# Stochastic differential inclusions of semimonotone type in Hilbert spaces

Document Type : Research Paper

Author

University of Kashan

Abstract

In this paper, we study the existence of generalized solutions for
the infinite dimensional nonlinear stochastic differential
inclusions $dx(t) \in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$
is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition.
We define the It\^{o} stochastic integral of operator set-valued stochastic processes
with respect to the cylindrical Brownian motion on separable Hilbert spaces.
Then, we generalize the existence results for
differential inclusions in [H. Abedi and R. Jahanipur, Nonlinear differential inclusions of semimonotone and
condensing type in Hilbert spaces,
\textit{Bull. Korean Math. Soc.},
{52} (2015), no. 2, 421--438.] to the corresponding stochastic differential inclusions
using the methods discussed in [R. Jahanipur, Nonlinear functional differential equations of monotone-type in
Hilbert spaces, {\it Nonlinear Analysis} {\bf 72} (2010), no. 3-4, 1393--1408,
R. Jahanipur, Stability of stochastic delay evolution equations with monotone
nonlinearity, {\it Stoch. Anal. Appl.}, {\bf 21} (2003), 161--181, and
R. Jahanipur, Stochastic functional evolution equations with monotone
nonlinearity: existence and stability of the mild solutions, {\it J. Differential Equations} {\bf 248} (2010), no. 5, 1230--1255.]

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### History

• Receive Date: 10 March 2013
• Revise Date: 30 December 2013
• Accept Date: 19 January 2014
• First Publish Date: 29 April 2015