Existence and approximation of solutions to fractional differential equations

In this paper we shall study a fractional order semilinear differential equation in an arbitrary Banach space X . We used the analytic semigroup theory of linear operators and fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of a global solution. Existence and convergence of an approximate solution to the given problem is also proved in a separable Hilbert space. Finally, we give an example to illustrate the applications of the abstract results.