本文考虑一类时滞依赖状态的集值抽象积分微分方程：

$ \begin{array}{l} \frac{d}{{dt}}\left[{x\left( t \right)-F\left( {t, {x_t}} \right)} \right] \in A\left[{x\left( t \right) + \int_0^t f \left( {t-s} \right)x\left( s \right)ds} \right] + G\left[{t, {x_{\rho \left( {t, {x_t}} \right)}}} \right], \\ t \in J = [0, a], \end{array} $

(1.1)

$ {x_0} = \varphi \in {\cal B}, $

(1.2)

其中A是Banach空间$\left( X, \Vert \cdot \Vert \right)$中紧分析预解算子$R\left( t\right), t>0$的无穷小生成元, $f(t), \ t\in J$是一个有界线性算子, $G:J\times \mathcal{B}\rightarrow P\left( X\right)$是有界闭凸值集值映射, $F:J\times \mathcal{B}\rightarrow X$, $\rho :J\times \mathcal{B}\rightarrow \left( -\infty, a\right]$是给定的函数, $\mathcal{B} $是一个抽象的相空间, $x_{t}:\left( -\infty, 0\right] \rightarrow X$, $% x_{t}\left( s\right) =x\left( t+s\right), s\leq 0$属于$\mathcal{B}$.

近年来，时滞依赖状态的微分方程温和解的存在性引起了许多学者的兴趣^[1-3]，具有预解算子的非线性积分微分方程作为偏积分微分方程的抽象形式出现在许多物理现象中^[4-6]，该预解算子类似于Banach空间中抽象微分方程的半群算子，并且可以从半群理论中提取(见文献[7-8]),，尽管如此，该预解算子不满足半群性质，受文献[9-10]启发，本文旨在利用预解算子理论建立(1.1)–(1.2) 温和解的存在性.

$ R\left(t\right), t>0$是由A产生的一个紧分析预解算子.设$\left(X, \| \cdot \|\right)$是Banach空间, $L\left(X;Y\right)$记为从X映射到Y的所有有界线性算子构成的Banach空间, 并且当$X=Y$时简记为$L\left(X\right)$.

定义2.1^[11] 若每个集合$\widetilde{B}_{i}$在空间$C\left(\left[t_{i}, t_{i+1}\right];X\right)$内是相对紧时，则称$B\subseteq \mathcal{P}\mathcal{C}$是相对紧的.

定义2.2 若对任意的$x\in X$有$G\left( x\right) $是凸(闭)的, 则称集值映射$G:X\rightarrow P\left( X\right) $是凸(闭)值的.对X的任一有界子集B, 如果$G\left( B \right) = \bigcup\limits_{x \in B} G \left( x \right)$有界, 则称G在有界集上有界(即$\mathop {\sup }\limits_{x \in B} \{ \sup \{ \left\| y \right\|:y \in G(x)\} \} < \infty $).

定义2.3  若对任意的$x_{0}\in X$, $G\left( x_{0}% \right)$是X的非空闭子集, 并且对包含$G\left( x_{0}\right)$的X的任一开子集$\Omega$, 存在$x_{0}$的一个开邻域V, 使得$G\left( V\right) \subseteq \Omega$, 则称G在X中是上半连续的.

定义2.4 如果对X的任一有界子集$\Omega$, 有$G\left( \Omega \right)$是相对紧的, 则称集值映射G是全连续的.

引理2.1^[12] 设集值映射^[13-14]G是全连续的, 并且有非空紧值, 则G是上半连续的当且仅当G有闭图像(即当$ x_{n}\rightarrow x_{\ast }, y_{n}\rightarrow y_{\ast }, y_{n}\in G\left( x_{n}\right)$时, 有$\ y_{\ast }\in G\left( x_{\ast }\right)$).

定义2.5 记$\mathcal{N}$是kuratowski非紧测度^[15], 如果集值映射$G:X\rightarrow P\left( X\right) $是上半连续的.且对所有满足$% \mathcal{N}\left( B\right) \neq 0$的X的子集B, 有$\mathcal{N}\left( G\left( B\right) \right) < \mathcal{N}\left( B\right)$, 则称G是凝聚的^[13].

$P_{bd, cp, cv}\left( X\right)$, $P_{bd, cl, cv}\left( X\right)$, $P_{cp, cv}\left( X\right)$分别表示由X中的所有有界紧凸, 有界闭凸, 紧凸子集组成的集类.若存在$x\in X$, 使得$x\in G\left( x\right)$, 则称x是集值映射G的一个不动点^[16].

下面介绍相空间$ \mathcal{B}$的公理化定义^[17].具体来说, $ \mathcal{B}$是由赋予半模$\Vert \cdot \Vert _{\mathcal{B}}$的从$\left( -\infty, 0\right] $映射到X的函数组成的线性空间, 并且符合以下公理:

(A) 如果$x:\left(-\infty, \sigma+a\right)\rightarrow X$, $a>0$, 使得$x|_{\left[\sigma, \sigma+a\right]}\in C\left(\left[% \sigma, \sigma+a\right], X\right)$且$x_{\sigma}\in \mathcal{B}$, 则对任意的$t\in\left[\sigma, \sigma+a\right]$以下条件成立: (ⅰ) $x_{t}\in\mathcal{B}$; (ⅱ) $\|x\left(t\right)\|\leq H\|x_{t}\|_{\mathcal{B}}$; (ⅲ) $\|x_{t}\|_{\mathcal{B}}\leq K\left(t-\sigma\right)\sup\left\{\|x\left(s\right)\|:\sigma \leq s \leq t\right\}+M\left(t-\sigma\right)\|x_{\sigma}\|_{\mathcal{B}}$, 其中$H>0$是常数; $K, M:\left[0, \infty\right)\rightarrow\left[1, \infty\right)$, $% K$连续, M局部有界, 并且H, K, M与$x\left(\cdot\right)$无关.

