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Fixed points and stability of neutral stochastic delay differential equations. (English) Zbl 1160.60020
The goal of the paper is to establish a necessary and sufficient condition for the mean square asymptotic stability of a linear scalar stochastic differential equation with time-depending delay using a fixed point theorem approach. Reviewer’s remarks: However, there are inconsistencies in the paper which makes it hard to understand. 1) The reviewer guesses that in (2.1), (2.2) $m(0)$ is meant to be negative, otherwise the space $C([m(0) , 0] \, , \, {\bbfR})$ is not well defined. On the other hand, if, for instance, $\tau(t) = \delta(t) = \frac{1}{2}t$ as in example 3.1, then $\tau(t) = \delta(t) \geq 0$ and $t -\tau(t) = t - \delta(t) = \frac{1}{2}t \to \infty$ (as $t \to \infty$ ) as required; but $\inf \{ s - \tau(s); s \geq 0 \} = \inf \{ s - \delta(s); s \geq 0 \} = 0$, therefore $m(0) = 0$. 2) If the Banach space $S$ consists of the processes $\psi: [m(0) ,\infty) \times \Omega \to {\bbfR}$ with $| \psi \|_{[0,t]}= \{ {\bold E} ( \sup_{s \in [0,t]} |\psi(s, \omega ) |^2 ) \}^{1/2} \to 0$ as $t \to \infty$, then $S = \{0\}$. Usually mean asymptotic square stability means $| \psi | = {\bold E} \{ \sup_{t\geq 0} | \psi(t; \phi) |^2 \} < \infty$ and $\lim_{\vert \phi \vert \to 0} {\bold E} \{ \sup_{t\geq 0} \vert\psi(t; \phi) \vert^{2} \} = 0$ (mean square stability) together with $$\lim_{T \to \infty} {\bold E} \{ \sup_{t\geq T} |\psi(t; \phi) |^{2} \} = 0.$$

##### MSC:
 60H10 Stochastic ordinary differential equations 34K20 Stability theory of functional-differential equations 34K50 Stochastic functional-differential equations 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47N20 Applications of operator theory to differential and integral equations
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##### References:
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