Riemann-Liouville Fractional SobolevSpaces On Time Scales And TheirApplications In Variational Methods

In this thesis,first,we establish the theories of left and right Riemann-Liouville fractional Sobolev spaces which are used to study Riemann-Liouville fractional boundary value problem on time scales via variational methods and critical point theory.Then,we discuss the existence and multiplicity for some Riemann-Liouville fractional boundary value problems on time scales by variational methods and critical point theory.Or rather,we first present the concepts of right Riemann-Liouville fractional operators and right Caputo fractional derivative operator on arbitrary time scales.Second,based on these notations,we establish some properties of those operators.Then,with the help of the given preliminaries,we define the left and right Riemann-Liouville fractional Sobolev spaces on time scales and give some their vital characteristics,such as completeness,reflexivity,separability,some embedding theorems and the continuously differentiable property of a class of functionals on these fractional Sobolev spaces.Finally,four applications are presented to illustrate the feasibility and effectiveness of our defined spaces when we use the variational methods and critical point theory to deal with Riemann-Liouville fractional boundary value problem on time scales.They include a system of over-determined Fredholm fractional integro-differential equations on time scales,a coupled system of p-Laplacian fractional system on time scales,impulsive fractional Dirichlet problem with Riemann-Liouville fractional derivative term on time scales and fractional Dirichlet problem of p-Laplacian with instantaneous and non-instantaneous impulses on time scales.