脉冲微分方程广泛应用于物理学、生物学和经济学模型, 其最突出的特点是, 可以准确地描述瞬时突变对系统状态的影响. 自1960年起, Mil'man和Myshsi^[15]开始研究脉冲微分方程以来, 得到了许多有趣的关于脉冲微分方程的定性理论的结果, 并形成了非线性分析的一个重要研究分支, 相关工作可参阅文献[3-4, 6, 11-12, 14, 20, 23]. 众所周知, 即使对于足够简单的脉冲方程, 也很难得到脉冲影响下的精确解. 一些学者致力于应用各种渐近方法来解决这一难题. 平均法作为渐近方法之一, 已广泛应用于各类脉冲微分系统中, 例如, 脉冲微分方程^{[1-2, 8-9, 25]}, 脉冲微分包含^{[5, 10, 16-17, 19, 24, 26-28]}等. 结果表明, 平均系统可以去掉脉冲系统中的脉冲项的影响, 从而克服脉冲项带来的困难, 并且使得平均系统的解在一定条件下接近原系统的解.

近年来, 半线性度量空间中的集值微分方程引起了研究人员的广泛关注. 平均法在集值微分系统中的应用结果, 可参见著作^[13], 文献[7, 18, 21]及其参考文献. 然而, 由脉冲微分方程与集值分析理论相结合得到的集值脉冲微分系统的研究结果相对较少, 参见文献[4, 22]. 本文的主要目的是将平均法推广到具有初值和多点边值问题的集值脉冲微分系统, 应用全局平均法和局部加法平均法, 得到原方程与平均方程之间解的近似关系.

首先给出本文需要用到的欧氏空间$ {{\Bbb R}} ^{n} $中关于集值微分方程的一些概念和符号, 参见文献[13].

令$ 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 $是$ K_{c}({{\Bbb R}} ^{n}) $的两个非空闭子集. 定义$ A $, $ B $的Hausdorff度量如下:

$ D[A, B] = \max\Big[\sup\limits_{a\in A}\inf\limits_{b\in B}\|a-b\|, \sup\limits_{b\in B}\inf\limits_{a\in A}\|a-b\|\Big], $

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 $为$ A $与$ B $的$ \rm Hukuhara $差集.

考虑下列集值微分方程

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

其中$ U\in C^{1}[I, K_c({{\Bbb R}} ^{n})] $, $ F\in C[I\times K_c({{\Bbb R}} ^{n}), K_c({{\Bbb R}} ^{n})] $, $ D_HU $为$ U(t) $在$ I $上的Hukuhara导数, $ I = [t_{0}, t_{0}+a] $, $ t_0\ge 0 $, $ a>0 $为常数.

定义2.2   若集值映射$ U:I\to K_{c}({{\Bbb R}} ^{n}) $连续可微, 并且满足下列积分

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

其中积分形式定义如下

$ \int F(s)\, {\rm d}s = \Big [ \int f(s){\rm d}s: f\; \mbox{是 $F$ 在 $I$ 上的任意可积选择}\Big], $

则称映射$ U:I\to K_{c}({{\Bbb R}} ^{n}) $是方程(2.1) 的一个解.

命题2.1   若$ F:I\to K_{c}({{\Bbb R}} ^{n}) $可积, 则

$ \begin{eqnarray*} \int_{t_{0}}^{t_{2}} \lambda F(s)\, {\rm d}s = \lambda\bigg(\int_{t_{0}}^{t_{1}} F(s)\, {\rm d}s +\int_{t_{1}}^{t_{2}}F(s)\, {\rm d}s\bigg), \; \; t_{0}\leq t_{1} \leq t_{2}, \; \; \lambda\in {{\Bbb R}} . \end{eqnarray*} $

命题2.2   若$ F, G:I\to K_{c}({{\Bbb R}} ^{n}) $可积, 则$ D[F(\cdot), G(\cdot)]:I\to {{\Bbb R}} $也可积, 并且

$ D \bigg[ \int_{t_{0}}^t F(s)\, {\rm d}s, \int_{t_{0}}^t G(s)\, {\rm d}s \bigg] \leq \int_{t_{0}}^t D[F(s), G(s)]\, {\rm d}s. $

接下来, 我们给出用于证明原始系统与平均系统之间解的近似关系的引理.

