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Nonlinear functional analysis is an important branch of mathematics, because it can well explain the nature of the various natural phenomena and by domestic and foreign mathematicians and natural scientific community attention. Nonlinear boundary value problem stems from applied mathematics, physics, control theory and other disciplines in a variety of applications, is the mathematical analysis of the most active areas of research. Among them, the multi-point boundary value problem from various fields of applied mathematics and physics model has important theoretical significance and application value. In this paper cone theory, fixed point theory, topological degree theory to study several types of nonlinear differential equations point boundary value problem situation, get some new results. This article is divided into the following three chapters based on the content: In the first chapter, we use Schauder fixed point theorem and topological degree theory, discusses a class of second order three Singular Impulsive Boundary Value existence, where a: (0,1) → [0, ∞) is continuous, and t = 0, t = 1 处 may be singular, 0 lt; t 1 lt; t 2 ... lt; t m lt; 1 is given. This equation ratio [2] in the equation is more extensive, in-band pulse conditions and bizarre circumstances, is not limited to ultra-linear or nonlinear term sublinear, using fixed point index equations obtained two positive solutions. In the second chapter, we use the Schauder fixed point theorem and topological degree theory, the following second-order differential equation n point nonhomogeneous boundary value the existence of which (2.1.3) satisfy the following conditions: (A 1 ) k i gt; 0 (i = 1,2 ..., n-2), 0 lt; ξ 1 lt; ..., lt; ξ n-2 lt; 1, Σ k = 1 n-2 k i lt ; 1. , a (t) is not identically zero. When n = 3, b = 0, the equation is (2.1.1) in the form, when ε = 0,, b = 0, the equation is (2.1.2) in the form of the equation of this study is more with a general, and get a good result. In the third chapter, the use of cones on the fixed point index theory, we study the following Strum-Liouville Boundary Value Problems of Impulsive Differential Equations Existence under certain conditions, to get this type of existence of at least two positive solutions, This result is more general, to a certain extent, the promotion of [22] results.
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