Fractional Stochastic Evolution Equations Research Articles

Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we first investigate the existence as well as the uniqueness of mild solutions for a class of α ∈ ( 1 , 2 ) -order Riemann–Liouville fractional stochastic evolution equations of Sobolev type in abstract spaces. Then the symmetrical technique is used to deal with the α ∈ ( 1 , 2 ) -order Caputo fractional stochastic evolution equations of Sobolev type in abstract spaces. Two examples are given as applications to the obtained results.

Transportation Inequalities for Coupled Fractional Stochastic Evolution Equations Driven by Fractional Brownian Motion

In this paper, we consider the existence and uniqueness of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H∈1/4,1/2. Our approach is based on Perov’s fixed-point theorem. Furthermore, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution.

Finite-approximate controllability of fractional stochastic evolution equations with nonlocal conditions

This paper deals with the finite-approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We establish sufficient conditions for the finite-approximate controllability of the control system when the compactness conditions or Lipschitz conditions for the nonlocal term and uniform boundedness conditions for the nonlinear term are not required. The discussion is based on the fixed point theorem, approximation techniques and diagonal argument. In the end, an example is presented to illustrate the abstract theory. Our result improves and extends some relevant results in this area.

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.

Approximate Controllability and Existence of Mild Solutions for Riemann-Liouville Fractional Stochastic Evolution Equations with Nonlocal Conditions of Order 1 < α < 2

Abstract In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 < α < 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 < α < 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.

Analysis and Numerical Solutions for Fractional Stochastic Evolution Equations With Almost Sectorial Operators

Fractional stochastic evolution equations often arise in theory and applications. Finding exact solutions of such equations is impossible in most cases. In this paper, our main goal is to establish the existence and uniqueness of mild solutions of the equations, and give a numerical method for approximating such mild solutions. The numerical method is based on a combination of subspaces decomposition technique and waveform relaxation method, which is called a frequency decomposition waveform relaxation method. Moreover, the convergence of the frequency decomposition waveform relaxation method is discussed in detail. Finally, several illustrative examples are presented to confirm the validity and applicability of the proposed numerical method.

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

This paper is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolution equations with time fractional differential operator α ∈ (0, 1). After establishing the well-posedness of the problem, and a result ensuring the existence and uniqueness of mild solutions globally defined in future, the existence of a minimal global attracting set is investigated in the mean-square topology, under general assumptions not ensuing the uniqueness of solutions. Furthermore, in the case of uniqueness, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. It is remarkable that the attraction property is proved in the usual forward sense, unlike the pullback concept used in the context of random dynamical systems, but the main point is that the model under study has not been proved to generate a random dynamical system.

The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay

We first prove the existence, uniqueness and continuous dependence of mild solutions to stochastic delay evolution equations with a Caputo fractional derivative:DtαCy(t)=Ay(t)+f(t,yt)+g(t,yt)dW(t)dt,12<α<1. Then, we investigate the asymptotic behavior of mild solutions to fractional stochastic delay evolution equations of the formDtαCy(t)=Ay(t)+It1−αf(t,yt)+[It1−αg(t,yt)]dW(t)dt,0<α<1. In particular, the existence of a global forward attracting set in the mean-square topology is established. A general theorem on the existence of mild solutions is obtained by using α-order fractional resolvent operator theory and the Schauder fixed point theorem.

Existence and Optimal Controls for Fractional Stochastic Evolution Equations of Sobolev Type Via Fractional Resolvent Operators

This paper is mainly concerned with controlled stochastic evolution equations of Sobolev type for the Caputo and Riemann–Liouville fractional derivatives. Some sufficient conditions are established for the existence of mild solutions and optimal state-control pairs of the limited Lagrange optimal systems. The main results are investigated by compactness of fractional resolvent operator family, and the optimal control results are derived without uniqueness of solutions for controlled evolution equations.

Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups

This paper deals with the existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions and and noncompact semigroups in a real separable Hilbert space. We assume that the linear part generates an equicontinuous -semigroup, and the nonlinear functions satisfy some appropriate growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic to the case that the corresponding solution semigroup is not compact. An example is also given to demonstrate the applications of our abstract result.

The regularity of fractional stochastic evolution equations in Hilbert space

ABSTRACTIn this paper, we concentrate on the regularity of the trajectories of the mild solution for the fractional evolution equations on the Hilbert space. By giving some critical estimates on the α-times resolvent generated by the operator A, we prove that under the condition of , the trajectories of the mild solution are Hölder continuous both on the Hilbert space and on the domain of the fractional power of the operator with some appropriate index.

Existence and regularity of mild solutions to fractional stochastic evolution equations

This study is concerned with the stochastic fractional diffusion and diffusion-wave equations driven by multiplicative noise. We prove the existence and uniqueness of mild solutions to these equations by means of the Picard’s iteration method. With the help of the fractional calculus and stochastic analysis theory, we also establish the pathwise spatial-temporal (Sobolev-Hölder) regularity properties of mild solutions to these types of fractional SPDEs in a semigroup framework. Finally, we relate our results to the selection of appropriate numerical schemes for the solutions of these time-fractional SPDEs.

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.

Nonlocal Problem for Fractional Stochastic Evolution Equations with Solution Operators

Abstract In this article, we are concerned with a class of fractional stochastic evolution equations with nonlocal initial conditions in Hilbert spaces. The existence of mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using fractional calculations, Schauder fixed point theorem, stochastic analysis theory, α-order fractional solution operator theory and α-resolvent family theory. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Sobolev type fractional stochastic integro-differential evolution

In this paper, we prove the existence of $\alpha$-mild solutions for a class of fractional stochastic integrodifferential evolution equations of sobolev type with fractional sobolev stochastic nonlocal conditions in a real separable Hilbert space. To establish our main results, we use the Banach contraction mapping principle, fractional calculus, stochastic analysis and an analytic semigroup of linear operators. An example is given to illustrate the feasibility of our abstract result.

Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces

In this paper, we study the approximate mild solutions of a class of fractional stochastic evolution equations in Hilbert spaces. By constructing Picard type approximate sequences, we present the approximate results under the non-Lipschitz and linear growth conditions.

On the initial value problem of fractional stochastic evolution equations in Hilbert spaces

In this article, we are concerned with the initial value problem of fractional stochastic evolution equations in real separable Hilbert spaces. The existence of saturated mild solutions and global mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using $\alpha$-order fractional resolvent operator theory, the Schauder fixed point theorem and piecewise extension method. Furthermore, the continuous dependence of mild solutions on initial values and orders as well as the asymptotical stability in $p$-th moment of mild solutions for the studied problem have also been discussed. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract results.

Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

Abstract We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s &lt; b. An example is provided to illustrate the application of the obtained results.

Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.