Right fractional Sobolev space via Riemann$-$Liouville derivatives on time scales and an application to fractional boundary value problem on time scales

TL;DR

This paper introduces right fractional Sobolev spaces on time scales using Riemann-Liouville derivatives, characterizes their properties, and applies variational methods to establish existence of solutions for fractional boundary value problems.

Contribution

It defines and analyzes fractional Sobolev spaces on time scales with Riemann-Liouville derivatives and applies variational techniques to fractional boundary value problems.

Findings

Established properties of fractional Sobolev spaces on time scales.

Proved existence of weak solutions for fractional boundary value problems.

Demonstrated the applicability of variational methods in this context.

Abstract

Using the concept of fractional derivatives of Riemann$-$Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some imbeddings. Finally, as an application, we propose a recent method to study the existence of weak solutions of fractional boundary value problems on time scales by using variational method and critical point theory, and by constructing an appropriate variational setting, we obtain two existence results of the problem.