Research On The Existence Of Solutions For Several Caputo Fractional Differential Equations

In this thesis,we mainly investigate the stability,boundedness,and exis-tence of the solutions of initial value and two-point boundary value problems involving the Caputo fractional derivatives with respect to another function.It is divided into 5 chapters.In the first chapter,we review the history of the development of the Caputo fractional derivatives with respect to another function and its current situation,and briefly summarizes the main efforts and the conclusion obtained in this thesis.The second chapter introduces some notations,basic concepts and lemmas which are useful in this thesis.In the framework of theory of the Caputo fractional derivatives with respect to another function,the third chapter studies the existence and uniqueness of the solutions of nonlocally initial value problems,and the global existence of solutions to the initial value problems,respectively.For the non-local initial value problems,we first obtain the sufficient conditions for the uniqueness of the solutions by using the Banach fixed point theorem,and then establish some criteria for the existence of at least one solutions to this problems by virtue of constructing an appropriate weighted norms,we yield soKrasnoselkii fixed point theorem.For the generally initial value problems,by me sufficient conditions for the global existence and uniqueness of the solutions by means of the Banach fixed point theorem.The fourth chapter treats with the existence of the solutions for two types of two-point boundary value problems involving the Caputo fractional deriva-tives with respect to a function.We derive several sufficient condition for the existence of at least one solutions to the first two-point boundary value problems by using the Schauder fixed point theorem.For the second one,we establish the criteria for the existence of at least positive solutions by using the classical norme-type cone expansion and cpmpression theorem.Based on the third and forth chapters,the fifth chapter discusses the boundedness of the solutions for the initial value problems and the Ulam-Hyer stability of the solutions for the two-point boundary value problems,respec-tively.By using the Gronwall inequality and the techniques in inequalities,we obtain some sufficient conditions ensuring the boundedness and the Ulam-Hyer stability of the solutionsThe main results obtained in the thesis are new in the setting of the Caputo fractional derivatives with respect to another function.