引理2.1^[13]   对于空间$ {{\Bbb R}} ^{n} $中任意的非空子集$ A, B, C, A', B' $及常数$ \lambda>0 $, 下列结论成立

$ D[A+C, B+C] = D[A, B], $

$ D[A, B] = D[B, A], $

$ D[\lambda A, \lambda B] = \lambda D[A, B], $

$ D[A, B]\leq D[A, C]+D[C, B], $

$ D[A+A', B+B']\leq D[A, B]+D[A', B']. $

引理2.2^[13]   对于任意的$ U(t), V(t)\in C[I, K_{c}({{\Bbb R}} ^{n})] $, 令

$ m(t) = D[U(t), V(t)], $

若

$ m(t)\leq M+K \int_{t_{0}}^t m(s)\, {\rm d}s, \; \; t\in I, $

则不等式

$ m(t)\leq M\exp\{K (t-t_{0})\} $

成立, 即

$ D[U(t), V(t)]\leq M\exp\{K (t-t_{0})\}, $

其中$ M, \; K $是非负常数.

考虑具有小参数的脉冲微分系统

(3.1) $ \begin{equation} \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. \end{equation} $

及在固定点$ 0\leq t_{0}<\cdot\cdot\cdot<t_{n}<\infty $处具有初值和Cancy-Nicoletti类型的多点边值问题

(3.2) $ \begin{equation} 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), \end{equation} $

其中$ X\in{{{\Bbb R}} ^{n}} $, $ F:{{\Bbb R}} ^{+}\times K_{c}({{\Bbb R}} ^{n})\to K_{c}({{\Bbb R}} ^{n}) $, $ \varepsilon\in(0, \varepsilon_{0}] $是小参数, $ \varepsilon_{0}>0 $为常数, 且$ I_{k}:K_{c}({{\Bbb R}} ^{n})\to K_{c}({{\Bbb R}} ^{n}) $是连续的, $ 0<\tau_{1}<\cdot\cdot\cdot<\tau_{m}<+\infty $, $ k = 1, \cdots , m $, $ \Phi:K_{c}({{\Bbb R}} ^{n})\times K_{c}({{\Bbb R}} ^{n})\times\cdots \times K_{c}({{\Bbb R}} ^{n})\times(0, \varepsilon_{0}]\to K_{c}({{\Bbb R}} ^{m}) $为连续函数, $ A_{0}, \; A_{i}(\varepsilon) $是$ m\times n $维矩阵, $ i = 1, \cdots , n $.

假设下列极限存在

(3.3) $ \begin{equation} \overline{F}(X(t)) = \lim\limits_{T\to\infty}\Big[\frac{1}{T}\int^{t+T}_{t}F(s, X(s)){\rm d}s+\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X(\tau_{k}))\Big], \end{equation} $

其中$ \overline{F}(X(t)) $是$ F(t, X(t)) $在$ [0, T] $上的平均.

则系统(3.1)–(3.2) 对应的平均系统如下

(3.4) $ \begin{equation} D_{H}Y(t) = \varepsilon \overline{F}(Y(t)), \end{equation} $

其初值和多点边值条件为

(3.5) $ \begin{equation} 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), \end{equation} $

其中$ \overline{F}: K_{c}({{\Bbb R}} ^{n})\to K_{c}({{\Bbb R}} ^{n}) $为连续函数.

令$ \Omega = \{(t, X(t))|\; t\geq0, \; X\in Q\subset K_{c}({{\Bbb R}} ^{n})\} $. 可以得到下列结论.

