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

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