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

Abstract

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 α-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.2000 Mathematics Subject Classification. 34A12, 35R11, 47J35, 60H15.

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

“…In recent years, stochastic differential equations have attracted great interest due to their successful applications to problems in mechanics, electricity, economics, physics, and several fields in engineering. For details, see [33][34][35][36][37][38][39] and the references therein. In particular, some researchers investigated controllability of stochastic dynamical control systems in infinite-dimensional spaces, see [26][27][28][29][30][31].…”

Existence and uniqueness for a kind of nonlocal fractional evolution equations on the unbounded interval

“…In [19], El-Borai obtained the existence results for fractional abstract equations of the following kind: In this paper, the equivalent integral equation in regard to the abstract equations is firstly described by means of some probability densities using Laplace transform and its inverse transform. Since then, many researchers have drawn on El-Borai's results to investigate fractional evolution equations, such as in [20][21][22][23][24][25][26][27][28].…”

On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces

“…One of the branches of stochastic differential equations is the theory of stochastic evolution equations. Since semilinear stochastic evolution equations are abstract formulations for many problems arising in the domain of engineering technology, biology, economic system, and so forth, stochastic evolution equations have attracted increasing attention in recent years and the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic evolution equations have been considered by many authors; see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. Taniguchi et al [6] discussed the existence, uniqueness, th moment, and almost sure Lyapunov exponents of mild solutions to a class of stochastic partial functional differential equations with finite delays by using semigroup methods.…”

“…Chang et al [12][13][14] studied the existence and uniqueness of Stepanovlike almost automorphic mild solutions, the existence of square-mean almost automorphic mild solutions, and the existence and uniqueness of quadratic mean almost periodic mild solutions to nonlinear stochastic evolution equations in real separable Hilbert spaces, respectively. Moreover, the existence of mild solutions of stochastic evolution equations in Hilbert spaces has also been discussed in [15][16][17][18][19][20].…”

“…However, to the best of the authors' knowledge, most of the existing articles (see, e.g., [6,[12][13][14][15][16][17][18][19]) are only devoted to studying the local existence of mild solutions for stochastic evolution equations, and there are no results yet present on the existence of saturated mild solutions and global mild solutions for stochastic evolution equations in Hilbert spaces. Motivated by the abovementioned aspects, in this paper, by using Schauder's fixed point theorem, compact semigroup theory, and piecewise extension method, we investigate the existence of saturated mild solution and global mild solution for the initial value problem to a class of semilinear stochastic evolution equations in real separable Hilbert spaces.…”