Implicit Functions and Solution Mappings
Implicit Functions and Solution Mappings
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DOI：
10.1007/978-0-387-87821-8
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发表时间：
2009-07
期刊：
影响因子：
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通讯作者：
A. Dontchev;R. Rockafellar
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文献类型：
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作者：
A. Dontchev;R. Rockafellar
本书第二版的准备工作是由一系列新的发展引发的，这些发展带来了许多有趣的结果。这一重要的新材料大大扩充了本书的内容，但也扩充了旧材料的内容，补充了第一版的成果，并使其得以进一步扩展。我们希望以这种方式提供了一个更全面的图片，我们的主题，从古典到最近。第1章有一个新的序言，更好地解释了我们的方法隐函数范式的方程，变分问题的解决方案映射，超越。在新的第1.8节[1H]中，我们针对仅连续但单调的函数提出了一个隐函数定理。大量的补充开始出现在第章。第四，引入广义分化。度量正则性的协导准则现在在4.3 [4C]中得到了证明。第4.4节[4D]被重新构造，以跟进度量正则性的严格导数条件，并立即继续Clarke和库默的反函数定理，以及非光滑广义方程的反函数定理，其证明被推迟到第10章。6.这样，所有基本的正则性性质都充分地提供了涉及广义导数的准则。第5章，处理理论的无限维分支，已经增加了很多。第5.7节[5G]提出了参数反函数定理，稍后将在第二章中使用。6.第5.8节[5 H]将结果推广到非线性度量空间，并给出了（推广的）Lyusternik-Graves定理的一个新证明。第5.9节[5I]链接了Lyusternik-Graves定理、不动点定理和集值分析中的其他结果。最后一节。5.11 [5 K]部署了一个反函数定理在Banach空间，它只依赖于选择逆方向导数。
The preparation of this second edition of our book was triggered by a rush of fresh developments leading to many interesting results. The text has significantly been enlarged by this important new material, but it has also been expanded with coverage of older material, complementary to the results in the first edition and allowing them to be further extended. We hope in this way to have provided a more comprehensive picture of our subject, from classical to most recent. Chapter 1 has a new preamble which better explains our approach to the implicit function paradigm for solution mappings of equations, variational problems, and beyond. In the new Sect. 1.8 [1H], we present an implicit function theorem for functions that are merely continuous but, on the other hand, are monotone. Substantial additions start appearing in Chap. 4, where generalized differentiation is brought in. The coderivative criterion for metric regularity has now a proof in 4.3 [4C]. Section 4.4 [4D] has been reconstituted to follow up with the strict derivative condition for metric regularity and immediately go on to the inverse function theorems of Clarke and Kummer, and an inverse function theorem for nonsmooth generalized equations whose proof is postponed until Chap. 6. In that way, all basic regularity properties are fully supplied with criteria involving generalized derivatives.Chapter 5, dealing with infinite-dimensional branches of the theory, has been augmented by much more. Section 5.7 [5G] presents parametric inverse function theorems which are later put to use in Chap. 6. Section 5.8 [5H] translates the result to nonlinear metric spaces and furnishes a new proof to the (extended) Lyusternik–Graves theorem. Section 5.9 [5I] links the Lyusternik–Graves theorem, fixed point theorems, and other results in set-valued analysis. The final Sect. 5.11 [5K] deploys an inverse function theorem in Banach spaces which relies only on selections of the inverse to the directional derivative.