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The abstract Cauchy-Kovalevskaya theorem in a weighted Banach space. (English) Zbl 0836.35004

A lot of authors have proved abstract versions of the classical Cauchy- Kovalevsky theorem. One of the most general versions was given by Nishida. The proof of Nishida’s theorem is basing on an iterative technique which generalizes the successive approximation to operator equations of the form \(u(t) = \int^1_0 F(u (\tau), \tau) d \tau\). Here is \(F\) a nonlinear operator acting in scales of Banach spaces. In the present paper the author gives a proof of Nishida’s theorem by using the contraction mapping principle. The suitable set of functions will be constructed by weighted norms containing all the norms of the scale. The weight expresses the typical property of conical evolution for the solutions of the above equation. For a similar reasoning see W. Walter [Am. Math. Mon. 92, 115-126 (1985; Zbl 0576.35002)] and W. Tutschke [Teubner-Texte zur Mathematik, Vol. 110 (Leipzig, 1989; Zbl 0677.30001)].
Remark: It would be an interesting problem to transfer this approach to systems of operator equations including Leray-Volevich effect [see the reviewer, Tsukuba J. Math. 18, No. 1, 193-202 (1994)].

MSC:

35A10 Cauchy-Kovalevskaya theorems
34G20 Nonlinear differential equations in abstract spaces
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