| 刘卫国,罗交晚.非自治随机时滞微分方程概周期解的存在唯一性[J].数学年刊A辑,2013,34(6):717~726 |
| 非自治随机时滞微分方程概周期解的存在唯一性 |
| Existence and Uniqueness of Almost Periodic Solutions to Non-autonomous Sto-chastic Differential Equations with Time Delay |
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| DOI： |
| 中文关键词: 非自治,时滞,概周期解,Acquistapace-Terreni条件 |
| 英文关键词:Non-autonomous, Time delay, Almost periodic solution,
Acquistapace-Terreni conditions |
| 基金项目:国家自然科学基金(No.11271093) |
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| 中文摘要: |
| 讨论以下非自治时滞随机微分方程:
\begin{align*}
\left\{\!\!\!\begin{array}{l}
\rmd[x(t)-h(t,x_t)]=[A(t)x(t)+f(t,x_t)]\rmd
t+g(t,x_t)\rmd W(t), \quad t\geq
t_0,\ x_{t_0}=\xi(\theta),\quad \theta\in[-r,0], \quad
r\geq0.
\end{array}\right.
\end{align*}
如果非自治线性算子$A(t)$满足Acquistapace-Terreni (简称为AT)条件,则能找到算子$\{U(t,s),t\geq s;t,s\in \mathbb R\}$与其存在某种对应关系,
然后根据算子$ \{U(t,s),t\geq s;t,s\in \mathbb R\}$的性质和Banach不动点定理,证明了以上方程存在唯一的均方概周期mild解. |
| 英文摘要: |
| The authors consider the following non-autonomous stochastic differential equation with time delay:
\begin{align*}
\left\{\!\!\!\begin{array}{l}
\rmd[x(t)-h(t,x_t)]=[A(t)x(t)+f(t,x_t)]\rmd t+g(t,x_t)\rmd W(t),\quad t\geq t_0,\ x_{t_0}=\xi(\theta), \quad \theta\in[-r,0],\quad
r\geq0.\end{array}\right.
\end{align*}
If the non-autonomous linear operator $A(t)$ satisfies Acquistapace-Terreni (or AT for short) conditions, there exists an operator $\{U(t,s),t\geq
s;t,s\in \mathbb R\}$ associated with it. By using the properties of $\{U(t,s),t\geq s;t,s\in \mathbb R\}$ and Banach fixed-point principle, the existence and uniqueness of almost periodic mild solution of the above equation are obtained. |
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