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Journal of Dynamical and Control Systems
Erschienen in: Journal of Dynamical and Control Systems 1/2020

11.02.2019

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

verfasst von: Pascal Remy

Erschienen in: Journal of Dynamical and Control Systems | Ausgabe 1/2020

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Abstract

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of \((0,0)\in \mathbb {C}^{2}\). In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.
Vorheriger Artikel Asymptotic Stability for a Viscoelastic Equation with Nonlinear Damping and Very General Type of Relaxation Functions
Nächster Artikel Stronger Forms of Transitivity and Sensitivity for Nonautonomous Discrete Dynamical Systems and Furstenberg Families

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Fußnoten
1
We denote \(\widetilde {q}\) with a tilde to emphasize the possible divergence of the series \(\widetilde {q}\).
 
2
See Remark 1.
 
3
See Remark 5.
 
4
In Appendix page 32, we present various results of the general theory of the Gevrey asymptotic expansions in the framework of the formal power series in \(\mathcal {O}(D_{\rho _2})[[t]]\).
 
5
A subsector Σ of a sector Σ′ is said to be a proper subsector and one denotes Σ ⋐Σ′ if its closure in \(\mathbb {C}\) is contained in Σ′∪{0}.
 
6
Of course, this case occurs if and only if i∗ < κ.
 
7
This set makes sense since, thanks to Remark 9, we have \(Ii^{\ast }>\kappa -i^{\ast }\geqslant i-i^{\ast }\).
 
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Metadaten
Titel
Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems
verfasst von
Pascal Remy
Publikationsdatum
11.02.2019
Verlag
Springer US
Erschienen in
Journal of Dynamical and Control Systems / Ausgabe 1/2020
Print ISSN: 1079-2724
Elektronische ISSN: 1573-8698
DOI
https://doi.org/10.1007/s10883-019-9428-0

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