The Existence Of The Third Order 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 theory as well as the fixed point index theory , to study several kinds of third-order three-point boundary value problems .The thesis is divided into three chapters according to contents.In chapter 1, we are concerned with the existence and multiplicity of positive solutions for third-order three-point nonlinear singular boundary value problems:whereλis a positive parameter, 0 <η< 1 and 1 <α< (?),αmay be singular at t = 0 and/or t = 1.First, the associated Green's function for the above problem is given. And then, some properties of the Green's function are discussed. Finally we establish intervals of the parameter A which yield the single and multiple positive solutions to the boundary value problems. The singularity may appear at t = 0 and/or t = 1. The Krasnoselkii-Guo theorem on cone expansion and compression is used. The main results improve and generalize the recent results.In chapter 2, we study the existence of at least one or two positive solutions to the third-order triple-point nonlinear boundary value problems with sign changing nonlinearities:whereη∈(0,1),α∈(0, (?)) are constants, h changes sign inη.First, Green's function is constructed. And then, by employing KrasnoselkiiGuofixed point theorem and Avery-Henderson fixed point theorem, we establish results on the existence of at least one or two positive solutions to the boundary value problems.In chapter 3, by constructing available operators and combining the method of the fixed point index theory in cones, we investigate the new existence theoremof positive solution to the third-order triple-point nonlinear boundary value problems with sign changing terms:where 0 <η< 1, 0 <α< 1, f∈C([0,1]×[0,∞),R) is continuous.Innovation of this article is: In the first chapter ,we investigate the existence of positive solutions under weaker conditions, and the main results contains and expand the results of the other article. In the second chapter ,for the third-order triple-point nonlinear boundary value problems with sign changing nonlinearities, we establish results on the existence of at least one or two positive solutions . In the third chapter , we weak the restrictions of nonlinearities, and still obtain a positive solution of it.