Journal ArticleDOI
Stability of Mild Solutions of Stochastic Evolution Equations with Variable Delay
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TLDR
In this paper, the existence and stability problems associated with semilinear stochastic evolution equations with variable delay in infinite dimensions were considered and compared under a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms.Abstract:
In this paper, we consider the existence and stability problems associated with semilinear stochastic evolution equations with variable delay in infinite dimensions. To be precise, we first study an existence result and then the exponential stability of a mild solution as well as asymptotic stability in probability of its sample paths. Such results are established employing a comparison principle under less restrictive hypothesis than the Lipschitz condition on the nonlinear terms. An application is included to illustrate the theory.read more
Citations
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Journal ArticleDOI
Exponential stability of second-order stochastic evolution equations with Poisson jumps
Rathinasamy Sakthivel,Yong Ren +1 more
TL;DR: In this paper, a set of sufficient conditions for the exponential stability of mild solutions to second-order nonlinear stochastic differential equations with infinite delay driven by Poisson jumps is derived.
Journal ArticleDOI
Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients
Jianhai Bao,Zhenting Hou +1 more
TL;DR: The existence and uniqueness of mild solutions to stochastic neutral partial functional differential equations (SNPFDEs) is investigated and some results in Govindan (2003, 2005) are generalized to cover a class of more general SNPFDEs.
Journal ArticleDOI
Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay
Yong Ren,Shiping Lu,Ningmao Xia +2 more
TL;DR: In this paper, the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((-~, 0];R^d) were obtained.
Journal ArticleDOI
Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays
TL;DR: In this paper, Caraballo and Liu established sufficient conditions for the exponential stability of mild solutions of stochastic partial differential equations with delays, by establishing an impulsive integral inequality.
Journal ArticleDOI
Existence, Uniqueness and Stability Results of Impulsive Stochastic Semilinear Neutral Functional Differential Equations with Infinite Delays
Annamalai Anguraj,A. Vinodkumar +1 more
TL;DR: In this article, the existence, uniqueness and stability of mild solutions of impulsive stochastic semilinear neutral functional differential equations without a Lipschitz condition were analyzed.
References
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Book
Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Book
Differential and integral inequalities : theory and applications
V. Lakshmikantham,S. Leela +1 more
Book
Exponential Stability of Stochastic Differential Equations
TL;DR: In this article, the authors present a systematic study of stochastic differential delay equations driven by nonlinear integrators, detailing various exponential stabilities for large-scale systems and large-dimensional systems.