(A1) 对(A)中的函数$x\left(\cdot\right)$, $t\rightarrow x_{t}$是从$\left[\sigma, \sigma+a\right]$映射到$\mathcal {B}$的连续函数.

(B) $\mathcal{B}$是完备的.

定义2.6  集值映射$G:J\times \mathcal{B}\rightarrow P_{cp, cv}\left( X\right)$称为Carathéodory集值映射, 如果:

(ⅰ) 对每个$\psi\in\mathcal{B}$, $G\left(\cdot, \psi\right):J\rightarrow X$是可测的;

(ⅱ) 对任意的$t\in J$, $G\left(t, \cdot\right):\mathcal{B}\rightarrow X$是上半连续的.

引理2.2^[12]  若G为Carathéodory集值映射, 且对固定的$\psi \in \mathcal{B}$, 集合$S_{G, \psi }=\left\{g\in L^{1}\left( J, X\right) \right. \left. :g\left( t\right) \in G\left( t, \psi \right) \ a.e.\ t\in J\right\}$是非空的, $\Gamma :L^{1}\left( J, X\right) \rightarrow C\left( J, X\right)$是线性连续映射, 则$ \Gamma \circ S_{G}:C\left( J, X\right) \rightarrow P_{cp, cv}\left( C\left( J, X\right) \right), \ y\rightarrow \left( \Gamma \circ S_{G}\right) \left( y\right) =\Gamma \left( S_{G}, _{y}\right)$是$C\left( J, X\right) \times C\left( J, X\right)$上的闭图算子.

考虑如下系统:

$ \frac{dx}{dt}=A\left[x\left( t\right) +\int_{0}^{t}f\left( t-s\right) x\left( s\right) ds\right], $

(2.1)

定义2.7^[7]  方程(2.1) 的预解算子^[4-5]是一个有界算子泛函$ R\left( t\right) \in L\left( X\right)$, 且具有下列性质:

(ⅰ) $R\left(0\right)=I$, I表示X上的恒等算子.

(ⅱ) 对所有的$u\in X$, $t\mapsto R\left(t\right)y$是连续的.

(ⅲ) 对$t\in J$, $R\left( t\right) \in B\left( Y\right), t\in J$, 其中Y是赋予图像范数$\Vert y\Vert_{Y}=\Vert Ay\Vert+\Vert y\Vert$的Banach空间$D\left( A\right)$.若$y\in Y$, 有$R\left( \cdot \right) y\in C^{1}\left( J, X\right) \cap C\left( J, Y\right)$并且

$ \begin{array}{l} \frac{d}{{dt}}R\left( t \right)y = A\left[{R\left( t \right)y + \int_0^t f \left( {t-s} \right)R\left( s \right)yds} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = R\left( t \right)Ay + \int_0^t R \left( {t -s} \right)AF\left( s \right)yds, t \in J. \end{array} $

令$0\in \rho (A)$, 当$0 < \alpha \leq 1$时，可以定义分数阶算子$% A^{\alpha }$, 此算子在其定义域$D\left( A^{\alpha }\right)$^[4]上是一个闭线性算子.并且, $D\left( A^{\alpha }\right)$在X上凝聚.表达式

$ \Vert x\Vert _{\alpha }=\Vert A^{\alpha }x\Vert, \ x\in D\left( A^{\alpha }\right) $

定义了$D\left( A^{\alpha }\right)$上的一个范数.此后，用$% X_{\alpha }$表示Banach空间$D\left( A^{\alpha }\right)$并赋予范数$\Vert x\Vert _{\alpha }$.

引理2.3 ^[4]  由上面条件可得：

(ⅰ)$A^{\alpha}:X_{\alpha}\rightarrow X_{\alpha}$，则当$0\leq \alpha \leq 1$时，$X_{\alpha}$是一个Banach空间.

(ⅱ)若预解算子A是紧的，则当$0 < \beta \leq \alpha$时，$X_{\alpha }\hookrightarrow X_{\beta }$是连续并且紧的.

(ⅲ)对每个$\alpha >0$，存在一个正常数$% C_{\alpha }$使得

$ \Vert A^{\alpha }R(t)\Vert \leq \frac{C_{\alpha }}{t^{\alpha }}, \ t\in J. $

引理2.4^[18]  假设(a)和(b)成立, 则存在一个常数$H=H(a)$, 使得

$ \Vert R(t+h)-R(h)R(t)\Vert_{L(X)}\leq Hh, 0\leq h\leq t\leq a. $

引理2.5^[7]  设B是Banach空间X中的有界凸子集, $\Gamma:B\rightarrow P\left(B\right)$是上半连续的凝聚集值映射, 如果对任意的$x\in B$, $\Gamma\left(x\right)$是B中的闭凸子集, 则$\Gamma$在B中有一个不动点.

