Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

Abstract

By the concept of fractional derivative of Riemann-Liouville on time scales, we first introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Then, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using fibering mapping and Nehari manifolds, the existence of weak solutions for a class of fractional boundary value problems on time scales is studied, and a result of the existence of weak solutions for this problem is obtained.