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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

An abstract nonlinear Cauchy-Kovalevska theorem


Author: François Trèves
Journal: Trans. Amer. Math. Soc. 150 (1970), 77-92
MSC: Primary 35.03
DOI: https://doi.org/10.1090/S0002-9947-1970-0274911-X
MathSciNet review: 0274911
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Abstract: A nonlinear version of Ovcyannikov's theorem is proved. If $ F(u,t)$ is an analytic function of $ t$ real or complex and of $ u$ varying in a scale of Banach spaces, valued in a scale of Banach spaces, the Cauchy problem $ {u_t} = F(u,t),u(0) = {u_0}$, has a unique analytic solution. This is an abstract version of the Cauchy-Kovalevska theorem which can be applied to equations other than partial-differential, e.g. to certain differential-convolution or, more generally, differential-pseudodifferential equations.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0274911-X
Keywords: Cauchy-Kovalevska, nonlinear, Cauchy problem, scale of Banach spaces, analytic, Ovcyannikov, Banach algebras, Fourier transform, analytic functionals, Gevrey class
Article copyright: © Copyright 1970 American Mathematical Society