Discrete admissibility and  exponential trichotomy  of dynamical systems

Adina Luminiţa Sasu; Bogdan Sasu

doi:10.3934/dcds.2014.34.2929

Article Contents

2014, Volume 34, Issue 7: 2929-2962. Doi: 10.3934/dcds.2014.34.2929

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Discrete admissibility and exponential trichotomy of dynamical systems

Adina Luminiţa Sasu^1, and
Bogdan Sasu^1,

1.
Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara

Received: February 28, 2013

Revised: September 30, 2013

Published: June 2014

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Abstract

The aim of this paper is to present a new and complex study on the asymptotic behavior of dynamical systems, providing necessary and sufficient conditions for the existence of the exponential trichotomy. We associate to a nonautonomous discrete dynamical system an input-output system and we define a new admissibility concept called $(l^\infty(\mathbb{Z}, X), l^1(\mathbb{Z}, X))$-admissibility. First, we prove that the admissibility is a sufficient condition for the existence of the trichotomy projections, for their uniform boundedness, for their compatibility with the coefficients of the initial dynamical system and for certain reversibility properties. Assuming that the associated input-output operators satisfy a natural boundedness condition, we deduce that the admissibility is a necessary and sufficient condition for the existence of the uniform exponential trichotomy. Next, based on admissibility arguments, we obtain, for the first time in the literature, that all the trichotomic properties of a nonautonomous system can be completely recovered from the trichotomic behavior of the associated discrete dynamical system. Finally, we apply the main results in order to obtain a new characterization for uniform exponential trichotomy of evolution families in terms of discrete admissibility.

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Mathematics Subject Classification: Primary: 34D09, 93C55; Secondary: 93D25.

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