Research On The Existence Of Solutions For Several Classes Of Fractional Differential Equations

In this p aper,we study the existence of solutions for several classes of fractional differential equations.The theorems of the existence of solutions for a class of nonlinear Conformable fractional differential equations and the existence theorems of solutions for fractional impulsive differential equations with integral initial conditions are obtained.This paper is divided into three chapters.In part one,we summarize the research background of fractional order calculus as well as the related definitions and theorems used in this paper.In part two,we study the existence of solutions for nonlinear fractional differential equation:Tax(t)+f(t,x(t))=0,0<t<1.The following boundary value conditions are satisfied:x(0)?x(1)?0,x(0)=x'(0)=x(1)?0,where 1<a<2(2<??3)is a real number,Tax(t)denotes the conformable derivative of x at t of order a.Based on the properties of the Green function,the existence theorems of solutions of fractional differential equation are obtained by using the Banach s contraction mapping principle and the Krasnoselskii's fixed point theorem in a cone.Finally,an example is given to demonstrate the feasibility of the obtained results.In part three,we investigate the existence of solutions to the following impulsive fractional differential equations with integral initial condition:where k= 1,…,m,J=[0,T],J':= J\{ti,...,tm},1<?<2,cD? is the Caputo fractional derivative.According to the definition of calculus and the related knowledge,the fractional differential equations are transformed into integral equations.The fixed point theorems are used to obtain the existence theorems of the solutions of the equation.At the same time,the corresponding examples are given.