Fractional Stochastic Evolution Equations Research Articles

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

In this paper, the notion of singular backward stochastic Volterra integral equations (singular BSVIEs for short) in infinite dimensional space is introduced, and the corresponding well-posedness is carefully established. A class of singularity conditions are proposed, which not only cover that of fractional kernel, Volterra Heston model kernel, completely monotone kernels, to mention a few, but also happen to be used in the forward stochastic Volterra integral with new conclusions arising. Motivated by mathematical physics problem such as the viscoelasticity/thermoviscoelasticity of materials, heat conduction in materials with memory, optimal control problems of abstract stochastic Volterra integral equations (including fractional stochastic evolution equations and stochastic evolutionary integral equations) are presented. At last, our BSVIEs are surprisingly used in maximum principle of controlled stochastic delay evolution equations. One advantage of this new standpoint is that the final cost functional can naturally depend on the past state.

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

Existence and asymptotic behavior of square-mean $ S $-asymptotically periodic solutions of fractional stochastic evolution equations

This paper investigates the abstract fractional stochastic evolution equations. A new existence result of the square-mean $ S $-asymptotically periodic mild solutions are obtained under the assumption that the nonlinear terms only satisfy some growth conditions. Moreover, the uniqueness and asymptotic stability results of the square-mean $ S $-asymptotically periodic solution are presented when the nonlinear functions satisfy the general Lipschitz condition. Finally, two examples are given to illustrate our main results.

The Averaging Principle for Hilfer Fractional Stochastic Evolution Equations with Lévy Noise

This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean square and of probability. Our results enable us to focus on the averaged system rather than the original, more complex one. Given that the existing literature on the averaging principle for Hilfer fractional stochastic differential equations has been established in finite-dimensional spaces, the novelty here is the derivation of the averaging principle for a class of HFSEEs in Hilbert space. Furthermore, an example is allotted to illustrate the feasibility and utility of our results.

The Existence of Mild Solutions for Hilfer Fractional Stochastic Evolution Equations with Order μ∈(1,2)

In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order μ∈(1,2) and type ν∈[0,1]. We prove the existence of mild solutions of Hilfer fractional stochastic evolution equations when the semigroup is compact as well as noncompact. Our approach is based on the Schauder fixed point theorem, the Ascoli–Arzelà theorem and the Kuratowski measure of noncompactness. An example is also provided, to demonstrate the efficacy of this method.

$$\alpha $$-Whittaker controllability of $$\vartheta $$-Hilfer fractional stochastic evolution equations driven by fractional Brownian motion

$$\alpha $$-Whittaker controllability of $$\vartheta $$-Hilfer fractional stochastic evolution equations driven by fractional Brownian motion

A note concerning to approximate controllability of Atangana–Baleanu fractional neutral stochastic integro‐differential system with infinite delay

This article is primarily targeting the approximate controllability outcomes for the Atangana–Baleanu fractional neutral stochastic integro‐differential equation with infinite delay. First, by using stochastic analysis, fractional calculus, and Krasnoselskii fixed point techniques, we demonstrate the existence of mild solutions for the fractional stochastic evolution equations. Then we present a sufficient condition that ensures the approximate controllability of the stochastic evolution equations. Our findings are then applied to nonlocal conditions. At last, an example is provided to define our primary results.

Analysis of fractional stochastic evolution equations by using Hilfer derivative of finite approximate controllability

<abstract><p>The approximate controllability of a class of fractional stochastic evolution equations (FSEEs) are discussed in this study utilizes the Hilbert space by using Hilfer derivative. For different approaches, we remove the Lipschitz or compactness conditions and merely have to assume a weak growth requirement. The fixed point theorem, the diagonal argument, and approximation methods serve as the foundation for the study. The abstract theory is demonstrated using an example. A conclusion is given at the end.</p></abstract>

Partial approximate controllability results for fractional order stochastic evolution equations using approximation method

The sufficient conditions for existence and partial approximate controllability of fractional stochastic evolution equations with nonlocal initial conditions have been discussed. The discussion is based on the variational method, fractional calculus, Schauder's fixed point theorem, and stochastic analysis. Contrary to the results available in the literature, the non-local function does not need to be compact or satisfy Lipschitz's condition. Moreover, the pertinent nonlinear function does not need to be uniformly bounded. At the end, an application has also been demonstrated.

