Riemann–Liouville Type Fractional Derivatives and Equivalences of Linear Riemann–Liouville Type Fractional Differential Equations

The first or higher order Riemann–Liouville (R‐L) type fractional derivatives are studied. The relations between the R‐L type fractional derivatives and the classical R‐L fractional derivatives are obtained and the spaces of functions on which the R‐L type fractional derivatives exist are provided. The identities for the compositions of the R‐L type fractional derivatives and the corresponding integral operators are obtained. Some of the above results are new and others generalize some previous known results. A continuity assumption used in the previous equivalence results of linear higher order fractional differential equations (FDEs) in the space of absolutely integrable functions will be dropped. The equivalences of linear higher order FDEs in the space of continuous functions are established and these equivalence results will be useful for further studying boundary value problems of nonlinear FDEs with a variety of boundary conditions.