An operator theoretical approach to a class of fractional order differential equations

We propose a general method to obtain the representation of solutions for linear fractional order differential equations based on the theory of (a, k)-regularized families of operators. We illustrate the method in case of the fractional order differential equation D t u ′(t) + μD t u(t) = Au(t) + t−α Γ(1− α) (u′(0) + μu(0)) + f(t), t > 0, 0 < α ≤ 1, where A is an unbounded closed operator defined on a Banach space X and f is a X-valued function.