Abstract
In this chapter we, roughly speaking, translate some principal results of the classical theory presented in the preceding chapter into purely metric language in which the key word is “distance” and words like “derivative” or “tangent space” make little sense.
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Notes
- 1.
It seems to be appropriate to quote here [82]: “in essence the whole history of the generalizations of the Lyusternik theorem reduces to finding new formulations from the standard process of proof”. But it soon became clear that certain results proved with the help of Ekeland’s principle (e.g. the general regularity criterion of Theorem 2.46) can hardly be obtained using Lyusternik–Graves iterations.
- 2.
We shall use the term “Aubin property” only in the local context, leaving the term “pseudo-Lipschitz” for non-local situations.
- 3.
To explain the terminology, consider a “control system” governed by the differential equation \(\dot{x}= f(x, u)\), where the control function u(t) is taken from a pool of admissible controls \({\mathcal U}\). Once a control u(t) and the initial state \(x_0\) at, say \(t=0\), of the system are given, the equation defines a trajectory x(t) of the system. Let \(x_1=x(1)\). The system is called locally controllable if small variations of u(t) allow us to transfer \(x_0\) to any point of a neighborhood of \(x_1\).
- 4.
See, for instance, Ekeland’s comments in [113], where the main result of [150] was interpreted as a nonsmooth mean value theorem.
- 5.
In [82] the authors considered systems of balls in X which are ‘full’ in the sense that all balls contained in an element of the system also belong to the system, and called a (single-valued continuous) mapping \(F: X\rightarrow Y\) an a-covering on the system if \(B(F(x),at)\subset F(B(x, t))\) whenever B(x, t) belongs to the system.
- 6.
In [154, 164] single-valued mappings between Banach spaces were considered. But the proof in [164] carries over to set-valued maps between metric spaces with minor changes.
- 7.
In [11] Aubin mentioned that a result close to his had been established in the thesis of G. Lebourg which, as I understand, was never published.
- 8.
See the footnote on p. 99.
- 9.
It is certain that Graves’ paper was not known to Milyutin in 1976. To the best of my knowledge the first reference to it in the Russian literature appeared in [82] in 1980. Moreover, I doubt that Graves’ theorem was known and/or used by the optimization community before 1980. It seems (curiously enough) that at the earlier stage of development of regularity theory all the main ideas appeared independently. Lyusternik in 1934 was likely unaware of the Banach open mapping theorem. Graves in 1950 knew Banach’s result but apparently not Lyusternik’s theorem. Tikhomirov and myself knew Lyusternik’s theorem while writing the 1974 book, but not the result of Graves, and Robinson in 1976 seemed to have been unaware of the papers by Lyusternik, Graves and our book.
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Ioffe, A.D. (2017). Metric Theory: Phenomenology. In: Variational Analysis of Regular Mappings. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-64277-2_2
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