Approximation of Solutions to Stochastic Neutral Fractional Integro-Differential Equation with Nonlocal Conditions

In this paper, we study a stochastic neutral fractional integro-differential equation with nonlocal conditions in separable Hilbert spaces. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We used the semigroup theory of linear operators and stochastic version of Banach fixed point theorem to study the existence and uniqueness of the mild solution for every approximate integral equation. Next, we show the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Moreover, the Faedo–Galerkin approximation of solution is established. In the last, an example is provided to illustrate the applications of the abstract results. © 2016, Springer India Pvt. Ltd.