Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space. (English) Zbl 1081.34083

The authors consider the following nonlinear neutral stochastic differential inclusion in a Hilbert space \[ d(X(t)-f(t,X_t)) \in AX(t)dt+G(t,X_t)dB(t). \] Here, \(X_t\) denotes the solution segment at time \(t\), \(f\) is a sufficiently regular nonlinear function, \(B\) is a Hilbert-space-valued Brownian motion, \(A\) is the infinitesimal generator of a strongly continuous semigroup and \(G\) is a suitable set-valued mapping. The main result is that this equation has – under certain regularity conditions – a mild solution. The main mathematical tool used in the proof is Martelli’s fixed-point theorem. Finally, an example of a neutral stochastic reaction-diffusion inclusion is discussed.


34K50 Stochastic functional-differential equations
34K40 Neutral functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K35 Control problems for functional-differential equations
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