The Fixed Point Theorem Based On The New Set With Applications To Fractional Differential Equations

In recent years,the fixed point theory of nonlinear operators has been widely used in algebraic equation,integral equation,differential equation,optimization theory and so on,which has attracted the attention and research interest of many scholars and accumulated many excellent research results.As an important tool to study the existence,uniqueness and multiplicity of equation solutions,it is of profound significance to continue to explore its theoretical knowledge.In this paper,by using cone theory and monotone iterative technique,we study three kinds of nonlinear operators in Banach space based on set Ph,e—increasing operator,ternary inverse mixed monotone operator,‘A+B+C' type sum operator,and obtain the existence and uniqueness of fixed point.As an application,this paper studies the existence of solutions for two-point boundary value problems and multi-point boundary value problems of nonlinear fractional differential equations.The research not only generalizes and improves the relevant conclusions in the literature,but also provides a new method for the study of fractional differential equations.The structure of this paper is as follows:The first chapter is the introduction,which mainly introduces the background,the current situation and the necessity to solve the problem,and expounds the main work of this paper.In the second chapter,based on the set Ph,e in Banach space,and without requiring operator compactness and continuity,using cone theory,monotone iterative technique and the properties of nonlinear operators with certain concavity,we study the fixed point theorems of increasing operator,ternary inverse mixed monotone operator,‘A+B+C' type sum operator.In the third chapter,by using the fixed point theorem of a class of sum operators in the set Ph,e,we study the two-point boundary value problem of nonlinear fractional differential equation,obtain the existence and uniqueness of the solution,and give an example to verify the correctness of the result.In chapter 4,we prove the existence and uniqueness of solutions for a class of nonlinear fractional differential equations with multi-point boundary value problems by using the fixed point theorem based on the set Ph,e of ternary inverse mixed monotone operator,and also verify the scientificity of the theorem with examples.