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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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References:

[1] Burton T.A., Stability by Fixed Point Theory for Functional Differential Equations (2006) · Zbl 1160.34001
[2] DOI: 10.1080/07362999908809633 · Zbl 0943.60050
[3] DOI: 10.3934/dcds.2007.18.295 · Zbl 1125.60059
[4] DOI: 10.1017/CBO9780511666223
[5] Govindan T.E., Stochastics 77 pp 139– (2005)
[6] DOI: 10.1016/0022-247X(82)90041-5 · Zbl 0497.93055
[7] DOI: 10.1201/9781420034820 · Zbl 1085.60003
[8] DOI: 10.1016/S0167-7152(00)00103-6 · Zbl 0966.60059
[9] DOI: 10.1016/j.jmaa.2006.12.058 · Zbl 1160.60020
[10] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations 44 (1983) · Zbl 0516.47023
[11] Taniguchi T., Stochastics 53 pp 41– (1995)
[12] DOI: 10.1016/j.jmaa.2006.08.055 · Zbl 1125.60063
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