A Class Of Nonlinear Differential Equations And Its Applications

Along with science's and technology's development, various nonlinear problems have aroused people's widespread interest day by day, the nonlinear analysis has become one important research directions in modern mathematics. The nonlinearfunctional analysis is an important branch in nonlinear analysis, because it can explain various natural phenomenon. The boundary value problem of nonlinear differential equations 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 studies at present.The boundary value problem of nonlinear differential equation with p-Laplacian is the hot spot at present [13,14,17,23,24,29].The equation with p-Laplacian arises in the modeling of different physical and natural phenomena, non-Newtonian mechanics, nonlinear flow laws[11, 12], nonlinear elasticty, combustion theory, population biology, many authors have studied the existence of the solution for p-Laplacian and obtained many good results [30,31].In this paper, we use the cone theory, Avery-Peterson's fixed point theory, as well as the fixed point index theory to study several kinds of boundary value problems for nonlinear differential equation with p-Laplacian and we apply the main results to integral boundary value problems for the differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we use Avery-Peterson's fixed point theorem to investigate the positive solutions of a class of boundary value problems for nonlinear differential equations with p-Laplacian in Banach spaces,We obtain the existenceof at least three positive solutions for boundary value problem (1.1.1).In chapter 2, we use the fixed point theory to investigate symmetric positive solutions of boundary problems,We obtain the existenceof at least three symmetric positive solutions for boundary value problem (2.1.1).In chapter 3, we use the fixed point index theory [3] to investigate positive solutions of boundary problems with sign changing nonlinearity,...