Existence And Pseudo Asymptotically Periodicity Of Solutions For The Fractional Differential Equations

In recent years,fractional differential equations play an important role in many fields of science,such as in biology,chemistry,engineering,playsics and so on.There has been a significant theoretical development in fractional differential equations.In this thesis,we study the existence of solution for boundary value problem and initial problem of two classes of the fractional differential equations,and the existence and uniqueness of pseudo asymptotically periodic solution for the fractional differential equations on unbounded in intervals.Firstly,we consider the following impulsive problem for fractional differential equa-tions with boundary value,where ? ?(1,2),?,?1,?2 ?(0,1),?-? ?(1,2).cD*? is the standard Caputo fractional derivative at the base points t = tk(k = 1,2,…,m).Ik,Ik ? C(R,R),The impulsive moments{tk} are given such that 0=t0<t1<…tm<tm+1 = T,?x(tk)reprepresents the jump of function x at tk,which is defined by ?x(tk)= x(tk+)-x(tk-),where x(tk+),x(tk-)represent the right and left limits of x(t)at t = tk respectively.?x'(tk)has the similar meaning for ?x(t).we study the existence of the solution to the problem by using the Schauder fixed point theorem.Secondly,we consider the following impulsive problems for fractional differential e-quations with nonlocal delay,where ?,T>0,cDt? is the Caputo fractional differential operator of order ? ?(0,1).A:D(A)(?)X?X is the generator of an analytic resolvent family {S?(t)}t?0 on a complex Banach space X.f:J × D × X ? X is a given function to be specified later.D = {??:[-?,0]? X is continuous everywhere except for a finite number of points s at which ?(s)and the right limit ?(s+)exist and ?(s-)ip(s-)= ?(s)}.g:D?X,? ?D.The impulsive moments {tk} are given such that 0=t0<t1<…<tp<tp+1=T,Ik:X?X(k=1,2,…,p)are appropriate functions,?x(tk)represents the jump of a function x at tk,which is defined by ?x(tk)=x(tk+)-x(tk-),where x(tk+)and x(tk-)are respectively the right and the left limits of x at tk.For any continuous function x defined on the interval[-?,T]\{t1,t2,…tp} and any t ?[0,T],we denote by xt the element of D defined by xt((?))= x(t +(?))for(?)?[-?,0].Thirdly,we consider the following initial valre problem of fractional differential equa-tions in Banach space,where q ?(0,1),cD0+q is the standard Capito fractional derivative,A is a closed linear operator in Banach space.We study the existence and uniqueness of the pseudo asymp-totically periodicity of the solution to the problem by using the semigroup of operator theory and fixed point theory.