Conformable Fractional Sobolev Spaces On Time Scales Applications In Variational Methods

In this dissertation, we establish the space which is used to study conformable fractional differential equations boundary value problem on time scales via vari-ational methods. Then, we study the existence and multiplicity for some con-formable fractional differential equations boundary value problem on time scales by variational methods. First, we improve some properties of the conformable fractional calculus on time scales. Second, using conformable fractional calculus on time scales, we define the conformable fractional Sobolev spaces on time scales and give some their important characteristics, such as completeness, reflexivity, uniform convexity, some embedding theorems and the continuously differentiable property of a class of functionals on these Sobolev spaces. Finally, as their ap-plication in the variational method, we construct the variational functionals of some boundary value problems for conformable fractional differential equation-s on time scales on these conformable fractional Sobolev spaces and study the existence and multiplicity for these problems by critical point theory. These con-formal fractional differential equations boundary value problems on time scales include conformable fractional P-Laplacian differential equations boundary value problem on time scales, conformable fractional Hamiltonian systems on time s-cales, conformable fractional Hamiltonian systems on time scales with impulsive effects, conformable fractional Hamiltonian system on time scales with forced ter-m and conformable fractional vibration problem on time scales with impulsive effects. Moreover, some examples are presented to illustrate the feasibility and effectiveness of the existence and multiplicity results.