Positive Solutions Of Nonlinear Second-order Neumann Boundary Value Problems

By using the fixed point index theory and the method of topological degree, the present paper is concerned with the existence and multiplicity of nontrivial positive solutions of nonlinear second-order Neumann boundary value problems (BVP) where M>0 is a constant,f is the nonlinear term, S is a Fredholm integral operator. According to contents, the thesis is divided into three sections. they are as follows:In Chapter 1, the research background and the main results of this thesis are exhibited. And some preliminaries on fixed point index theorem of completely continuous mapping, measure of non-compactness and its theorem and fixed point index theory of condensing mapping are also displayed in this chapter.Chapter 2 is divided into two parts. In the first part, the existence of nontriv-ial positive solutions of BVP(P) is studied by using the fixed point index theory of completely continuous mapping under nonsingular case. At first, we obtain an existence result of nontrivial positive solutions of BVP(P) under limits conditions. Then we extend the limits conditions into inequality conditions, and discuss the existence and multiple of nontrivial positive solutions of BVP(P). The results ob-tained in this chapter improve and extend the main results in [8-10]. In the second part, we study the existence of nontrivial positive solutions of BVP(P) by using the fixed point index theory of completely continuous mapping under singular case. The results obtained in this chapter improve and extend the corresponding results in [21,22].In Chapter 3, we discuss the existence of positive solutions of BVP(P) in or-dered Banach spaces. By using the fixed point index theorem of condensing mapping, we obtain two existence theorems under more general ordered conditions, which ex-tend some existing results.