scholarly journals Controllability of quasilinear stochastic evolution equations in Hilbert spaces

2001 ◽  
Vol 14 (2) ◽  
pp. 151-159 ◽  
Author(s):  
P. Balasubramaniam
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Stochastic Version

Controllability of the quasilinear stochastic evolution equation is studied using semigroup theory and a stochastic version of the well known fixed point theorem. An application to stochastic partial differential equations is given.

scholarly journals Controllability of semilinear stochastic delay evolution equations in Hilbert spaces

2002 ◽  
Vol 31 (3) ◽  
pp. 157-166 ◽  
Author(s):  
P. Balasubramaniam ◽  
J. P. Dauer
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Fixed Point Theorem ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Banach Fixed Point Theorem ◽  
Stochastic Delay ◽  
Stochastic Version

The controllability of semilinear stochastic delay evolution equations is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. An application to stochastic partial differential equations is given.

scholarly journals Controllability of semilinear stochastic evolution equations in Hilbert space

2001 ◽  
Vol 14 (4) ◽  
pp. 329-339 ◽  
Author(s):  
P. Balasubramaniam ◽  
J. P. Dauer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Evolution Equations ◽  
Stochastic Partial Differential Equation ◽  
Semigroup Theory ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations

Controllability of semilinear stochastic evolution equations is studied by using stochastic versions of the well-known fixed point theorem and semigroup theory. An application to a stochastic partial differential equation is given.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

CONTINUOUS DEPENDENCE ON INITIAL DATA FOR SOLUTIONS OF NONLINEAR STOCHASTIC EVOLUTION EQUATIONS

2002 ◽  
Vol 05 (03) ◽  
pp. 333-350 ◽  
Author(s):  
THOMAS DECK
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Initial Data ◽  
Point Theorem ◽  
Evolution Equations ◽  
Noise Analysis ◽  
Reaction Diffusion Equations ◽  
Volterra Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations

We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spaces arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction–diffusion equations and to anticipating stochastic differential equations.

scholarly journals Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Li Xi-liang ◽  
Han Yu-liang
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Banach Contraction ◽  
Stability Results ◽  
Stochastic Functional

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

scholarly journals An Inverse Problem for a Class of Linear Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25
Author(s):  
Yuhuan Zhao
Keyword(s):  
Inverse Problem ◽  
Evolution Equation ◽  
Evolution Equations ◽  
Optimal Parameter ◽  
Optimization Method ◽  
Necessary Condition ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Fréchet Differentiable

An inverse problem for a linear stochastic evolution equation is researched. The stochastic evolution equation contains a parameter with values in a Hilbert space. The solution of the evolution equation depends continuously on the parameter and is Fréchet differentiable with respect to the parameter. An optimization method is provided to estimate the parameter. A sufficient condition to ensure the existence of an optimal parameter is presented, and a necessary condition that the optimal parameter, if it exists, should satisfy is also presented. Finally, two examples are given to show the applications of the above results.

scholarly journals Existence uniqueness of mild solutions for ψ-Caputo fractional stochastic evolution equations driven by fBm

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min Yang
Keyword(s):  
Fixed Point ◽  
Time Delay ◽  
Stochastic Analysis ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Varying Time Delay ◽  
Fixed Point Technique ◽  
Noncompact Measure

AbstractIn 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.

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