Neutral Stochastic Evolution Equations Research Articles

New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.

Second-order neutral impulsive stochastic evolution equations with infinite delay: existence, uniqueness and averaging principle

In this paper, a class of second-order neutral impulsive stochastic evolution equations with infinite delay driven by fractional Brownian motion and Poisson jumps is considered. We utilise the Carathéodory approximation technique, the Burkhölder–Davies–Gundy inequality and the Gronwall inequality to present the existence and uniqueness of mild solutions for the addressed system under Carathéodory conditions. Furthermore, we derive an averaging principle for the proposed system which is approximated by the simplified equation in the sense of mean square. Finally, an application is used to illustrate the obtained theoretical results.

Mixed Neutral Caputo Fractional Stochastic Evolution Equations with Infinite Delay: Existence, Uniqueness and Averaging Principle

The aim of this article is to consider a class of neutral Caputo fractional stochastic evolution equations with infinite delay (INFSEEs) driven by fractional Brownian motion (fBm) and Poisson jumps in Hilbert space. First, we establish the local and global existence and uniqueness theorems of mild solutions for the aforementioned neutral fractional stochastic system under local and global Carathéodory conditions by using the successive approximations, stochastic analysis, fractional calculus, and stopping time techniques. The obtained existence result in this article is new in the sense that it generalizes some of the existing results in the literature. Furthermore, we discuss the averaging principle for the proposed neutral fractional stochastic system in view of the convergence in mean square between the solution of the standard INFSEEs and that of the simplified equation. Finally, the obtained averaging theory is validated with an example.

Qualitative analysis of A.P.A. solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion

In this paper, we will analyses the square mean almost pseudo automorphic mild solution for fractional order equation, (1) c 0 D α γ [ℵ(γ)−D(γ,ℵ(γ))] = [Aℵ(γ) +φ(γ,ℵ(γ))]dγ + ϕ(γ,ℵ(γ))d<B>(γ) + ψ(γ,ℵ(γ))dB(γ), γ∈R where A(γ) : D(A(γ)) ⊂ L 2 G (F) → L 2 G (F) is densely closed linear operator and the functions D, φ, ϕ and ψ: L 2 G (F) → L 2 G (F) are jointly continuous. We drive square mean almost pseudo automorphic mild solution for fractional order neutral stochastic evolution equations driven by G-Brownian motion is obtain by using evolution operator theorem and fixed point theorem. Moreover, we prove that this mild solution of equation (1) is unique.

Global attractiveness and exponential stability for impulsive fractional neutral stochastic evolution equations driven by fBm

This paper is concerned with a class of fractional neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion (fBm). First, by means of the resolvent operator technique and contraction mapping principle, we can directly show the existence and uniqueness result of mild solution for the aforementioned system. Then we develop a new impulsive-integral inequality to obtain the global attracting set and pth moment exponential stability for this type of equation. Worthy of note is that this powerful inequality after little modification is applicable to the case with delayed impulses. Moreover, sufficient conditions which guarantee the pth moment exponential stability for some pertinent systems are stated without proof. In the end, an example is worked out to illustrate the theoretical results.

Existence and exponential stability of almost pseudo automorphic solution for neutral stochastic evolution equations driven by G-Brownian motion

This paper mainly concerns the quasi sure exponential stability of square mean almost pseudo automorphic mild solution for a class of neutral stochastic evolution equations driven by G-Brownian motion. By means of evolution operator theorem and fixed point theorem, existence and uniqueness of square mean almost pseudo automorphic mild solution is obtained. Also, a series of sufficient conditions on exponential stability and quasi sure exponential stability are established.

Neutral stochastic functional differential evolution equations driven by Rosenblatt process with varying-time delays

Hermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process with index H ∈ ( 1/2 , 1) which is a special case of a self-similar process with long-range dependence. More precisely, we prove the existence and uniqueness of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.

