Nonlinear functional analysis and convex analysis problem by using fixed point theory

利用不动点理论研究了关于非线性泛函分析和凸分析的一些问题。我们首先考虑了Hilbert空间或Banach空间中由无穷族非扩张映象给出的迭代格式，然后证明了非扩张映射族的强收敛定理。利用这些结果，我们还考虑了寻找无限非扩张映象的公共不动点的可行性问题。其次，介绍了索洛多夫和斯瓦伊特提出的迭代格式所提出的两种近似点算法，以求T^∧{-1}0$中$v的解，其中$T$是极大单调算子。在强收敛的情况下，利用度量投影和广义投影建立了主要结果。我们还将这些结果应用于寻找Banach空间中下半连续凸函数的极小点。最后，我们介绍了寻找非扩张映象不动点集的公共元素和逆-强单调映象的变分不等式解集的迭代格式。利用这些结果，我们考虑了寻找极大单调映象的零点集和逆-强单调映象的零点集的公共元的问题。

We studied some problems concerning nonlinear functional analysis and convex analysis by using fixed point theory. We first considered iteration schemes given by an infinite family of nonexpansive mappings in Hilbert spaces or Banach spaces and then proved strong convergence theorems for the family of nonexpansive mappings. Using these results, we also considered the feasibility problem of finding a common fixed point of infinite nonexpansive mappings. Next, we introduced two proximal point algorithms suggested by the iterative schemes introduced by Solodov and Svaiter in order to find a solution of $v \in T^∧{-1}0$, where $T$ is a maximal monotone operator. Main results were established by using metric projections and generalized projections in the case of the strong convergence. We also applied these results to find a minimizer of a lower semicontinuous convex function in a Banach space. Finally, we introduced iteration schemes of finding a common element of the set of fixed points of nonexpansive mappings and the set of solutions of the variational inequality for inverse-strongly-monotone mappings. Using these results, we considered the problem of finding a common element of the set of zeros of a maximal monotone mapping and the set of zeros of an inverse-strongly-monotone mapping.