×

zbMATH — the first resource for mathematics

A new higher order chain rule and Gevrey class. (English) Zbl 0719.58002
From the author’s abstract: “A new higher order chain rule both in the sense of ordinary functions and in the sense of Fréchet derivatives is given. It is applied to prove that a composite map of Gevrey maps is again Gevrey. An inverse map theorem etc. are also given in the framework of Gevrey maps.”
Reviewer: N.Jacob (Erlangen)

MSC:
58C25 Differentiable maps on manifolds
46G05 Derivatives of functions in infinite-dimensional spaces
58J99 Partial differential equations on manifolds; differential operators
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
58C15 Implicit function theorems; global Newton methods on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abraham, R.-Robbin, J.: Transversal mappings and flows. W. A. Benjamin, Inc., New York 1967. · Zbl 0171.44404
[2] Dieudonné, P.- Foundations of Modern Analysis. Academic Press, New York 1960. · Zbl 0100.04201
[3] Duchateau, P.-Tréves, F.: An abstract Cauchy-Kovalevska Theorem in scales of Gevrey classes. Symposia Math., vol.7 (1971), pp. 135-163, Academic Press. · Zbl 0226.35003
[4] Faa’di Bruno, F.: Sullo sviluppo delle funzioni. Annali di Scienze Matematiche e Fisiche di Tortolini, 6 (1855), 479-480.
[5] Kajitani, K.: Non-linear Leray-Volevich systems and Gevrey class. Comm. in Partial Differential Equations, 6 (1981), 1137-1162. · Zbl 0495.35026
[6] Koike, M.: An abstract nonlinear Cauchy problem with a vector valued time variable. Funkcialaj Ekvcioj 32 (1989), 1-22. · Zbl 0694.35004
[7] Komatsu, H.: The implicit function theorem for ultra-differentiable mappings. Proc. Japan Acad., Ser. A, 55 (1979), 69-72. · Zbl 0467.26004
[8] Lusternik, L. A.-Sobolev, V. J.: Elements of Functional Analysis. Hindustan Publ. Co., Delhi, 1961. · Zbl 0293.46001
[9] Perron, O.: Über eine Formel des Herrn Schwatt. Math. Zeitschr. 31 (1930), 159-160. · JFM 55.0132.09
[10] Persson, J.: New proofs and generalizations of two theorems by Lednev for Goursat problems. Math. Annalen 178 (1968), 184-208. · Zbl 0159.14701
[11] Persson, J.: Quasianalytic classes, Holmgren’s uniqueness theorem, Hamilton-Jacobi equations and the inverse function theorem. ?Current topics in partial differential equations? (ed. Ohya et al.), pp. 27-53, Kinokuniya, Tokyo, 1986.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.