Neutral Stochastic Evolution Equations Research Articles

Existence and Stability Results for Second-Order Stochastic Equations Driven by Fractional Brownian Motion

Fractional Brownian motion has been widely used to model a number of phenomena in diverse fields of science and engineering. In this article, we investigate the existence, uniqueness and stability of mild solutions for a class of second-order nonautonomous neutral stochastic evolution equations with infinite delay driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1) in Hilbert spaces. More precisely, using semigroup theory and successive approximation approach, we establish a set of sufficient conditions for obtaining the required result under the assumption that coefficients satisfy non-Lipschitz condition with Lipschitz condition being considered as a special case. Further, the result is deduced to study the second-order autonomous neutral stochastic equations with fBm. The results generalize and improve some known results. Finally, as an application, stochastic wave equation with infinite delay driven by fractional Brownian motion is provided to illustrate the obtained theory.

Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay and Poisson jumps (SNSEEIPs), in which the initial value belongs to the abstract space \documentclass[12pt]{minimal}\begin{document}$\mathcal {B}$\end{document}B. We establish the existence and uniqueness of mild solutions for SNSEEIPs under non-Lipschitz condition with Lipschitz condition being considered as a special case by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial data by means of a corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given to illustrate the theory.

Successive approximation of neutral stochastic evolution equations with infinite delay and Poisson jumps

In this paper, we study the existence and uniqueness of mild solutions of neutral stochastic evolution equations with infinite delay and Poisson jumps in real separable Hilbert spaces. We study the continuous dependence of solutions on the initial value. The nonlinear term in our equations are not assumed to Lipschitz continuous. The results of this paper generalize and improve some known results.

Existence results for impulsive neutral second-order stochastic evolution equations with nonlocal conditions

In this paper we consider a class of impulsive neutral second-order stochastic evolution equations with nonlocal initial conditions in a real separable Hilbert space. Sufficient conditions for the existence of mild solutions are established by operator theory and the Sadovskii fixed point theorem. An example is provided to illustrate the theory.

Second-order Neutral Stochastic Evolution Equations with Infinite Delay under Carathéodory Conditions

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay, in which the initial value belongs to the abstract space ℬ. We establish the existence and uniqueness of mild solutions for SNSEEIs under global and local Caratheodory conditions by means of the successive approximation. An application to the stochastic nonlinear wave equations with infinite delay is given to illustrate the theory.

Second-order neutral impulsive stochastic evolution equations with delay

In this paper, we study the second-order neutral stochastic evolution equations with impulsive effect and delay (SNSEEIDs). We establish the existence and uniqueness of mild solutions to SNSEEIDs under non-Lipschitz condition with Lipschitz condition being considered as a special case by the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial data by means of corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with impulsive effect and delay is given to illustrate the theory.

Approximating solutions of neutral stochastic evolution equations with jumps

In this paper, we establish existence and uniqueness of the mild solutions to a class of neutral stochastic evolution equations driven by Poisson random measures in some Hilbert space. Moreover, we adopt the Faedo-Galerkin scheme to approximate the solutions.

Second‐order neutral stochastic evolution equations with heredity

Existence, continuous dependence, and approximation results are established for a class of abstract second‐order neutral stochastic evolution equations with heredity in a real separable Hilbert space. A related integro‐differential equation is also mentioned, as well as an example illustrating the theory.