Pth mean almost periodic solutions to neutral stochastic evolution equations with infinite delay and Poisson jumps

In this paper, we investigate the pth mean almost periodic solution to a neutral stochastic evolution equation with infinite delay and Poisson jumps. We give a sufficient condition for the existence and uniqueness of pth mean almost periodic solution and the condition depends on the continuity of coefficients and the power of a fractional operator. We give an example to illustrate the abstract results.

Controllability of neutral stochastic evolution equations driven by fBm with Hurst parameter less than 1/2

ABSTRACTIn this paper, we investigate the controllability for a class of neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter in a Hilbert space. By using the stochastic analysis of fractional Brown motion with Hurst parameter , the properties of operator semigroup and the Banach fixed point theorem, the sufficient conditions for the controllability of the considered equations are obtained. In the end, one example is given to illustrate the feasibility and effectiveness of the obtained results.

Existence and stability of square-mean almost automorphic solution for neutral stochastic evolution equations with Stepanov-like terms on time scales

This paper is devoted to study the existence, uniqueness and stability of the square-mean almost automorphic solution for neutral stochastic evolution equation with Stepanov-like terms on time scales. We introduce time scale version of the Stepanov like square mean almost automorphic functions. At the end, an example is given to illustrate the effectiveness of the analytic results.

Exponential stability for nonautonomous impulsive neutral partial stochastic evolution equations with delay

ABSTRACTIn this paper, we give sufficient conditions for the exponential stability of pseudo almost periodic in distribution mild solutions for nonautonomous impulsive neutral partial stochastic evolution equations with delay in a Hilbert space. Our working tools are based on the interpolation theory, the stochastic analysis techniques and fixed point theorem. To illustrate our main result, we study the exponential stability of a pseudo almost periodic in distribution solution for the neutral stochastic model arising in physical systems.

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

ABSTRACTIn this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.

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.

Controllability of neutral stochastic evolution equations driven by fractional brownian motion

In this paper, we investigate the controllability for neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter H ɛ (½, 1) in a Hilbert space. We employ the α-norm in order to reflect the relationship between H and the fractional power α. Sufficient conditions are established by using stochastic analysis theory and operator theory. An example is provided to illustrate the effectiveness of the proposed result.

Controllability of neutral stochastic integro-differential evolution equations driven by a fractional Brownian motion

We establish sufficient conditions for the controllability of a certain class of neutral stochastic functional integro-differential evolution equations in Hilbert spaces. The results are obtained using semigroup theory, resolvent operators and a fixed-point technique. An application to neutral partial integro-differential stochastic equations perturbed by fractional Brownian motion is given.

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.

Second-order neutral impulsive stochastic evolution equations with infinite daelay

In this paper, we study a class of second-order neutral impulsive stochasticevolution equations with infinite delay (SNISEEIs in short), in which theinitial value belongs to the abstract space . Sufficient conditions for the existence of themild solutions for SNISEEIs are derived by means of the Krasnoselskii-Schaeferfixed point theorem. Two examples are given to illustrate the obtainedresults.

Stability of Neutral Impulsive Nonlinear Stochastic Evolution Equations with Time Varying Delays

AbstractIn this paper, we consider the stability of mild solutions to neutral impulsive nonlinear stochastic evolution equations with time varying delays. With the non‐Lipschitz condition, the Lipschitz condition being taken as a special case, on the impulsive term, the existence, uniqueness and sufficient conditions for the exponential stability in the pth moment and the almost sure exponential stability of the mild solutions are derived by employing a fixed point approach. An example is provided to illustrate the efficiency of the obtained theorems.

A note on the second-order non-autonomous neutral stochastic evolution equations with infinite delay under Carathéodory conditions

In this note, we aim to study a class of second-order non-autonomous neutral stochastic evolution equations with infinite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion with Hurst parameter H∈(1/2,1), in which the initial value belongs to the abstract space B. We establish the existence and uniqueness of mild solutions for this kind of equations under some Carathéodory conditions by means of the successive approximation. The obtained result extends some well-known results. An example is proposed to illustrate the theory.

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.