Research On Solutions Of Several Types Of Fractional Differential Equations (groups

Nonlinear function analysis is a research direction in modern mathematics with profound theoretical relevance and broad practical value.It uses numerous abstract issues from natural disciplines as a starting point and develops a set of theories and methods to address the resulting nonlinear problems.These research results mainly include partial order methods,topology degree methods,critical point theory,analytic methods and monotone mapping theory.Nonlinear functional analysis techniques and theories are crucial in the study of various nonlinear differential equations,integral equations,and partial differential equations.Fractional order differential equations are an important branch of differential equations,and many of their new models have found successful applications in mechanics,(bio)chemistry,electrical engineering,medicine,economics,etc.This is based on the fact that fractional order derivatives are an excellent tool for describing the memorability and heredity of various materials and processes,and that the advantages of fractional derivatives become apparent when simulating the mechanical and electrical properties of real materials,as well as in describing the rheological properties of rocks,among many other fields.Due to the accuracy and widespread application of the fractional differential equation model,more and more experts and scholars have been drawn to do more systematic and in-depth research on fractional order calculus and fractional order-differential,integral equations and have achieved a series of achievements.For the reasons stated above,we intend to further explore the properties of fractional order calculus and the initial and boundary value problems of nonlinear fractional order differential equations based on the findings of previous research.In this paper,we study the existence,uniqueness and stability of solutions to several classes of Riemann-Liouville and Hadamard type fractional order differential equations with boundary or initial value conditions using various methods,including Sch?uder fixed point theorem,Leray-Sch?uder nonlinear alternative theorem,Boyd-Wong contraction principle.Guo-Krasnoselskii fixed point theorem,Banach contraction mapping principle and a new fixed point theorem on a complete metric space.Simultaneously,we investigate important properties of fractional order integral operators.such as boundedness and continuity,on function spaces consisting of particular functions defined on finite or infinite intervals,and obtain some novel and meaningful results.The paper is divided into seven chapters.In Chapter Ⅰ,we present an overview of the research background and development of fractional order differential equations,some basic theoretical knowledge about nonlinear functional analysis and fractional order differential equations(including basic definitions and properties),and list the existence or uniqueness lemmas of fixed points that will be used in subsequent chapters.Furthermore,we introduce several required function spaces as well as other relevant notations.In Chapter Ⅱ,we investigate the existence of positive solutions to a class of higher-order fractional differential equations with generalized boundary conditions,where the nonlinear term of the equation comprises various range of lower-order derivatives,especially one low order derivative whose difference from the order is less than 1.We also obtain the continuity properties of Riemann-Liouville integral operator on the space of weighted Lebesgue integrable functions and use them to establish the existence of positive solutions to the boundary value problem under weaker conditions using the method of order reduction for fractional order and Guo-Krasnselskii fixed point theorem.Finally,several examples are given to verify the validity of the main results.In Chapter Ⅲ,we discuss a class of nonlinear fractional-order differential equations with a new type of boundary value condition.Using Leray-Sch?uder nonlinear alternative theorem and Boyd-Wong contraction principle,we obtain existence,uniqueness,and various Hyers-Ulam type stability results for positive solutions of boundary value problem.We also demonstrate the validity of the results by means of an example.In Chapter Ⅳ,we first study the properties of Mellin convolution operation on the space of integrable functions with exponential and power weights on an infinite interval,and then obtain properties such as boundedness and continuity of Hadamard fractional order integral operators.Finally,we prove the existence of unique positive solutions for a class of Hadamard fractional order differential equations with interference parameters and integral boundaries in two different spaces by using a fixed point theorem on complete metric space and Banach compression mapping principle.In Chapter Ⅴ,we investigate the existence of solutions to singular Hadamard fractional order differential equations with integral boundary values on an infinite interval,where the nonlinear term of the equation has two lower order derivatives.On the constructed space of weighted continuous functionsusing Schauder fixed point theorem we obtain the existence of solutions to the boundary value problem.In Chapter VI,we study the initial value problem of a class of nonlinear fractional order differential equations on the infinite interval.where the differential operator takes Hadamard type fractional order derivative and the nonlinear term involves two lower order fractional order derivatives of the unknown function.To establish the global existence theorem,we first prove the existence of a unique positive solution to an integral equation by using a new singular Gronwall-type integral inequality.Next,applying Sch?uder fixed point theorem,we obtain the existence of at least one solution to the initial value problem on a metrizable and complete locally convex space.Chapter Ⅶ summarizes the main work of the full text and proposes the exploration direction of the next research work.