Flow-invariant sets
Flow-invariant sets
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流不变集
DOI：
10.1285/i15900932v9n2p221
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
C. Terp
中科院分区：
文献类型：
--
作者：
C. Terp
设S是赋范空间E的一个子集，S是定义在包含S的开子集U上的一个连续函数。我们考虑满足微分方程的所有功能，其中T是一个正数，取决于z。这样的函数将被称为（1）的解。我们说S是不变量C，满足条件fo（1），如果（1）的每一个解都满足条件S，即。关于有限维情况下不变集的特征的早期结果是由Nagumo [Na 42]得到的。进一步的研究由Bony [Bo69]，Brezis [BGO]，Crandall [Cr72]，Hartman [Ha721和Yorke Tyo673，[Yo70]独立完成。Bony和Brezis分别对S和x施加的不同条件由Redheffer [Re72]相互关联。他阐明并概括了他们的结果。甚至在无限维空间的情况下也有了一些进展。Redheffer和Walter [ReWa75]是第一个调查这一领域的人。他们的结果被Republmann [Vo73]，[Vo75]推广。最后，Martin [Ma73]和Reynmann [Vo76]在E是Banach空间的情况下得到了意义深远的推广。这说明提供了一种替代方法的基础上的一个基本引理理论的功能的一个真实的变量。作为推论，我们得到了布雷齐斯不变性定理的一个稍微更一般的形式.以这种方式，它变得明显，后者是一个更高的维推广的引理从一维微积分。
Let S be a subset of some normed space E and let be a continuous function defined on an open subset U containing S. We consider functions which satisfy the differential equation for all , where T is a positive number, depending on z. Such functions will be called solutions of (1). We say that S is invarianC with reqhzt fo (l), if every solution of (1) with remainsin S,that is . Early results conceming the characterization of invariant sets in the finite dimensional case were obtained by Nagumo [Na42]. Further research was independently done by Bony [Bo69], Brezis [BGO], Crandall [Cr72], Hartman [Ha721 and Yorke Tyo673, [Yo70]. Different conditions imposed on S and x by Bony and Brezis respectively were related with one another by Redheffer [Re72]. He elucidated as well as generalized their results. There was even some progress in the case of infinite dimensional spaces. Redheffer and Walter [ReWa75] were the first to investigate this field. Their results were extended by Volkmann [Vo73], [Vo75]. Finally, Martin [Ma73] and Volkmann [Vo76] came to farreaching generalizations in case E is a Banach-space. This note offers an alternative approach based on an elementary lemma from the theory of functions of one real variable. As a corollary we obtain a slightly more general version of Brezis’ Invariance Theorem. In this fashion it becomes evident that the latter is a higher dimensional generalization of the lemma from one-dimensional calculus.