Boundary Value Problems Of Nonlinear Differential Equations

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. The singular nonlinear differential equation boundary value problem is also the hot spot which has been discussed in recent years. So it become a very important domain of differential equation research at present. In this paper, we use the cone theory, the fixed point index theory, the topological degree theory to study several kinds of boundary value problems for nonlinear singular differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we use the cone theory and fixed point index theorem under some conditions concerning the first eigenvalue corresponding to the relevant linear operator to investigate the positive solutions of nonlinear singular m-point boundary value problemwhereξ_i∈(0,1),α_i∈(0,1), i = 1,2,…,m-2 are given constant, and 0 <∑_i=1^m-2α_iφ₁(ξ_i)<1(φ₁ will be given in the second part of chapter 1). f∈C([0,1]×(0, +∞), [0, +∞)), and h(t) is allowed to be singular at t = 0, 1, and besides f is singular at u=0. This paper generalize and improve the results in [5, 14].In chapter 2, we use the cone theory and fixed point index theorem to investigatethe solution of problem (1.1.1). In this paper h(t) is allowed to be singular at t = 0,1 and f is not necessary to be nonnegative . we obtain the existence results of nontrivial solutions, in particular, the existence results of at least one positive solution and one negative solution. This paper generalize and improve the results in [5, 13].In chapter 3. we use the cone theory and fixed point index theorem to investigatethe positive solutions of nonlinear Sturm-Liuouville boundary value problemwhere (Lφ)(x)=-(p(x)φ'(x))+q(x)φ(x). and the nonlinearity depend on the derivative. Inspiered by the papers of [7, 10, 12, 15, 16] , this paper generalize and improve the results in [7, 10].