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)(a,k)-regularized families of operators. We illustrate the method for the case of the fractional order differential equation Dtαu′(t)+μDtαu(t)=Au(t)+t−αΓ(1−α)(u′(0)+μu(0))+f(t),t>0,0<α≤1,μ≥0, where AA is an unbounded closed operator defined on a Banach space XX and ff is an XX-valued function.