定理3.1   假设在区域$ \Omega $上下列条件成立:

($ A_{3.1} $)   集值映射$ F:G\to K_{c}({{\Bbb R}} ^{n}) $, $ I_{k}:Q\to K_{c}({{\Bbb R}} ^{n}) $连续, 且存在正数$ M_{1} $, $ M_{2} $, $ \lambda_{1} $和$ \lambda_{2} $, 使得下列不等式

$ D\Big[\int^{t+T}_{t}F(s, X(s)){\rm d}s, \{0\}\Big]\leq M_{1}, \; \; D[I_{k}(X(\tau_{k})), \{0\}]\leq M_{2}; $

$ D[F(s, X_{1}(s)), F(s, X_{2}(s))]{\rm d}s\leq\lambda_{1}D[X_{1}(t), X_{2}(t)]; $

$ D[I_{k}(X_{1}(t)), I_{k}(X_{2}(t))]\leq\lambda_{2}D[X_{1}(t), X_{2}(t)] $

成立;

($ 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

$ \begin{eqnarray*} &&D[\Phi(X_{0}(t_{0}), \cdots , X_{n}(t_{n}), \varepsilon), \Phi(X_{0}'(t_{0}), \cdots , X_{n}'(t_{n}), \varepsilon)]\\ &\leq& \mu_{0}(\varepsilon)D[X_{0}(t_{0}), X_{0}'(t_{0})]+\sum\limits_{i = 1}^{n}\mu_{i}(\varepsilon)D[X_{i}(t_{i}), X_{i}'(t_{i})], \end{eqnarray*} $

其中$ \mu_{i}(\varepsilon)>0, \; i = 0, 1, \cdots , n $, $ l(\varepsilon) = \max\limits_{1\leq i\leq n}\mu_{i}(\varepsilon) $为连续函数, 并且对任意的$ \varepsilon\in(0, \varepsilon_{0}] $, $ \lim\limits_{\varepsilon \to 0}l(\varepsilon) = 0 $;

($ A_{3.3} $)

$ \begin{eqnarray*} \Big\|\Big[A_{0}+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)\Big]^{-1}\Big\| \Big[\mu_{0}+\sum\limits_{i = 1}^{n}\mu_{i}(\varepsilon)\Big]<1, \; \; \varepsilon\in(0, \varepsilon_{0}], \end{eqnarray*} $

其中$ A_0 $为常数矩阵, 且$ \det A_0\neq 0 $; 对任意的矩阵$ A = (a_{jk})_{m, n} $, $ \parallel A \parallel = \Big[\sum\limits_{k = 1}^{n}\sum\limits_{j = 1}^{m}a^2_{jk}\Big]^{\frac{1}{2}} $, $ \mu_{i}(\varepsilon)>0, \; i = 0, 1, \cdots , n $, $ h(\varepsilon) = \max\limits_{1\leq i\leq n}\parallel A_{i}(\varepsilon)\parallel $为连续函数, 且$ \lim \limits_{\varepsilon \to 0}h(\varepsilon) = 0 $;

($ 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

$ \frac{1}{T}k(t, t+T)\leq d_1<\infty, $

其中$ k(t, t+T) $是在区间$ (t, t+T] $上序列$ \tau_{k} $的点数;

($ 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) $为$ Y(t) $的$ \rho $邻域, $ \rho>0 $为常数.

则存在$ \varepsilon_{0}(\eta, L)>0 $, 使得对任意的$ \varepsilon\in(0, \varepsilon_{0}] $及$ t\in[0, L\varepsilon^{-1}] $, 下列不等式

$ D(X(t), Y(t))<\eta $

成立, 其中$ \eta>0 $, $ L>0 $是常数, $ X(t) $为系统$ \rm(3.1) $–$ \rm(3.2) $的解.

证  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

(3.6) $ \begin{equation} \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. \end{equation} $

且

(3.7) $ \begin{equation} 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), \end{equation} $

其中$ X_{0} = X(t_{0}), \; \zeta_{i} = \int^{t_{i}}_{t_{0}}F(s, X(s)){\rm d}s+\sum\limits_{t_{0}\leq\tau_{k}<t_{i}}I_{k}(X(\tau_{k})) $, $ i = 1, 2, \cdots , n $.

同理, 具有初值和多点边值条件的系统(3.4)–(3.5) 的解满足下列积分方程

(3.8) $ \begin{equation} \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. \end{equation} $

且

(3.9) $ \begin{equation} 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), \end{equation} $

其中$ Y_{0} = Y(t_{0}), \; \overline{\zeta}_{i} = \int^{t_{i}}_{t_{0}}\overline{F}(Y(s)){\rm d}s $, $ i = 1, 2, \cdots , n $.

