The Application Of Critical Point Theory In Solving The Dirichlet Boundary Value Problems Of Fractional Differential Equations

In this thesis, the existence and multiplicity for boundary value problems of two class fractional differential equations are mainly studied. The sufficient conditions of the existence and multiplicity of solutions for the corresponding problems are obtained by using the critical point theory via building different variational structures. The thesis Is divided into five chapters.In the first chapter, the research background for the fractional order cal-culus including the important progress, research status and main conclusions of this thesis is briefly introduced.In the second chapter, the basis knowledge of relevant for this thesis, basic definitions of the fractional calculus and critical point theory such as nature, lemma and the theorem are expounded.In the third chapter, the following fractional differential equation is studied by applying variational method and critical point theory.When the pulse meets superquandratic linear case, the sufficient conditions of the existence and multiplicity of solutions are gained by using the mountain pass lemma and the symmetric mountain pass lemma of the critical point theory, moreover some examples are given to demonstrate the accuracy of the results.In the fourth chapter, the following fractional differential equation with parameters is discussed.The sufficient conditions of the existence and multiplicity of solutions are obtained via the critical point theory and iterative technique, and an example is provided to illustrate the efficiency of the obtained results.The fifth chapter summarizes the main research content of this thesis, and the future research is prospected.