Luo, Jiaowan Fixed points and stability of neutral stochastic delay differential equations. (English) Zbl 1160.60020 J. Math. Anal. Appl. 334, No. 1, 431-440 (2007). 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 \[ \lim_{T \to \infty} {\mathbf E} \{ \sup_{t\geq T} |\psi(t; \phi) |^{2} \} = 0. \] Reviewer: Volker Wihstutz (Charlotte) Cited in 51 Documents 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 Keywords:stochastic differential equations mean square stability mean square asymptotic stability delay equation; unbounded delay fixed point theorem PDFBibTeX XMLCite \textit{J. Luo}, J. Math. Anal. Appl. 334, No. 1, 431--440 (2007; Zbl 1160.60020) Full Text: DOI References: [1] Brayton, R. 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