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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],) is not well defined. On the other hand, if, for instance, τ(t)=δ(t)=1 2t as in example 3.1, then τ(t)=δ(t)0 and t-τ(t)=t-δ(t)=1 2t (as t ) as required; but inf{s-τ(s);s0}=inf{s-δ(s);s0}=0, therefore m(0)=0.

2) If the Banach space S consists of the processes ψ:[m(0),)×Ω with |ψ [0,t] ={𝐄(sup s[0,t] |ψ(s,ω)| 2 )} 1/2 0 as t, then S={0}.

Usually mean asymptotic square stability means |ψ|=𝐄{sup t0 |ψ(t;ϕ)| 2 }<

and lim |ϕ|0 𝐄{sup t0 |ψ(t;ϕ)| 2 }=0 (mean square stability) together with

lim T 𝐄{sup tT |ψ(t;ϕ)| 2 }=0·

MSC:
60H10Stochastic ordinary differential equations
34K20Stability theory of functional-differential equations
34K50Stochastic functional-differential equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47N20Applications of operator theory to differential and integral equations