Existence Of Positive Solutions For Boundary Value Problems Of Several Class Of Nonlinear Fractional Differential Equations

Nonlinear functional analysis is a research field of mathematics with profound theories and extensive applications.It constructs many general theories and methods to deal with nonlinear problems on the basis of the study of the nonlinear problems which appeared in mathematics and the natural sciences.Because it can explain all kinds of natural phenomena commendably,its rich theory and advanced method have provided the effective theory tool for solving the nonlinear problem which emerges incessantly in the technical domain one after another.At present,nonlinear functional analysis mainly cover topology degree theory,critical point theory,partial order method,analysis method,monotone mapping theory,and so on.The nonlinear theory and method play an important role in dealing with various nonlinear integral equation,differential equation and partial differential equations which come from practical problems,so the study of nonlinear problems has gradually become and one of the hot topics of natural science at home and abroad in recent years.The study of nonlinear fractional differential equations is an important branch of nonlinear analysis.Fractional-order models have been shown to be more accurate and realistic than integer-order models,this is because fractional-order models is any order differential,and it is the generalization of classical nonlinear integer order differential equation.Fractional order differential equations boundary value problem arouse more and more attention in recent years,this is not only the application in mathematics,but also the wide application in fluid mechanics,viscoelastic mechanics,the conduction of biological systems,nerve score mode and score regression model and so on.Due to its universality application and effectiveness,so the research of fractional differential equations boundary value problems is very important in both theory and applications.Moreover,more and more mathematics research workers have researched nonlinear fractional differential equation boundary value problems in recent years and a series of excellent research results have been obtained.This paper study mainly the existence,multiplicity and uniqueness of positive solutions of several nonlinear singular fractional differential equation,the main method of a series of excellent results obtained include: cone theory,Krasnosel'skii fixed point theorem,Avery-Peterson fixed point theorem,mixed monotone operator fixed point theorem,monotonic iterative method,the theory of spectrum analysis,Leray-Schauder nonlinear alternative?Schauder fixed point theorem,Sadovskii's fixed point theorem and Banach fixed point theorem and so on.The dissertation is divided into seven chapters.In Chapter I,the background of nonlinear functional analysis and nonlinear fractional differential equation have been introduced,some preliminary definitions and properties of nonlinear nonlinear functional are given,also several lemmas on the existence of fixed point,which play an important role in the next chapters.In Chapter II,by using the Avery-Peterson fixed point theory on a special cone on a space,we obtained the multiplicity of positive solutions for a class of infinite point boundary value problems.The nonlinearity term of this paper including derivative term which bring a lot of difficulties for us.In Chapter III,we study the existence of positive solutions of a class of infinite point fractional differential equation,and n-1 terms fractional order derivative are contained in the nonlinear term and the nonlinear term is singular not only with respect to time variable but also space variable.We get a better result than the relevant references under a weaker conditions.At last,an example is given to confirm our main results.In Chapter IV,we get the uniqueness of iterative positive solutions for the singular fractional differential equations with integral boundary conditions,the fractional orders are involved in the nonlinearity of the boundary value problem and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable.The existence of uniqueness of positive solution is mainly obtained by fixed point theorem of mixed monotone operator and the positive solution of equation system is dependent on .An iterative sequence and convergence rate are given which are important for practical application and an example is given to demonstrate the validity of our main results.In Chapter V,we get the uniqueness of iterative positive solutions for -Laplacian singular fractional differential equations,an iterative sequence and convergence rate are also given.In Chapter VI,we obtain the existence of positive solutions for fractional differential system with integral boundary conditions via spectral analysis.Fractional orders are involved in the nonlinearity of the boundary value problem and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable.In S 7.1 we obtain the iterative solution for a class of nonlinear impulsive fractional reaction-diffusion equation;In S 7.2 we get the existence of mild solution to impulsive fractional reaction-diffusion equations with delay.