Faedo-Galerkin approximate solutions of a neutral stochastic fractional differential equation with finite delay

In this work, we study a class of neutral stochastic fractional differential equation in an arbitrary separable Hilbert space H. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by projection of considered associated integral equation onto finite dimensional space. The existence and uniqueness of solutions to every approximate integral equation are obtained by using Banach fixed point theorems and analytic semigroup theory. We show the convergence of the solutions by using Faedo-Galerkin approximations and give an example to show the effectiveness of the main theory. Finally, we provide the conclusion at the end.