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An application of the Nash-Moser theorem to ordinary differential equations in Fréchet spaces. (English) Zbl 0949.34052
Summary: A general existence and uniqueness result of Picard-Lindelöf type is proved for ordinary differential equations in Fréchet spaces as an application of a generalized Nash-Moser implicit function theorem. Many examples show that the assumptions of the main result are natural. Applications are given for the Fréchet spaces \(C^\infty(K)\), \({\mathcal S}(\mathbb{R}^N)\), \({\mathcal B}(\mathbb{R}^N)\), \({\mathcal D}_{L_1}(\mathbb{R}^N)\), for Köthe sequence spaces, and for the general class of subbinomic Fréchet algebras.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
46A04 Locally convex Fréchet spaces and (DF)-spaces
58C15 Implicit function theorems; global Newton methods on manifolds
46A45 Sequence spaces (including Köthe sequence spaces)
34G10 Linear differential equations in abstract spaces
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