Nonlinear evolution equations and singular perturbations of variational inequalities

自牛顿发明微分方程以来，微分方程（即含有未知函数及其导数的方程）一直是并仍然是描述和研究物理学和工程学中各种数学现象的最有用的工具。从抽象的非线性发展方程的观点出发，考虑了由不等式给出约束条件的微分方程的变分不等式问题，并证明了含有一个参数的问题的奇异极限的存在性，该参数的极限改变了方程的类型.在研究过程中，我们综合了非线性分析的主要工具：变分原理和不动点分析，提出了拟变分分析的概念，并引入了拟次微分算子的概念，这是一类新的抽象非线性算子。

Since Newton's invention, differential equations, that are equations containing unknown functions and their derivatives, have been and remain the most useful tool to describe and study mathematically various phenomina arising in physics and engineerings. From the viewpoint of abstract nonlinear evolution equations, we considered variational inequalities, that are problems of differential equations with constrains given by inequalities and imposed upon the unknown functions, and proved the existence of the singular limit of a problem containing a parameter whose limit changes the type of equation. Also in the course of the study, we proposed the concept of quasi-variational analysis by synthesizing the principal tools of nonlinear analysis: variational principles and fixed point analysis, and introduced the concept of a quasi-subdifferential operator which is a new class of abstract nonlinear operators.