Existence Of Solutions For Nonlinear Second-Order Differential Equations Boundary Value Problems

Nonlinear functional analysis are research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general theories and methods to handle nonlinear problem. Since it can commendably explain many kinds of natural phenomena, in recent years, which has received widely attention in domestic and foreign mathematics and natural science field, and formed an important discipline gradually. Its rich theory and advanced method have provided the effective theory tool for solving the nonlinear problems one after another appearing in scientific and technical areas. To handle all kinds of nonlinear integral equations, the differential equations and the partial differential equations in actual problems, it plays role which can not be substituted.The dissertation is divided into four chapters: The first chapter introduces the background, methods of the related problems and the major work for this paper.The second chapter, by means of the Avery-Peterson fixed point theorem on cone and some skills and techniques of analysis, we consider the positive solutions for nonlinear second order boundary value problem, which the nonlinear term contains the first order derivative explicitly and with a generalized Sturm-Liouville condition. Some sufficient conditions for the existence of at least three positive solutions are obtained, which improve and generalize some related results in the references.The third chapter, by using the fixed point theorem of cone expansion-compression of functional type, we firstly obtain some sufficient conditions of the existence of at least one positive solution for two class of nonlinear second order multipoint boundary value problem under the weaker hypothesis.The fourth chapter studies the existence of at least one nontrivial solution for nonlinear second order generalized differential equation, which satisfy some different class of multipoint boundary conditions and at resonance. The main tool is the coincidence degree theory of Mawhin type.