The Singular Nonlinear Differential Equations And Pulse Phenomena And Their Applications

Nonlinear functional analysis is a 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. Because it can commendably explain all kinds of natural phenomenal, in recent years it has received highly attention of the 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 appearing in the technical domain one after another incessantly. 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 method to research nonlinear problems mainly has variations method, partial order method, topology degree method, analysis method and so on. The main questions to research are the existence and uniqueness of solution for nonlinear operator equation, multi-solution, structure of solution, approximate solution, divergent theory of solution, iteration arithmetic, nonlinear operator theory as well as the application for partial differential equation, differential equation, integral equation and differential-integral equation. These problems all are the active domain in analyze mathematics at present. Among them, the problems in the space both having the algebra structure and the topology structure (for example, the Banach space) have already studied quite full. But the another mathematical structure- order, its research is quite slower than other in recent years. The Banach space with the ordered structure accommodates three most basic mathematical frameworks (the algebra, topology, order) in a body, there is the vital significance in the theory and the application to study for them. Therefore, using many kinds of advanced analysis tool developed in nonlinear analysis for several years to study the singular boundary value problem or impulsive initial value problem for nonlinear ordinary differential equation is also a research topic to have a strong interest and maybe obtain the new significance achievement.The aim of this article is, on basis of developing partial order theory, to study the existence and uniqueness of solution, iterative scheme and an error estimate for nonlinear integro-differential equations with or without impulse and boundary value problem of differential equations in Banach space by using nonlinear functional analysis method. We are concert with the study of singular and impulsive phenomena, including some singular semipositone problems, the eigenvalue problems of high singularity, necessary and sufficient condition of existence of positive solutions for singular problem and unique solution for impulsive initial value problems. By deep study, we obtained many new results, all these results have published in the domestic and foreign famous journal, such as,《J. Math. Anal.Appl.》(SCI),《Nonlinear Anal.》(SCI),《Appl. Math. Comput.》(SCI),《Appl. Math. Lett.》(SCI),《Math. Computer Modelling》(SCI),《Dynamic of Continuous, Discrete and Impulsive Systems》(SCI),《Nonlinear Funct. Anal. Appl.》and《Systems Sci. Math. Sci.》,《Math. Acta. Sci.》ect.The paper is divided into five chapters. In the chapter one, we mainly introduce the developing history of nonlinear analysis. The chapter two, we carry out study for the singular semipositone problem with profound background and give a new method to solve this type problems. In the chapter three, we study the existence of positive solution for singular differential system, by means of monotone iterative technique and the upper and lower solution method, we give necessary and sufficient condition of the existence of positive solution for serval class of differential systems and the iterative sequence of solution, error estimation and so on. The fourth chapter focus on the study of eigenvalue problems of high singularity, in this section we firstly obtain some sufficient conditions of existence of positive solutions for elastic beam equation with Sturm - Liouville boundary condition and three point boundary value by the upper and lower solution and Schauder fixed point theorem, and then study the existence of eigenvalue for third-order three point differential equation by using Leray - Schauder continuation theorem. In the fifth chapter, we are concern with the existence and uniqueness of solution, iterative scheme and an error estimate for nonlinear first and second order impulse integro-differential equations. In this part we obtained many varied and colorful results, these results have the vital significance whether in the theory or in nature.