Stochastic Functional Evolution Equations Research Articles

Poisson stable solutions for stochastic functional evolution equations with infinite delay

This work is devoted to Poisson stable (including stationary, periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent) solutions for stochastic functional evolution equations (SFEEs) with infinite delay. First, we prove the existence of bounded mild solutions for SFEEs with infinite delay. Then, according to the relationship between the solution and (drift and diffusion) coefficients, we obtain such Poisson stable solutions. Because the solutions of the delay equations are not Markov, we employ the solution maps in some phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability.

NEWTON S METHOD FOR STOCHASTIC FUNCTIONAL EVOLUTION EQUATIONS IN HILBERT SPACES

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

Pseudo Almost Periodicity and Its Applications to Impulsive Nonautonomous Partial Functional Stochastic Evolution Equations

Abstract In this paper, we establish a new composition theorem for pseudo almost periodic functions under non-Lipschitz conditions. We apply this new composition theorem together with a fixed-point theorem for condensing maps to investigate the existence of p $p$ -mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial functional stochastic evolution equations in Hilbert spaces, and then, the exponential stability of p $p$ -mean piecewise pseudo almost periodic mild solutions is studied. Finally, an example is given to illustrate our results.

Abstract functional second-order stochastic evolution equations with applications

We investigate a class of abstract second-order damped functional stochastic evolution equations driven by a fractional Brownian motion in a separable Hilbert space. The global existence of mild solutions is established under various growth and compactness conditions. The case of a nonlocal initial condition is addressed. A related convergence result is discussed, and the theory is applied to stochastic wave and beam equations, as well as a spring-mass system, for illustrative purposes.

Mild solutions of local non-Lipschitz neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching

ABSTRACTIn this article, we initiate a study on neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching in real separable Hilbert spaces. Our goal here is to derive the existence and uniqueness of mild solutions to equations of this class under local non-Lipschitz condition proposed by Taniguchi [J. Math. Anal. Appl. 340:(2009)197–208] by means of stopping time technique and Banach fixed-point theorem. The results obtained here generalize the main results from Luo and Taniguchi [Stoch. Dyn. 9:(2009)135–152] and Jiang and Shen [Comput. Math. Appl. 61:(2011)1590–1594]. Finally, an example is worked out to illustrate the obtained results.

Practical Stability of Stochastic Delay Evolution Equations

In this paper we investigate the almost sure practical stability for a class of stochastic functional evolution equations. We establish some sufficient conditions based on the construction of appropriate Lyapunov functional. The abstract results are then applied to some illustrative examples.

On almost periodic mild solutions for neutral stochastic evolution equations with infinite delay

In this paper, a class of neutral stochastic functional evolution equations with infinite delay is investigated. Under some suitable assumptions, the existence and uniqueness of quadratic mean almost periodic mild solutions for these equations are discussed by means of semigroups of operators and the fixed point method. Moreover, two examples are given to illustrate our results.

Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.

Controllability of Second‐Order Semilinear Impulsive Stochastic Neutral Functional Evolution Equations

We consider a class of impulsive neutral second‐order stochastic functional evolution equations. The Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators are used to investigate the sufficient conditions for the controllability of the system considered. An example is provided to illustrate our results.

Boundedness and continuity of the mild solutions of semilinear stochastic functional evolution equations

Boundedness and continuity of the mild solutions of semilinear stochastic functional evolution equations

The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps

In this paper, we consider the existence and uniqueness of the energy solutions to the following non-Lipschitz stochastic functional evolution equation driven both by Brownian motion and by Poisson jumps where is a linear or nonlinear bounded operator, , and are measurable functions. We also investigate the almost sure exponential stability of energy solutions by using the energy equality for this equation.

Stochastic functional evolution equations with monotone nonlinearity: Existence and stability of the mild solutions

In this paper, we study a class of semilinear functional evolution equations in which the nonlinearity is demicontinuous and satisfies a semimonotone condition. We prove the existence, uniqueness and exponentially asymptotic stability of the mild solutions. Our approach is to apply a convenient version of Burkholder inequality for convolution integrals and an iteration method based on the existence and measurability results for the functional integral equations in Hilbert spaces. An Itô-type inequality is the main tool to study the uniqueness, p-th moment and almost sure sample path asymptotic stability of the mild solutions. We also give some examples to illustrate the applications of the theorems and meanwhile we compare the results obtained in this paper with some others appeared in the literature.

ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.

Existence and Stability of Solutions for Nonautonomous Stochastic Functional Evolution Equations

We establish the results on existence and exponent stability of solutions for a semilinear nonautonomous neutral stochastic evolution equation with finite delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic partial differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences.

Stability of nonlinear functional stochastic evolution equations of second order in time

Stochastic differential delay equations and their asymptotic behavior have received much attention in recent years (see [1, 2, 5, 7–10, 12, 13] and the references therein), since these retarded problems often appear in physics, biology, engineering, etc. The delays can enter into the formulations in very different ways, e.g., as a constant or variable delay or as a distributed one; some of them can even appear in the model at the same time. However, it is possible to consider all of them under a unified formulation by using appropriate differential functional equations. In this sense, we will carry out our analysis in a functional framework that will cover a wide variety of situations containing finite delays. There exists a vast literature concerning pathwise exponential stability of parabolic stochastic evolution equations (with and without delays). We mention here, among many others, [4, 11–13] and the references therein. In the case without hereditary characteristics, the problem of the asymptotic stability of delay stochastic partial differential equations of second order in time has been considered by Curtain in [6], where one can find sufficient conditions for the exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths, when the main operator generates a strongly continuous contraction semigroup. In this paper, we shall develop the theory in a variational framework for nonlinear operators in general. We are going to consider the following model:

On duality between estimation and control for linear stochastic functional evolution equations in Hilbert spaces

This paper treats a linear least-squares estimation problem in which both the state and observation processes are governed by infinite-dimensional linear stochastic functional evolution equations. The main result obtained in this paper is expressed in terms of a duality principle between the estimation and a linear-quadratic control problem. As a by-product, the representation of the solution process of the equation is also obtained.