保持问题和Banach-Stone型定理是泛函分析中非常重要的研究课题，是属于Banach空间理论、算子理论以及代数学等学科相互交叉的理论。本项目旨在研究Lipschitz函数以及连续可微函数空间上的保持算子与相应的Banach-Stone型定理。特别地，本项目将着重研究保持零集非空与保序这两大类保持算子，并且将证明它们可以写成由（Lipschitz或者可微）同胚映射所诱导的加权复合算子的形式。同时，本项目还将给出有界Lipschitz函数空间上的加权复合算子是紧算子的充分必要条件，并给出它的谱。我们将综合运用Banach空间理论、Lipschitz函数理论和Frechet可微函数理论解决本项目的所有问题。

Preserver problem and Banach-Stone type theorems are important topics of functional analysis, and they will combine Banach space theory, operator theory and algebra. This project will study the preservers and related Banach-Stone type theorems in the Lipschitz functions and continuous differentiable functions. In particular, we will focus on the nonvanishing preserving maps and order isomorphisms, and we will show that they are weighted composition operators induced by (Lipschitz or differentiabe) homeomorphisms. This project also will give the sufficient and necessary conditions such that the weighted composition operators on the bounded Lipschitz functions are compact; and we will present its spectrum. We will use the methods of Banach space theory, Lipschitz function theory and Frechet differentiable theory to solve all the problems.