On exponential stability of mild solution of a stochastic integrodifferential equation in a complex Hilbert space

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Résumé

In this work, we consider a system of stochastic integrodifferential
equations in a complex Hilbert space. We first establish the existence
and uniqueness of mild solutions for equation (1) under non-Lipschitz
conditions. Then we show under certain assumptions that the found
mild solution is exponentially stable on average of order n. Note that
the same equation was studied in [10] where the authors found the
solution in a real Hilbert space. We now provide a generalization of
this result in a complex Hilbert space. We obtain existence and
uniqueness results by using the Lipschitz global and growth conditions

Mots-clés

Exponential stability, Hilbert Space, analytical semigroups, analytical resolving operator, Stochastic Integrodifferential Equation, mild solution