由条件$ (A_{3.1}) $和$ (A_{3.4}) $, 可得

(3.10) $ \begin{eqnarray} D[\overline{F}(X(t)), \{0\}]&\leq& D\Big[\overline{F}(X(t)), \frac{1}{T}\int^{t+T}_{t}F(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}F(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}F(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+M_{1}+d_1M_{2}, \end{eqnarray} $

(3.11) $ \begin{eqnarray} D[\overline{F}(X_{1}(t)), \overline{F}(X_{2}(t))] &\leq &D\Big[\overline{F}(X_{1}(t)), \frac{1}{T}\int^{t+T}_{t}F(s, X_{1}(s)){\rm d}s +\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X_{1}(\tau_{k}))\Big]{} \\ &&+D\Big[\frac{1}{T}\int^{t+T}_{t}F(s, X_{1}(s)){\rm d}s+\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X_{1}(\tau_{k})), {} \\ &&\frac{1}{T}\int^{t+T}_{t}F(s, X_{2}(s)){\rm d}s+\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X_{2}(\tau_{k}))\Big]{} \\ &&+D\Big[\frac{1}{T}\int^{t+T}_{t}F(s, X_{2}(s)){\rm d}s +\frac{1}{T}\sum\limits_{t\leq\tau_{k}<t+T}I_{k}(X_{2}(\tau_{k})), \overline{F}(X_{2}(t))\Big]{} \\ &\leq&2\delta+\frac{1}{T}\int^{t+T}_{t}D[F(s, X_{1}(s)), F(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+(\lambda_{1}+d_1\lambda_{2})D[X_{1}(t), X_{2}(t)], \end{eqnarray} $

其中通过选择合适的$ T $, 可使$ \delta $充分小. 因此, 可得下列不等式

(3.12) $ \begin{equation} D[\overline{F}(X(t)), \{0\}]\leq M_{1}+d_1M_{2}, \end{equation} $

(3.13) $ \begin{equation} D[\overline{F}(X_{1}(t)), \overline{F}(X_{2}(t))]\leq(\lambda_{1}+d_1\lambda_{2})D[X_{1}(t), X_{2}(t)]. \end{equation} $

进一步, 可得

(3.14) $ \begin{eqnarray} D[X(t), Y(t)]& = &D\Big[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})), Y_{0} +\varepsilon \int^{t}_{t_{0}}\overline{F}(Y(s)){\rm d}s\Big] {}\\ &\leq &D[X_{0}, Y_{0}]+\varepsilon D\Big[\int^{t}_{t_{0}}F(s, X(s)){\rm d}s {}\\ &&+ \sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), \int^{t}_{t_{0}}F(s, Y(s)){\rm d}s+\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k}))\Big]{}\\ &&+\varepsilon D\Big[\int^{t}_{t_{0}}F(s, Y(s)){\rm d}s +\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k})), \int^{t}_{t_{0}}\overline{F}(Y(s)){\rm d}s\Big] {}\\ &\leq& D[X_{0}, Y_{0}]+\varepsilon(\lambda_{1}+d_1\lambda_{2}) \int^{t}_{t_{0}} D[X(s), Y(s)]{\rm d}s{}\\ &&+\varepsilon D\Big[\int^{t}_{t_{0}}F(s, Y(s)){\rm d}s +\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k})), \int^{t}_{t_{0}}\overline{F}(Y(s)){\rm d}s\Big]. \end{eqnarray} $

分析上述不等式, 将区间$ [0, L\varepsilon^{-1}] $进行$ m $等分, 具体表示如下

$ \begin{eqnarray*} t_{0} = 0, \; t_{1} = \frac{L}{\varepsilon m}, \cdots , \; t_{i} = \frac{iL}{\varepsilon m}, \cdots , \; t_{m} = \frac{L}{\varepsilon}, \; \; m\in \mathbb{N}. \end{eqnarray*} $

