Class Of Nonlinear Differential Equation Boundary Value Problems And Applications

Along with science's and technology's development, various non-linear problemhas aroused people's widespread interest day by day, and so the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and each kind of application discipline. It is one of most active domains of functional analysis studiesin at present. So it become a very important domain of differential equation research at present. In this paper, we use the cone theory, the fixed point theory and combined with a iterative technique,to study several kinds of boundary value problems for nonlinear differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we use the Krasnoselsii's fixed point theorm to study the existenceand multiplicity of symmetric positive solutions for a class of p-laplacian fourth-order nonlocal boundary value problem:The arguments are based on a speciallyconstructed cone and the fixed point theorey for cones.The nonexistence of a positive solution is also studied.In chapter 2,we consider the four point boundary value problem for onedimension p-laplacian1. Applying the Avery and Peterson fixed point ,we study the existence of at least three symmetric positive solutions, we provide sufficent conditions for the problem,the interesting point is that nonlinear term f contains the first-order derivative and the boundary condition is of Sturm-Liouville type.In chapter 3, we obtain the existence of pseudo-symmetric monotone positive solutions and establish a corresponding iterative scheme for the following fourpoint boundary value problem:the main tool is the monotone iterative technique .The interesting point is that the nonlinear term involves the first-order derivative.In chapter 1,we improve the results in [5],by changing second-order into fourth-order,we get the symmetric positive solutions for fourth-order boundary value problem; In chapter 2,In [33],by putting first-order derivative into nonlinearterm and making the boundary value condition into Sturm-Liouville type;In chapter 3,we improve the results in [46],by monotone iterative technique,we get the pseudo-symmetric positive solutions for four-point boundary value problem.