Asymptotically almost periodic solutions for a class of stochastic functional differential equations
type: Item
instance of: scholarly article
title: Asymptotically almost periodic solutions for a class of stochastic functional differential equations
zbMATH Open document ID: 1468.34110
DOI: 10.1155/2014/934534
author: Aimin Liu
author: Yongjian Liu
author: Qun Liu
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations \[ \mathrm dx(t) = [A(t)x(t) + F (t, x(t), x_t)]\mathrm dt + H(t, x(t), x_t) \circ \mathrm dW(t). \] A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup \(\{T(t)\}_{t\geq 0}\) is essentially removed, which is generated by the linear densely defined operator \(A: D(A) \subset L^2 (\mathbb {P,H}) \rightarrow L ^ 2 (\mathbb {P,H})\), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow's domain. An example is also given to illustrate our results.
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 34K14
zbMATH DE Number: 7023344
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/934534
OpenAlex ID: W1987003772
Wikidata QID: Q59042947
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description: scientific article; zbMATH DE number 7023344