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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 \( C([m(0) , 0] \, , \, {\mathbb{R}})\) 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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References:

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