Averaging principle for non‐Lipschitz fractional stochastic evolution equations with random delays modulated by two‐time‐scale Markov switching processes

In the paper, we consider the averaging principle for a class of fractional stochastic evolution equations with random delays modulated by a two‐time‐scale continuous‐time Markov chain under the non‐Lipschitz coefficients, which extends the existing results: from Lipschitz to non‐Lipschitz case, from classical to fractional equations, from constant to random delays. Using ‐order fractional resolvent operator theory and stopping time technique, a general theorem on the existence and uniqueness of mild solutions is obtained first; further, strong averaging principle for fractional stochastic delay evolution equation is investigated, which simplifies the original system.

Hilfer fractional stochastic evolution equations on infinite interval

Abstract This paper concerns the global existence of mild solutions for a class of Hilfer fractional stochastic evolution equations on infinite interval (0, +∞), while the existing work were considered on finite interval. The main difficulties here are how to construct suitable Banach spaces, proper operator relations, and then how to formulate the new criteria to guarantee the global existence of mild solutions on the previous constructed spaces under non-Lipschitz conditions. We mainly rely on the generalized Ascoli–Arzela theorem we established, Wright function, Schauder’s fixed point principle, and Kuratowski’s measure of noncompactness to handle with the infinite interval problems. Moreover, we give two examples to demonstrate the feasibility and utility of our 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.

Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle

<p style='text-indent:20px;'>In this paper, we obtain a closed-form representation of a mild solution to the fractional stochastic degenerate evolution equation in a Hilbert space using the subordination principle and semigroup theory. We study aforesaid abstract fractional stochastic Cauchy problem with nonlinear state-dependent terms and show that if the Sobolev type resolvent families describing the linear part of the model are exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions for nonlinearity. We also establish conditions for stabilizability and prove that the stochastic nonlinear fractional Cauchy problem is exponentially stabilizable when the stabilizer acts linearly on the control systems. Finally, we provide applications to show the validity of our theory.</p>

Nonlocal fuzzy fractional stochastic evolution equations with fractional Brownian motion of order (1,2)

<abstract><p>In this manuscript, we formulate the system of fuzzy stochastic fractional evolution equations (FSFEEs) driven by fractional Brownian motion. We find the results about the existence-uniqueness of the formulated system by using the Lipschitizian conditions. By using these conditions we have also investigated the exponential stability of the solution for the above system driven by fractional Brownian motion. Finally, the applications in financial mathematics are presented and the use of financial mathematics in the fractional Black and Scholes model is also discussed. An example is propounded to show the applicability of our results.</p></abstract>

Existence uniqueness of mild solutions for \u03c8-Caputo fractional stochastic evolution equations driven by fBm

In this paper, we investigate the existence uniqueness of mild solutions for a class of ψ-Caputo fractional stochastic evolution equations with varying-time delay driven by fBm, which seems to be the first theoretical result of the ψ-Caputo fractional stochastic evolution equations. Alternative conditions to guarantee the existence uniqueness of mild solutions are obtained using fractional calculus, stochastic analysis, fixed point technique, and noncompact measure method. Moreover, an example is presented to illustrate the effectiveness and feasibility of the obtained abstract results.

Riemann\u2013Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion

This article is devoted to the study of the existence and uniqueness of mild solution to a class of Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. Our results are obtained by using fractional calculus, stochastic analysis, and the fixed-point technique. Moreover, an example is provided to illustrate the application of the obtained abstract results.

Approximate controllability of fractional stochastic evolution equations with nonlocal conditions

Abstract This paper deals with the approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We delete the compactness condition or Lipschitz condition for nonlocal term appearing in various literatures, and only need to suppose some weak growth condition on the nonlocal term. The discussion is based on the fixed point theorem, diagonal argument and approximation techniques. In the end, an example is presented to illustrate the abstract theory.

Continuous Dependence on the Coefficients for Mean-Field Fractional Stochastic Delay Evolution Equations

Continuous Dependence on the Coefficients for Mean-Field Fractional Stochastic Delay Evolution Equations

Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups

AbstractThis article deals with the exact controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space under the assumption that the semigroup generated by the linear part is noncompact. Our main results are obtained by utilizing stochastic analysis technique, measure of noncompactness and the Mönch fixed point theorem. In the end, an example is presented to illustrate that our theorems guarantee the effectiveness of controllability results in the infinite dimensional spaces.

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.