为了证明问题(1.1)–(1.2) 的温和解的存在性, 假设$\rho :J\times \mathcal{B}\rightarrow \left( -\infty, a\right]$连续, 另外, 假定以下条件成立:

H$_{\varphi }$ 函数$t\rightarrow \varphi _{t}$从$\mathcal{R}% \left( \rho ^{-}\right) =\left\{ \rho \left( s, \psi \right) :\rho \left( s, \psi \right) \leq 0, \left( s, \psi \right) \in J\times \mathcal{B}\right\}$到$\mathcal{B}$上是连续的, 且存在一个连续有界函数$J^{\varphi }:\mathcal{R}\left( \rho ^{-}\right) \rightarrow \left( 0, \infty \right)$, 使得对每个$t\in \mathcal{R}\left( \rho ^{-}\right)$, 有$\Vert \varphi _{t}\Vert _{% \mathcal{B}}\leq J^{\varphi }\left( t\right) \Vert \varphi \Vert _{\mathcal{B% }}$.

H1 存在常数$M_{1}, M_{2}>0, $使得$\Vert R\left( t\right) \Vert \leq M_{1}, \Vert f\left( t\right) \Vert \leq M_{2}, t\in J$.

H2 $F:J\times \mathcal{B}\rightarrow X$是连续的, 存在$% \beta \in \left( 0, 1\right)$, $L>0$使得对所有的$\ \left( t_{i}, \psi _{i}\right) \in J\times \mathcal{B}, \ i=1, 2.$，F是$X_{\beta }$-值并满足

$ \Vert A^{\beta }F\left( t_{1}, \psi _{1}\right) -A^{\beta }F\left( t_{2}, \psi _{2}\right) \Vert \leq L\left( \left\vert t_{1}-t_{2}\right\vert +\Vert \psi _{1}-\psi _{2}\Vert _{\mathcal{B}}\right) $

而且, 存在正常数$L_{1}$使得

$ \Vert A^{\beta }F\left( t, \psi \right) \Vert \leq L_{1}\left( \Vert \psi \Vert _{\mathcal{B}}+1\right), \ \text{for }t\in J, \psi \in \mathcal{B}. $

H3 函数$G:J\times \mathcal{B}\rightarrow P_{bd, cp, cv}\left( X\right)$满足如下性质：

(a) 对所有的$\psi\in\mathcal{B}$，函数$G\left(\cdot, \psi\right):J\rightarrow X$是强可测的.

(b) 对每个$t\in J$，函数$G\left(t, \cdot\right):\mathcal{B}\rightarrow X$是连续的，且对固定的$\psi \in \mathcal{B}$, 集合$S_{G, \psi }=\left\{ g\in L^{1}\left( J, X\right) :g\left( t\right) \in G\left( t, \psi \right)\ a.e.\ t\in J\right\}$是非空的.

(c) 对每个正常数$r, $存在正函数$\Theta _{r}\in L^{1}\left( J, \mathbb{R}_{+}\right)$使得

$ \left\| {G\left( {t,\psi } \right)} \right\| = \mathop {\sup }\limits_{{{\left\| \psi \right\|}_{\cal B}} \le r} \left\{ {\left\| g \right\|:g(t) \in G(t,\psi )} \right\} \le {\Theta _r}\left( t \right),{\rm{ }}\mathop {\lim \inf }\limits_{r \to + \infty } \frac{1}{r}\int_0^a {{\Theta _r}} \left( t \right)dt = \delta < + \infty . $

注3.1^[17]  设$\varphi \in \mathcal{B}, t\leq 0$. $\varphi _{t}$表示定义为$\varphi _{t}\left( \theta \right) =\varphi \left( t+\theta \right)$形式的函数.因此, 如果公理A中的函数$x\left( \cdot \right)$满足$ x_{0}=\varphi$, 则$x_{t}=\varphi _{t}$.

定义3.1  函数$x:(-\infty, a]\rightarrow X$称为问题(1.1)–(1.2) 的温和解, 如果对任意的$s\in J$, 有$x_{0}=\varphi$, $% x_{\rho \left( s, x_{s}\right) }\in \mathcal{B}$, 函数$AR\left( t-s\right) F\left( s, x_{s}\right), s\in \left[ 0, a\right]$Bochner可积，$x(\cdot )$在区间$(t_{i}, t_{i+1}]\ (i=0, 1, \cdots, n)$上是连续的，并且满足

$ \begin{array}{l} x\left( t \right) = R\left( t \right)\left[{\varphi \left( 0 \right)-F\left( {0, \varphi } \right)} \right] + F\left( {t, {x_t}} \right) + \int_0^t A R\left( {t -s} \right)F\left( {s, {x_s}} \right)ds\\ \;\;\;\;\;\;\;\; = + \int_0^t A R\left( {t -s} \right)\int_0^s f \left( {s -\tau } \right)F\left( {\tau, {x_\tau }} \right)d\tau ds + \int_0^t R \left( {t - s} \right)G\left( {s, {x_{\rho \left( {s, {x_s}} \right)}}} \right)ds. \end{array} $

引理3.1^[17]  如果$x:\left( -\infty, a\right] \rightarrow X$是一个使得$x_{0}=\varphi$且$x|_{J}\in \mathcal{P}% \mathcal{C}\left( J, X\right)$成立的函数, 则

$ \Vert x_{s}\Vert _{\mathcal{B}}\leq \left( M_{a}+\widetilde{J^{\varphi }}% \right) \Vert \varphi \Vert _{\mathcal{B}}+K_{a}\sup \left\{ \Vert x\left( \theta \right) \Vert ;\theta \in \left[0, \max \left\{ 0, s\right\} \right] \right\}, \ s\in \mathcal{R}\left( \rho ^{-}\right) \cup J, $

其中$\widetilde{J^{\varphi }}=\mathop {\sup }\limits_{t\in \mathcal{R}\left( \rho ^{-}\right) }J^{\varphi }\left( t\right)$, ${M_a} = \mathop {\sup }\limits_{t \in J} M\left( t \right)$，${K_a} = \mathop {\max }\limits_{t \in J} K\left( t \right)$.

