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Fixed points and stability of stochastic neutral partial differential equations with infinite delays. (English) Zbl 1177.93094

Summary: In this article, we study the existence and the asymptotical stability in mean square of mild solutions to stochastic neutral partial differential equations with infinite delays
\[ \begin{cases} d[X(t)+f(t,X(t-\tau)))]=[AX(t)+a(t,X(t-\delta(t)))]\,dt+b(t,X(t-\rho(t)))\,dW(t),\quad t\geq 0,\\ X_0=\varphi\in D^b_{{\mathcal F}_0}([m(0),0],H),\end{cases} \]
where \(t-\tau(t)\), \(t-\delta(t)\), \(t-\rho(t)\to\infty\) with delays \(\tau(t)\), \(\delta(t)\), \(\rho(t)\to\infty\) as \(t\to\infty\). Our method for investigating the stability of solutions is based on the fixed point theorem.

MSC:

93E15 Stochastic stability in control theory
60H20 Stochastic integral equations
34K50 Stochastic functional-differential equations
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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