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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 $$\cases d[X(t)+f(t,X(t-\tau)))]=[AX(t)+a(t,X(t-\delta(t)))]\,dt+b(t,X(t-\rho(t)))\,dW(t),\quad t\ge 0,\\ X_0=\varphi\in D^b_{{\cal F}_0}([m(0),0],H),\endcases$$ 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.

93E15Stochastic stability
60H20Stochastic integral equations
34K50Stochastic functional-differential equations
47N10Applications of operator theory in optimization, convex analysis, programming, economics
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