定理3.1  假设H1-H3和H$_{\varphi }$成立，如果

$ K_{a}\left[\left\Vert A^{-\beta }\right\Vert L_{1}+\frac{C_{1-\beta }}{% \beta }a^{\beta }L_{1}\left( 1+aM_{2}\right) +M_{1}\delta \right] <1, $

(3.1)

$ K_{a}\left[\left\Vert A^{-\beta }\right\Vert L+\frac{C_{1-\beta }}{\beta }% a^{\beta }L\left( 1+aM_{2}\right) \right] <1, $

(3.2)

则系统(1.1)–(1.2) 至少存在一个温和解.

证  在赋予一致收敛范数的空间$\mathbb{Y}=\left\{ u\in \mathcal{P}\mathcal{C}:u\left( 0\right) =\varphi \left( 0\right) \right\}$上定义算子$\Gamma :\mathbb{Y}% \rightarrow P\left( \mathbb{Y}\right)$，定义如下

$ \Gamma \left( x\right) =\left\{ u\in \mathbb{Y}:u\left( t\right) =R\left( t\right) \left[\varphi \left( 0\right) -F\left( 0, \varphi \right) \right] +F\left( t, \overline{x}_{t}\right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, \overline{x}_{s}\right) ds\right. \\ \;\;\;\;\;\;\;\left. +\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, \overline{x}_{\tau }\right) d\tau ds+\int_{0}^{t}R\left( t-s\right) g\left( s, \overline{x}_{\rho \left( s, \overline{x}_{s}\right) }\right) ds, \right. \\ \;\;\;\;\;\;\;\left.\ \ t\in J, \ g\in S_{G, \overline{x}_{_{\rho }}}, t\in J\right\}, $

其中

$ S_{G, \overline{x}_{_{\rho }}}=\left\{g\in L^{1}\left( J, X\right) :g\left( t\right) \in G(t, \overline{x}_{\rho \left( t, x_{t}\right) }), \ {\rm{a}}{\rm{.e}}.\ t\in J\right\}, $

$\overline{x}:\left( -\infty, a\right] \rightarrow X$满足$ \overline{x}_{0}=\varphi$, 且在J上$\overline{x}=x$.

设$\overline{\varphi }:\left( -\infty, a\right) \rightarrow X$是$\varphi$的延拓使得在J上$\overline{% \varphi }\left( \theta \right) =\varphi \left( 0\right)$, 且

$% \widetilde{J^{\varphi }}=\sup \left\{ J^{\varphi }\left( s\right) :s\in \mathcal{R}\left( \rho ^{-}\right) \right\}.$

由公理A和$R\left( t\right)$的强连续性, 可知当$x\in B_{r}\left( 0, \mathbb{Y}% \right)$时，$\Gamma x\in \mathcal{P}\mathcal{C}$.又由引理2.3、H2和引理3.1, 可得

$ \left\Vert AR\left( t-s\right) F\left( s, \overline{x}_{s}\right) \right\Vert =\left\Vert A^{1-\beta }R\left( t-s\right) A^{\beta }F\left( s, \overline{x}% _{s}\right) \right\Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\le \frac{C_{1-\beta }}{\left( t-s\right) ^{1-\beta }}L_{1}\left[ rK_{a}+\left( M_{a}+\widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{% \mathcal{B}}+1\right], $

然后由Bocher's定理^[19]可知$AR\left( t-s\right) F\left( s, \overline{x}_{s}\right)$是可积的.所以$\Gamma$在$B_{r}\left( 0, \mathbb{Y}\right)$上有明确定义.

分下面几步来证明温和解的存在性.

第一步  存在$r>0$, 使得$\Gamma \left( B_{r}\left( 0, % \mathbb{Y}\right) \right) \subset B_{r}\left( 0, \mathbb{Y}\right)$.

对任意的$r>0$, 易知$B_{r}$是$\mathbb{Y}$中的一个有界闭凸子集.下面证明存在$r>0$, 使得$\Gamma \left( B_{r}\right) \subset B_{r}$, 其中$\Gamma \left( {{B_r}} \right) = \bigcup\limits_{x \in {B_r}} \Gamma \left( x \right)$.采用反证法, 事实上, 如果上述结果不成立, 则对任意的$r>0$, 存在$ x^{r}\in B_{r}$, 使得$u^{r}\in \Gamma \left( x^{r}\right) $但$\Vert u^{r}\Vert _{\infty }>r$, 且对某些$g^{r}\in S_{G, \left(\overline{x^{r}}\right)_{\rho }}$, 有

$ u^{r}\left( t\right) =R\left( t\right) \left[\varphi \left( 0\right)-F\left( 0, \varphi \right) \right] +F\left( t, x_{t}\right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, x_{s}\right) ds \\ \;\;\;\;\;\;+\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, x_{\tau }\right) d\tau ds+\int_{0}^{t}R\left( t-s\right) G\left( s, x_{\rho \left( s, x_{s}\right) }\right) ds, $

故$r < \Vert u^{r}\Vert _{\infty }=\max_{t\in J}\left\vert u^{r}\left( t\right) \right\vert$.

