Nodal Solution For Boundary Value Problems Of Four-order Ordinary Differential Equations

In this doctoral dissertation, we firstly obtain the results of spectrum of the fourth-order two-point boundary value problem of linear generalized eigenvalue problem by the Disconjugate theory. In the following, the global structure and the existence of nodal solutions and positive solutions of some fourth-order two-point boundary value problems with parameters were gained by the bifurcation techniques and the above results of spectrum for the fourth-order two-point of linear problem. In addition, we also study the existence of positive solutions and the dependence of the solutions on the parameters to the fourth-order two-point nonhomogeneous boundary value problem by employing the fixed point theory on the cones. This thesis consists of six chapters.In chapter one, the background and some preliminaries are introduced. The main problems as well as the corresponding results are also presented.In chapter two, by the the Disconjugate theory, we overcome some difficulties brought by linear operator containing kx"(compared to [85]) and make L[x]:=χ″″+κχ″+lχ become L[χ]=14(l3(l2(l1(l0χ)')')')'. It follows that the results of spectrum of the following generalized eigenvalue problem are gained by the Disconjugate theory.In chapter three, when f0∈(0,∞), by using global bifurcation theorems and the results of spectrum of the problem (0.1), we obtain the global structure and the existence and multiplicity of nodal solutions of the problem Furthermore, similarly, the results are extended to the boundary value problem (0.2) corresponding to nonlinear terms with f(t, χ). The results in this chapter improve and generalize some main results of [85](2010).Chapter four, when f0=∞, by using global bifurcation theorems and topological approach, we study the global structure and the existence and multiplicity of nodal solutions of the problem (0.2). We overcome some difficulties brought by the nonlinear term to satisfy the conditions f0=∞and obtain the conclusions by a skill of connected branch by limitting. The results in this chapter improve and generalize some main results of [85](2010).In fifth chapter, because the derivative of Frechet for f in0and∞don't exist, the approach used in the previous two chapters to solve the problem is no longer adapt to this chapter. By using topological degree theory and the bifurcation theorems from an interval [67], we overcome above difficulties and obtain global structures of positive solutions for the boundary value problem (0.2) corresponding to nonlinear terms with f(t,χ).In chapter six, by employing the fixed point theory in cones, we study the exis-tence of positive solutions and the dependence of the solutions on the parameters to the fourth-order two-point nonhomogeneous boundary value problem...