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

He Yang

doi:10.3390/sym12061031

Abstract

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.

Highlights

Since fractional differential equations can describe many problems in the fields of physical, biological and chemical and so on, some properties of solutions for the fractional differential equations have been considered by many authors, see [1,2,3,4,5,6,7,8]
In [2], when the nonlinearity satisfies non-Lipschitz conditions, Wang studied the existence of mild solutions of α ∈ (0, 1)-order fractional stochastic evolution equations with Caputo derivative in abstract spaces
Sobolev type differential equation arises in various areas of physical problems, see [4,5], it has been investigated by researchers recently, see [4,5,6,7]

Summary

Introduction

Since fractional differential equations can describe many problems in the fields of physical, biological and chemical and so on, some properties of solutions for the fractional differential equations have been considered by many authors, see [1,2,3,4,5,6,7,8]. In [6], by means of the operator semigroup theory, fractional calculus and stochastic analysis technique, Benchaabane et al established a group of sufficient conditions to guarantee the existence as well as the uniqueness of solutions for the α ∈ (0, 1)-order fractional stochastic evolution equations of Sobolev type. As far as we know, the existence as well as the uniqueness of mild solutions for the Sobolev type fractional stochastic evolution equations of order α ∈ (1, 2) have not been extensively discussed yet. We consider the existence as well as the uniqueness of mild solutions for two classes of the initial value problems (IVPs) of fractional stochastic equations of Sobolev type in a Hilbert space X. We have to emphasize that we do not assume the compactness of the (α, α − 1)-resolvent family and the (α, 1)-resolvent family in our main results

Preliminaries

Main Results

Applications

Conclusions