Deterministic and Stochastic Differential Equations in Hilbert Spaces Involving Multivalued Maximal Monotone Operators

Abstract

This work deals with a Skorokhod problem driven by a maximal operator: \begin{aligned} &du(t)+Au(t)(dt)\ni f(t)dt+dM(t), \; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} which is a multivalued deterministic differential equation with a singular inputs $dM(t)$, where $t\to M(t)$ is a continuous function. The existence and uniqueness result is used to study an Itô's stochastic differential equation \begin{aligned} &du(t)+Au(t)(dt)\ni f(t,u(t))dt+B(t,u(t))dW(t),\; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} in a real Hilbert space $H$, where $A$ is a multivalued ($\alpha$-)maximal monotone operator on $H$, and $f(t,u)$ and $B(t,u)$ are Lipschitz continuous with respect to $u$. Some asymptotic properties in the stochastic case are also found.

arXiv commentThis is an electronic reprint of the original article published by the Panamer. Math. J. 6 (1996), no. 3, 83--119, MR1400370. This reprint differs from the original in pagination and typographic detail. The article is posted on ArXiv.org because the online version is not available on the web page of the PanAmerican Mathematical Journal