## 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 $$C([m(0) , 0] \, , \, {\mathbb{R}})$$ 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 $$\psi: [m(0) ,\infty) \times \Omega \to {\mathbb{R}}$$ with $$| \psi \|_{[0,t]}= \{ {\mathbf E} ( \sup_{s \in [0,t]} |\psi(s, \omega ) |^2 ) \}^{1/2} \to 0$$ as $$t \to \infty$$, then $$S = \{0\}$$.
Usually mean asymptotic square stability means $$| \psi | = {\mathbf E} \{ \sup_{t\geq 0} | \psi(t; \phi) |^2 \} < \infty$$
and $$\lim_{| \phi | \to 0} {\mathbf E} \{ \sup_{t\geq 0} |\psi(t; \phi) |^{2} \} = 0$$ (mean square stability) together with $\lim_{T \to \infty} {\mathbf E} \{ \sup_{t\geq T} |\psi(t; \phi) |^{2} \} = 0.$

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34K20 Stability theory of functional-differential equations 34K50 Stochastic functional-differential equations 47H10 Fixed-point theorems 47N20 Applications of operator theory to differential and integral equations
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