The Existence And Multiplicity Of Solutions Of Impulsive Differential Equations

Nonlinear functional analysis is an important branch of modern mathematics, because of the wide variety of natural phenomena are well explained in nature that is widely concerned by more and more researchers, and in physics, applied mathematics, aerospace,biological and other fields has been widely used.The thesis is divided into three chapters. The chapter 1, we consider three-point boundary value problems of impulsive equations where ?u|t=tk=u(tk+)-u(tk-),?u'|t=tk= u'(tk+)-u'(tk-),k= 1,2,3,...,0 = t0 < t1 <t2 <...<1,0 <?<1, 0<tk <?<1,k=1,2,3,...,m.This chapter combines the impulsive term of [1] with the boundary value conditions of[2] to study equation (1.1.1). Compared with the document[1][2] we change finite pulse point to infinite pulse point, and use Lerray-Schauder alternative theorem, the fixed point index instead of cone compression and expansion and the fixed point index to gain the existence of solutions; Adding impulsive term in [3] studys impulsive differential equations.In addition, considering [4][5] equations all study the existence of positive solutions, this chapter not only gains that, but also has the existence of sign-changing solutions.The chapter 2, we study multipoint BVPs of impulsive fractional differential equations on half infinite interval where ? are parameters, q ?(0,1), 0 = s0<t1?s1<?1?t2<…ti?…,as i??,ti?1.f:R+×R×R×R?R is continuous and gi: [ti, si] × R ? R,i = 1,2,3,....The symbol cDsi,t q denotes the Caputo fractional derivative of the order q.This chapter is based on [6], changing finite intervals to infinite intervals to study infinite impulsive point; Adding the conditions of u(si) in [7], where 0 = s0<t1?s1? t2 <...<ti?...; We generalize integer order to fractional order and also add the conditions of u(si)in [8]. In addition, [9][10][11][12] consider boundary value problem of impulsive fractional differential equations on finite interval, but this chapter consider boundary value problem of impulsive fractional differential equations on infinite interval,using Banach contractive mapping to gain the unique solution.The chapter 3, we study nonlocal boundary value problems of fractional differential equations where ? ? (2,3], ? > 0, 0 <? <?< 1,k> 0 is a parameter,a is a real constant,f: [0,1] × R?R is continuous, cD? denote the Caputo fractional derivative of order ?.This chapter is based on [13],?- 1 will be improved the first-order derivative; Com-pared with the docume1t[14],we improves the first-order derivative in equation (3.1.1)and use Banach contractive mapping to gain the unique solution; Also compared with the document [15], generalizing local boundary value condition to nonlocal boundary value condition makes integral boundary value condition be more extensive which it is u(?) =a?0?(?-s)?-1/?(?)u(s)ds and using the type of Caputo insteads of the type ofRiemann-Liouville.