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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2002-12-05
Accepted: 2003-05-20
Published online: 2003-06-19

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

Author's affiliations:

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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     title = {Generalized {Riesz} basis property in the analysis of neutral type systems},
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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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