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Along with development of modern physics and applied mathematics, vari-ous nonlinear problem has aroused people’s widespread attention day by day.The nonlinear functional analysis as an important branch in nonlinear analysis has be-come an important tool for study mathematics, pysics,chemistry, biology’s tech-nology. The nonlinear functional analysis offers effective theoretic meaning for these problems,and it is a reaserch subjct for profound theories and broad applica-tion. The nonlinear functional analysis bases on nonlinear problems of math and all kinds of scientific field,and constructs general theories and methods.it could solve various the natural phenomenon and problem.However,higher-ordr differ-ential equation and nonlinear differential equation are the hot spot which have discussed in recent years,and the singular differential equation boundary value problem is one ofmost hot field of nonlinear functional analysis at present. In this thesis, we used the cone theory, the fixed point theory and the fixed point in-dex theory, to study the existence of solution for higher-ordr nonlinear impulsive integro-differential equation and singular differential equation boundary problem. The thesis is divided into three chapters: In Chapter 1, we study the following higher-ordr nonlinear impulsive integro-differential boundary problem. where E is real Banach space.f∈C[J x En+1,E],w∈C(J, [0,+∞)), J [0,1], Ik∈C[E, E], Tk∈C[E x E, E], g∈L1 [0,1] is nonegtive. (Ax)(t)=(?)0tk(t,s)x(n-2)(s)ds, (Bx)(t)=(?)0lh(t,s)x(n-2)(s)ds.k∈C[D, R+],D= {(t,s)∈J×J:t≥s},h∈C[J×J, R+].In abstract space,by the method of reducing order,we turn higher-ordr impulsive integro-differential equation into second-order.And by the fixed point theory for strict set contraction opera- tors,we obtain the existence of solution.so this chapter improve the main results of [4],[5](see Remark 1.3.1 on page 14). In Chapter 2,we study the existence of positive solutions of second-order singular integral boundary value problem with a parameter. (?) whereλ>0 is a parameter, a(t) may be singular at t=0 and/or t=1,the nonlinear f(t,x) is allowed to have singularity at x=0. By the fixed point index theory and Leggett-williams theory, we get the existence and multiplic-ity of positive solutions for the above boundary value problem concerning the first eigenvalue corresponding to the relevant linear operator.Compare with the [26],[29] the above boundary problemunder the singularity,generalize off,and the main results improve[26],[29](see Remark 2.3.1 on page 33).In Chapter 3,we considered the existence of positive solutions for the fol-lowing third-order m point singular nonhomogeneous boundary value problem. wherea(t)∈C((0,1),[0,+∞)),f(t)∈C([0,+∞), [0,+∞)).a(t)may be sin-gular at t=0 and/or t=1.and 0<(?)01(1-s)sa(s)ds<∞.By using the Guo-Kransnoselskii’s fixed point index theorem,we argue the scope of A in order to obtain the the existence of positive solutions.for the above boundary problem,corresponding to the superlinear or sublinear of f.
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