Nonlinear Stability Of θ-methods For Neutral Functional Differential Equations In Banach Space

Neutral functional differential equations (NFDEs) can be found in many scientific and technological fields such as biology, physics, control theory, engineering and so on. It is important for the development of these fields to study theory and application of numerical methods for NFDEs. In the last four decades, it have been widely discussed by many researchers such as Barwell, Bellen, Watanabe, Roth, in't Hout, Koto, Shoufu Li, Jiaoxun Kuang, Mingzhu Liu, Chengming Huang, Chengjian Zhang, Hongjiong Tian, Guangda Hu, Siqing Gan. However, because of the difficulty of the research, its numerical solutions is still limited to linear problems and several classes of special nonlinear problems so far. For more general nonlinear NFDEs, there have little research in literature. But for some stiff problems, it may happen that the one-side Lipschitz constant is very large. Therefore, it is urgent and meaningful to break through the restriction of the inner product and the corresponding norm.This paper is concerned with the numerical solution to initial value problems of nonlinear neutral functional differential equations in Banach space. Base onθ-methods, a series of stability results are obtained. Numerical examples are given to conform the theoretical results in the end.