Holder type conditions for stability of stochastic functional differential equations

in this article, we consider stochastic differential equations with heredity in a Hilbert space. Our goal here is to study the asymptotic stability in probability and exponential stability of solutions for such class of systems by employing the Holder type conditions instead of the Lipschitz and linear growth conditions on the nonlinear terms. These stability results are obtained as by-products of our existence results. The method used is an interplay of the method of successive approximations and a comparison principle. An application is included to motivate this study.