The Existence And Hyers-Ulam Stability For Two Classes Of Linear Fractional Delay Differential Equations With Variable Coefficients

In this thesis,we mainly study a class of linear fractional differential equations with variable coefficients and infinite delay as well as such equations with variable coefficients and ψ-Caputo fractional derivative.There are the following four chapters.In chapter 1,we introduce the background of fractional calculus theory,its research significance,the current state of the studies on its existence and the Hyers-Ulam stability of its solutions,and the primary research topics.In chapter 2,we present some key theorems and pertinent theories that are necessary for this paper.It introduces the fundamental ideas and pertinent natures of Riemann-Liouville,Caputo and ψ-Caputo fractional calculus,as well as the basics of Picard operator theory and phase space,certain widely used fixed point theorems and several kinds of Gronwall inequalities.In chapter 3,we study a class of Caputo linear fractional differential equations with variable coefficients and infinite delay,and provide the right form of solution by using the phase space defined by axiomatic method.Based on Banach contracting mapping principle and Leray-Schauder fixed point theorem,we determine the existence and uniqueness of fractional differential equation solutions.Besides,the comparative nature of a fractional integral and a kind of Gronwall inequation are used to further prove Hyers-Ulam stability of this kind of fractional differential equation delay solution.As an application,the existence and Hyers-Ulam stability of the solution are also confirmed using a specific numerical case.In chapter 4,we investigate the existence and stability of solutions of a class of linear fractional delay differential equations solution with variable coefficients and ψ-Caputo fractional derivative.By Leray-Schauder fixed point theorem and Banach contracting mapping principle,we obtain the existence and uniqueness of solutions to such equations.Besides,because this kind of equations has a feature of composite function,Picard operator theory and a kind of Gronwall inequation are used to discuss the Hyers-Ulam stability and Hyers-Ulam-Riassias stability of solutions to such equations.This chapter gives a specific numerical example to verify the existence and stability of solution,which can be deemed as an application.