The Existence And Uniqueness Of The Solution Of The Fractional Differential Equation With Parameter

This thesis discusses the existence and uniqueness of solutions of initial value problem of fractional differential equation problem of the following form: where 0<n<1,f(x,λ,y)is a function with parameters of λ,D*0n is a standard Caputo fractional differential.In the first two chapters of the thesis,we show the background and the current situation of the fractional differential equations,as well the definition and the related properties needed in this thesis.The conclusion related to the main content of this thesis are also introduced here.In the first section of the third chapter,we uses the Picard iteration method to prove the existence and uniqueness of solution for initial val-ue problems(*)under assuming thatf(x,λ,y)is continuous and satisfy Lipschitz conditions.If f(x,λ,y)do not meet the Lipschitz conditions, the problem(*)was discussed by by Schauder fixed point theorem.Finally this thesis proves the existence of solutions for initial value problems(*) when f(x,λ,y)is continuous,and satisfies the following inequality: |f(x,λ,y)|≤a+b|y|μ,(?)(x,λ)∈G,-∞<y<+∞In the second section of the third chapter,the existence of the initial value problem(*)is studied by using the Schauder fixed point theorem.