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, Rabah1 ; Sklyar, Grigory M.2 ; Rezounenko, Alexander V.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, Tome 337 (2003) no. 1, pp. 19-24.

On étudie une équation différentielle fonctionnelle de type neutre. Nous considérons le modèle opérationnel dans l'espace de Hilbert M 2 =C n ×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 ×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.

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DOI : 10.1016/S1631-073X(03)00251-6
Rabah, Rabah 1 ; Sklyar, Grigory M. 2 ; Rezounenko, Alexander V. 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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Rabah, Rabah; Sklyar, Grigory M.; Rezounenko, Alexander V. Generalized Riesz basis property in the analysis of neutral type systems. Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 19-24. doi : 10.1016/S1631-073X(03)00251-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00251-6/

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