Qualitative Research Of Nonlinear Fractional Differential Equations

Using Banach fixed point theorem, Schauder fixed point theorem, Krasnosel-skii fixed point theorem, nonlinear Leray-Schauder fixed point theorem, Leggett-Williams fixed point theorem on a convex cone, M¨onch fixed point theorem, Thisthesis mainly studies some qualitative properties of solutions to the initial valueproblem and boundary value problem of fractional difierential equations. Somenumerical examples are given to confirm the correctness of the results.This thesis is divided into six chapters.The first chapter presents some history background of fractional difierentialequations, development situation and application prospect, also gives the mainworks, main conclusions and the basic knowledge.The second chapter, using Schauder fixed point theorem, considers the ex-istence and uniform stability of solutions to initial value problems of fractionalimpulsive functional difierential equations. Some suficient conditions are ob-tained and some numerical examples are given to illustrate the correctness of theconclusions.The third chapter obtains the suficient conditions of the existence of three positive solutions to three-point boundary value problem of nonlinear fractionaldifierential equations, by utilizing Leggett-Williams fixed point theorem on aconvex cone, a numerical example is also given to ensure the efiectiveness of mainresults.The fourth chapter researches the existence of positive solutions and multiplepositive solutions to m point boundary value problems of nonlinear fractionaldifierential equations with p Laplace operator. Two numerical examples are givento verify the main results of this chapter.The fifth chapter obtains the suficient conditions of the existence of solu-tions to anti-periodic boundary value problems of Langevin fractional difierentialequation with two fractional derivatives using Krasnoselskii's fixed point theoremand Banach fixed point theorem.The sixth chapter discusses the existence of the mild solutions to the initial-value problem of fractional ordinary difierential equations, neutral fractional func- tional difierential equations and neutral fractional functional difierential-integralequations with Riemann-Liouville fractional derivative, using Banach contractionmapping principle, Schauder fixed point theorem, and the non-compactness Haus-dorfi measure, M¨onch fixed point theorem, a series of the suficient conditions areobtained.