In the present paper, a nonlinear fractional Schrodinger integro-differential equation is considered in a Hilbert space. Operator approach is applied on multidimensional problems with nonlinearity that deserve a studious treatment. In this paper, theorems on existence and uniqueness of a bounded solution for the abstract problem are achieved. Additionally, existence theorems are obtained for first and second orders of accuracy difference schemes of the abstract problem. Furthermore, theorems are applied on a one-dimensional problem with nonlocal condition and a multidimensional problem with Dirichlet boundary condition. Numerical results and illustrations are presented to show the effectiveness of the theoretical results.