则对于式(3.14) 的最后一项, 由引理2.1, 我们可得

$ \varepsilon D\Big[\int^{t}_{t_{0}}F(s, Y(s)){\rm d}s +\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k})), \int^{t}_{t_{0}}\overline{F}(Y(s)){\rm d}s\Big]\leq L_{1}+L_{2}+L_{3}, $

其中

$ \begin{eqnarray*} L_{1}& = &\varepsilon\sum^{m-1}_{i = 0}D\Big[\int^{t_{i+1}}_{t_{i}}F(s, Y(s)){\rm d}s+ \sum\limits_{t_{i}\leq\tau_{k}<t_{i+1}}I_{k}(Y(\tau_{k})), \int^{t_{i+1}}_{t_{i}} F(s, Y(t_{i})){\rm d}s\\ &&+\sum\limits_{t_{i}\leq\tau_{k}<t_{i+1}}I_{k}(Y(\tau_{k}))\Big], \\ L_{2}& = &\varepsilon\sum^{m-1 }_{i = 0}\int^{t_{i+1}}_{t_{i}}D[\overline{F}(Y(s)), \overline{F}(Y(t_{i}))]{\rm d}s, \\ L_{3}& = &\varepsilon\sum^{m-1}_{i = 0}D\Big[\int^{t_{i+1}}_{t_{i}}F(s, Y(t_{i})){\rm d}s +\sum\limits_{t_{i}\leq\tau_{k}<t_{i+1}}I_{k}(Y(\tau_{k})), \int^{t_{i+1}}_{t_{i}}\overline{F}(Y(t_{i})){\rm d}s\Big]. \end{eqnarray*} $

根据条件($ A_{3.1} $), 我们有

$ \begin{eqnarray*} D[Y(t_{1}), Y(t_{2})] & = &D\Big[Y_{0}+\varepsilon \int^{t_{1}}_{t_{0}}{\overline F}(Y(s)){\rm d}s, Y_{0} +\varepsilon \int^{t_{2}}_{t_{0}}{\overline F}(Y(s)){\rm d}s\Big]\\ & = &D\Big[\varepsilon\int^{t_{2}}_{t_{1}}{\overline F}(Y(s)){\rm d}s, \{0\}\Big]\\ &\leq& \varepsilon (M_{1}+d_1M_{2})(t_{2}-t_{1}), \; \; \; \; \; t_{2} \geq t_{1}, \end{eqnarray*} $

因此对于所有的$ s \in[t_{i}, t_{i+1}] $, $ i = 0, \cdots , m-1 $, 以下估计

$ D[Y(s), Y(t_{i})]\leq \varepsilon (M_{1}+d_1M_{2}) (s-t_{i}) $

成立. 进一步计算可得

(3.15) $ \begin{eqnarray} L_{1}&\leq&\varepsilon\lambda_{1}\sum^{m-1}_{i = 0}\int^{t_{i+1}}_{t_{i}}D[Y(s), Y(t_{i})]{\rm d}s {}\\ &\leq& \varepsilon^{2} (M_{1}+d_1M_{2})\lambda_{1}\sum^{ m-1}_{i = 0}\int^{t_{i+1}}_{t_{i}}(s-t_{i}){\rm d}s{}\\ &\leq&\frac{\lambda_{1} (M_{1}+d_1M_{2}) L^{2}}{2m}. \end{eqnarray} $

由(3.13) 式, 类似可得

$ \begin{eqnarray*} L_{2}\leq \varepsilon(\lambda_{1}+d_1\lambda_{2})\sum^{m-1}_{i = 0}\int^{t_{i+1}}_{t_{i}}D[Y(s), Y(t_{i})]{\rm d}s \leq\frac{(\lambda_{1}+d_1\lambda_{2})(M_{1}+d_1M_{2})L^{2}}{2m}. \end{eqnarray*} $

因此

(3.16) $ \begin{equation} L_{1}+L_{2}\leq\frac{\lambda_{1} (M_{1}+d_1M_{2}) L^{2}}{2m} +\frac{(\lambda_{1}+d_1\lambda_{2})(M_{1}+d_1M_{2})L^{2}}{2m} = \gamma(m), \end{equation} $

且$ \lim \limits_{m\to\infty}\gamma(m) = 0 $.

