An inverse function theorem in Fréchet spaces. (English) Zbl 1256.47037

Ekeland, Ivar

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28, No. 1, 91-105 (2011).

In this very nice paper, the author presents a local inverse function theorem for equations of the type \(F(x)= y\), where \(F\) maps a graded Fréchet space into another one. Among the permitted Fréchet spaces are, for example, the intersections of scales of \(C^k\)-spaces or of Sobolev spaces.
In contrast to the classical hard implicit function theorems, where the solution is found by a Newton iteration, here the solution is found by Ekeland’s variational principle. Therefore, the smoothness assumptions on \(F\) are essentially weaker (\(F\) has to be continuous and Gâteaux-differentiable, but the derivative need not to be continuous), and the conclusion is weaker (there is local existence, but no local uniqueness). The possibly multivalued inverse map \(F^{-1}\) is shown to be Lipschitz continuous.
Just as in the classical hard implicit function theorems, the main difficulty is to overcome the allowed loss of derivatives.

Reviewer: Lutz Recke (Berlin)

MSC:

47J07	Abstract inverse mapping and implicit function theorems involving nonlinear operators
46A61	Graded Fréchet spaces and tame operators
49J52	Nonsmooth analysis

Keywords:

Ekeland’s variational principle; no local uniqueness; loss of derivatives; inverse function theorem; implicit function theorem; Fréchet space; Nash-Moser theorem

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