Types Of Nonlinear Boundary Value Problems Existence Study

Along with science's and technology's development, various non-linear problem has aroused people's widespread interest day by day, because it can well explain various the natural phenomenon. So, the mathemaatical world and the natural science world attach importance to the nonlinear functional analysis. They have obtained some new results for the nonlinear functional analysis and its applications. The singular nonlinear differential equation boundary value problem is also the hot spot which has been discussed in recent years.In this paper, we use the cone theory, the fixed point theory, the topological degree theory, Leray-Schauder Continuation Principle as well as the fixed point index theory and combined with Lower and upper solutions, to study several kinds of boundary value problems for nonlinear singular differential equation and we apply the main results to the boundary value problem for the singular integral differential equation, and a necessary and sufficient condition for the existence for a nonlinear singular second-order boundry problem.The thesis is divided into three chapters according to contents.In chapter 1, It is concerned with the Fourth-order singular BVPUnder a general assumption, the existence of symmetric positive solution are obtained by the method of upper-lower solutions and Schauder's fixed point theorem. Fredholm alternative theorem and minimax theorems are invalid here and our results generalize many recent studies.In chapter 2, we consider the following nonlinear third-order boundry problem(?), (2.1.1)(?), (2.1.2)where f: [0,1]×R~3�'R is L_p-Carath(?)ordory, 1≤p <∞. we can obtain the existence of at least one positive solution, the discussion is based on the Leray-Schauder Continuation Principle by separately considering the case p = 1 and P>1.In chapter 3, by using the tensile and the compression fixed-piont theorems in cone, A necessary and sufficient condition for the existence for the following nonlinear singular second-order boundry problemwhereα,β,γ,δ≥0,ρ=αβ+αδ+γβ> 0, f(t, u) is singular at u = 0 and t = 0 or t = 1. The first order derivative positive solutions to a class of second-order boundary value problems is investigated. A necessary and sufficient condition for the existence of the first- order derivative positive solutions is obtained under the condition that nonlinear term is super- linear or sub-linear, or nonlinear term decomposed into super-linear and sub-linear, which imposed some known results.