Existence Of Solutions Of Nonlinear Boundary Value Problems

Along with science's and technology's development, various non-linear problem has 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 studies at present. In this paper, we use the cone theory, the fixed point theory , as well as the fixed point index theory, to consider the existence of solutions of several kinds problems for nonlinear differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we consider the existence of symmetric positive solutions of the following singular fourth order boundary value prodlem:Whereα,β,γ＞0,α+β＞γ/2,η∈(0, 1/2] ,λ∈(0, 1),φ_p(s) = |s|^p-2s,φ_p^-1=φ_q, p＞1, 1/p+1/q=1, h : (0,1)â†'[0,∞), is symmetric on (0, 1) and may be singular at t=0 and t=1 . First, the Green's function for associated linear boundary value problem is constructed, and some useful properties of the Green's function are obtained. Then by applying the fixed-point index theory, we establish some optimal criteria for the the existence of one or two symmetric positive solutions which involve the principal eigenvalue of a related linear operator. optimal criteria for the existence of one or two symmetric positive solutions which involve the principal eigenvalue of a related linear operator.In chapter 2, we apply a fixed point index to consider the existence of pseudosymmetricpositive solutions of the following four-point second-order p-Laplacian boundary value problem:In chapter 3, By constructoing a particular closed convex set and appling the M(?)nch fixed-point theorem, we study the existence of positive solutions of the singular boundary value problem:in Banach space, singularities occurring at t = 0,1, and x =θ.