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The boundary value problems for nonlinear ordinary differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory, it’s at present one of the active fields in nonlinear functions analysis. This paper uses the cone theory, topological degree theory and the corresponding fixed point theory to discuss the existence of solutions for nonlinear ordinary differential equations boundary value problems.Chapter 1 is the introduction of this paper, which introductions the main contents of this paper.In chapter 2 we considers the existence of positive solution and multiple solutions for the second order integral boundary value problems whichα,β,γ,δ≥0 are constant andρ=βγ+αγ+αδ>0, a,b∈[0,1]×[0,∞) are Lebesgue integrals, h(t) is allowed to be singular at t=0 and t=1. The existence of positive solution and multiple solutions are obtained by using the generalized cone expansion and compression fixed point theorem of cone and the five functional fixed point theorem. The conclusions extend and improve the main result of Zhang Guowei and Sun Jingxian.In chapter 3 we considers the existence of nontrivial solutions and multiple solutions for the nonlinear singular n-order three point boundary value problem which 0<η<1,1<α<1/η, n is a positive integer and n≥3, h(t) is allowed to be singular at t= 0 and t= 1,f is not necessary to be nonnegative. The existence of nontrivial solutions and multiple solutions are obtained by using the topological degree and fixed point index theory on the cone. The conclusions extend and improve the main result of Guo Lijun.
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