Stability Of Nonlinear Neutral Functional Differential Equations

There is a long history on theory of differential equation and a great and better results are researched so far. Neutral functional differential equation can be used to describe many natural phenomena. In the physical, biological, ecological and other areas, it has have a very wide range of applications. Thus both in theory and in practice qualitative research of the neutral differential functional equation has very important meaning, and the stability of differential equation reflects the study the stability of the structure which has high practical value.Based on the above reasons, this paper discusses two kinds of neutral differential equation asymptotic stability of the zero solution.Chapter1introduces first the research dynamic of neutral differential equation at home and abroad and summarizes research methods and its limitation of solutions for neutral differential equation stability. Then, this chapter also presents contents and methods.Changing the integral equation into the differential equation and using Banach fixed point theorem, Chapter2shows necessary and sufficient conditions of asymptotically stable of the zero solution to neutral differential equations x’(t)=-a(t)x(t)+c(t)x’(t-r(t))+q(t,x(t),x(t-σ1(t)), x(t-σ2(t)),…,x(t-σn(t))) and the results in this chapter generalize the existing results.Using Banach fixed point theorem, Chapter3shows sufficient condition of asymp-totically stable of the zero solution to neutral differential equations with impulses In addition, this chapter not only shows asymptotically stable of the zero solution but also gives sufficient condition of exponential stable. Thus, the results in this chapter are more valuable in the applications.