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.


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
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.