赋序Banach空间上的序性质及其应用

John Nash由于开创Nash均衡理论而获得1994年度诺贝尔经济学奖。经典均衡问题所涉及的效益函数都是实函数，而许多经济、军事、环境等问题中参与者的效益函数值域取值于赋序Banach空间。因此，要解决其均衡问题就必须在赋序Banach空间中考虑。另外，考虑到某些偏微分方程解空间存在着特定序结构，我们预期可以利用赋序Banach空间上序性质解决某些非线性偏微分方程解的存在性。基于以上两个原因，本项目将对赋序Banach空间上序性质及相关性态进行深入研究，拟在赋序Banach空间上建立FKKM 定理、Ekeland 变分原理、Leray-Schauder 连续定理和一些不动点定理。以此作为工具，解决一些赋序均衡问题和某些偏微分方程的可解性。本项目主要是应用基于赋序Banach空间上序结构和映射序性质的新技巧，其研究成果将丰富Banach空间理论，而且解决某些偏微分方程解的存在性。

John Nash won the 1994 Nobel Prize in Economics for his pioneering　Nash equilibrium theory. The utility functions involved in the classical equilibrium problems are real functions, and the values of the utility functions of the participants in many economic, military and environmental problems are in ordered Banach spaces. Therefore, to solve their equilibrium problems must be considered in ordered Banach spaces. In addition, considering the existence of some specific ordering structures in some solutions of partial differential equations, we expect that the existence of solutions of some nonlinear partial differential equations can be solved by using the order properties in ordered Banach spaces. For the above two reasons, this project will carry on the thorough research on the order property and the correlation state in the ordered Banach space, and construct the FKKM theorem, the Ekeland variational principle, the Leray-Schauder continuous theorem and some fixed point theorems, and solve some of the order equilibrium problems and some partial differential equation solvability. This project is mainly to apply the new technique based on the order property of ordered Banach space and the order property of the mapping. The research results will enrich the Banach space theory and solve the existence of solutions of some partial differential equations.