A Study Of Boundary Value Problems Of Fractional Differential Equtions With Caputo Fractional Derivatives

In recent years,many fields such as physics,chemistry and engineering can be described by fractional differential equations,the field of fractional calculus and its applications have gained a lot of attention.In this thesis,we investigate the existence and uniqueness of solution for the nonlinear multiple base points fractional impulsive anti-periodic boundary value problem and impulsive boundary value problems for the nonlinear multiple base points fractional differential equations,and the uniqueness of solution for impulsive boundary value problems with two Caputo fractional derivatives.Firstly,we study the following impulsive anti-periodic boundary value problem for the nonlinear multiple base points fractional differential equations,where 0 < α,γ < 1,λ > 0,cD·*is the standard Caputo fractional derivative,cD·*|(tk,tk+1]x(t)=cD·t+k x(t)for all t ∈(tk,tk+1].Iγ0+denotes the fractional integral of order γ,Ik∈ R,The impulsive moments {tk} are given such that t0= 0 < t1< · · · < tm< tm+1= 1,?x(tk)represents the jump of function x at tk,which is defined by ?x(tk)= x(t+k)-x(tk),where x(t+k),x(tk)represent the right and left limits of x(t)at t = tkrespectively.we investigate the existence of solutions of the problem by using the Banach’s contraction mapping principle.Secondly,we study the following impulsive boundary value problems for the nonlinear multiple base points fractional differential equations,where α,γ,δ ∈(0,1),α > γ,α > δ,λ > 0.cD·*is the standard Caputo fractional derivative at the base points t = tk(k = 0,1,2,· · ·,m);that is,cD·*|(tk,tk+1]x(t)=cD·t+k x(t)for all t ∈(tk,tk+1],Iγ0+denotes the fractional integral of order γ,The impulsive moments{tk} are given such that 0 < t1< · · · < tm-1< 1,?x(tk)represents the jump of function x at tk,which is defined by ?x(tk)= x(t+k)-x(tk),where x(t+k),x(tk)represent the right and left limits of x(t)at t = tkrespectively.we investigate the existence of solutions of the problem by using the Krasnoselskiˇi fixed point theorem.Thirdly,we consider the following impulsive boundary value problems with two Caputo fractional derivatives for the nonlinear multiple base points fractional differential equations,where 0 < α ≤ β < 1,0 < α + β < 1,0 < γ1,γ2< β,λ > 0,cD·*is the standard Caputo fractional derivative,cD·*|(tk,tk+1]x(t)=cD·t+k x(t)for all t ∈(tk,tk+1].Ik∈ R,The impulsive moments {tk} are given such that t0= 0 < t1< · · · < tm< tm+1= 1,?x(tk)represents the jump of function x at tk,which is defined by ?x(tk)= x(t+k)-x(tk),where x(t+k),x(tk)represent the right and left limits of x(t)at t = tkrespectively.we study the existence of solutions of the problem by using the Banach’s contraction mapping principle.