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 XX. 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.