Two kinds of fixed point theorems and reverse mathematics
Two kinds of fixed point theorems and reverse mathematics
复制标题
两种不动点定理及逆向数学
DOI：
10.1002/malq.201600096
复制
发表时间：
2017
影响因子：
0.3
通讯作者：
Takeshi Yamazaki
中科院分区：
文献类型：
--
作者：
W. Peng;Takeshi Yamazaki
在本文中，我们研究了逆向数学背景下两种类型不动点定理的逻辑强度。一是涉及巴拿赫收缩原理的扩展。在这类定理中，我们主要证明 Caristi 不动点定理等价于 RCA0 上的 ACA 。另一个致力于拓扑不动点定理，例如布劳威尔不动点定理。我们介绍 Fan-Browder 不动点定理和 Kakutani 不动点定理的一些变体，我们分别称为 FBFP 和 KFP 。然后我们证明，在 RCA0 上，FBFP 相当于 WKL，KFP 相当于 ACA。此外，我们还研究了 Fan-Browder 不动点定理在游戏系统中的应用。
In this paper, we investigate the logical strength of two types of fixed point theorems in the context of reverse mathematics. One is concerned with extensions of the Banach contraction principle. Among theorems in this type, we mainly show that the Caristi fixed point theorem is equivalent to ACA over RCA0 . The other is dedicated to topological fixed point theorems such as the Brouwer fixed point theorem. We introduce some variants of the Fan‐Browder fixed point theorem and the Kakutani fixed point theorem, which we call FBFP and KFP , respectively. Then we show that FBFP is equivalent to WKL and KFP is equivalent to ACA , over RCA0 . In addition, we also study the application of the Fan‐Browder fixed point theorem to game systems.