根据条件($ A_{3.4} $), 存在单调递减的函数$ \delta(t) $, 使得

(3.17) $ \begin{equation} D\Big[\int^{t}_{t_{0}}F(s, X(t)){\rm d}s+\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), \int^{t}_{t_{0}}\overline{F}(X(t)){\rm d}s\Big]\leq t\delta(t) \end{equation} $

成立. 其中$ \lim\limits_{t \to\infty}\delta(t) = 0 $. 因此

(3.18) $ \begin{eqnarray} L_{3}&\leq&\varepsilon\sum^{m-1}_{i = 0}D\Big[\int^{t_{i+1}}_{t_{0}}F(s, Y(t)){\rm d}s +\sum\limits_{t_{i}\leq\tau_{k}<t_{i+1}}I_{k}(Y(\tau_{k})), \int^{t_{i+1}}_{t_{0}}\overline{F}(Y(t_{i})){\rm d}s\Big]{}\\ &&+\varepsilon\sum^{m-1}_{i = 0}D\Big[\int^{t_{i}}_{t_{0}}F(s, Y(t)){\rm d}s +\sum\limits_{t_{}\leq\tau_{k}<t_{i+1}}I_{k}(Y(\tau_{k})), \int^{t_{i}}_{t_{0}}\overline{F}(Y(t_{i})){\rm d}s\Big]{}\\ &\leq&2m\theta(\varepsilon), \end{eqnarray} $

其中$ \theta(\varepsilon) = \sup\limits_{s\in[0, L]}s\delta\Big(\frac{s}{\varepsilon}\Big), s = \varepsilon t $, 且$ \lim\limits_{\varepsilon \to 0}\theta(\varepsilon) = 0. $

由式(3.7)和(3.9) 可得

(3.19) $ \begin{eqnarray} D[X_{0}, Y_{0}]&\leq &\Big[A_{0}+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)\Big]^{-1}\Big[\mu_{0} +\sum\limits_{i = 1}^{n}\mu_{i}(\varepsilon)\Big]D[X_{0}, Y_{0}]{}\\ &&+\varepsilon\Big[A_{0}+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)\Big]^{-1}\Big[\mu_{0}D[\zeta_{0}, \overline{\zeta}_{0}] +\sum\limits_{i = 1}^{n}\mu_{i}(\varepsilon)D[\zeta_{i}, \overline{\zeta}_{i}]\Big]{}\\ &&-\varepsilon\Big[A_{0}+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)\Big]^{-1} \Big[A_{0}D[\zeta_{0}, \overline{\zeta}_{0}]+\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)D[\zeta_{i}, \overline{\zeta}_{i}]\Big], \end{eqnarray} $

即

(3.20) $ \begin{equation} D[X_{0}, Y_{0}]\leq\varepsilon B(\varepsilon)\sum\limits_{i = 1}^{n}D[\zeta_{i}, \overline{\zeta}_{i}], \end{equation} $

其中

(3.21) $ \begin{equation} B(\varepsilon) = \Big[l(\varepsilon)+h(\varepsilon)\Big]\bigg\|\Big[A_{0} +\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)\Big]^{-1}\bigg\|\times\bigg[1-\bigg\|\Big[A_{0} +\sum\limits_{i = 1}^{n}A_{i}(\varepsilon)\Big]^{-1}\bigg\|\Big[\mu_{0}+\sum\limits_{i = 1}^{n}\mu_{i}(\varepsilon)\Big]\bigg]^{-1}. \end{equation} $

类似于不等式(3.17) 和(3.18) 的估计, 可得

(3.22) $ \begin{equation} \varepsilon D[\zeta_{i}, \overline{\zeta}_{i}] = \varepsilon D\Big[\int^{t_{i}}_{t_{0}}F(s, X(t)){\rm d}s +\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), \int^{t_{i}}_{t_{0}}\overline{F}(Y(t)){\rm d}s\Big] \leq\theta(\varepsilon), \end{equation} $

由式(3.14), (3.16), (3.20) 及(3.22) 可得

$ D[X(t), Y(t)]\leq \rho(\varepsilon)+\varepsilon(\lambda_{1}+d_1\lambda_{2}) \int^{t}_{t_{0}} D[X(s), Y(s)]{\rm d}s, $

其中$ \rho(\varepsilon) = \gamma(m)+2m\theta(\varepsilon)+2n B(\varepsilon)\theta(\varepsilon) $.

