ON EXPONENTIAL STABILITY OF MILD SOLUTION OF A STOCHASTIC INTEGRODIFFERENTIAL EQUATION IN A COMPLEX HILBERT SPACE

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 mild solution found is exponentially stable on average of order k. We assume in our work that the delayed part of our equation admits in the sense of Grimmer [11] an analytic resolving operator which is locally non-Lipschitzian. We obtain existence and uniqueness results by using the Lipschitz global and growth conditions applying the properties of the analytic semigroup. The application of Gronwall’s lemma together with the properties of the stochastic integral gives the exponential stability. Received: June 3, 2024 Accepted: July 31, 2024