Higher Order Nonlinear Differential And Singular Impulsive Boundary Value Problems,

Along with science's and technology's development, various non-linear prob-lem has aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. It has profound theories and broad applications. For example, it plays an important role in the fields of physics, chemistry, mathematics, biology, medicine, economics, engineering, cybernetics and so on. Through constructing all kinds of nonlinear integral equations, nonlinear differential equations and partial differential equa-tions, it explains various natural phenomenon and solves lots of problems. The high order nonlinear differential equation boundary value problem and period im-pulsive differential equation are the hot spots discussed in recent years. Singular impulsive differential equation also received a great deal of attention recently. So it has become an important research directions in modern mathematics and aroused more and more people's widespread attention day by day.In this paper, we use the cone theory, the fixed point theory and the fixed point index theory, to study some kinds of boundary value problems for nonlinear high order differential equation, singular period impulsive differential equation.The thesis is divided into three chapters.In Chapter 1, we investigate the positive solutions of boundary problems for the following high order nonlinear differential equations. whereαi,βi,δi,γi,≥0 are constants such thatρi=βiδi+αiδi+αiγi>0, 0≤i≤n-1, a(t)∈C((0,1), [0,+∞)), a(t) may be singular at t=0 and or t=1. f(t,v1,v2,…,vn)∈C([0,1]×(0,+∞)×(-∞,0)×…×(-1)n-1(0,+∞),[0,+∞)), f(t, v1,v2,…,vn) may also be singular at vi=0, i=1,2,..., n, where (-1)n-1(0,+∞) =(0,+∞) if n is an odd number, (-1)n-1(0,+∞)=(-∞,0) if n is an even num-ber. The existence of the positive solutions of BVP (1.1.1) is obtained by fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator.This chapter generalizes and improves the results of[32,33](page 14,Remark 1.3.2).In Chapter 2,we talk about the positive solutions of singular periodic bound-ary value problem with impulse effects. Here J=[0,2π],00,f∈C(J×R+,R+),Ik∈C(R+,R),Jk∈C(R+,R+),R+=[0,+∞),R+=(0,+∞) with-(?)Jk(u)