因此, 对某些$t\in \left[0, a\right]$, 有

$ r <\Vert R\left( t\right) \left[\varphi \left( 0\right)-F\left( 0, \varphi \right) \right] \Vert +\Vert F\left( t, \left( \overline{x^{r}}% \right) _{t}\right) \Vert \\ \;\;\;\;\;+\int_{0}^{t}\Vert AR\left( t-s\right) F\left( s, \left( \overline{% x^{r}}\right) _{s}\right) \Vert ds \\ \;\;\;\;\;+\left\Vert \int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, \overline{x^{r}}_{\tau }\right) d\tau ds\right\Vert \\ \;\;\;\;\;+\int_{0}^{t}\Vert R\left( t-s\right) G\left( s, \overline{x^{r}}_{\rho \left( s, \overline{x}_{s}\right) }\right) \Vert ds \\ \;\;\;\;\;\leq M_{1}\left[H\Vert \varphi \Vert _{\mathcal{B}}+\left\Vert A^{-\beta }\right\Vert \left( L_{1}\Vert \varphi \Vert _{\mathcal{B}}+1\right) \right] \\ \;\;\;\;\;+\left\Vert A^{-\beta }\right\Vert L_{1}\left[rK_{a}+\left( M_{a}+% \widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+1\right] \\ \;\;\;\;\;+\frac{1}{\beta }C_{1-\beta }a^{\beta }L_{1}\left[rK_{a}+\left( M_{a}+% \widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+1\right] \\ \;\;\;\;\;+aM_{2}\int_{0}^{t^{r}}\frac{C_{1-\beta }}{\left( t^{r}-s\right) }L_{1}% \left[rK_{a}+\left( M_{a}+\widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+1\right] ds \\ \;\;\;\;\;+M_{1}\int_{0}^{t^{r}}\Theta _{rK_{a}+\left( M_{a}+\widetilde{J^{\varphi }}% \right) \Vert \varphi \Vert _{\mathcal{B}}}\left( s\right) ds \\ \;\;\;\;\;\leq M_{1}\left[H\Vert \varphi \Vert _{\mathcal{B}}+\left\Vert A^{-\beta }\right\Vert \left( L_{1}\Vert \varphi \Vert _{\mathcal{B}}+1\right) \right] \\ \;\;\;\;\;+\left\Vert A^{-\beta }\right\Vert L_{1}\left[rK_{a}+\left( M_{a}+% \widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+1\right] \\ \;\;\;\;\;+\frac{1}{\beta }C_{1-\beta }a^{\beta }L_{1}\left[rK_{a}+\left( M_{a}+% \widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+1\right] \\ \;\;\;\;\;+\frac{1}{\beta }C_{1-\beta }a^{1+\beta }M_{2}L_{1}\left[rK_{a}+\left( M_{a}+\widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+1% \right] \\ \;\;\;\;\;+M_{1}\int_{0}^{a}\Theta _{rK_{a}+\left( M_{a}+\widetilde{J^{\varphi }}% \right) \Vert \varphi \Vert _{\mathcal{B}}}\left( s\right) ds. $

所以

$ 1\leq K_{a}\left[\left\Vert A^{-\beta }\right\Vert L_{1}+\frac{C_{1-\beta }% }{\beta }a^{\beta }L_{1}\left( 1+aM_{2}\right) +M_{1}\delta \right], $

这与(3.1) 矛盾.因此存在$r>0$使得$\Gamma \left( B_{r}\left( 0, \mathbb{Y}\right) \right) \subset B_{r}\left( 0, \mathbb{% Y}\right)$.

第二步  对每个$x\in \mathbb{Y}, \Gamma\left(x\right)$是凸的.

如果$u_{1}, u_{2}\in \Gamma \left( x\right)$, 则存在$ g_{1}, g_{2}\in S_{G, \overline{x}_{\rho }}$, 使得对任意的$t\in J$, 有

$ u_{i}\left( t\right) =R\left( t\right) \left[\varphi \left( 0\right)-F\left( 0, \varphi \right) \right] +F\left( t, x_{t}\right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, x_{s}\right) ds \\ \;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, x_{\tau }\right) d\tau ds +\int_{0}^{t}R\left( t-s\right) g_{j}\left( s\right) ds, \ j=1, 2. $

设$0\leq \lambda \leq 1$, 则对任意的$t\in J$, 有

$ \left( \lambda u_{1}+\left( 1-\lambda \right) u_{2}\right) \left( t\right) =R\left( t\right) \left[\varphi \left( 0\right)-F\left( 0, \varphi \right) \right] +F\left( t, x_{t}\right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, x_{s}\right) ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, x_{\tau }\right) d\tau ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t}R\left( t-s\right) \left[\lambda g_{1}\left( s\right) +\left( 1-\lambda \right) g_{2}\left( s\right) \right]ds. $

因为$S_{G, \overline{x}_{\rho }}$是凸的(因为$G$有凸值), 所以.

$ \left( \lambda u_{1}+\left( 1-\lambda \right) u_{2}\right) \in \Gamma \left( x\right) $

第三步  对每个$x\in \mathbb{Y}, \Gamma\left(x\right)$是闭的.

