The Studies Of Positive Solutions For Two Kinds Of Boundary Value Problems Of Nonlinear Fractional Ordinary Differential Equations

The boundary value problem(BVP)of nonlinear fractional ordinary differential equations(ODEs)is an important component of the theory of ODEs.Compared with integer-order ODEs,fractional ODEs can characterize the properties of things and forecast their development trends more accurately.Therefore,it is of great theoretical significance and application value to study the BVPs of nonlinear fractional ODEs.In this thesis,the existence and multiplicity of positive solutions for two kinds of BVPs of nonlinear fractional ODEs are explored.Firstly,the BVPs of nonlinear fractional ODEs are transformed into the problems for finding fixed points of the corresponding integral operators by Green functions;then,the existence and multiplicity of positive solutions for the BVPs of nonlinear fractional ODEs are studied by using related fixed point theorems;finally,some concrete examples are given to verify the obtained results.The thesis is divided into the following four chapters.In the first chapter,we briefly summarize the development history and research status of the BVPs of nonlinear fractional ODEs.Moreover,some relevant mathematical concepts and lemmas are presented.Then,we clarify how to transform BVPs of nonlinear fractional ODEs into the problems for finding fixed points of the corresponding integral operators,and two examples are given.In the second chapter,we mainly study the existence and multiplicity of positive solutions for a class of higher-order integral BVP of nonlinear fractional ODEs.By analyzing the properties of Green function corresponding to the integral BVP and using Guo-Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem,we prove the existence and multiplicity of positive solutions for the integral BVP.Finally,examples are presented to show the applications of our results.In the third chapter,we mainly investigate the existence and multiplicity of positive solutions for a class of lower-order multi-point BVP of nonlinear fractional ODEs.The existence theorem and multiplicity theorem of positive solution for the multi-point BVP are presented via analyzing the properties of Green function corresponding to the multi-point BVP and using Guo-Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem.Finally,some concrete examples are still given to show the applications of our results.The fourth chapter is the summary and prospect of the thesis.