根据引理2.2可知, 下列不等式

(3.23) $ \begin{equation} D(X(t), Y(t))\leq\rho(\varepsilon)\exp\{L(\lambda_{1}+d_1\lambda_{2})\} \end{equation} $

成立. 因此, 对于不等式(3.23), 给定$ m $及任意的$ \eta>0 $, 使得对任意$ \varepsilon\in(0, \varepsilon_{0}] $, 有不等式

$ \rho(\varepsilon)\exp\{L(\lambda_{1}+d_1\lambda_{2})\}<\eta $

成立, 因此$ D(X(t), Y(t))< \eta $. 定理3.1证毕.

在本节中, 我们应用局部加法平均法, 研究具有初值和Cancy-Nicoletti类型的多点边值的集值脉冲微分方程问题.

考虑以下带有小参数的集值脉冲微分方程

(4.1) $ \begin{equation} \left\{ \begin{array}{ll} D_{H}X(t) = \varepsilon [G_{1}(t, X(t))+G_{2}(t, X(t))], \; \; \; &t\neq\tau_{k}, \\ X(\tau_{k}^{+})-X(\tau_{k}^{-}) = I_{k}(X(\tau_{k})), \; \; \; &t = \tau_{k}, \end{array} \right. \end{equation} $

及在固定点$ 0\leq t_{0}<\cdot\cdot\cdot<t_{n}<+\infty $处具有初值和Cancy-Nicoletti类型的多点边值问题

(4.2) $ \begin{equation} 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), \end{equation} $

其中$ G_{1}, \; G_{2}:{{\Bbb R}} ^{+}\times K_{c}({{\Bbb R}} ^{n})\to K_{c}({{\Bbb R}} ^{n}) $, 其他假设与第3节中相同.

以下考虑对于$ G_{1}(t, X(t)) $的部分平均问题(对于$ G_{2}(t, X(t)) $可以同理考虑).

假设下列极限存在

(4.3) $ \begin{equation} \overline{G}_{10}(X(t)) = \lim\limits_{T\to\infty}\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}))\Big], \end{equation} $

其中$ \overline{G}_{10}(X(t)) $为区间$ [0, T] $上$ G_{1}(t, X(t)) $的平均, 且是连续函数.

则与系统(4.1) 和(4.2) 对应的局部平均系统如下

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

且初值和多点边值条件为

(4.5) $ \begin{equation} 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). \end{equation} $

定理4.1   假设条件($ A_{3.2} $)–($ A_{3.3} $) 成立, 且在$ \Omega $上满足下列条件.

($ 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

$ D\Big[\int^{t+T}_{t}G_{1}(s, X(s)){\rm d}s, \{0\}\Big]\leq N_{1}, \; \; D[I_{k}(X(\tau_{k})), \{0\}]\leq N_{2}; $

$ D[G_{i}(s, X_{1}(s)), G_{i}(s, X_{2}(s))]{\rm d}s\leq\nu_{i}D[X_{1}(t), X_{2}(t)], \; \; i = 1, 2; $

$ D[I_{k}(X_{1}(t)), I_{k}(X_{2}(t))]\leq\nu_{3}D[X_{1}(t), X_{2}(t)] $

成立;

($ 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) $是$ Y(t) $的$ \rho $邻域, $ \rho>0 $是常数.

则存在$ \varepsilon_{0}(\eta, L)>0 $, 使得对任意的$ \varepsilon\in(0, \varepsilon_{0}] $, 不等式

$ D(X(t), Y(t))<\eta, \ \ t\in[0, L\varepsilon^{-1}] $

成立, 其中$ \eta>0 $, $ L>0 $为常数. $ X(t) $是系统$ \rm(4.1) $和$ \rm(4.2) $的解.

