Research On The Properties Of Solutions For Variable Fractional Order And Hadamard Fractional Order Equations

Fractional order equation is a generalization of integer order equation.Using fractional order equations to establish models can accurately describe natural phenomena and actual situations in life,and can effectively solve some problems where integer order equation models contradict scientific experimental results.Variable fractional order equations and Hadamard fractional order equations are important components of fractional order equations.The study of their initial and boundary value problems has practical significance and has gradually become a research hotspot.In this paper,we will study the existence and uniqueness of solutions and Ulam-Hyers stability of several kinds of variable fractional order equations and Hadamard fractional differential equations.The main contents are as follows:In Chapter 1,we summarize the research background of variable fractional order and Hadamard fractional calculus,as well as the relevant definitions and lemmas used in this paper.In Chapter 2,we consider the boundary value problems of variable fractional integrodifferential equations.By using --contractions and -contractions,we obtain sufficient conditions for the existence of solutions of boundary value problems for fractional equations with variable Riemann-Liouville fractional integrals.In Chapter 3,we discuss the existence of positive solutions for variable fractional equations with p-Laplacian operators.Through Green’s function,we obtain the integral equation satisfied by the solution of the equation.We use the fixed point theorem on the cone to obtain relevant conclusions.In Chapter 4,we study the existence of solutions for higher order Hadamard fractional neutral differential equations and differential inclusion problems.According to the properties of Hadamard fractional differential and integral,the differential equation is transformed into an integral equation.By using Banach contraction mapping principle,Boyd and Wong fixed point theorem,Leray-Schauder nonlinear alternative theorem,we obtain the existence and uniqueness of the solution of the equation,and study the UlamHyers stability of the solution.By using the Bohnenblust-Karlin fixed point theorem,the Martelli fixed point theorem and the nonlinear alternative theorem on the Kakutani mapping,we obtain the sufficient conditions for the existence of solutions of fractional differential inclusion problems when multi-valued mappings have convex values.In Chapter 5,we analyze the existence and stability of solutions of Hadamard fractional impulsive differential equations.We obtain the existence and uniqueness of the solution of the equation by using the Kuratowski noncompact measure,the topological degree theory of condensing mapping and the Banach contraction mapping principle.We study the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the problem by constructing inequalities.