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Existence and approximation of solutions to fractional differential equations. (English) Zbl 1165.34304
Summary: We 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.

##### MSC:
 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 26A33 Fractional derivatives and integrals 34K30 Functional-differential equations in abstract spaces
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