Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

First, we show the equivalence of two definitions left Riemann–Liouville fractional integral on time scales. Then, establish and characterize Sobolev space with help notion derivative At same time, define weak derivatives demonstrate that they coincide ones Next, prove kinds norms in introduced derive its completeness, reflexivity, separability, some embedding. Finally, as an application, by constructing appropriate variational setting, using mountain pass theorem genus properties, existence solutions for a class Kirchhoff-type p-Laplacian systems scales boundary conditions is studied, three results this problem obtained.