Generalized Riesz basis property in the analysis of neutral type systems

Rabah Rabah; Grigory M. Sklyar; Alexander V. Rezounenko

doi:10.1016/S1631-073X(03)00251-6

Ordinary Differential Equations

Generalized Riesz basis property in the analysis of neutral type systems
[Bases généralisées de Riesz pour les systèmes de type neutre]

Présenté par : Pierre-Louis Lions

Rabah Rabah ¹ ; Grigory M. Sklyar ² ; Alexander V. Rezounenko ³

¹ IRCCyN UMR 6597, 1, rue de la Noë, PB 92101, 44321 Nantes cedex 3, France
² Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Wielkopolska 15, Poland
³ Department of Mechanics and Mathematics, Kharkov University, 4 Svobody sqr., Kharkov, 61077, Ukraine

Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 19-24.

Résumé
Abstract

On étudie une équation différentielle fonctionnelle de type neutre. Nous considérons le modèle opérationnel dans l'espace de Hilbert $M_{2} = C^{n} \times L_{2} (- 1, 0; C^{n})$ et montrons qu'il existe dans cet espace une base de Riesz de sous-espaces de dimensions finies invariants par l'opérateur générateur infinitésimal du système. Nous donnons également un exemple précisant qu'il n'existe pas de base de Riesz de sous-espaces propres.

The functional differential equation of neutral type is studied. We consider the corresponding operator model in Hilbert space $M_{2} = C^{n} \times L_{2} (- 1, 0; C^{n})$ and prove that there exists a sequence of invariant finite-dimensional subspaces which constitute a Riesz basis in M₂. We also give an example emphasizing that the generalized eigenspaces do not form a Riesz basis.

Reçu le : 2002-12-05
Accepté le : 2003-05-20
Publié le : 2003-06-19

DOI : 10.1016/S1631-073X(03)00251-6

Affiliations des auteurs :

Rabah Rabah ¹ ; Grigory M. Sklyar ² ; Alexander V. Rezounenko ³

¹ IRCCyN UMR 6597, 1, rue de la Noë, PB 92101, 44321 Nantes cedex 3, France
² Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Wielkopolska 15, Poland
³ Department of Mechanics and Mathematics, Kharkov University, 4 Svobody sqr., Kharkov, 61077, Ukraine

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     author = {Rabah Rabah and Grigory M. Sklyar and Alexander V. Rezounenko},
     title = {Generalized {Riesz} basis property in the analysis of neutral type systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {19--24},
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     volume = {337},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00251-6},
     language = {en},
}

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Rabah Rabah; Grigory M. Sklyar; Alexander V. Rezounenko. Generalized Riesz basis property in the analysis of neutral type systems. Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 19-24. doi : 10.1016/S1631-073X(03)00251-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00251-6/

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