设$\left\{ x_{n}\right\} _{n\geq 0}\in \Gamma \left( y\right)$, 使得在$\mathbb{Y}$中$ x_{n}\rightarrow x$, 则$x\in \mathbb{Y}$, 且存在$g_{_{n}}\in S_{G, \overline{x}_{_{\rho }}}$, 使得

$ x_{n}\left( t\right) =R\left( t\right) \left[\varphi \left( 0\right)-F\left( 0, \varphi \right) \right] +F\left( t, x_{t}\right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, x_{s}\right) ds \\ \;\;\;\;\;\;\;\;\;+\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, x_{\tau }\right) d\tau ds+\int_{0}^{t}R\left( t-s\right) g_{_{n}}\left( s\right) ds. $

由于G有非空紧值, 则在$L^{1}\left( J, X\right) $中存在一个序列$g_{_{n}}$使得$g_{_{n}}\rightarrow g$, 则$g\in S_{G, \overline{x}_{_{\rho }}}$.故

$ x_{n}\left( t\right) \rightarrow x\left( t\right) =R\left( t\right) \left[\varphi \left( 0\right)-F\left( 0, \varphi \right) \right] +F\left( t, x_{t}\right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, x_{s}\right) ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, x_{\tau }\right) d\tau ds+\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds, $

从而得到$x\in \Gamma \left( x\right)$.

第四步  从$B_{r}\left( 0, \mathbb{Y}\right) $到$B_{r}\left( 0, \mathbb{Y}\right)$上，$\Gamma$是凝聚映射.

为证明$\Gamma$是凝聚映射，令$\Gamma =\Gamma _{1}+\Gamma _{2}$，其中

$ \Gamma _{1}x\left( t\right) =F\left( t, \overline{x}_{t}\right) -R\left( t\right) F\left( 0, \varphi \right) +\int_{0}^{t}AR\left( t-s\right) F\left( s, \overline{x}_{s}\right) ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}f\left( s-\tau \right) F\left( \tau, \overline{x}_{\tau }\right) d\tau ds, \ t\in J, \\ \;\;\;\;\;\;\;\Gamma _{2}x=\left\{ u\in\mathbb{ Y}:u\left( t\right) =R\left( t\right) \varphi \left( 0\right) +\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds, \ g\in S_{G, \overline{x}_{\rho }}\right\} . $

下面证明$\Gamma _{1}$压缩且$\Gamma _{2}$是全连续算子.为了证明$\Gamma _{1}$压缩, 任取$x^{\ast }, x^{\ast \ast }\in B_{r}\left( 0, \mathbb{Y}% \right)$.对每个$t\in J$，有

$ \Vert \left( \Gamma _{1}x^{\ast }\right) \left( t\right) -\left( \Gamma _{1}x^{\ast \ast }\right) \left( t\right) \Vert \leq \Vert F\left( t, % \overline{x^{\ast }}_{t}\right) -F\left( t, \overline{x^{\ast \ast }}% _{t}\right) \Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;+\Vert \int_{0}^{t}AR\left( t-s\right) \left[F\left( s, \overline{x^{\ast }% }_{s}\right)-F\left( s, \overline{x^{\ast \ast }}_{s}\right) \right] ds\Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;+\Vert \int_{0}^{t}AR\left( t-s\right) \int_{0}^{s}M_{1}\left[F\left( \tau, \overline{x^{\ast }}_{\tau }\right)-F\left( \tau, \overline{x^{\ast \ast }}_{\tau }\right) \right] d\tau ds\Vert \\ \\ \;\;\;\;\;\;\;\;\;\;\;\;\leq \left\Vert A^{-\beta }\right\Vert L\Vert \overline{x^{\ast }}_{t}-% \overline{x^{\ast \ast }}_{t}\Vert _{\mathcal{B}}+L\int_{0}^{t}\frac{% C_{1-\beta }}{\left( t-s\right) ^{\beta }}\Vert \overline{x^{\ast }}_{s}-% \overline{x^{\ast \ast }}_{s}\Vert _{\mathcal{B}}ds \\ \;\;\;\;\;\;\;\;\;\;\;\;+aM_{2}L\int_{0}^{t}\frac{C_{1-\beta }}{\left( t-s\right) ^{\beta }}\Vert \overline{x^{\ast }}_{s}-\overline{x^{\ast \ast }}_{s}\Vert _{\mathcal{B}}ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\leq \left\Vert A^{-\beta }\right\Vert LK_{a}\sup \left\{ \Vert \overline{% x^{\ast }}\left( \theta \right) -\overline{x^{\ast \ast }}\left( \theta \right) \Vert, \ 0\leq \theta \leq t\right\} \\ \;\;\;\;\;\;\;\;\;\;\;\;+\frac{C_{1-\beta }}{\beta }a^{\beta }LK_{a}\sup \left\{ \Vert \overline{% x^{\ast }}\left( \theta \right) -\overline{x^{\ast \ast }}\left( \theta \right) \Vert, \ 0\leq \theta \leq t\right\} \\ \;\;\;\;\;\;\;\;\;\;\;\;+\frac{C_{1-\beta }}{\beta }a^{\beta +1}M_{2}LK_{a}\sup \left\{ \Vert \overline{x^{\ast }}\left( \theta \right) -\overline{x^{\ast \ast }}\left( \theta \right) \Vert, \ 0\leq \theta \leq t\right\} \\ \;\;\;\;\;\;\;\;\;\;\;\;\leq L^{\ast }\mathop {\sup }\limits_{0 \le s \le a}\Vert \overline{x^{\ast }}\left( s\right) -\overline{x^{\ast \ast }}\left( s\right) \Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;=L^{\ast }\mathop {\sup }\limits_{0 \le s \le a}\Vert x^{\ast }\left( s\right) -x^{\ast \ast }\left( s\right) \Vert \text{ (since }\overline{x}=x\text{ on }J\text{ )}, $

其中

$ L^{\ast }=K_{a}\left[\left\Vert A^{-\beta }\right\Vert L+\frac{ C_{1-\beta }}{\beta }a^{\beta }L\left( 1+aM_{2}\right) \right] <1 $

(见(3.2) 式).因此

$ \Vert \Gamma _{1}x^{\ast }-\Gamma _{1}x^{\ast \ast }\Vert _{\mathcal{P}% \mathcal{C}}\leq L^{\ast }\Vert x^{\ast }-x^{\ast \ast }\Vert _{\mathcal{P}% \mathcal{C}}, $

即$\Gamma _{1}$是压缩的.

