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

We propose a general method for obtaining 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 for the case of the fractional order differential equation D t α u ′ ( t ) + μ D t α u ( t ) = A u ( t ) + t − α Γ ( 1 − α ) ( u ′ ( 0 ) + μ u ( 0 ) ) + f ( t ) , t > 0 , 0 < α ≤ 1 , μ ≥ 0 , where A is an unbounded closed operator defined on a Banach space X and f is an X -valued function.