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

2020 ◽  
Vol 18 (1) ◽  
pp. 616-631
Author(s):  
Yonghong Ding ◽  
Yongxiang Li
Keyword(s):  
Evolution Equations ◽  
Initial Conditions ◽  
Linear Part ◽  
Nonlocal Conditions ◽  
Exact Controllability ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
Nonlocal Initial Conditions ◽  
Analysis Technique

Abstract This 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.

Approximate controllability of fractional stochastic evolution equations with nonlocal conditions

2020 ◽  
Vol 21 (7-8) ◽  
pp. 829-841
Author(s):  
Yonghong Ding ◽  
Yongxiang Li
Keyword(s):  
Approximate Controllability ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Nonlocal Conditions ◽  
Nonlocal Term ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Approximation Techniques ◽  
Nonlocal Initial Conditions ◽  
The Fixed Point Theorem

AbstractThis 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.

Nonlocal problem for fractional stochastic evolution equations with solution operators

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Pengyu Chen ◽  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Nonlocal Problem ◽  
Growth Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Nonlocal Initial Conditions ◽  
Analysis Theory

AbstractIn 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,

scholarly journals Existence of Solutions in Some Interpolation Spaces for a Class of Semilinear Evolution Equations with Nonlocal Initial Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Jung-Chan Chang ◽  
Hsiang Liu
Keyword(s):  
Evolution Equations ◽  
Existence Of Solutions ◽  
Initial Conditions ◽  
Linear Part ◽  
Nonlocal Conditions ◽  
Strong Solutions ◽  
Interpolation Spaces ◽  
Nonlocal Initial Conditions ◽  
Densely Defined ◽  
Semilinear Evolution Equations

This paper is concerned with the existence of mild and strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The linear part is assumed to be a (not necessarily densely defined) sectorial operator in a Banach spaceX. Considering the equations in the norm of some interpolation spaces betweenXand the domain of the linear part, we generalize the recent conclusions on this topic. The obtained results will be applied to a class of semilinear functional partial differential equations with nonlocal conditions.

Random invariant manifolds for ill-posed stochastic evolution equations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050013
Author(s):  
Alexandra Neamţu
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Linear Part ◽  
Time Behavior ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Stable And Unstable Manifolds ◽  
Long Time ◽  
Ill Posed ◽  
Unstable Manifolds

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.

Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2

2019 ◽  
Vol 22 (4) ◽  
pp. 1086-1112 ◽  
Author(s):  
Linxin Shu ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Approximate Controllability ◽  
Stochastic Systems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative

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.

P-th moment growth bounds of infinite-dimensional stochastic evolution equations

1998 ◽  
Vol 128 (1) ◽  
pp. 107-121 ◽  
Author(s):  
Kai Liu
Keyword(s):  
Evolution Equations ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Infinite Dimensional Space ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
Sample Paths ◽  
Logarithmic Growth

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.

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