下面证明$\Gamma _{2}$是全连续的.

(ⅰ)显然$\Gamma _{2}\left( B_{r}\left( 0, \mathbb{Y}\right) \right) =\left\{ \Gamma _{2}x:x\in B_{r}\left( 0, \mathbb{Y}\right) \right\}$是有界的.

(ⅱ)$\Gamma _{2}\left( B_{r}\left( 0, \mathbb{Y}\right) \right) =\left\{ \Gamma _{2}x:x\in B_{r}\left( 0, \mathbb{Y}\right) \right\}$是等度连续的.

令$t_{1}, t_{2}\in J, \ t_{1} < t_{2}$.设$x\in B_{r}, u\in \Gamma _{2}\left( x\right)$, 则存在$g\in S_{G, \overline{x}_{\rho }}$, 使得

$ u\left( t\right) =R\left( t\right) \varphi \left( 0\right) +\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds, $

则

$ \Vert u\left( t_{2}\right) -u\left( t_{1}\right) \Vert \leq \Vert \left[R\left( t_{2}\right)-R\left( t_{1}\right) \right] \varphi \left( 0\right) \Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\Vert \int_{0}^{t_{1}-\varepsilon }\left[R\left( t_{2}-s\right)-R\left( t_{1}-s\right) \right] G\left( s, \overline{x}_{\rho \left( s, \overline{x}% _{s}\right) }\right) ds\Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\Vert \int_{t_{1}-\varepsilon }^{t_{1}}\left[R\left( t_{2}-s\right) -R\left( t_{1}-s\right) \right] G\left( s, \overline{x}_{\rho \left( s, % \overline{x}_{s}\right) }\right) ds\Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\Vert \int_{t_{1}}^{t_{2}}R\left( t_{2}-s\right) G\left( s, \overline{x}% _{\rho \left( s, \overline{x}_{s}\right) }\right) ds\Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq \Vert \left[R\left( t_{2}\right)-R\left( t_{1}\right) \right] \varphi \left( 0\right) \Vert \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t_{1}-\varepsilon }\left\Vert R\left( t_{2}-s\right) -R\left( t_{1}-s\right) \right\Vert \Theta _{r^{\ast }}\left( s\right) ds \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2M_{1}\int_{t_{1}-\varepsilon }^{t_{1}}\Theta _{r^{\ast }}\left( s\right) ds+M_{1}\int_{t_{1}}^{t_{2}}\Theta _{r^{\ast }}\left( s\right) ds, $

其中

$ r^{\ast }=\left( M_{a}+\widetilde{J^{\varphi }}\right) \Vert \varphi \Vert _{\mathcal{B}}+K_{a}r. $

因为$R\left( t\right)$是紧且解析的，所以$R\left( t\right)$在$\left( 0, a\right]$上是一致算子拓扑连续的.又因$t_{2}\rightarrow t_{1}$且$\varepsilon$充分小, 所以上述不等式的右端趋于零且与$x\in B_{r}$无关.

(ⅲ) $\left( \Gamma _{2}B_{r}\left( 0, \mathbb{Y}\right) \right) \left( t\right) =\left\{ \Gamma _{2}x\left( t\right) :x\in B_{r}\left( 0, \mathbb{Y}\right) \right\}$是相对紧的$% t\in J$.

当$t=0$时, 易证$\Gamma _{2}\left( B_{r}\right) \left( t\right)$是相对紧的.令$0 < t\leq a$是固定的, 且$0<\epsilon <t$, 对$% y\in B_{r}, u\in \Gamma _{2}\left(x\right)$, 存在泛函$ g\in S_{G, \overline{x}_{_{\rho }}}$, 使得

$ u\left( t\right) =R\left( t\right) \varphi \left( 0\right) +\int_{0}^{t-\epsilon }R\left( t-s\right) g\left( s\right) ds+\int_{t-\epsilon }^{t}R\left( t-s\right) g\left( s\right) ds. $

当$t=0$时, 显然$\Gamma _{2}\left( B_{r}\right) \left( t\right) $在X中是相对紧的.假如$0 < t\leq a$是固定的且$0<\epsilon <t$, 对任意的$% z\in B_{r}$和$\mathcal {K}_{1}\in \Gamma_{2}\left( x\right)$, 存在$% g\in S_{G, z_{_{\rho }}}$, 使得

$ \mathcal {K}_{1}\left( t\right) =\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds. $

显然, 即需要证明集合

$ U(t)=\left\{\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds\right\} $

在X中是相对紧的.

