Left fractional Sobolev space via Riemann$-$Liouville derivatives on time scales and its application to a fractional boundary value problem on time scales

TL;DR

This paper develops fractional Sobolev spaces on time scales using Riemann-Liouville derivatives, proves their properties, and applies them to establish the existence of solutions for fractional boundary value problems.

Contribution

It introduces and characterizes fractional Sobolev spaces on time scales, linking Riemann-Liouville derivatives with weak derivatives, and applies variational methods to fractional boundary value problems.

Findings

Established equivalence of fractional integral definitions on time scales

Proved properties like completeness and reflexivity of the spaces

Demonstrated existence of solutions for Kirchhoff-type fractional systems

Abstract

We first prove the equivalence of two definitions of Riemann-Liouville fractional integral on time scales, then by the concept of fractional derivative of Riemann-Liouville on time scales, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Next, 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 the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary condition is studied, and three results of the existence of weak solutions for this problem is obtained.