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Impulsive perturbation of \(C_ 0\)-semigroups and stochastic evolution inclusions. (English) Zbl 1039.34055

The author considers the following class of stochastic impulsive systems where the principal operator is the generator of a \(C_{0}\)-semigroup \[ dx(t)-Ax(t)\,d\beta(t)- F(t,x)\,d\mu(t)\in C(t,x)\,dW, \quad x(0)=\xi, \] where \(W\) is a cylindrical Brownian motion taking values in a Hilbert space \(U\). It is proven that, under certain assumptions on the pair \((A,\beta(\cdot))\), the nonlinear map \(F\) and the vector measure \(\mu\) and the multivalued operator \(C\), the stochastic inclusion has solutions. Some topological properties of the solutions set are presented.

MSC:

34G25 Evolution inclusions
34F05 Ordinary differential equations and systems with randomness
47D06 One-parameter semigroups and linear evolution equations
49J27 Existence theories for problems in abstract spaces
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