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

使用不动点理论的非线性泛函分析和凸分析问题

基本信息

批准号：
12640157
负责人：
TAKAHASHI Wataru
金额：
$ 2.24万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640157/
关键词：
Nonlinear Functional Analysis Nonlinear Operators Nonlinear Ergodic Theorem Nonlinear Evolution Equation Convex Analysis Fixed Point Theorem Mini-max Theorem Nonlinear Variational Inequality

项目摘要

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.

我们利用不动点理论研究了非线性泛函分析和凸分析的一些问题，首先考虑了希尔伯特空间或巴纳赫空间中无限族非扩张映射给出的迭代方案，然后证明了该族非扩张映射的强收敛定理。这些结果，我们还考虑了寻找无限非扩张映射的公共不动点的可行性问题接下来，我们介绍了由 Solodov 和提出的迭代方案提出的两种近点算法。 Svaiter 为了找到 $v \in T^∧{-1}0$ 的解，其中 $T$ 是最大单调算子，主要结果是在强收敛的情况下使用度量投影和广义投影建立的。我们还应用这些结果来寻找 Banach 空间中的下半连续凸函数的最小化函数。最后，我们介绍了寻找非扩张映射不动点集合的公共元素的迭代方案。逆强单调映射的变分不等式的解集使用这些结果，我们考虑了寻找最大单调映射的零集和逆强单调映射的零集的公共元素的问题。单调映射。