由引理2.4和算子$R\left(\epsilon\right)$的紧性可知

$ U_{\epsilon}(t)=\left\{R(\epsilon)\int_{0}^{t-\epsilon }R\left( t-s-\epsilon\right)g\left( s\right) ds\right\} $

在X中是相对紧的, 而且, 由引理2.4, 有

$ \left\Vert R(\epsilon)\int_{0}^{t-\epsilon }R\left( t-s-\epsilon\right) g\left( s\right) ds-\int_{0}^{t-\epsilon}R\left( t-s\right)g\left( s\right) ds\right\Vert\\ =\left\Vert \int_{0}^{t-\epsilon }R(\epsilon)R\left( t-s-\epsilon\right) g\left( s\right) ds-\int_{0}^{t-\epsilon}R\left( t-s\right)g\left( s\right) ds\right\Vert\\ \leq \int_{0}^{t-\epsilon }\Vert R(\epsilon)R\left( t-s-\epsilon\right)-R\left( t-s\right)\Vert_{L(X)} \Vert g\left( s\right)\Vert ds\\ \leq \epsilon H \int_{0}^{t-\epsilon} \Vert g\left( s\right)\Vert ds. $

所以集合$\left\{\int_{0}^{t-\epsilon }R\left( t-s\right) g\left( s\right) ds\right\}$在X中是相对紧的.对任意的$t\in J$, 有

$ \left\Vert \int_{0}^{t}R\left( t-s\right)g\left( s\right) ds-\int_{0}^{t-\epsilon}R\left( t-s\right) g\left( s\right) ds\right\Vert =\left\Vert \int_{t-\epsilon}^{t}R\left( t-s\right)g\left( s\right) ds\right\Vert\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq M_{1}\int_{t_{1}-\varepsilon }^{t_{1}}\Theta _{r^{\ast }}\left( s\right) ds, $

不等式右端当$\epsilon \rightarrow 0$时, 一致收敛于零.由此可知, 存在相对紧集序列任意逼近于集合$ \left\{\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds\right\}$.所以集合$\left\{\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds\right\}$是X中的相对紧集.

(ⅳ)$\Gamma_{2}$有闭图.设$y^{n}\rightarrow y^{\ast }$，$y^{n}\in B_{r}, u_{n}\in \Gamma_{2}\left( y^{n}\right)$及$u_{n}\rightarrow u^{\ast }$, 需要证明$% u^{\ast }\in \Gamma _{2}\left( y^{\ast }\right)$.

事实上, 若$u_{n}\in \Gamma _{2}\left( y^{n}\right)$, 则存在$f_{n}\in S_{G, % \left(\overline{y^{n}}\right)_{\rho }}$, 使得对任意的$t\in J$, 有

$ u_{n}\left( t\right) =R\left( t\right) \varphi \left( 0\right) +\int_{0}^{t}R\left( t-s\right) g_{n}\left( s\right) ds. $

下面证明存在$g^{\ast }\in S_{G, \left(\overline{y^{\ast }}% \right)_{\rho }}$, 使得

$ u^{\ast }\left( t\right) =R\left( t\right) \varphi \left( 0\right) +\int_{0}^{t}R\left( t-s\right) g^{\ast }\left( s\right) ds. $

易知当$n\rightarrow \infty$时

$ \left\Vert \left[u_{n}-R\left( t\right) \varphi \left( 0\right) \right] -% \left[u^{\ast }-R\left( t\right) \varphi \left( 0\right) \right] \right\Vert \rightarrow 0. $

考虑如下线性连续算子

$ \Gamma^{*} :L^{1}\left( J, X\right) \rightarrow C\left( J, X\right), \ f\rightarrow \Gamma^{*} \left( g\right) \left( t\right) =\int_{0}^{t}R\left( t-s\right) g\left( s\right) ds. $

由引理2.2可得, $\Gamma^{*} \circ S_{G}$是闭图算子, 且有

$ u_{n}\left( t\right) -R\left( t\right) \varphi \left( 0\right) \in \Gamma^{*} \left( S_{G, \left(\overline{y^{n}}\right)_{\rho }}\right) . $

由于$y^{n}\rightarrow y^{\ast }$, 从引理2.2可得

$ u^{\ast }\left( t\right) -R\left( t\right) \varphi \left( 0\right) \in \Gamma^{*} \left( S_{G, \left(\overline{y^{\ast }}\right)_{\rho }}\right) . $

即一定存在$g^{\ast }\left( t\right) \in S_{G, \left(\overline{y^{\ast }}\right)_{\rho}}$, 使得

$ u^{\ast }\left( t\right) -R\left( t\right) \varphi \left( 0\right) =\Gamma^{*} \left( g^{\ast }\left( t\right) \right) =\int_{0}^{t}R\left( t-s\right) g^{\ast }\left( s\right) ds. $

所以$\Gamma _{2}$是上半连续的.从而$\Gamma =\Gamma_{1}+\Gamma_{2}$是上半连续且凝聚的.由引理2.5知, 具有分数阶算子的时滞依赖状态的脉冲多值积分微分方程问题(1.1)–(1.2) 至少有一个温和解.

推论3.1 假设H1, H2, H3(a)-(b), H$_{\varphi }$以及下面条件成立

H3(c')存在一个可积函数$m:J\rightarrow % \left[0, +\infty \right)$和常数$\upsilon \in \lbrack 0, 1)$使得

$ \Vert G\left( t, \psi \right) \Vert \leq m\left( t\right) \left( 1+\Vert \psi \Vert _{\mathcal{B}}^{\upsilon }\right) \text{ for each }\left( t, \psi \right) \in J\times \mathcal{B}\text{, } $

则当

$ K_{a}\left[\max \{L, L_{1}\}\left( \left\Vert A^{-\beta }\right\Vert +\left( 1+aM_{2}\right) \frac{C_{1-\beta }}{\beta }\right) \right] <1 $

时, 问题(1.1)–(1.2) 至少存在一个温和解.