On Time Fractional Derivatives in Fractional Sobolev Spaces and Applications to Fractional Ordinary Differential Equations

In this article, we formulate two kinds of time fractional derivatives of the Caputo type with order α in fractional Sobolev spaces and prove that they are isomorphisms between the corresponding Sobolev space of order α and the L2 -space. On the basis of such fractional derivatives, we formulate initial value problems for time fractional ordinary differential equations and prove the well-posedness. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.