On quantitative versions of theorems due to
On quantitative versions of theorems due to
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关于定理的定量版本
DOI：
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
U. Kohlenbach
中科院分区：
文献类型：
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作者：
F. Browder;R. Wittmann;U. Kohlenbach
本文是逻辑分析证明以提取新的（通常有效的）信息（“证明挖掘”）程序的另一个案例研究。我们提取明确的一致率的亚稳性（在意义上的T。Tao）对F.E. Browder关于非扩张映象不动点逼近的收敛性以及R。Wittmann定理，它可以看作是平均遍历定理的非线性推广。第一个速率是从Browder的原始证明中提取的，该证明基于弱序列紧性的应用（除了投影参数之外）。Wittmann的证明遵循类似的推理路线，我们调整我们对Browder证明的分析，以获得Wittmann定理的定量版本。在这两种情况下，人们也得到了这些定理的完全初等的证明（甚至对于加强的数量形式），它们既不使用弱紧性，也不存在投影。通过这种方式，本文还讨论了从基于弱紧性的证明中提取有效信息的一般特征。然后，我们从Browder定理的另一种证明中提取另一种亚稳定性（性质相似），这主要是由于Halpern已经避免了任何弱紧性的使用。本文的结论是一般性评论的逻辑分析的基础上证明弱紧性以及定量形式的所谓的半封闭性原则。在随后的文件中，这些结果将被用于定量分析Baillon的非线性遍历定理。
This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder’s original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann’s proof follows a similar line of reasoning and we adapt our analysis of Browder’s proof to get a quantitative version of Wittmann’s theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder’s theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon’s nonlinear ergodic theorem.
序贯弱紧性的统一定量形式及Baillon非线性遍历定理
影响因子：
1.6
作者：
Kohlenbach
通讯作者：
Kohlenbach