Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space X. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem.Wealso prove the existence of global solution. The existence and convergence of the Faedo Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator A. Finally we give an example to illustrate the applications of the abstract results.