证  带有初值和多点边值条件的系统(4.1)–(4.2) 的解满足如下积分方程

(4.6) $ \begin{equation} \left\{ \begin{array}{ll} { } X(t) = X_{0}+\varepsilon \Big[\int^{t}_{t_{0}}G_{1}(s, X(s)){\rm d}s+\int^{t}_{t_{0}}G_{2}(s, X(s)){\rm d}s\Big] +\varepsilon\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), \\ X(0) = \Psi(0), \end{array} \right. \end{equation} $

且

(4.7) $ \begin{equation} 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), \end{equation} $

其中$ X_{0} = X(t_{0}), \; \zeta_{i} = \int^{t_{i}}_{t_{0}}G(s, X(s)){\rm d}s + \sum\limits_{t_{0}\leq\tau_{k}<t_{i}}I_{k}(X(\tau_{k})) $, $ i = 1, 2, \cdots , n $.

带有初值和多点边值条件的系统(4.4)–(4.5) 的解满足如下积分方程

(4.8) $ \begin{equation} \left\{ \begin{array}{ll} { } Y(t) = Y_{0}+\varepsilon \Big[\int^{t}_{t_{0}}\overline{G}_{10}(Y(s)){\rm d}s+\int^{t}_{t_{0}}G_{2}(s, Y(s)){\rm d}s\Big], \\ Y(0) = \Psi(0), \end{array} \right. \end{equation} $

且

(4.9) $ \begin{equation} 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), \end{equation} $

其中$ Y_{0} = Y(t_{0}), \; \overline{\zeta}_{i} = \int^{t_{i}}_{t_{0}}\overline{G}(t, Y(s)){\rm d}s $, $ i = 1, 2, \cdots , n $.

由条件$ (A_{4.1}) $和$ (A_{4.2}) $, 下列不等式

(4.10) $ \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} $

(4.11) $ \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} $

成立. 其中通过选择合适的$ T $, 可使得$ \delta $充分小. 因此, 可得不等式

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

(4.13) $ \begin{equation} 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)]. \end{equation} $

进一步, 可得

$ \begin{eqnarray*} D[X(t), Y(t)]& = &D\bigg[X_{0}+\varepsilon \Big[\int^{t}_{t_{0}}G_{1}(s, X(s)){\rm d}s+\int^{t}_{t_{0}}G_{2}(s, X(s)){\rm d}s\Big] \\ && +\varepsilon\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), Y_{0}+\varepsilon \Big[\int^{t}_{t_{0}}\overline{G}_{10}(Y(s)){\rm d}s+\int^{t}_{t_{0}}G_{2}(s, Y(s)){\rm d}s\Big]\bigg]\\ &\leq &D[X_{0}, Y_{0}]+\varepsilon D\Big[\int^{t}_{t_{0}}G_{1}(s, X(s)){\rm d}s \\ &&+\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(X(\tau_{k})), \int^{t}_{t_{0}}G_{1}(s, Y(s)){\rm d}s+\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k}))\Big]\\ &&+\varepsilon D\Big[\int^{t}_{t_{0}}G_{1}(s, Y(s)){\rm d}s +\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k})), \int^{t}_{t_{0}}\overline{G}_{10}(Y(s)){\rm d}s\Big]\\ &&+\varepsilon D\Big[\int^{t}_{t_{0}}G_{2}(s, X(s)){\rm d}s, \int^{t}_{t_{0}}G_{2}(s, Y(s)){\rm d}s\Big]\\ &\leq& D[X_{0}, Y_{0}]+\varepsilon(\nu_{1}+\nu_{2}+d_2\nu_{3})\int^{t}_{t_{0}} D[X(s), Y(s)]{\rm d}s\\ &&+\varepsilon D\Big[\int^{t}_{t_{0}}G_{1}(s, Y(s)){\rm d}s +\sum\limits_{t_{0}\leq\tau_{k}<t}I_{k}(Y(\tau_{k})), \int^{t}_{t_{0}}\overline{G}_{10}(s, Y(s)){\rm d}s\Big]. \end{eqnarray*} $

后续证明类似于定理3.1的证明过程, 在此省略. 定理4.1证毕.