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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) $m\left(0\right)$ is meant to be negative, otherwise the space $C\left(\left[m\left(0\right),0\right]\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}ℝ\right)$ is not well defined. On the other hand, if, for instance, $\tau \left(t\right)=\delta \left(t\right)=\frac{1}{2}t$ as in example 3.1, then $\tau \left(t\right)=\delta \left(t\right)\ge 0$ and $t-\tau \left(t\right)=t-\delta \left(t\right)=\frac{1}{2}t\to \infty$ (as $t\to \infty$ ) as required; but $inf\left\{s-\tau \left(s\right);s\ge 0\right\}=inf\left\{s-\delta \left(s\right);s\ge 0\right\}=0$, therefore $m\left(0\right)=0$.

2) If the Banach space $S$ consists of the processes $\psi :\left[m\left(0\right),\infty \right)×{\Omega }\to ℝ$ with ${|\psi \parallel }_{\left[0,t\right]}=\left\{𝐄\left({sup}_{s\in \left[0,t\right]}{|\psi \left(s,\omega \right)|}^{2}{\right)\right\}}^{1/2}\to 0$ as $t\to \infty$, then $S=\left\{0\right\}$.

Usually mean asymptotic square stability means $|\psi |=𝐄\left\{{sup}_{t\ge 0}|\psi \left(t;\varphi \right){|}^{2}\right\}<\infty$

and ${lim}_{|\varphi |\to 0}𝐄\left\{{sup}_{t\ge 0}{|\psi \left(t;\varphi \right)|}^{2}\right\}=0$ (mean square stability) together with

$\underset{T\to \infty }{lim}𝐄\left\{\underset{t\ge T}{sup}{|\psi \left(t;\varphi \right)|}^{2}\right\}=0·$

##### 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