The Study of Nonlinear Functional Analysis and Convex Analysis and its Applications Based on Optimization Theory and Fixed Point Theory

本文利用最优化理论和不动点理论，得到了非线性泛函分析和凸分析中非线性问题的许多新的重要定理。例如，我们解决了Ray在1980年提出的Banach空间几何中的一个开放问题，证明了自反的严格凸光滑Banach空间的非空闭凸子集到自身的每一个非扩展映射都有一个不动点当且仅当该集合是有界的。

In this study, we obtain many new and important theorems for nonlinear problems in nonlinear functional analysis and convex analysis by using optimization theory and fixed point theory. For example, we solved an open problem in geometry of Banach spaces which has been posed by Ray in 1980 by proving that every nonspreading mapping of a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space into itself has a fixed point if and only if the set is bounded.