The Nonoscillation Solution Of Neutral Differential Equations

There is a long history on theory of differential equation and many results are researched so far. In the information age, with the rapid development of economic and science, ordinary differential equations are widely used in various fields. It merges constantly in Physics, Mechanics, Biology, Demography, Chemistry, Economics etc. In addition, several urgent problems to be solved are raised. Therefore, the research on ordinary differential equations is an important subject of theory and practical appli-cations. However, with the research work of the existence of the neutral differential equation, a lot of work has been given conditions of the existence of of nonoscillatory solutions for low-order neutral differential equations, but rarely has given high-order differential equations.In the study on the existence of neutral delay differential equation, most scholars applied the Knaster-Tarski fixed point theorem, and in this article we use the Ba-nach contraction mapping principle to study the existence of nonoscillatory solutions of higher-order neutral differential equations.In the real world, the mathematical model on practical problems can come down to the existence of nonoscillatory solutions for neutral differential equations system. Consequently, the research on high-order differential equations is full of applications value in mathematical theory. In this thesis, we shall study existence of two types of neutral differential equations. The full text structure is as follows:The first chapter introduces the setting of the research, the main work in this paper, and at the same time the main results are obtained.Using Banach fixed point theorem, Chapter 2 shows the existence of nonoscillatory solutions for high-order neutral differential equations.Chapter 3 using the mathematical induction and the constructor method to study the asymptotic behavior of delay differential equations(3.3.1).