Existence And Approximate Representation Of Nonoscillatory Solutions For Neutral Functional Differential Equation

There is a long history on theory of differential equation and a great results are researched so far. With the development of modern society, it is widely used in engineering, ecology, economics, finance etc. However, with the research work of the existence for the functional differential equation, a lot of work has been only given sufficient conditions for existence of the solution, but nobody has give approximate representation for the solution. In fact, we will not be more valuable in the applications until we give approximate representation for the existence of the functional differential equation. In this thesis, we shall study two types of existence and approximation representation for nonoscillatory solutions of neutral delay differential equations.Chapter 1 introduces first the research background and current situation of func-tional differential equation. In particular, we summarize the sufficient conditions and the limitations of existence of the nonoscillatory solutions for the functional differential equation. Then, this chapter also presents contents and methods.Using Banach fixed point theorem and Lebegue's control convergence theorem, Chapter 2 show sufficient conditions of existence of nonoscillatory solutions for high order functional differential equations In addition, this chapter not only establishs the existence of uncountably many nonoscil-latory solutions but also gives iterative approximate sequence and error estimate. Thus, the results in this chapter are more valuable in the applications.Using the Krasnoselskii fixed-point theorem and Schauder fixed-point theorem, chapter 3 show sufficient conditions of existence of nonoscillatory solutions for high order functional differential equations In addition, this chapter not only establishs the existence of uncountably many nonoscil-latory solutions but also gives iterative approximate sequence and error estimate. Thus, the results in this chapter are more valuable in the applications.