, \cdots , Y_{0}+\varepsilon \overline{\zeta}_{n}, \varepsilon),$$$

$$$D_{H}Y(t) = \varepsilon [\overline{G}_{10}(Y(t))+G_{2}(t, Y(t))],$$$

$$$Y(0) = \Psi(0), \ \ A_{0}Y(t_{0})+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)Y_{i}(t_{i}) = \Phi(Y_{0}(t_{0}), \cdots , Y_{n}(t_{n}), \varepsilon).$$$

($A_{4.1}$)    $F: \Omega\to K_{c}({{\Bbb R}} ^{n})$, $I_{k}:Q\to K_{c}({{\Bbb R}} ^{n})$连续, 且存在正数$N_{1}$, $N_{2}$, $\nu_{1}$, $\nu_{2}$, $\nu_{3}$, 使得

($A_{4.2}$)   对任意的$X\in Q$, 极限$\rm(4.3)$一致存在, 并且$\frac{1}{T}k(t, t+T)\leq d_2<\infty,$其中$k(t, t+T)$是在区间$(t, t+T]$上序列$\tau_{k}$的点数;

($A_{4.3}$)   对任意的$0<\varepsilon\leq\varepsilon_{0}$, $0\leq t\leq L\varepsilon^{-1}$, $X(0)$, $Y(0)\in Q\subset K_{c}({{\Bbb R}} ^{n})$, 系统(4.4)–(4.5) 存在解$Y(t)$, $O(Y(t), \rho)\subset P$, 其中$O(Y(t), \rho) $$Y(t)$$ \rho$邻域, $\rho>0$是常数.

$\begin{eqnarray} D[\overline{G}_{10}(X(t)), \{0\}]&\leq &D\Big[\overline{G}_{10}(X(t)), \frac{1}{T}\int^{t+T}_{t}G_{1}(s, X(s)){\rm d}s +\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X(\tau_{k}))\Big]{}\\ &&+D\Big[\frac{1}{T}\int^{t+T}_{t}G_{1}(s, X(s)){\rm d}s+\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X(\tau_{k})), \{0\}\Big]{}\\ &<&\delta+\frac{1}{T}D\Big[\int^{t+T}_{t}G_{1}(s, X(s)){\rm d}s, \{0\}\Big] +\frac{1}{T}D\Big[\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X(\tau_{k})), \{0\}\Big]{}\\ &\leq&\delta+N_{1}+d_2N_{2}, \end{eqnarray}$

$\begin{eqnarray} D[\overline{G}_{10}(X_{1}(t)), \overline{G}_{10}(X_{2}(t))] &\leq&2\delta+\frac{1}{T}\int^{t+T}_{t}D[G_{1}(s, X_{1}(s)), G_{1}(s, X_{2}(s))]{\rm d}s{}\\ &&+\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}D[I_{k}(X_{1}(\tau_{k})), I_{k}(X_{2}(\tau_{k}))]{}\\ &\leq&2\delta+(\nu_{1}+d_2\nu_{2})D[X_{1}(t), X_{2}(t)] \end{eqnarray}$

$$$D[\overline{G}_{10}(X(t)), \{0\}]\leq N_{1}+d_2N_{2},$$$

$$$D[\overline{G}_{10}(X_{1}(t)), \overline{G}_{10}(X_{2}(t))]\leq(\nu_{1}+d_2\nu_{2})D[X_{1}(t), X_{2}(t)].$$$

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